Dimension Theory in Dynamical Systems: Contemporary Views and Applications (Chicago Lectures in Mathematics)

G-Structures Associated with Dynarmcal Systems: Thermodynamic Formalism

65

Pz(cp) = CPz(cp) = CPz(cp). In the latter case, the common value coincides with the "classical" topological pressure introduced by Bowen, Ruelle, and Walters. We stress that for an arbitrary subset Z, one has three, in general distinct, quantities Pz(cp), CPz(cp), and CPz(cp) to be used as a generalization of the classical notion of topological pressure. In view of the variational principle (see Appendix II below) a crucial role is played by the quantity Pz(cp) while lower and upper capacity topological pressures are also often used in computing dimension of invariant sets for dynamical systems (see Chapters 5 and 7). There is an important particular case when cp = 0. We call the quantities Pz(O), CPz(O), and CPz(O) the topological entropy and lower and upper capacity topological entropies and we use the generally accepted notations ht(Z), Ch 1(Z), and Cht(Z). The topological entropy is a well-known invariant of dynamical systems and plays a key role in topological dynamics. For example, for subshifts of finite type (and hence for Axiom A diffeomorphisms) the topological entropy is the exponential growth rate of the number of periodic points. The first definition of the topological entropy for compact invariant sets was given by Adler, Konheim, and McAndrew [AKMJ. In [Bol], Bowen extended it to non-compact invariant sets and pointed out the "dimensional" nature of this notion. We emphasize that the straightforward generalization of the AdlerKonheim-McAndrew definition of the topological entropy for non-compact sets leads to the quantities Ch 1(Z) and Cht(Z). On the other hand, we show that Bowen's topological entropy coincides with ht(Z). Let J.L be a Borel probability measure on X. In Section 10 we will show that the C-structure T generates the Caratheodory dimension and lower and upper CaratModory capacities of J.L· We denote them by P,_.(cp,U), CP1..(cp,U), and CP,_.(cp,U) respectively. We show that these quantities have limits as diamU tends to zero and these limits coincide and are equal to h,..(f) + fx cpd!J., where hp..(!) is the measure-theoretic entropy off The expression h,..(f) + fx cpd!J. is the potential function in the variational principle (see below Appendix II). In the case cp = 0 our approach produces, in particular, a "dimension" definition of the measure-theoretic entropy. 10. A Modification of the General Caratbeodory Construction

We describe a modification of a general Caratheodory construction. Let X and S be arbitrary sets and F = {U. : s E S} a collection of subsets in X. We assume that there exist two functions TJ, 'lj!. S-+ JR+ satisfying the following conditions: Al. there exists s0 E S such that U,0 == 0; if U. == 0 then 71(s) == 0 and 'lj!(s) = 0; if U8 =!= 0 then 11(s) > 0 and 'f/!(s) > 0; A2. for any 'lj!(s):::; e;

{J

> 0 one can find

e

> 0 such that

17(s) :::; 6 for any s E S with

66

Chapter 4

A3. for any e > 0 there exists a finite or countable subcollection g covers X (i.e., U Us:::) X) and t/1(9) ~f sup{t/J(s): s E S}:::; e.

c

S which

•E9

Let {. S -+ JR+ be a function. We say that the set S, collection of subsets :F, and the functions {, q, t/J, satisfying Conditions A1, A2, and A3, introduce the Caratheodory dimension structure or C-structure r on X and write T = (S, :F, {, f1, '1/J). If the maps f-t Us is one-to-one then the functions{, 17, and 'ljJ can be considered as being defined on the set :F and thus, the above C-structure coincides with the C-structure introduced in Section 1. In the general case one can still follow the approach, described in Chapter 1, to define the Caratheodory dimension and lower and upper CaratModory capacities generated by the C-structure. We shall briefly outline this approach. Given a set Z c X and numbers a E lR, e > 0, we define

Mc(Z,a,e) =

i~f{ Ee(s)q(s)"}, sell

where the infimum is taken over all finite or countable subcollections g c S covering Z with '1/J(fi):::; e. By Condition A3 the function Mc(Z, a, e) is correctly defined. It is non-decreasing as e decreases. Therefore, the following limit exists:

mc(Z,a)

=

limMc(Z,a,e:).

<--+0

One can show that the function mc(Z, a) satisfies Propositions 1.1 and 1.2. We define the Caratheodory dimension of the set Z by (1.3) It has properties stated in Theorem 1.1. Let X and X' be sets endowed with C-structures r = (S, :F, t,, q, t/1) and r' = (S, :F', t;,', 17', t/J') respectively. One can show that the Caratheodory dimension of sets is invariant with respect to a bijective map x: X -+ X' which preserves the C-structures rand~ (compare to Theorem 1.3). We shall now assume that the following condition holds.

A3'. there exists E > 0 such that for any 0 < e :::; f there exists a finite or countable subcollection g c S covering X such that 'ljJ ( s) = e for any s E g. Given a E R. and e > 0, let us consider a set Z C X and define

Rc(Z,a,e:)

= i~f { L

e(s)1J(s)a}'

oE9

where the infimum is taken over all finite or countable subcollections g C S covering Z such that '1/J(s) = e for any s E g, According to A3', Rc(Z,a,e:) is correctly defined. We set

!:c(Z,a)

= limRc(Z,a,e), e--+0

i"c(Z,a) =lim Rc(Z,a,e:). <--+0

C-Structures Associated with Dynanucal Systems: Thermodynamic Formalism

67

One can show that the functions r.c(Z, a) and rc(Z, a) satisfy Proposition 2.1. We define the lower and upper Caratheodory capacities of the set Z, CapcZ and CapcZ, by (2.1). They have properties stated in Theorem 2.1. For any e > 0 and any set Z c X, let ns put

where the infimum is taken over all finite or countable subcollections g c S covering Z for which '1/J(s) = e for all sEQ. Let us assume that the function 77 satisfies the following condition: A4. 77(s 1 ) = rt(sz) for any S1> s2 E S for which '1/J(sl) = 'ljJ(s2). One can now correctly define the function 17(e) of a real variable e by setting 7J(e) = ?J(s) if '1/J(s) =e. One can prove that, provided Condition A4 holds, the lower and upper Caratheodory capacities of sets satisfy Theorem 2.2, 2.3, and 2.4.

Let X and X' be sets endowed with C-structures r = (S,:F,~,7J,'I/J) and r' = (S, :F', ~', r/, '1/J') respectively. One can show that the lower and upper Caratheodory capacities of sets are invariants with respect to a bijective map x: X --t X' which preserves C-structures rand r' (compare to Theorem 2.5). Let (X, I') be a Lebesgue space with a probability measure I' endowed with a C-structure r = (:F,f.,1f,'I/J). Assume that any set Ua E :F is measurable. We define the Caratheodory dimension of the measure "' dimH '-'• and lower and upper Caratheodory capacities of the measure '-'• Cape'-' and Cap 0 !-', by (3.1) and (3.2) respectively. We have that

We shall now assume that the following condition holds· A5. for !-'-almost every x E X and any e > 0, if s E S and U8 3 x is a set with '1/J(s) ~ e then 1-'(U.) > 0 and f.(s) > 0. For each point x EX and a number e, 0 < e e, we chooses= s(x,e) E S 1 such that x E U, and '1/J(s) = e (this is possible in view of Condition A3 ). Once this choice is made we obtain the subcollection

s

S'

= {s(x,e) E S: x EX, 0 < e S

~:}.

Given a E R and x E X, we define now the lower and upper a-Caratheodory pointwise dimensions of I' at x by odogJI.(U•(z,e)) ( ) _lim d !!!C,,..,a x - e->O log (f.(s(x, e))77(s(x, e))")'

a log JI.(Us(z,e)) ( ) -1'd c,,..,o x = .~ ,....lo_g_,(~-(,_s.,...(x:....,-:e)..,..)ry-'-("-s-':-(x"'",-:e)-:-)-:-")

Chapter 4

68

(see (3.5)). We have that .!k-,JA,<>(x) ::; dc,JA,a(x) for any x E X. It is a simple exercise to prove that the conclusion of Theorems 3.1, 3.2, 3 3, 4.1, 4.2, 5.1, and 5.2 hold (Wlth obvious modifications in the formulations). We also define ..,..,

.!:!.0

( ) I' . f o:logJ,t(U.) 1m m ,JA,a x = e->0 zEU, log (~(s)T1(S) 0 )' tb(•)=<

( ) 'De ' P ' a x

(10.1)

-. o: log J,t(U8 ) = e->0 hm :.eu. sup 1og (t'( .. s )T1 (s )a ) · ,P(s)=<

It is easy to check that conclusions of Theorem 3.5 and 3.6. hold. 11. Dimensional Definition of Topological Pressure; Topological and Measure-Theoretic Entropies Topological Pressure Let (X, p) be a compact metric space with metric p, f: X--+ X a continuous map, and cp: X --+ R a continuous fUllction. Consider a finite open cover U of X and denote by Sm(U) the set of all strings U = {Uio ... U•m-l : U,j E U} of length m = m(U) We putS= S(U) = Um>oSm(U). To a g~ven string U = {U'" ... U;,_,} E S(U) we associate the set

X(U) = {x EX: Ji(x) E U;, for j = 0, ... , m(U) -1}.

(11.1)

Define the collection of subsets

:F = :F(U)

= {X(U):

U

(11.2)

E S(U)}

and three functions{, '711/1: S(U)--+ R as follows m(U)-1

{(U) =exp

sup

( :cEX(U)

Tl(U) =exp(-m(U)),

L

)

cp(fk(x))

,

(11.3)

k=O

'1/;(U) = m(u)-

1

•

It is straightforward to verify that the setS, the collection of subsets :F, and the functions 1/, ~. and '1/J satisfy Conditions Al, A2, A3, and A3' in Section 10 and hence they determine a C-structure r = r(U) = (S,:F,{,.,,'I/J) on X. The corresponding Caratheodory function mc(Z,o:) (where Z c X and o: E R; see Section 10) depends on the cover U (and the function cp) and is given by

mc(Z,o:)

= N->oo lim M(Z,o:,cp,U,N),

where

M(Z,o:,cp,U,N) = inf { (i

L UE(i

1

exp (-o:m(U)

+

sup ;oEX(U)

m~- cp(fk(x)))} k=O

(11.4)

C-Structures Associated with Dynamical Systems: Thermodynarmc Formalism

69

and the infimum is taken over all finite or countable collections of strings 9 c S(U) such that m(U) ;:::: N for all U E 9 and 9 covers Z (i e., the collection of sets {X(U) : U E Q} covers Z). Furthermore, the Caratheodory functions r.c(Z, a) and 'i'c(Z, a) (where Z C X and a E lR; see Section 10) depend on the cover U and are given by rc(Z,a)

= N-4oo lim R(Z,a,!p,U,N),

rc(Z,a)

= N-+oo lim R(Z,o.,!p,U,N),

where R(Z,o.,!p,U,N) = inf { g

L

exp (-aN+

UEQ

sup

~'P(fk(x)))}

(11.4

1 )

xEX(U)k=O

and the infimum is taken over all finite or countable collections of strings 9 c S(U) such that m(U) N for all U E 9 and Q covers Z. According to Section 10, given a set Z c X, the C-structure T generates the Caratheodory dimension of Z and lower and upper Caratheodory capacities of Z specified by the cover U and the map f. We denote them by Pz('P,U), CPz('P,U), and CPz('P,U) respectively. We have that (compare to (1.3) and (2.1))

=

Pz('P,U) = inf{a: mc(Z,a) = 0} =sup{ a: mo(Z, a)= oo}, CPz(I(),U) = inf{a · rc(Z, a)= 0} =sup{ a.: ro(Z,o) = oo}, CPz('P,U) = inf{o. 'i'c(Z,o.) = 0} = sup{o: 'i'c(Z,a) = oo}.

Let lUI= max{diamU,:

u, c U} be the diameter of the cover U.

Theorem 11.1. For any set Z C X the following limits

ex~st.

Pz('P) ~f lim Pz('P,U), IUI-+0

CPz('P) ~r lim CPz('P,U),

-

IUI-+0-

CPz('P) ~f lim CPz('P,U). IUI-+0

Proof. Let V be a finite open cover of X with diameter smaller than the Lebesgue number of U. One can see that each element V E V is contained in some element U(V) E U. To any string V = {V; 0 ••• Vim} E S(V) we associate the string U(V) = {U(V;.) ... U(V; .. )} E S(U). If 9 c S(V) covers a set Z c X then U(9) = {U(V) : V E Q} c S(U) also covers Z. Let 'Y = 'Y(U) = sup{I'P(x)- 'P(Y)I: x,y E U for some U E U}. One can verify using (11.4) that for every a E JR. and N > 0 M(Z,a,I{),U,N):::; M(Z,a-'"(,I{),V,N)

(11 5)

Chapter 4

70

This implies that

Pz(cp,U)- "Y :S: Pz(cp, V). Since X is compact it has finite open covers of arbitrarily small diameter. Therefore, Pz(IP,U)- "Y :S: lim Pz(IP, V). lVI -tO

If /U/ -+ 0 then -y(U) -+ 0 and hence lim Pz(IP,U) :-:; lim Pz(cp, V). lUI-tO

IVI--+0

This implies the existence of the first limit. The existence of two other limits can be proved in a similar fashion by using the inequality which is an analog of (11.5)

(11 6)

R(Z,a,cp,U,N)::; R(Z,a--y,ip, V,N). This complett>-S the proof of the theorem.

•

We call the quantities Pz(cp), CPz(ip), and CPz(fP), respectively the topological pressure and lower and upper capacity topological pressures of the function IP on the set Z (with respect to !). Sometimes more explicit notations Pz,J(cp), CPz.J(cp), and CPz.J(IP) will be used to emphasize the dependence on the map f. We emphasize that the set Z can be arbitrary and need not be compact or invariant under the map f. If f is a homeomorphism then for any set Z c X its topological pressure coincides with topological pressure on the invariant hull of Z (i.e., the set U r(Z); this follows from Theorem 11.2 below) However, this nEZ

may not be true for lower and upper capacity topological pressures (see Example 11 2 below). We formulate the basic properties of topological pressure and lower and upper capacity topological pressures. They are immediate corollaries of the definitions and Theorems 1.1 and 2.1.

Theorem 11.2. (1) P0(1P) :S: 0. (2) Pz, (IP)::; Pz.(cp) if Z1 c Z2 c x. (3) Pz(IP) =SUP;> I Pz, (cp), where = ui>lzi and (4) Iff is a homeomorphism then Pz(cp),::::: P1(z)(cp).

z

z, c X, i =

1, 2, ....

Theorem 11.3. (1) CPG(IP) :-:; 0, CP0(IP) :-:; 0. (2) CPz, (I/') :S: CPz,(IP) and CPz,(cp) :-:; CPz,(IP) if Z1 C Z2 C X. (3) CPz(IP) 2: supi>I QEz,(cp) and CPz(IP) 2: supi>l CPz,(cp), where Z = U;> 1 z, and Zi X, i = 1, 2,. . . (4) If ii: X-+ X is a homeomorphism which commutes with f (i.e., f o h = ho f} then

c

Pz(cp)

= Ph(Z)(cp o h- 1 ),

CPz(IP) = CPh(Z)(cpoh- 1 ),

CPz(IP) = CPh(Z)(cpoh- 1 )

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

71

Obviously, the functions 11 and 'lj; satisfy Condition A4 in Section 10. Therefore, by Theorems 2.2 and 11.1, we have for any Z C X that

CPz(cp)

=

lim

1 lim N log A(Z,cp,U,N),

IUI-+0 N-+oo

-

CPz(cp) = lim

(11 7)

-1

lim Nlog A(Z,cp,U,N),

IUI-+ON-+oo

where in accordance with (2.3), (11.1), and (11.3)

A(Z, cp,U, N) = inf { g

L

UeQ

exp ( sup "I:\p(fk(x)))}

(11.8)

xEX(U) k=O

and the infimum is taken over all finite or countable collections of strings 9 C S(U) such that m(U) = N for all U E g and g covers Z. We also point out the continuity property of the topological pressure and lower and upper capacity topological pressures.

Theorem 11.4. For any two continuous functions cp and 'lj; on X IPz(cp)- Pz('I/J)I ::S:

llcp -1/111. llcp- 1/JII, ::S: II'P- 1/JII

ICPz(cp)- CPz(1/J)I ::S: ICPz(cp)- CPz(1/J)I where

11·11

denotes the supremum norm in the space of continuous functions on X.

Proof. Given N > 0, we have that

~sup

N-1

L icp(f'"(x))- 1/J(fk(x))l ::S: II'P- 1/JII.

:tEX k=O

It follows that

M(Z,a:+

llcp- 1/JII,'Ij;,U,N):::; M(Z,a.,cp,U,N):::; M(Z,a:-llcp- 1/JII,'Ij;,U,N)

This implies that

Pz('lj;,U)

-II'P -1/111

::S: Pz(cp,U) ::S: Pz('I/J,U)

+ II'P- 1/JII

and concludes the proof of the first inequality The proof of the other two in• equalities is similar.

72

Chapter 4 One can easily see that

Pz(cp):::; CPz(cp):::; CPz(cp).

(11.9)

Below we will give an example where the strict inequalities occur (see Examples 11.1 and 11.2}. The situation for invariant and compact sets is different. Theorem 11.5. (1) For any f-invanant set Z C X we have CPz(cp) = CPz(cp); moreover, for any open coverU of X, we have CPz{cp,U) = CPz(cp,U). (2) For any compact invanant set Z c X we have Pz(cp) = CP z(cp) CPz(cp); moreover, for any open cover U of X, we have Pz(cp,U) CPz(cp,U) = CPz(cp,U). Proof. Let Z c X be an !-invariant set. Choose two collections of strings 9m C Sm (U) and 9n C Sn (U) which cover Z and consider

9m,n ~f {UV: U E 9m, V E 9n} C Sm+n(U). Since Z is /-invariant the collection of strings 9m,n also covers Z. We wish to estimate A(Z, cp, U, m + n) using (11.8). We have

This implies that A(Z,cp,U,m + n):::; A(Z,cp,U, m) x A(Z,cp,U,n). Let am = logA(Z,cp,U,m). Note that A(Z,cp,U,m) 2 e-mll'l'll. Therefore, infm;?:l ~ 2 -II'PII > -oo. The desired result is now a direct consequence of (11.7) and the following lemma (we leave its proof to the reader; see Lemma 1.18 in [Bo2]) Lemma. Let am, m = 1, 2,.. be a sequence of numbers satisfying infm>l l!m. > -oo and am+n :::; am + an for all m, n 2 1. Then the limit limm->oo ~ e~sts and coincides with infm?:l ~. Choose any o > Pz(cp,U). There exist N > 0 and covers Z and Q(Z,o,Q) ~r

L exp UEQ

(

-om(U)

+

g C S(U)

m(U)-1

sup

L

zEX(U)

k=O

such that

)

cp(Jk(x))

<1.

g

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

Since Z is compact we can choose

73

g to be finite and hence M

9

c

U Sm(U) m=l

for some M 2 1. Put 00

gn={U1

.

u,.:UiE9}andr=U9". n=l

Since Z is invariant r covers Z It is a simple exercise (which we leave to the reader) to check that for every n > 0,

Q(Z,a,9")

~

Q(Z,a,9)".

Hence, 00

Q(Z,a,r) = LQ(Z,a,9") < oo. n=l

Let us fix some N > 0 and consider a point x E Z Since r covers Z there exists a string U E r such that x E X(U) and N ~ m(U) < N + M. Denote by U* the substring that consists of the first N symbols of the string U. We have that N-1

sup

m(U)-1

L'P(fk(y)) ~

yEX(U•) k=O

sup yEX(U)

L

rp(fk(y)) + Mllrpll·

k=O

If fN denotes the Collection of all SUbstrings U• COnstructed above then N-1

e-"'N L U•ErN

exp

sup

L

rp(fk(y)) ~ max{l,e-"'M}eMIJ'I'Il Q(Z,a,r) < oo.

yEX(U•) k=O

By (11.7) we obtain that a> CPz(rp), and hence the desired result follow (11.9) Theorem 11.5 shows that for a compact invariant set Z the topological pre&sure and lower and upper capacity topological pressures coincide and the common value yields the classical topological pressure (see, for example, [Bo2]). It is worth pointing out that this common value is a topological invariant (i.e , Px(rp) = Px(rp a h), where h is a homeomorphism which commutes with f). This means that the pressure does not depend on the metric on X. If a set Z is neither invariant nor compact one has three, in general distinct, quantities. the topological pressure, Pz(rp), and lower and upper capacity topological pressures, CPz(rp) and CPz(rp). The latter coincide if the set Z is invariant and may not otherwise (see Example 111 below). Furthermore, they are defined by formulae (11.7) and (11.8) which are a straightforward generalization of the classical definition of the topological pressure In view of the

74

Chapter 4

variational principle, the topological pressure Pz(cp) seems more adapted to the case of non-compact sets and plays a crucial role in the thermodynamic formalism (see Appendix II).

Remarks. (1) We describe another approach to the definition of topological pressure. Let (X,p) be a compact metric space with metric p, f:X---+ X a continuous map, and cp: X ---+ lR a continuous function. Fix a number li > 0. Given n > 0 and a point x E X, define the (n, /i)-ball at x by

Bn(x,o) Put S

=X

= {y EX: p(l(x),ti(y))::; 6,

for 0::;

i::; n}.

(11.10)

x N. We define the collection of subsets

:F = {Bn(x,o): x EX, n EN} and three functions

~. 7J, 1/J: S---+

lR as follows

~(x,n) = exp ( 71(x, n)

=

sup

I:cp(fk(y))),

yeB.(z,6) k=O

exp( -n),

1/J(x, n)

= n- 1

One can directly verify that the setS, the collection of subsets :F, and functions 71, and 1/J satisfy Conditions AI, A2, A3, and A3' in Section 10 and hence determine a C-structure T = (S,:F,f;.,rJ,t/J) on X. According to Section 10, given a set Z C X, this C-structure generates the Caratheodory dimension of Z and lower and upper CaratModory capacities of Z which depend on o. We denote them by Pz('fJ,O), CPz(cp,/i), and CPz('fJ,fJ) respectively. Let U be a finite open cover of X and li(U) its Lebesgue number. It is easily seen that for every x EX, if x E X(U) for some U E S(U) then

e,

(11.11)

It follows now from Theorem 11.1 that

Pz(cp) = limPz(cp,o), 6-+0

(2) If the map f: X ---+ X is a homeomorphism we can consider the topological pressure and lower and upper capacity topological pressures for the map f as well as for the inverse map f- 1 • If Z is an invariant subset of X then for any continuous function '{J: X ---+ R,

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

75

These equalities hold no matter whether Z is compact or not but may fail to be true if Z is not invariant (see Example 11.3 below). If Z is invariant and compact then in addition we have that

and this may fail if Z is not compact (although still invariant; see Example 11.3 below). Topological Entropy We consider the special case t.p = 0 Given a set Z C X, we call the quantities

hz(f) ~ Pz(O),

Chz(f) ~ CPz(O),

Chz(J) ~rCPz(O)

respectively, the topological entropy and lower and upper capacity topological entropies of the map f on Z. We stress again that the set Z can be arbitrary and need not be compact or invariant under f. It follows from (11.9) that (1112) If the set Z is /-invariant, we have

Chz(f) = Chz(f) ~r Chz(f). By (11.7) we obtain for an invariant set Z that 1

Chz(f) = lim lim N log A(Z, O,U, N) IUI-+ON-+oo

=lim

1 lim N log A(Z,O,U,N),

(11.13)

IUI-+ON-+oo

where, in accordance with (11 8), A( Z, 0, U, N) is the smallest number of strings U of length N, for which the sets X(U) cover Z. Formula (11.13) reveals the meaning of the quantity Chz(f): it is the exponential rate of growth inN of the smallest number of strings U of length N, for which the sets X (U) cover Z. For a compact invariant set Z we have by Theorem 11.5, that hz(f) = Chz(f) = Chz(f). The topological entropy and lower and upper capacity topological entropies have properties stated in Theorems 11.2 and 11.3 (applied to t.p = 0). In particular, they are invariant under a homeomorphism of X which commutes with f. We now proceed with the inequalities (11.12). In examples below we consider the symbolic dynamical system (Ep, fl), where Ep is the space of two-sided infinite sequences on p symbols and fl is the (two-sided) shift. We recall that the cylinder set ci,. il consists of all sequences w = (jk) for which im = im, ... , it = it (see more detailed description in Appendix II below).

76

Chapter 4

Example 11.1. There eXIsts a compact non-invanant set Z C Ea for which Chz(a) < Chz(a). Proof. Let Z

nk

be a strictly increasing sequence of integers. Define the set

= {w == (wn) E l:3 :wn =

1 or 2

if n2t

:s; n < n2t+1 and n < nzt+2 for some l}.

Wn = 1,2, or 3 if n2t+1 :s;

Obviously, the set Z is compact. Consider the sequence an defined as follows: n < nu+1 and~= 3 if n2t+1 :s; n < nze+2· Set Sn = TI~=t ak· We choose the sequence nk growing so fast that

an= 2 if nu :s;

where C1 > 0 is a constant independent of l. Given m 2: 0, consider the cover Um of l:a by cylinder sets C;_.,. ;.,. . Notice that for every string U the set X(U) is a cylinder. Therefore, in accordance with (11.8), A(Z, O,Um, N) is the smallest number of cylinders of length m + N needed to cover the set Z. It follows that if nk :s; N < nk+l• then

where C2(m) > 0 is a constant independent of n. Applymg (11.13) with U = Um (and lUI-+ 0 as m-+ 0) yields Chz(u)

= log2,

Chz(a)

= log3.

The desired result follows.

•

Example 11.2. There is an invariant set Z C l:2 for which hz(u) < Chz(u). Proof. Define the sets

zk = {w = (wn)

E E2

Wn

= 1 for alllnl2: k}, z = uzk· kEZ

It is easy to see that the set Z is invariant and everywhere dense in l:2. Therefore, Chz(u) = ChE 2 (a) = log2. Given m 2: 0, consider the cover Um of E2 by cylinder sets C;_.,. ;.,.. It is easy to see that A(Zk, 0, Um, N) :s; C, where C > 0 is a constant independent of N. Therefore, Chz"' (u) = Chz.,. (u) = 0 and hence hz.,. (u) =0 for all m. This • implies that hz(u) = 0.

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

77

Note that the set Z in this example is the invariant hull of the set Zo and Chz0 (u) = Chz0 (u) = 0 while Chz(u) = log2.

Example 11.3. (1) There is an invanant (non-compact) set Z C E 2 for which hz(u) = log2 while hz(u- 1 ) = 0. (2) There is a compact (non-invanant) set Z c E 2 for which hz(u)

= Chz(u) = Chz(o-) = log2

whue

Proof. Define the sets

Z~r. = {w = (wn)

E E2:

Wn = 1 for all n::,:: k},

Z = Uz~c. kEZ

Obviously, the set Z is mvariant (but not compact) and the set Zo is compact (but not invariant). We leave it as an exercise to the reader to show that Z fulfills requirements in Statement 1 and so does Zo in Statement 2.

Remark. Let U be a finite open cover of X. Given a set Z hz(f,U) ~r Pz(O,U),

Chz(f,U) ~ CPz(O,U),

c X, the quantities Chz(J,U) ~r CPz(O,U)

are called the topological entropy and lower and upper capacity topological entropies off on Z with respect to U. By Theorem 11.5, if Z is invariant, then Q!.z(J,U} = Chz(J,U) ~f Chz(J,U) and if, in addition, Z is compact then hz(f,U) = Chz(f,U). Let V be a finite open cover of X whose diameter does not exceed the Lebesgue number of U. Applying (11.5) with rp = 0 we obtain that hx(f,U) ::; hx(J, V).

In [BGH], Blanchard, Glasner, and Host obtained a significantly stronger statement. Namely, let € be a finite Borel partition of X such that each element of { is contained in an element of the cover U. Then there eXUlts an f-invanant ergodic measure J-1 on X for which hx(f,U)::; hp(f,€). Measure-Theoretic Entropy Let p. be a Borel probability measure on X (not necessarily invariant under f). Consider a finite open cover U of X. According to Section 10, the C-structure 7 = (S, :F, f,, T/, ¢) on X, introduced by (11.1), (11.2), and (11.3), generates the Caratheodory dimension of p, and lower and upper CaratModory

Chapter 4

78

capacities of /J specified by the cover U and the map f. We denote them by P11 (1p,U), ~(IP,U), and GP11 (1p,U) respectively. We have that

= inf {Pz(!p, U) : p(Z) =

P 11 (1p, U)

1},

GP 11 (1p,U)

= j~ inf {CPz(!p,U):

p,(Z) ~ 1- o},

GP 11 (1P,U)

= j~ inf {CPz(IP,U):

p,(Z) ~ 1- o}.

(11.14)

It follows from Theorem 11.1 that there exist the limits

P11 (1p) ~r lim P,.(IP,U), IUI->0

CPiiP)

~r lim CP,.(IP,U),

-

de£.-

-

CP,.(IP)

(11.15)

IUI->0-

= lUI-tO hm CP,.(ip,U).

Given a point x EX, we set in accordance with (10.1)

_

V ~,,.,a ( X, !p, U) -

. . f 1tm 11} N->oo

alogp,(X(U))

-Na + sup

N-l

L

cp(!k(y))

yEX{U) k=O

( U) -. V c,,.,a x, !p, = 1tm sup N-+oo u

alogp(X(U)) N-l

-Na + sup

L

1p(jk(y))

yEX{U) k:O

where the infimum and supremum are taken over all strings U with x E X(U) and m(U) = N.

Proposition 11.1. If p, is a Borel probalnlity measure on X invanant under the map f and ergodic, then for every a E R and p,-almost every x E X lim 12c,."'(x,cp,U) = lim Dc,.o.(X,!{),U) =

IUI-+0

' '

IUI-+0

' '

a7(f)d ,

a- x 'P

/.1-

where h,. (f) is the measure-theoretic entropy off.

Proof. We need the following statement known as the Brin-Katok local entropy formula (see [BK]).

Lemma. For IJ.-almost every x

E X we have

. -log~J.(B,.(x,o)) _ . lim -logp,(Bn{x,o)) _ . 1 1 -1m , 1 h ,. (/) -tmrm 6->0 ..~ n 6->0 n ..... oo n where Bn(x,o)

iB

the (n,o)-ball at x (see (11.10}).

Proof of the lemma. For the sake of reader's convenience we present a simplified version of the proof in [BK] which exploits the fact that the measure

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism p. is ergodic.

Fix

o>

79

0 and consider a finite meaBurable partition { with 0 ~ o. Denote by q..(x) the element of the

diam~ ~f max{diamCe- : Ce E partition

~n =~V

r

1

{V···V

rn€

containing x. Obviously, Oen (x) C Bn(x, o). By the Shannon-McMillan-Breiman theorem the following limit exists for p.-almost every x E X:

where h,..(f, ~) is the measure-theoretic entropy off with respect to{. It follows that lim lim -logi'(Bn(x,o)) ~ h (!,{) ~ h (!). 6--+0 n--+oo n ,.. ,.. We proceed now with the estimate from below. Fix c > 0. One can show that there exists a finite measurable partition (. of X satisfying (1) h,..(J, ~) ;::: h11 (J)- e; (2) Jt(o~) = 0, where{)~ denotes the boundary of the partition{. Foro>Olet U5({)

= {x EX:

the ball B(x,o) is not contained in Cdx)}.

n

Since 6 >0 U6({) = 8(. we obtain that Jt(U6({)) -+ 0 as o-+ 0. Therefore, one can choose oo > 0 such that Jt(U6(~)) ~ e for any 0 < o : =:; oo. Hence, by the Birkhoff ergodic theorem, for p.-almost every x EX there exists N 1 (x) such that for any n;::: N 1 (x), l

n-1

.

-L:xu (e)U'(x)) ::=:;e. ni=O 6

Let At= {x EX: N 1 (x) ~ £}. Clearly, the sets At are nested and exhaust X up to a set of measure zero. Therefore, there exists £0 > 1 such that I'( At) ;::: 1 - e for any l;::: lo. Fix l;::: £0 . Given a point x EX, we call the collection

(Ce(x), Ce(J(x)), ... , Ce(f"- 1 (x))) the (~,n)-name of x. If y E Bn(x,o) then for any 0 ::=:; i ::=:; n- 1 either f'(x) and J•(v) belong to the same element of~ or J•(x) E U6 ((.). Hence, if X E At and y E Bn(x, o), then the Hamming distance between ({, n)-names of x and y does not exceed e (recall that the Hamming distance is defined aB follows: ~ I;~,:-01 Jt(Ce(/i(x)}ACe(!i(y)))). Furthermore, for x E At, Bowen's ball Bn(x,c5) is contained in the set of points y whose (~,n)-names are e-close to the ({, n)-name of x. It can be shown that the total number Ln of such ({, n)-names admits the following estimate:

80

Chapter 4

where K 1 > 1 is a constant independent of x and n. We wish to estimate the measure of those points in At whose (~, n )-names have an element of the partition ~n of measure greater than exp((-h"(f,~) + 2K1 e)n) in therr Hamming .:-neighborhood. Obviously, the total number of such elements does not exceed exp ((hi'(J, ~)- 2K1e)n). Hence, the total number Qn of elements in their Hamming .:-neighborhood satisfies

By the Shannon-McMill an-Breiman theorem for JL-almost every x E X there exists N2(x) such that for any n 2: N2(x),

Let Bk = {x EX: N 2 (x) ~ k}. Clearly, the sets Bk are nested and exhaust X up to a set of measure zero. Therefore, there exists k > 1 such that JL(Bk) 2: 1-.: for any k 2: /co. Fix such a number k and consider those of the Qn elements of whose intersection with At n Bk have positive measure To estimate their total measure Sn we multiply their number by the upper bound of their measure

en

This implies that for any sufficiently small e > 0 and 5 > 0 lim -IogJL(Bn(x,o)) >h (1")-Kc:>h (1)-c:-K.: p. ,.. 2 I' 2

-

n--+oo

n

for every point x in a subset D c At n Bk of measure 2: JL(At n Bk) - Kae (here K2 > 0 and K 3 > 0 are constants independent of x, and e). Therefore,

o,

This completes the proof of the statement. We continue the proof of the proposition. Let U be a finite open cover of X and o(U) its Lebesgue number. Since o(U) -t 0 as lUI -t 0 it follows from (11.11) and the lemma that h"(f)

=

lim

lim inf -logJL(X(U)) N . -li -logJL(X(U)) =- 11m m sup , IUI-+0 N-+oo U N lUi-tON-too U

(11.16}

where the infimum and supremum are taken over all strings U for which x E X(U) and m(U) = N. Let us fix a number c: > 0. Since

0 such that j
C-Structures Associated with DynaiWcal Systems: Thermodynamic Formalism

81

o.

x,y EX with p(x,y) S: Therefore, if lUIS: t5 then by view of Birkhoff ergodic theorem, we obtain for J.t-almost every x E X that

I

L

1 N-1 cp(f"(y))lim inf sup N

N-->oo U y€X(U)

L

lim sup sup N1 N-1 cp(fk (y)) -

-

N-->oo U 1

y€X(U)

1 J

I

cpd~-t S: o:,

X

k=O

cp d~-t I S: c,

X

k=O

where the infimum and supremum are taken over all strings U for which x E X(U) and m(U) = N Since cis arbitrary this implies that N-1

lim lim inf sup Nl Lcp(J"(y)) IUI-->ON-->co U yEX(U)

k=O

(11.17)

N-1

=

lim lim sup sup Nl IUI-->ON-->oo u

y€X(U)

L cp (!" (y)) = k=O

{ cp d~-t. Jx

The desired result follows immediately from (11.16) and {11.17). We now use Proposition 11.1 to prove the following result. Theorem 11.6. Let f be a homeomorphism of a compact metric space X and p a non-atomic Borel ergodic measure on X. Then

Proof. Set h = h,.(f) 2: 0 and a= fx cpd~-t. We first assume that a > 0. We wish to use Theorems 3.4 and 3.5 to obtain the proper lower bound for P,.(cp) and upper bound for CP,.(cp) To do so we need to find estimates of 12c,p,a(x, cp,U) and 'Dc,,.,a (x, cp, U) from below and above respectively which do not depend on

a. Fix c, 0 < c < ~. By Proposition 11.1 one can choose p-almost every x E X,

ah 'l2c,,.,a(x,cp,U) 2: - a-a

o > 0 such that for

-o;.

Note that the function g(a) = ah(a- a)- 1 - o: is decreasing. Assuming that a varies on the interval [h +a - c, h + a], we obtain that for p-almost every x E X,

12c,p,a(x,cp,U) 2: h+a- 2c. We conclude, using Theorem3.4, that Pp(cp,U) 2: h+a-2e- and hence Pp(cp,U) 2: h +a. Since this holds for every finite open cover U by (11.15) we obtain that

P,.(cp)

~

h +a.

82

Chapter 4

We now show that CP,_.(cp):::; h+ a. Fix c > 0. Let~= {Ct.·· .,Cv} be a finite measurable partition of X with jhi-'(J,{)- hj :::; c and U = {U1 , ... , Up} a finite open cover of X of diameter :::; c for which C; C U;, i = 1, . ,p. By the Birkhoff ergodic theorem for J.'-almost every x E X there exists a number N1(x) > 0 such that for any n:?: N 1 (x),

'

~I:cp(f"(y))- al S c.

(11.18)

k=O

By the Shannon-McMillan-Breiman theorem for J.t-almost every x E X there exists a number N2(x) > 0 such that for any n :::: N2(x),

~~Jog/-I(Cen(x))+h,.(f,~)l ::;c.

(11.19)

Let !:!.. be the set of points for which (11.18) and (11.19) hold. Given N > 0, consider the set I:!..N = {x E !:!.. : Nt(x) :::; Nand N2(x) :::; N}. We have that I:!..N c I:!..N+1 and !:!.. = UN>oi:!..N Therefore, given 5 > 0, one can find No > 0 for which J.'(l:!..n 0 ) :?: 1 - 5. Fix a number N :?: No and a point x E I:!..N. Let U be a string of length m(U) = N for which x E X(U). It follows from (11.18) that

I~

sup tcp(f"(y))-a,Se+/,

(11.20)

yEX(U) k=O

where 1 = -y(U). Furthermore, using (11.19) we obtain that 1-!(C{N(x))? exp(-h- 2e:)N. This implies that the number of elements of the partition ~N that have non-empty intersection with the set I::J.N does not exceed exp(h + 2e:)N. To each element C{N of the partition f.N we associate a string U of length m(U) = N for which C{N c X(U) The collection of such strings consists of at most exp(h + 2e:)N elements which comprise a cover Q of I:!..N. By (11 8) and (11.20) we obtain that

A(I::J.N,cp,U,N):::;

L

UEQ

exp ( sup tcp(f"(y))) yEX(U)k=O

:::; exp(a + h + 3e + 1)N. In view of (11.7) this means that

CPil.N(cp,U) :::;a+h+3e+"f· This implies that CP,.(cp,U) S a+h+3c+"f. Passing to the limit as diamU -t 0 yields that CP,.(r.p) :::; a+ h + 3c. It remains to note that £ can be chosen arbitrarily small to conclude that C P,. ( cp) :::; a + h In the case a:::; 0, let us consider a function 1/J = cp + C, where C is chosen such that fx'I/Jd~-t > 0. Note that P,.(,P,U) = P"(cp,U) +C and CP~-'(1/!,U) = CP"(r.p,U) +C, and the desired result follows. •

C-Structures Associated Wlth Dynamical Systems: Thermodynamic Formalism

83

As an immediate consequence of Theorem 11.6 we obtain that

h,..(f) == P,..(O) == CP,.. (0)

= CP,..(0).

These relations reveal the "dimension" nature of the notion of measuretheoretic entropy, introduced by Kolmogorov and Sinai within the framework of general measure theory. One can obtain another "dimension" interpretation of measure-theoretic entropy using Proposition 11.1. Namely,

h,..(f) = lim f1c,.. "'(x, O,U) = lim de~-' "'(x, O,U). IUI-+0

' '

IUI-+0

' '

In conclusion, we point out a remarkable application of relations (11.14) and (11.15) known as the inverse variational principle for topological pressure:

h~-'(f)+ LVJdp.==inf{Pz(rp) In particular, when rp topological entropy

p.(Z)=l}.

= 0 this gives the inverse variational principle for

hi-' (f)= inf {hz(J): p.(Z) = 1}.

This result was first established by Bowen [Bol]. Let us also point out that the requirement in Proposition 11.1 and Theorem 11.6, that f1- is ergodic, is crucial; they may not hold true otherwise.

12. Non-additive Thermodynamic Formalism Let (X, p) be a compact metric space with metric p, f: X -t X a continuous map, and r.p = { rpn: X -t lR} a sequence of continuous functions. Consider a finite open cover U of X and definE' for each n ~ 1

"fn(r.p,U) == sup{lrpn(x)- rpn(Y)I: x,y E X(U) for some U E Sn(U)}. We assume that the following property holds: lim lim 'Yn(r.p,U) n

=0

IUI-+On-+oo

(12.1)

(since 'Yn(r.p,U) 2: 0 one can show that the limit exists as IUI-t 0). We define now the collection of subsets F by (11.1) and (11.2) and three functions (., 1], 'ljJ: S(U)-+ lR as follows

€(U)

= exp(

sup rpm(U)(x) ),

TJ(U) = exp(-m(U)),

zEX(U)

.,P(U) = m(U)- 1 •

(12 2)

Chapter 4

84

One can verify using (12.1) that the collection of subsets :F and the functions 1 in Section 10 and hence determine a C-structure r = (S,:F,e, 17, 1/;) on X. The corresponding Caratheodory function mc(Z, a) (Z c X and a E R) depends on the cover U and is given by

1/,

e, and 1/; satisfy Conditions AI, A2, A3, and A3

mc(Z,a) = lim M(Z,a,ip,U,N), N-too

where M(Z,a,ip,U,N)

= inf { E exp (-am(U) + g

ueg

sup zEX(U)

V'm(U)(:v))}

and the infimum is taken over all finite or countable collections of strings g C S(U) such that m(U)? N for all U E g and g covers Z Furthermore, the Caratheodory functions r.c(Z, a) and rc(Z, a) (where Z C X and a E IR) depend on the cover U and are given by r.c(Z,a)

= N-+oo lim R(Z,a,ip,U,N),

rc(Z,a)

= N-+oo lim R(Z,a,ip,U,N),

where R(Z, a,ip,U,N) =in£ { C

E exp (-aN+

UEQ

sup xEX(U)

V'N(x))}

and the infimum is taken over all finite or countable collections of strings g c S(U) such that m(U) = N for all U E g and g covers z. According to Section 10, given a set Z C X, this C-structure generates the Caratheodory dimension of Z and the lower and upper Caratheodory capacities of Z specified by the cover U and the map I. We denote them, respectively, by Pz(rp,U), CPz(rp,U), and GPz(rp,U). Repeating arguments in the proof of Theorem 11.1, one can show that for any Z C X the following limits exist: Pz(rp) ~£ lim Pz(rp,U), IUI-+0

GPz(rp) ~ lun CE.z(rp,U), IUI-+0

CPz(rp) ~ lim CPz(rp,U). !UI-+O

We call the quantities Pz (rp), C P z (rp), and G P z (rp), respectively, the nonadditive topological pressure and non-additive lower and upper capacity topological pressures of the sequence of functions rp on the set Z (with respect to f). They were introduced by Barreira in [Bar2]. We emphasize that the set Z can be arbitrary and need not be compact or invariant under the map I.

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

85

The quantities Pz(cp), CPz(cp), and CPz(cp) have the properties stated in Theorems 11.2 and 11.3. One can check that the functions 7] and 1/J satisfy Condition A4 in Section 10. Therefore,

CPz(cp) -

CPz(cp)

1 = IUlim lim N Iog A(Z,cp,U,N), l-+0 n-+co -1 = IUI-+0 lim lim Nlog A(Z, cp,U, N), n-+co

where, in accordance with (2.3) and (12 2), A(Z,cp,U,N)==inf{Lexp( sup rpN(x))} Q

UEQ

(12.3)

"'EX(U)

and the infimum is taken over all finite or countable collections of strings g c S(U) such that m(U) = N for all U E 9 and 9 covers Z. The use of the adjective "non-additive" is due to the following observation. A sequence of functions cp = { rpn} is called additive if

rpn+m(x) = rpn(x)

+ rpm(r(x))

for any n, m 2: 1, and x E X. One can verify that the sequence cp is additive if and only if n-1

rpn(x)

=L

rp(fk(x)),

k=O

where rp is a function. It is not difficult to check that an additive sequence satisfies (12.1) if the function rp is continuous. Furthermore, in this case, for any ZcX,

Pz(cp)

= Pz(rp), CPz(cp) = CPz(rp), CPz(cp)

= CPz(rp).

In the second part of the book we will often deal with non-additive sequences of functions One can naturally associate such sequences with geometric constructions of a general type in dimension theory (including Moran-like geometric constructions; see Chapter 5) as well as with dynamical systems of hyperbolic type (including smooth expanding maps and Axiom A diffeomorphisms; see Chapter 7). The non-additive topological pressure will be used as an essential tool in computing the Hausdorff dimension of the limit sets of geometric constructions as well as of invariant sets for hyperbolic dynamical systems. We note that the sequences of functions we will deal with, being in general non-additive, often satisfy a special property which we now describe. We call a sequence of functions cp = { rpn} sub-additive if for any n, m ~ 1, and x E X,

rpn+m(x) $ rpn(x)

+ rpm(r(x))

(12 4)

The proof of the following result is similar to the proof of Theorem 11.5.

86

Chapter 4

Theorem 12.1. [Bar2] (1) If a set Z c X lS f -invariant and a sequence of functions r.p is sub-additive then for any finite open coverU of X we have CPz(cp,U) = CPz(cp,U). (2) If a set Z C X is f -invanant and compact and a sequence of functions r.p is sub-additive and sattsfies 'Pn :=:; 'Pn+l

for some K Pz(r.p,U)

+K

> 0 then for any finite open cover U of X,

1 = CPz(cp,U) = -CPz(cp,U) = n-+oo lim Nlog A(Z, cp,U, N),

where A(Z, r.p,U, N) is gwen by {12.2).

(12.5)

Appendix II

Variational Principle for Topological Pressure; Symbolic Dynamical Systems; Bowen's Equation

The mathematical foundation of the thermodynamic formalism, i.e., the formalism of equilibrium statistical physics, has been led by Ruelle [Rl] Bowen, Ruelle, and Sinai have used the thermodynamic approach to study ergodic properties of smooth hyperbolic dynamical systems (see references and discussion in [KH)). The main constituent components of the thermodynamic formalism are: (a) the topological pressure of a continuous function cp which determines the "potential of the system"; (b) the variational principle for the topolog1eal pressure, which establishes the variational property of the "free energy" of the system (which is defined as the sum of the measure-theoretic entropy and the integral of cp with respect to a probability distribution in the phase space of the system); (c) existence, uniqueness, and ergodic properties of equilibrium measures (which are extremes of the variational principle). Ruelle's version of the thermodynamic formalism is based on the classical notion of topological pressure for compact invariant sets. In this appendix we outline more general versions of the variational principle by considering topological pressure on non-compact sets and non-additive topological pressure. We use the thermodynamic formalism to describe Gibbs measures for symbolic dynamical systems. For the reader's convenience we also provide a brief description of basic notions in symbolic dynamics which are widely used in the second part of the book. One of the main manifestations of the thermodynamic formalism in dimension theory is different versions of Bowen's equation. Its roots often provide optimal estimates (and sometimes the exact value) of the dimension of an invariant set. We describe some properties of the pressure function and study roots of Bowen's equation. In the second part of the book the reader will find many applications of these results to dimension theory as well as to the theory of dynamical systems.

Variational Principle for Topological Pressure Let (X, p) be a compact metric space with metric p, f: X-+ X a continuous map, and cp: X -+ R a continuous function. Denote by !m(X) the set of all/invariant Borel ergodic measures on X. Given an /-invariant (not necessarily

87

88

Appendix II

compact) set Z C X, denote also by !m(Z) C !m(X) the set of measures I' for which ~J.(Z) = 1. For each x EX and n?: 0 we define a probability measure 1-'z,n on X by

where 5y is the a-measure supported at the point y. Denote by V(x) the set of limit measures (in the weak topology) of the sequence of measures (tJ..,,n)nEN· It is easy to see that 0 :f. V(x) C rol(X) for each x E X. Put .C(Z) = { x E Z : V(x) nrol{Z) :f. 0}. It is easy to check that .C(Z) is a Borel /-invariant set. The following statement establishes the variational principle for the topological pressure on non-compact sets. It was proved by Pesin and Pitskel' in (PP].

Theorem A2.1. Let Z function

c X be an f-invanant set. Then for any continuous

Pc(z)(t.p)

=

sup (hJJ{f) + f cpdl£) JJE!JJ!(Z) lz

(recall that Py(t.p) is the topological pressure of the function

One can prove the following statement.

Lemma 1. For any e > 0 there exists o, 0 < o :=:; s, a finite Borel partition ~ = {C1, .. ,Cm}, and a finite open coverU = {U1, . ,Uk},k?: m of X such that (1) diamU;:::; s, diamCi:::; s, i = 1, . . k, j = 1,. ,m; (2) U; C C;, i = I, ... , m (where A denotes the closure of the subset A C Y in the induced topology of Y }; (3) tL(C; \ U;):::;

o, i =

k

1, .. , m and!-'( UUi) S r5; i=m+l

(4) 2r5logm:::; s. Given y E Y, let tn(Y) denote the number of those i, 0:::; f.< n, for which Ji(y) E U; for some i = m + 1, ... , k. It follows from Lemma 1 (see Statement 3) and the Birkhoff ergodic theorem that there exist N 1 > 0 and a set A 1 c Y such that ~J.(A 1 ) ?: I - fJ and for any y E A1 and n?: N 1 , (A2.2)

89

Variational Principle for Topological Pressure

Set ~n = ~ V /- 1 ~ V · · · V f-n~. It follows from the Shannon-McMillan-Breiman theorem that there exist N2 > 0 and a set A2 C Y such that J.t(A2) ~ 1- 5 and for any y E A 2 and n ~ Nz, J.t(C~Jy))::; exp(-(h,..(f,~)- il)n),

(A2.3)

where 0<,. (y) denotes the element of the partition ~n containing y. At last, using the Birkhoff ergodic theorem one can find N3 > 0 and a set A3 C Y such that J.t(As) ~ 1 - tl and for any y E A3 and n ~ N3,

1~ ~~u·(y)) _ l ~dills o.

(A2.4)

Set N = max{N1 , N2, N3} and A= A 1 n A2 n Aa. We have that J.t(A) ~ 1- 3o. Choose any n

~

(A2.5)

N and any

A<

h,_.(f,~) + [ ~dJ.t- -y(U)

(we recall that -y(U) = sup{l~(x)- ~(Y)i: x,y E U; for some U; E U}). By the defimtion of the topological pressure there exists a finite or countable collection of strings g C S(U), which covers Y, (i.e., the collection of sets {X(U): U E Q} covers Y) such that m(U) ~ N and

L exp (-Am(U) + ueg

1

sup "'EX(U)

m~-
s 6.

k=O

(A2.6) Let Qe c g be a subcollection of strings for which m(U) = f. and X (U) n A f. 0. Denote by Pe the cardinality of Qt. Set Yt = U X(U). UE!.lt

Lemma 2. We have Pe

~

J.t(Ye n A)exp((h,_.(/,~)- o- 2ologm)f.).

PllOof of the lemma. Let Le be the number of those elements of the partition for which (A2.7)

~t

It is easy to see that (A2.8) where the sum is taken over all elements of the partition ~t for which Condition (A2.7) holds. Since 0<,, n A 2 =f. 0 we obtain by (A2.3) and (A2.8) that

Lt

~

J.t(Yt n A) exp ((hp.(f, ~)- o)e).

(A2.9)

Let us fix U E gt· Since X(U) n A1 =f. 0 the inequality (A2.2) implies that the number Se(U) of those elements q, of the partition ~e, for which X(U) n Cc,, n A f. 0, admits the following estimate: Se(U)::; m 20t

= exp(28elogm).

The lemma now follows from (A2.9) and (A2.10).

(A2.10)

90

Appendix II

We continue the proof of the inequality (A2.1). Using Lemma 2, (A2.4), and (A2.5} we obtain that

L UEQ

2:

f: L 2: f:

exp

(-,\£ +

Ptexp (( -,\ +

p(Yt n A) exp

m~-I cp(fk(x)))

sup

k=O

xEX(U)

l=N

f:

k=O

xEX(U)

t=N UE!It

2:

m~- cp(fk(x))) 1

exp (-,\m(U) + sup

((h,.(f,~) + {

}y

l=N

{ cpdJL- o- "f(U))t)

}y

cpdp,- 2o- 2r5logm- -y(U)- .\)t)

00

2:

I: p,(Yt n A) = p,(A) 2: 1 - 36. l=N

We used here the fact that for sufficiently small e

h,_.(f,e) + [ cpdp,-

2: h,_.(f, {) +

i

2o- 2ologm- "t(U)-)..

cpdJL- 2£- 2£ log m- "f(U)-).. > 0.

By (A2.6) we have that M(Y,>.,cp,U,N) 2: 1- 4o 2: 1/2 if e: (and hence 6) is sufficiently small. Therefore, Py(cp,U) >)..and hence

Py(cp,U) 2: h,.(f,{) +

i

cpdp- "t(U).

Letting e: --+ 0 yields -y(U) --+ 0 and diam{ --+ 0. The latter also implies that h,_.(!,f,) approaches h,_.(f) and (A2.1) follows. Consider a Borel /-invariant subset Z c X. Given a measure 1-' E !m(Z) denote by = {x E Z: V(x) = {j.t}}. It is easy to see that JL(Z,.) = 1 and that c C(Z). Therefore, by (A2.1),

z,.

z,.

Pc.(ZJ(cp) 2: Pz,.(cp) 2: h,.(f)

+

l

cpdJL.

(A2.11)

We now prove that for any Borel /-invariant (not necessarily compact) subset Y c X with the property that V(x) n !m(Y) i 0 for any x E Y we have

Py (cp) ::;

sup

,.e!l11(Z)

(h,. (f)

+ f

}y

cp dJL)

(A2.12)

Let E be a finite set and.!!= (a 0 , ... , ak-Il E Ek. Define the measure /-tg,_ onE by

/Lg,_(e)

= ~(the number of those j

for which ai =e).

Variational Principle for Topological Pressure

91

Set H(gJ =- 'L:tt!!.(e) logp!!.(e). eEE

Consider the set R(k, h,E) = {!! E Ek· H(g_) ~ h}.

The following statement describes the asymptotic growth in k of the number of elements in the set R(k, h, E). The proof is based on rather standard combinatorial arguments and is omitted.

Lemma 3. (See Lemma 2.16 in [Bol]}. We have -1 lim -k logiR(k,h,E)i ~h.

k--too

Let U = {U1 , .•• , Ur} be an open cover of X and c > 0.

Lemma 4. Given x E Y and IJ. E V(x) n rot(Y), there exzsts a number m > 0 such that far any n > 0 ane can find N > n and a strmg U E S(U) with m(U) = N satisfying: (1) X E X(U); (2) m(U)-1

sup xEX(U)

E

Cl'(x))

~ N(h~-<(f,~)+ f

}y

k=O


(3) the strmg U contains a substrmg U' af length m(U') = km ~ N - m which, being wntten as J! (ao, ... , ak_ 1 ), satisfies the inequality

=

1 -H(g.) ~ h,.(!) + e:.

m

(A2.13)

Proof. There exists a Borel partition ( = { C1 , •.. , Cr} of X such that

C; c

U;, i = 1, ... , r. By the definition of measure-theoretic entropy there exists a number m > 0 such that

Assume that IJ.z,n; converges to the measure n 1 > n we write n n'-n f..£x,n'

=

n'J.tx,n

J.t

for some subsequence n 1. For

+ ~JI.fn(x),n'-n·

This shows that if we replace the number n; by the closest integer wluch is a factor of m then the new subsequence of measures will still converge to p. Thus, we can assume that nj = mk;. Let D 1 , .•• , Dt be the non-empty elements of the partition ~ = ( V · · · V f-{m+ll( Fix {3 > 0. For each D; one can find a compact set K; C D; such that f..£(D; \ K;) ~ {3. Each element D; is contained in an element of the cover

92

Appendix II

V = U V · · · V f-(m+l)U which we denote by B;. One can find disjoint open subsets V; such that K, C V; c B,. Moreover, there exist Borel subsets V;* comprising a Borel partition of X such that Vi c V;* c B,. Given nj = mki, we denote by M,(j) the number of those s E [0, ni ), for which f8(x) E V;*, and by M;~~ the number of those s E M;(j), for which s = q (mod m). Define pi?) t,q

= M~i) /k3'· t,q

pf.?l •

= M.(j) /n J· = .!. m (P~j) t,O + · · · + p~i) ,,m-1 ) I

Since J-Lz,ni converges to the measure J-L we obtain that hm ppl ~ J-L(K;) ~ J.L(D;) - (3, j-+oo

If j is sufficiently large and (3 is sufficiently small we find that

! (- ¥~j) logp~il)

~ (- ~J.L(D;) log J-L(Di)) + ~ ::; h,.(f) +e.

::;

Since the function g(x) = -x log x is convex we obtain that m-1

g(pJil)

~ ~ 2:;uCP~~JJ q=O

and hence

Therefore, the inequality

L9(Pl:~)::; i

L9CPlj)) i

should hold for some q E [0, m). This implies that

Let N = ni + q. For some sufficiently large j we choose a string U E S(U) with m(U) = N in the following way. For s < q we choose u. E U which contains f"(x) Further, for every V;* we choose a string U; = Uo,i .. Um-l,i such that

93

Variational Principle for Topological Pressure

Then, for s ;::: q we write s = q + mp + e with p ;=:: 0 and m > e ;::: 0 and set u. = Ue,i• where i is chosen such that r+mP(x) E Vi"· Set ap = Uo,;Ul,i ... Um-l,i , ak;-1), the and consider the string U0 ... Uq-laoa 1 ... ak,-l· For !! = (ao, measure 11-1}_ is given by probabilities vi~~, i = 1 ... t and it satisfies

This proves the first and the third statements of the lemma. Since 11-z,n; converges to the measure 11- we obtain for sufficiently large N that

This implies the second statement and completes the proof of the lemma. Given a number m > 0, denote by Ym the set of points y E Y for which Lemma 4 holds for this m and some measure J.l. E V(x) n !m(Y) We have that Y = Um:>O Ym. Denote also by Ym,u the set of points y E Ym for which Lemma 3 holds for some measure 11- E V(x) n !m(Y) satisfying Jy cpdiJ, E [u- e, u + e]. Set c = sup (h~'(f) + f cpdl-£).

}y

JSE!m(Y)

Note that if x E Ym,u then the corresponding measure 11- satisfies h,_.(j) :s; c-u+e. Let Qm,u be the collection of all strings U described in Lemma 4 that correspond to all x E Ym,u and all N exceeding some number N 0 • It follows from (A2.13) that for any x E Ym,u the substring constructed in Lemma 3 is contained in R(k,m(h + e),Um), where h = c- u +e. Therefore, the total number of the strings constructed inLemma4 does not exceed b(N) = IUimiR(k, m(h+e),Um)l. By Lemma 3 we obtain that -. Iogb(N) 1rm N

N-too

h

(A2.14)

:s; +e.

Since the collection of strings Qm,u covers the set Ym,u we conclude using Lemma 4 and (A2.14) that (

oo

M(Ym,u,>..,cp,U,No):S

L

b(N)exp

N=No

:s;

)

m(U)-1

->..m(U)+ sup zEX(U)

L

lp(jk(x))

lo=O

N~o b(N) exp ( ->.m(U) + N ([ cpdl-£ + 'Y(U) +e)) .

If N 0 is sufficiently large, we have that b(N)

:s; exp(N(h + 2e)). Hence, (3No

M(Ym,u,>., cp,U,No) :S:: l - (3'

(A2.15)

Appendix II

94

where {3 = exp

(-A+ h+ [

!pdJL+'Y(U)

+3e).

It follows from (A2.15) that if>.> c + ')'(U} + 4e then mc(Ym,u, >.) = 0. Hence, >.;:::: Py,,,.(~P,U). Assume that points U1, ... ,ur form an e-net of the interval HI~PII, II~PI!]. Then m=li=l

We have that>. 2:: Py"'·"• (~P,U) for any m and i. Therefore,

). 2::

SUJ?PYm,u; (~P,U)

m,•

= Py(~P,U)

This implies that c + 'Y(U) + 4e;:::: Py(~P,U). Since e can be chosen arbitrarily small it follows that c + 'Y(U) ;:::: Py(~P,U). Taking the limit as lUI -+ 0 yields • c ;:=:: Py (
fz

C

X, any con-


(2) classical vanational pnnciple for topological pressure on compact sets: for any continuous function

E!m(X)

(h,..(f) +

f
(3) classical vanational principle for topological entropy on compact sets: hx(f)

=

sup h,.(f); !'E!m(X)

(4) for any set Z

c X,

any continuous function cp on X, and any measure

JL E !JJt(Z), (A2.16) where G,. is the set of all forward generic points of the measure p (i.e., the points for which the Birkhoff ergodic theorem holds for any continuous function on X); applying this equality to cp = 0 yields Bowen's formula for the measure-theoretic entropy of f

h,.(f) = ha,.(f); (5) let Z

c

X be

an invariant set; if V(x) n !JJt(Z) i= 0

Pz(~P) =

sup i'E!m(Z)

(h,.(f) +

j cpd~-t) . Z

for every x E Z, then

Variational Principle for Topological Pressure

95

In [PP], Pesin and Pitskel' described an example that shows that the assumption "V (x) n VJZ( Z) =1- 0 for every x E Z" is crucial and that the variational principle for the topological pressure on non-compact sets may fail otherwise. It also reveals some new and interesting phenomena associated with the topological entropy in the case of non-compact sets.

Proposition A2 .I. Let (E2, 0') be the full shift on two-sided sequences of two symbols (the classical Bernoulli scheme) and Z ={wE E 2 w ¢. Gp. for any measure J.t E VJZ(E2)}

= E2 \

U

Gp,

pE!)]I(:Eo)

where Gp is the set of points which are both forward and backward genenc with respect to the measure J.l· Then hz(a) = hE 0 (a) = log2.

Remark. Note that

~-t(Z)

= 0 for any J.l E !m(E 2 ). hz(a) = hE 2 (a)

Hence,

>0=

sup

hp.(a)

p.E!)]I(Z)

and the variational principle for the topological pressure fails (note that in this case .C(Z) = 0). The set Z is an example of so-called metrically irregular sets for dynamical systems which we describe in detail in Appendix IV.

Proof of the proposition. Consider a Bernoulli measure J.t on E 2 such that ~-t(Co) = p and J.L(CI) = 1 - p = q, where Co = {w = (i,) : io = 0} and C 1 = {w = (in) : io = 1} are cylinders and pi- q. Given 6, 0 < o ~ 1, we can choose p such that log2- h11 (a) ~ 6.

Let {n,.} be a sequence of positive integers satisfying n,. < nk+ 1 and nk -+ oo as k -+ oo. We decompose the set of all integers into two disjoint subsets Q1 and Q2 in the following way: i E Ql if n2" ~ JiJ < n21<+ 1 and i E Q2 otherwise. Consider the map 'if;: E 2 -t E 2 given as follows in

('if;(w))n = { in+ 1 (mod 2) where w = (... L 1 i 0 i 1 .•• ). Note that 'if; is a homeomorphism (even hi-Lipschitz) but it does not commute with the shift. Set Y = 'if;(G1,.).

Lemma I. If the sequence {nk} grows sufficiently fast then Y C Z.

Proof of the lemma. Let

x be

the indicator of the set C0 • If the sequence

{ nk} increases sufficiently rapidly then by the Birkhoff ergodic theorem for any w E Y we obtain that

1

lim 2

k--+oo

Since p =1- q it follows that the Birkhoff sum ~ hence wE Z.

n-1

n2k

n ..

+

1

.L

X(O''(w)) = q.

s=-n2k

_E x(ai(w)) does not converge and

•=O

96

Appendix II

Let (be the partition of I:2 by the sets Co and C 1 . Given m > 0 and n > 0, denote by Tfrn = . V uif, and f.n = ('7m)n = . V <Jj'7m J=-m

J=-n

Lemma 2. For every wE:: lim

n-+00

6,.

(-~log p(Cen (.,P(x)))) n

== hiL(u).

=

Proof of the lemma. Set Q(i, m, n) Q; n [-m- n + 1, m + n- 11, where i = 1, 2, m > 0, and n > 0. By the law of large numbers for every wE GIL there exists the limit

1

-- lim )/ n-+oo/Q(l, m, n .

~ L

log_u(C,.;{(?f(w}}} = h,.(u),

;EQ(l,m,n)

where fAI denotes the number of elements in the set A. Since the involution 0 o-+ 1, 1 o-+ 0 transfers the measurt~ J.L into a Bernoulli measure with the same entropy we obtain that . .,...,-,...---:-:1 - lun n-+oo

fQ(2, m, n)f.

L

JEQ(3,m,ro}

For any n > 0 we write l 1 -~ logJ.L(C~. (.,P(x))) =-;:;log

m+n-1

IT

p(Cu>((1,b(w)))

i=-m-n+l

=

IQ(1, m, n)l n

~

1

)Q(l, m, n)/.

logp(Co-J{(lb(w)})

L

JEQ(l,m,n)

IQ(2,m,n)l n

1

IQ(2,m,n)f.

"

~

logp(CuJ{(.,P(w))).

JEQ{2,m,n)

•

The desired result follows

We proceed with the proof of the proposition. Fix m > 0 and consider the partition fJm which is also a finite cover of I:2. It follows from Lemma 2 that for any r > 0 there exib"tl. a set D c GIL and a number N > 0 such that J.L(D) 2: 1-"( and for any wED and n 2: N,

J.L(C(,..,). (.,P(w))):::; cxp ( -n(h,.(u)- -y)). Fix n 2: N and choose a coll~>ction of strings Q c S(7Jm) which cover Y (i.e., the sets {E2(U): U E Q} cover Y) and satisfy. m(U) 2: nand

I

M(Y,a,0,7]m,n)-

L ueQ

exp{-om)l:::; "(

97

Variational Principle for Topological Pressure

Let 9t

= {U E g: m(U) = £} and Kt be the number of elements in 9t·

Set

Since the sets E2(U') and E2(U") are disjoint for any distinct U', U" E 9t we obtain that

Therefore, 00

L exp(-cm) = :L:exp(-al)Kt 2: ~-£(D)exp((-a+ hp(u)- 1)£) ueg

f=n 00

2::: /!(·r (Ee) n D) exp ((-a+ h (u)- 7)£) 1

11

i=n

2: {1- 21) cxp ({-a+ hp.(u)- 1)n) ifn is sufficiently big (such that J..t(1/l- 1 (En)nD) 2: (1-27)) For this implies that 1 M(Y, a, 0, 1Jm, n) 2: 1 - ')' 2: .

a<

h,..(u) -')'

2

Taking the limit as n--+ oo we obtain that mc(Z, a)= oo and hence

Since diam11m--+ 0 as m---+ oo this implies that hy(u) = Py(O) 2: log2- o- 'Y· Taking into account that the numbers 5 and 'Y can be chosen arbitrarily small we conclude that hz(u) 2: hv(u) 2: log2. Thib completes the proof of the proposition.

Equilibrium Measures Let rp be a continuous function on X. Given an /-invariant set Z C X we call a measure J..t = 1-'"' an equilibrium measure on Z corresponding to the function rp if 1-''1' E rot( Z) and hi'., (f)

+

r rp d/).op

1Z

=

sup

t~E!!71(Z)

(h,..(f) + 1rZ rp dj.£) •

The following statement establishes the existence of an equilibrium measure for a continuous function rp. Note that, in general, this measure may not be unique. We recall that a homeomorphism f of X is called expansive if there exists c > 0 such that for any two points x, y E X if p(fk(x), Jk(y)) ~ £ for all k E Z then x = y.

98

Appendix II

Theorem A2.2. (PP] Assume that the followmg conditions hold: (1) f is a homeomorphism of X; (2) f 'IS expansive; (3) the set !.Dt(Z) 'IS closed in !.Dt(X) (in the weak• -topology). Then for any contmuous function fP there e:nsts an equilibrium measure ~., on Z. Proof. Let~ be a partition of X with diam~ ~e. Set ~n = . is expansive we obtain that diamen -t 0. Therefore, h,..(f)

the other hand, it is easy to see that

h,_.(f,~n)

= h,_.(f,e).

V Ji~. Since f J=-n = n-+oo lim h,..(f,t;n)· On Therefore, h,_.(f) =

h,_.(f, t;). We now show that the map~>-+ h,_.(f) is upper semi-continuous on !.Dt(Z). Then the map /.1. >-+ h,_.(f) + f 1pd~ is also upper semi-continuous on !m(Z) and the desired result follows from Theorem A2.1 and the fact that an upper semicontinuous function on a compact set attains its supremum. Fix~ E !.Dt(Z), a> 0, and a partition t; = {C1 .•. Cn} of X with diamt;, ~e. For a sufficiently large m we have that

!_H,..(~ V · · · V rm+l~) ~ h,_.(f) +a.

m

Let us fix such an m. Given {3, choose compact sets

nrk(C;.)

m-1

K; 0

;.,._,

c

k=O

such that

This implies that L; ~£

m-1

U UJi(K;o

in.-1) C C;.

]=Oi;-i

Since L; are disjoint and compact one can find a partition with diam £;,' ~ e such that Li c intC;. We have that

~'

= {C~ . .

n

m-1

K; 0

;"'_ 1

c int

rk(c:.).

k=O

If a measure

and

11

E !.Dt(Z) is close to J1. in the weak•-topology then

C~}

of X

99

Variational Principle for Topological Pressure If {3 is sufficiently small it follows that

This completes the proof of the theorem. As a direct consequence of Theorem A2.2 we obtain the following statement. Theorem A2.3. Assume that a map f satisfies Conditions 1 and 2 of Theorem A2.2. Then for any compact f -invanant set Z C X and any continuous function rp on Z there exists an equilibrium measure Jl,

h,.~ (!) +

L

cpdp."'

= Pz(cp)

In particular, if a map f satisfies Conditions 1 and 2 of Theorem A2.2 then for any compact /-invariant set Z c X there exists a measure of maximal entropy, i.e., an equilibrium measure corresponding to the function cp = 0 for which hz(f) = sup h,.(f). I'E!m(Z)

Let us notice that, in view of {11.14), (1115), and Theorems 11.6 and A2.1 for any compact invariant set Z we have that

Pz(cp)

= supPp(rp) I'

(provided conditions of Theorem 11.6 hold). Therefore, the variational principle for topological pressure can be viewed as a variational principle for Caratheodory drmension (where the dimension is generated by the C-structure defined by (11.1)-(11.3); see Sections 1 and 5). Moreover, any equilibrium measure is a measure of full Caratheodory dimension (with respect to the space rot of invariant measures; see (5.4)). In particular, any measure of maximal entropy is a measure of full Caratheodory dimension. Non-additive Variational Principle We now state a non-additive version of the variational principle for topological pressure established by Barreira in [Bar2]. Let I()= {cp,.} be a sequence of continuous functions. Theorem A2.4. Let Z C X be an f -invariant set. Assume that there exists a continuous function 1/r X -4 lR such that Cf!n+l - Cf!n 0

f

-4 'if;

(A2.17)

Appendix II

100

uniformly on Z as n --+ oo. Then

As an immediate consequence of the above statement we have that if Z is an f -invanant compact set, then

Pz(t.p)= sup p.Erot(Z)

cX

(h,..(J)+lt/Jd/L),

where the function 1/J satisfies ( A2.17} Falconer [F3] established another version of the variational principle for topological pressure assuming that the sequence of functions t.p is sub-additive (it also should satisfy some other additional requirements) - the so-called "subadditive» variational principle for topological pressure. Symbolic Dynamical Systems We briefly describe some basic concepts of symbolic dynamics which are used in the second part of the book. For each p E N we denote the space of right-sided infinite sequences of p symbols by

We call the number ii the j-coordinate of the point w (we also use another notation wj)· We write w+ for points in ~t to stress that we are dealing with a right-sided infinite sequence. A cylinder (or a cylinder set) is defined as

We also use more explicit notation C~ in. Given /3 > 1, we endow the space Et with the metric I

_

00 """

1·~k - ~k., I

d{J(w,w)- ~ ~· k=O

(A2.18)

where w = (i 0 i 1 ... ) and w' = (i~ii ... ). It induces the topology on ~t such that the space is compact and cylinders are disjoint open (as well as closed) subsets. The (one-sided) shift on ~t is defined by

(we also use more explicit notation a+). It is easily seen to be continuous. A subset Q c ~t is said to be a-invariant if u(Q) = Q. When the set Q c ~t is compact and a-invariant, the map uiQ is called a (one-sided) subshift

101

Variational Principle for Topological Pressure Let A be a p x p matrix whose entries and O"-invariant subset

a;j

are either 0 or 1. The compact

is called a topological Markov chain with the transfer matrix A. The map 0"1~1 is called a (one-sided) subshift of finite type. It IS topologically transitive (i.e., for any two open subsets U, V c ~1 there exists n > 0 such that O"n(U) n V #= 0) if the matrix A is irreducible, i.e., for each entry O..i there exists a positive integer k such that a~j > 0, where a~i is the (i,j)-entry of the matrix Ak The map O"~~~ is topologically mixing (i.e., for any two open subsets U, V C ~~ there exists N > 0 such that O"n(U) n V #= 0 for any n > N) if the transfer matrix A is transitive, i.e., Ak > 0 for some positive integer k. We call (Q, O") a so fie system if Q C ~t is a finite factor of some topological Markov chain ~1, i.e , there exists a continuow; surjective map (: ~~ -+ Q such that O"IQ o ( = ( oO". An example is the even system, i.e., the set Q of sequences of 1's and 2's, where the 2's are separated by an even number of 1's Similarly to the above, we consider the space of left-sided infinite sequences

We also write w- for points in~;. A cylinder in~; is denoted by Cs_. more exphcitly C;:_,. "'). The (one-sided) shift is defined by

io

(or

(we also use more explicit notation O"-). It is continuous. Further, given a transfer matrix A= (a;j), we set ~A=

The map

0"1~:4

{w =

( ... Ltio) E ~;: a._.;_,.H = 1 for all

n EN}.

is a (one-sided) subshift of finite type.

We also consider the space of two-sided infinite sequences of p symbols

A cylinder (or a cylinder set) is defined as

C;m ;,. = where m

:<::::

{w =

( ..

·i-doil· .. ) E ~P :jk = ik, k = m, ... ,n},

n. Given f3 > 1, we endow the space E, with the metric (A2.18')

102

Appendix II

where w = (... Ltioi 1 ... ) and w' = (... i~ 1 i~ii ... ). It induces the compact topology on Ep with cylinders to be disjoint open (and at the same time closed) subsets. The (two-sided)shift u: Ep -t Ep is defined by u(w)k = WTo+l· It is an expansive homeomorphism. Given a compact u-invariant set Q C E,, we call the map uiQ a (two-sided) subshift. Let A be a p x p transfer matrix with entries 0 and 1. Consider the compact u-invariant subset

EA ={wEE,:

awnWn+l

=

1 for all n E Z}.

The map uiEA is called a (two-sided) subshift of finite type. Let us notice that given a point w E EA, the set of points w' E EA having the same past as w (i.e., w; = w~ for i ~ 0) can be identified with the cylinder c E~. Similarly, the set of points w' E EA having the same future as w (i.e., W; = w~ for i ~ 0) can be identified with the cylinder c;;, c E::i. Thus, the cylinder C;o c EA can be identified with the direct product C;~ X c;;,. For symbolic dynamical systems the definitions of topological pressure and lower and upper capacity topological pressure can be simplified based on the following observation (which we have already used in the proof of Proposition A2.1). Let Un be the open cover of Et by cylinder sets C; 0 ; ... Notice that IUnl -t 0 as n ~ oo and for any U E S(Un) the set X(U) is a cylinder set. Therefore, the function M(Z, a, cp,Un, N) can be rewritten according to (11 4) as M(Z,et,({),Un,N) =

c,:;;

i~f {

0; 0

2: im

exp (-a(m + 1) eg

+w

~up .

E •o

•m

(A2.19)

f:({)(uk(w)})} k=O

and the infimum is taken over all finite or countable collections of cylinder sets C; 0 im with m ;::: N > n which cover Z. Furthermore, the function R(Z, Ct, ({),Un, N) (N > n) can be rewritten according to (11.4') as R(Z, et, cp,Un, N)

c,o

~eQ exp (-a(N + 1) + we~~p •N~cp(uk(w}))

and the sum is taken over the collection of all cylinder sets C;0

z.

iN

(A2.19')

intersecting

Let (Q,u) be a symbolic dynamical system, where Q is a compact u-invariant subset of and cp a continuous function on Q A Borel probability measure J.1. = J.l.'P on Q is called a Gibbs measure (corresponding to ~P) if there exist constants D 1 > 0 and D2 > 0 such that for any n > 0, any cylinder set C; 0 .i .. , and any w E C; 0 ; .. we have that

Et

(A2.20)

Variational Principle for Topological Pressure

103

where P = Pq(cp). Note that if Condition (A2.20) holds for some number P then P = Pq. Indeed, in this case for every e > 0 by (A2.19) we obtain that 1

M(Z,P+e,cp,Un,N) S D1 i~f

~ 0, 0

p,(Cio i,.)exp(-e+r(Un))

imE9

S D1 1 exp( -e + r(Un)). Letting n -+ oo yields that Pq S P + e and hence Pq S P since e is arbitrary. The opposite inequality can be proved in a similar fashion (I thank S. Ferleger for pointing out this argument to me). Any Gibbs measure is an equilibrium measure but not otherwise. It is known that the spectjication property (see (KH] for definition) of a topologically mixing symbolic dynamical system (Q, u) ensures that any equilibrium measure corresponding to a Holder continuous function is Gibbs. It is known that any subshift of finite type (Et u) satisfies the specification property. Therefore, an equilibrium measure f.lcp, corresponding to a Holder continuous function cp, is a Gibbs measure provided the transfer matrix A is transitive. In this case it is also a Bernoulli measure. For an arbitrary transfer matrix A, by the Perron-Frobenius theorem one can decompose the set E~ into two shift-invariant subsets: the wandering set QI (correspondmg to the nonrecurrent states) and the non-wandering set Q2 (corresponding to the recurrent states). The latter can be further partitioned into finitely many shift-invariant subsets of the form E1,, where each matrix A; is irreducible and corresponds to a class of equivalent recurrent states (see [KHJ for details). Moreover, for each i there exists a number n; such that the map un• is topologically nuxing. Note also that any sofic system satisfies the specification property. We define the notion of Gibbs measures for two-sided subshifts. Let Q be a compact u-invariant subset of Ep and cpa continuous function on Q. A Borel probability measure p, = f.lcp on Q is called a Gibbs measure (corresponding to cp) if there exist constants D 1 > 0 and D 2 > 0 such that for any m < 0, n > 0, any cylinder set C;,. in, and any w E C;"' in we have that (A2.20')

where P = Pq(cp). Again, any Gibbs measure is an equilibrium measure and the specification property ensures otherwise. In the case of subshifts of finite type there is a deep connection between Gibbs measures for one-sided and two-sided subshifts. In order to describe this connection consider a two-sided subshift of finite type (EA, u) and a Holder continuous function

w~•> = i fori= 1, . . ,p and set n = (w< 1>, ... ,w
Appendix II

104

provided

Wo

= i. Further, we define the function

o
=

o&")

on EA by

00

o<">(w) =
E [<;?(ITi+l(rn(w)))-
Given a Holder continuous function ifJ on EA and a positive integer n, we define the n-variatwn of the function ¢> by var.. ¢ = sup{i¢(w)- ¢(w')l: w;

= w;, 0:::; i < n}.

If w and w' are two points whose i-eoordinates coincide for all i between -n and oo then varnifJ :::; C{P', where C > 0 is a constant and f3 is the coefficient in the d13-metric on EA (see (A2.18'). Lemma A2.1. The junction o

Proof. We have that 00

o<">cwJ =
2:::

[<;?
j=-1 00

+

L [
j=O

where

00

u(w) =

L

[IP(ui(w))-
j=O

In order to complete the proof of the lemma we need only to show that the function u(w) is Holder continuous. Since the i-coordinates of the points ui(w) and ui(rn(w)) coincide for all i between -j and oo we obtain that.

IIP(cri(w))-
w~

for Iii :::; n then for

and Therefore, [n/2]

L IIP(ui(w)}-
j=O

j>(n/2]

:::; 2C

((i:J

pn-j

j=O

This completes the proof of the lemma.

+

L j>(n/2]

w) <

oo.

•

Variational Principle for Topological Pre55ure

105

It follows from the lemma that the measure p. is the Gibbs measure corresponding to ()(u) Let rr+: EA-+ E! and 1r-: EA-+ EA: be the projections 1r+( .. Ltioit·· )=(ioit ... ),

11"-(... Ltioit ... )=( ... Ltio)

One can easily check that (J("l( .. Ltioi 1 ... ) = ()(")( i~ 1 i 0 i~ .. ) whenever i1 = ij for every j :2:: 0. This means that there is a function tp(u) onE! such that (J(u) = tp(u) o 7r+ on EA. In a similar fashion, we define the function ()(•) = (}~) on EA, and find a function tp<•l on E::i such that o<•> = tp<•> o 1r-. The functions on E::i defined by log 1/>(u) =
log¢<•> =


PE- (). A

Lemma A2.2. We have that log¢
for each w+

= (ioit ... ) E E!

p.<">(c:+-0 ' p.(ul(Ct

in

)

;J

= lim log IL n-+oo

(C

)

io in f.L(Ci 1 in)

{with uniform convergence}, and

for each w- = ( ... Ltio) E EA: (with uniform convergence). Proof. The statement is an immediate corollary of the following property of Gibbs measures (see Proposition 3.2 in (PaPo]). let f.L be the Gibbs measure corresponding to a Holder continuous function


- nl-~.l )< 'I' "' -

exp ( a

p.(C;! ;Jexp(-
Here a is the Holder exponent of r/> and 1: n :2:: 0}).

lrl>la

< ( nl-~.l - exp a 'I' "' )•

is the Holder norm of


Appendix II

106

The lemma implies that the functions 1/;(v.) and ¢<•) do not depend on the choice of the point n. The following statement shows that the measure p. is the "direct product" of measures p.(u) and p.(•).

Proposition A2.2. The following properties hold: (1) ~A (
= P1;+A (
PE- (); A positive constants A 1 and A2 such that for every integers n, m ~ 0, and any ( . L1ioi1 ... ) E EA,

(2) there

ex~st

Proof. Since the functions
=

=

has a unique root. In [Bo3j, Bowen discovered this equation while studying the Hausdorff dimension of quasi-circles. This equation is known now in dimension theory as Bowen's equation.

If Z c X is an arbitrary set (not necessarily invariant or compact) one can consider three pressure functions 1/;(t) = Pz(t
(2) there

1/;(t) = 0,

Y!_(t)

= o,

7/}(t)

= 0;

(3) 0 ::; s ::; §. S 8 and 8 < oo if hx(f) < oo; (4) s = 0 if and only if hz(f) = 0; 8 = 0 if and only ifChz(f) = 0.

§.

= 0 if and only if Chz(f)

= 0; and

Variational Principle for Topological Pressure

107

Proof. Lipschitz continuity of the functions '1/J(t), '1/J(t), and ?ji(t) follows from Theorem 11.4. If t' 2: t, we obtain that -

-(t'- t)C1 s; '1/!(t') - '1/!(t) s; -(t1 - t)C2, where C1 and C2 are positive constants. This implies that the function '1/!(t) is strictly decreasing. Similar arguments show that the other two functions are also strictly decreasing The proof of convexity of these functions is straightforward. Since t/!(0) 2: 0, t/!(0) 2: 0, and ?ji(O) 2: 0 the second statement follows The last two statements are consequences of Theorems 11.2 and 11.3. • We notice that by Theorem 11.5, if the set Z is invariant then Y!_(t) = ?ji(t) for all t and hence§.= s; if, in addition, the set Z is compact then '1/J(t) = '1/!(t) = ?ji(t) for all t and hence 8 = §. = 8. Following Barreira [Bar2], we consider a non-additive version of Bowen's equation. Let <.p = {'Pn: X ---t IR} be a sequence of continuous functions satisfying (12.1). We assume, in addition, that the following condition holds:

there e:t?;St negative constants B 1 and B 2 such that for any sufficiently large n (A2.22) This condition is satisfied if cp.,. is additive. The following statement is an extension of Theorem A2.5 to the non-additive case.

Theorem A2.6. [Bar2) If a sequence of functions <.p satisfies Condition (A2.22} then the pressure functions '1/!(t) = Pz(tcp), Y!_(t) = CPz(tcp), and "jf(t) = CPz(t<.p) satisfy Statements {1)-(4) of Theorem A2.5. In particular, there exist unique roots s, §., and 8 of the equations '1/!(t) = 0, '1/J(t) = 0, and ?ji(t) = 0 respectively, which satisfy 0 s; 8 s; §. s; s. We notice that by Theorem 12.1, if the sequence of functions <.p is sub-additive (see (12.4)) and the set Z is invariant then '1/J(t) = ?ji(t) for all t and hence§.= s; if, in addition, the set Z is compact and thesequence of functions {'Pn} satisfies (12.5) then ¢(t) = Y!_(t) = ?ii(t) for all t and hence s = !i = 8. One can use the construction in Example 15.1 below to show that there exists a sequence of functions <.p = { cpn} defined on a compact invariant set Z such that: 1) it satisfies conditions (12.1) and (A2.22) (and hence there exist unique roots~ and s of the equations '1/!(t) = 0 and ?ji(t) = 0); 2) it is not sub-additive; 3) !!. < 8 (see detailed description in [Bar2)). We apply the above results to a symbolic dynamical system (Q,u), where Q is an invariant compact subset of~:. Assume that for any (n + 1)-tuple (io ... in) we have a positive number a., '". Consider a sequence of functions <.p = {'Pn}, where 'Pn ( w) = log a., . in for w = (i 0 i 2 ... ) One can verify that the sequence of functions <.p satisfies (12.1). The non-additive topological pressure and non-additive lower and upper capacity topological pressures corresponding to <.p admit the following explicit description established in [Bar2).

108

Theorem A2.7. For every t E R we have: (1) if the sequence of junctions cp is sub-additive then 1 CPq(tcp) ::: -CPq(tcp) ::: lim -log n-too n

" L.,' (io

in)

Q-admiBI!!lible

and the limit exists as n-+ oo.

(2) if the sequence of functions cp

UJ

sub-additive and satisfies {12.5} then

PQ(tcp)::: CPQ(tcp) = CPQ(tcp). Assume that the numbers aio

in

satisfy the following condition·

e-an ~ a;o

in ~

e-f3n,

(A2.23)

where a and f3 are positive constants. Clearly, (A2.23) implies (A2.22), and hence the equations

Pq(tcp) = 0,

~(tcp)

= 0, CPq(tcp) = 0

have unique roots s, §., and s winch satisfy 0 ~ s ~ §. ~ s < oo. We consider an important particular case Let 0 < a 1 , . , ap < 1 be numbers. Define the function lp on Et by l{)(w) = loga; 0 , where w = (ioh ... ).

Theorem A2.8. (1) /fQ C Et is a compact u-invanant set, then Bowen's equation Pq(tl{)) = 0 has a unique root. (2) lf(Q,CJ) is the full shift, i.e., Q= Et. then Bowen's equationPq(tl{))::: 0 is equivalent to the equation p

~a/= 1. i=l

EX

(3) If (Q,a) is a su.bshift of finite type, i e., Q = with a transfer matrix A, then Bowen's equation Pq(tcp) = 0 is equivalent to the equation p(AMt(a)) = 1, where p(B) denotes the spectral radius of the matnx B and Mt(a) = diag (a 1 t, · , apt). Proof. Let

J . EX -+ lR be a continuous function.

(Lq,f)(w) ~r

~

Consider the transfer operator

exp(({)(w'))J(w') = I:exp(({)(k))J(kioil ... )A(k,io),

w'Ea- 1 (w)

k

where w = (ioi1 .. ) and the function 'f'(w) = loga; 0 = 'f'(io) depends on the first coordinate only The eigenvalue equation (corresponding to an eigenvalue 'TJ) is Eexp(cp(k))h(k)A(k,j) = .,h(j). k

According to [R1], the largest eigenvalue of Lrp is exp(P(cp)). Hence, exp(P(cp)) is the spectral radius of the matrix A*~, where~ denotes the (pxp) diagonal matrix diag( erp(l), e'P( 2 l, · · , e'P(P)). This implies the third statement. The second one is an immediate consequence of the third statement. The first statement follows from Theorem A2.5. •

Variational Principle for Topological Pressure

109

We conclude the appendix with formulae for the topological entropy of a subshift of finite type and the Hausdorff dimension of a topological Markov cham.

Theorem A2.9. Let A be a tmnsfer matrix. Then (1) hE+(U) = hE-(u) = hEA(u) = p(A); A A

(2) dimH E! = dimH E_A = loFog<;l, where {3 is the C<Jefficient in the dpmetnc (see (A2.18}}; (3) dimH EA = 2 1'1,~~)' where (3 is the coefficient in the dp-metnc (see (A2.18

1

)).

Proof. The first statement follows from the third statement of Theorem A2.8. The second and the third statements can be proved by straightforward calculation which we leave to the reader. Note that the second statement lS essentially a corollary of Theorem 13.2 (see Statement 2; see also self-similar constructions in Section 13) while the third statement is a corollary of Theorem 22.2 (see also linear horseshoes in Section 23). •

Appendix III

An Example of Caratheodory Structure Generated by Dynamical Systems

In this Appendix we briefly discuss an example of C-structure generated by dynamical systems This C-structure was introduced by Barreira and Schmeling [BS] in their study of metrically irregular sets for dynamical systems. See Appendix N. Let f: X -t X be a continuous map of a compact metric space X and cp: X -t R a continuous stnctly positive function. Consider a finite open cover U of X and define the collection of subsets :F = :F(U) by (11.2) and three functions t;, 1J, '1/J: S(U) -t R as follows ~(U)

= 1,

17(U)

= exp(-

sup Sm(U)'P(x)),

'ljJ(U)

= m(U)- 1 ,

xEX(U)

where

m(U)-1 Sm(U)'P

L

=

tp 0

fk.

k=O

One can directly verify that the collection of subsets :F, and the functions 71, t;, and 'ljJ satisfy Conditions AI, A2, A3, and A3' in Section 10 and hence determine a C-structure r = r(U) = (S,:F,f.,T/,1/J) on X. The corresponding Caratheodory function mc(Z,a) (where Z C X and a E lR; see Section 10) depends on the cover U (and the function cp) and is given by

mc(Z,a) = lim M(Z,a,cp,U,N), N-too

where

M(Z, a, tp,U, N) = inf { (}

L

exp( -a sup Sm(u)tp(x))}

UE(}

xEX(U)

and the infimum is taken over all finite or countable collections of strings g c ~ N for all U E 9 and 9 covers Z. Furthermore, the CaratModory functions rc(Z,a) and rc(Z,a) (where Z c X and a E R; see Section 10) depend on the cover U and are given by

S(U) such that m(U)

rc(Z,a) = lim R(Z,a,cp,U,N), N-too

rc(Z,a) = lim R(Z,a,cp,U,N), N-too

110

An Example of Caratheodory Structure

111

where R(Z,n,rp,U,N)

= inf { L t;

exp(-n sup SNrp(x))}

UEt;

"'EX(U)

and the infimum is taken over all finite or countable collections of strings g c S(U) such that m(U) = N for all U E g and g covers Z. According to Section 10, given a set Z c X, the C-structure r generates the Caratheodory dimension of Z and lower and upper Caratheodory capacities of Z specified by the cover U and the map f. We denote them by BScp,u(Z), BS
BScp(Z) ~r lim BS'(Iu(Z), IUI-+0

-

'

def.-

BScp(Z) =

hm BScp u(Z).

IUJ-+0

'

We call the quantities BScp(Z), BScp(Z), and BBcp(Z), respectively the BSdimension of a set and lower and upper BS-box dimensions of the set Z (specified by the function rp and the map f; after Barreira and Schmeling). We emphasize that the set Z can be arbitrary and need not be compact or invariant under the map f. The main properties of BS-dimension and lower and upper BS-box dimensions are described below. They are immediate corollaries of the definitions and Theorems 1.1 and 2.1. Theorem A3.1. (1) BScp0 = 0; BScp(Z) ~ 0 for any non-empty set Z C X. (2) BS"'(Z!) ~ BS'(I(Z2) if Zt C Z2 C X. (3) BS'(I(Z) = SUP;;:: 1 BS"'(Z;), where Z = U;;:: 1Z; and Z; C X, i = 1, 2, .... (4) M,p0 = BS"'0 = 0; BS"'(Z) ~ 0 and BS"'(Z) ~ 0 for any non-empty set Z C X (5) Jl..S..p(Z1 ) ~ Jl..S..p(Z2) and BScp(Zt) ~ BScp(Z2) if Zt c Z2 c X. (6) BScp(Z) ~ supi;?:l Jl..S..p(Z,) and BS"'(Z) ~ supi;?:l BScp(Z;), where Z = U,> 1 Z, and c X, i = 1,2, .... (7) If h: X -+ X is a homeomorphtSm which commutes unth f (i.e., f o h = ho f) then

z,

BScp(Z)

= BSopoh-•(h(Z)),

Jl..S..p(Z) = BScpoh-,(h(Z)),

BScp(Z) = BScpoh-,(h(Z)).

(8) For any two continuous functions rp and 1/J on X

llrp -1/JII, ~ IJrp -1/JII, ~ llrp -1/JII,

JBScp(Z)- BS.p(Z)I ~ JBS'~'(Z)- M.p(Z)I

IBS..,(Z)- BS.p(Z)I

112

Appendix III where II on X.

II denotes the supremum norm in the space of continuous functions

Remarks. (1) In the case t.p = 1, the BS-dimension of a set Z c X and lower and upper BS-box dimensions of Z coincide with the topological entropy off on Z and lower and upper capacity topological entropies of f on Z respectively. (2) Compare the definitions of topological pressure and BS-dimension one can obtain that for any set Z c X the BS-dimension of Z is a unique root of Bowen's equation Pz( -st.p) = 0, i.e., s = BScp(Z). Let p. be a Borel probability measure on X which is invariant under f. According to Section 10, the C-structure r = (S,F,(.,T/,1/J) on X generates the CaratModory dimension of p. and lower and upper Caratheodory capacities of p. specified by the coverU and the map f. We denote them by BScpi-J(P.), ~.u(P.), and BScp,u(P.) respectively. It follows from what was srud above that there exist the limits BS"'(p.) ~ lim BS"'u(P.), IUI-+0

'

BS"'(p.) ~ lim BS"'u(P.), IUJ-+0

-

BS"'(JJ)

def

•

'

-

= iUJ-+0 lim BS"' u(p.). '

We call these quantities the BS-dimension of a measure and lower and upper BS-box dimensions of the measure p. (specified by the function cp and the map f). Given a point x EX, we set in accordance with (10.1) . . logp.(X(U)) J2.cpa(X,t.p,U) = hm mf S ( )' ' ' N-+co u - sup N'fJ Y yEX(U)

-. Dc,"'·"'(x,t.p,U) = hm sup N .... oo

u -

logp.(X(U)) ( ), sup 8 N'P y yEX(U)

where the infimum and supremum are taken over all strings U with x E X(U) and m(U) = N. We leave the proof of the following statement to the reader (compare to Proposition 11.1). Theorem A3.2. If p. is a Borel probabzlity measure on X invartant under the map f and ergodic, then ( 1) for every a E lR and p,-almost every x E X, . ( U) hm J2.c~-'"' x,t.p,

IUI-+0

' '

. ;n ( ) hp(!) def = IUJ-+0 hm vcpa I,t.p,U = J d = d, ' ' X t.p p,

where hp (f) is the measure-theoretic entropy off. {2) BS"'(p,) = BS"'(p,) = BScp(p,) =d.

An Example of CaratModory Structure

113

As a consequence of Theorem A3.2 we establish the vanationa/ pnnciple for the BS-dimension: (A3.1) BS'I'(G) = sup BScp(/1-), I'E!m{X)

where G

=

U

Gl' and Gl' is the set of all forward generic points of the

I'E!m{X)

measure /1- (see (A2 16)) Indeed, by Theorem A3 2 and (A2.16),

This implies that BSrp(/1-) = BS'I'(G,.) and hence

This Page Intentionally Left Blank

Part II

Applications to Dimension Theory and Dynamical Systems

This Page Intentionally Left Blank

Chapter 5

Dimension of Cantor-like Sets and Symbolic Dynamics

It is now the prevailing opinion among experts in dimension theory that the comcidence of the Hausdorff dimension and lower and upper box dimensions of a set is a ''rare" phenomenon and can occur only if a set has a "rigid" geometric structure. Nevertheless, there exists a broad class of subsets in lRm, known as Cantor-like sets, for which the coincidence usually takes place. These sets were a traditional object of study in dimension theory for a long time and also served as a touchstone for different techniques. Computing their Hausdorff dimension and box dimension was a great challenge to experts in the field who had to create a number of highly non-trivial methods of study. These sets often were the source for exciting examples that demonstrated different phenomena in dimension theory. The Cantor-like sets are defined by geometric constructions of different types. We begin with the most basic geometric construction. Starting from p arbitrary closed subsets Cl. 1 , ... , Cl.p of lR.ffl we define a Cantor-like set by

n 00

F

=

U

n=O(io

Cl.;o

in,

in)

where the basic sets on the nth step of the geometric construction, 6.; 0 in, ik = 1, ... ,p (n 2: 0) are closed and satisfy the following conditions: (CGl) 6.; 0 ini C Cl.; 0 in for J = 1, ... ,p; (CG2) diamCJ.io. in --+ 0 as n--+ oo; (CG3) (separation condition): Cl.; 0 in nA-;0 in nF = 0 for any ('!o· .. in)=/= (jo · · in)· The set F is perfect, nowhere dense, and totally disconnected. See Figure 1. The description of a geometric construction includes the description of its symbolic representation and its geometry.

Et

Symbolic representation. Let be the set of all one-sided infinite sequences (i 0i 1 ... ) of p symbols (pis the number of basic sets on the first step of the construction) endowed with the d13 metric given by (A2.18) (see Appendix II). Given w = (ioil .. ) E Et the intersection n::'=o Cl.; 0 in is non-empty and by Condition (CG2), consists of only one point x. Thus, the formula x(w) = x defines correctly a continuous map X· F which we call the coding map. By the separation condition this map is one-to-one and hence (since is compact) it is a homeomorphism.

Et-+

Et

117

118

We consider Chapter 5 a geometri namical sy st em (Q ,o ), w c construction modeled he by re Q C E t is m ap o, i.e., a general sy a compact mbolic dyu(Q) Q. Namely, admissible nwe allow only se t invariant under th e sh tuples (io .. . ift ba si c in ) with resp such th at io ect to Q, i.e., se ts th at correspond to = io, J1 = il> there exists g eo m et ri c .. . ,j n = in (i co n st ru ct io n s. T he li m Such constructions are ca o h .. . ) E Q it se t F is de lled sy m b o li fined by c

=

n 00

F=

n= O

u

{so i, ) Q ·a dm iss ib

le O ne ca n clas si fy ge om et sentation. A ric constructi ons accordin construction g to their sy is called a si modeled by mbolic reprem p le g eo m th e full shift, et ri c co n st i.e., Q = E t of geometric ru ct io n if it (see Figure 1) construction is s are construc . Namely, th e tions modeled Another im po rt an t class geometric co nstruction is by subshifts ti o n if Q = E of finite type called a M ar 1 (we remind th e re . k o v g eo m et admissible w ader th at E~ ri c co n st ru it h respect to co ns cists of all sequ th e transfer m A (i j, i; + l) = ences (ioi1 .. at ri 1 .) Finally, a ge for j = 0, 1 .. . ; see Appen x A with entries A( i, j) "" 0 or 1, i.e., ometric cons di x II ; se e also Figure truction is ca (Q, o ) is a so 2 in Section lled a so fi c fic sy st em (i 13). g eo .e., a finite fa ctor o f a subs m et ri c co n st ru ct io n if hift o f finite type).

F ig u re 1. A

S IM P LE G E O M E TR

IC C O N ST

R U C TI O N . G eo m et ry o f th e co n st ru ct io n . Thi ment of th e s includes th basic sets, th e informatio ei r geometri control over n on th e plac c shapes, an th e sh ap e an ed "sizes". If d sizes of ba one has st ro sic sets, th en ng th e placemen t can be fair ly

Dimension of Cantor-like Sets and Symbolic Dynamics

119

arbitrary, and vice-versa. To illustrate this we first consider constructions whose geometry seems to be most simple where one has complete control over the shape and sizes of basic sets which are balls. Denote by r; 0 ; . the radius and by X; 0 in E ntm the center of the basic set (ball) .6.,0 in on the nth step of the construction. The collection of numbers r; 0 •· and coordinates of points Xio ;. provide complete information on the geometry of the construction and, in particular, is sufficient to compute the Hausdorff dimension and box dimension of the limit set. Surprisingly, one can often use only the numbers r; 0 in to obtain refined estimates for the Hausdorff dimension (from below) and the upper box dimension (from above) of the limit set (see Section 15) The main tool to study this geometric construction is the thermodynamic formalism, developed in Chapter 4 and Appendix II (in particular its non-additive version), applied to the underlying symbolic dynamical system. Namely, consider the sequence of functions 'Pn(w) = logr; 0 in on Q, where w = (ioii ... ). Clearly it satisfies Condition (12.1). According to Appendix II (see Theorem A2.6) the equations Pq(s{cpn}) = 0, CPq(s{cpn}) = 0 (where Pq and CPq denote the non-additive topological pressure and nonadditive upper capacity topological pressure specified by the sequence of functions {'Pn}) have unique roots provided the sequence of radii admits the asymptotic estimate (A2.23). We denote these roots by!!. and 8 respectively. Clearly, !!. :::::; 8. In Section 15 we will show that, under some mild additional assumptions, the number !!. provides a lower bound for the Hausdorff dimension and the number 8 provides an upper bound for the upper box dimension of the linrit set (see Theorem 15.1). We emphasize that the above approach works well for an arbitrary collection of numbers rio in which may depend on the whole "past" and may not admit any asymptotic behavior as n -t oo. Yet, the Hausdorff dimension and lower and upper box dimensions may not coincide and their exact values may depend on the placement of basic sets (see Example 15.1). However, if, for instance, the sequence of functions 'Pn is sub-additive (see Condition (12.4)), then the Hausdorff dimension and lower and upper box dimensions coincide and are completely determined by the sizes of basic sets only, i.e., the numbers r; 0 in. In this case !1. = s ~r sand regardless of the placement of basic sets, the Hausdorff dimension and lower and upper box dimensions of the limit set coincide and are equal to s. Moreover, s is the unique root of the equation

PQ(s{ cp,.})

= 0.

An equation of this type was first discovered by Bowen in his study of the Hausdorff dimension of quasi-circles (see [Bo2]) and now bears his name. Bowen's equation is one of the main manifestations of the general Caratheodory construction developed in Chapter 1: it expresses intimate relations between two Caratheodory dimension characteristics: the Hausdorff dimension of the linrit set of a geometric construction and non-additive topological pressure specified by the sequence of functions { 'Pn} for the underlying symbolic dynamical system

120

Chapter 5

(Q, u) Bowen's equation contains all necessary information on the geometric construction that matters in computing the Hausdorff dimension of the limit set: the information on the underlying symbolic dynamical system and its realization into the Euclidean space by basic sets Bowen's equation seems to be "universal" in dimension theory: we will demonstrate that the Hausdorff dimension of limit sets for various classes of geometric constructions as well as invariant sets for various classes of dynamical systems can be computed as roots of Bowen equations written with respect to the underlying symbolic dynamical systems. In 1946 Moran, in his seminal paper (Mo], introduced and studied geometric constructions with basic sets satisfying the following conditions· (CMl) each basic set t:-; 0 ; .. is the closure of its interior; (CM2) t:-;0 ; .. ; C t:-; 0 in for j = 1, .. ,p; (CM3) each basic set t:-; 0 ini is geometrically similar to the basic set. t:-; 0 in for every j; (CM4) for any (io ... in) =/= (jo . -Jn) the basic sets t:-; 0 ; . and t:-;0 1.. do not overlap, i.e., their interiors are disjoint; (CM5) diamt:-; 0 ini = A;diamt:-; 0 ;,., where 0 < A; < 1 for j = 1, ... ,pare constants. Such constructions are called Moran geometric constructions (the name coined by Cawley and Mauldin (CM]). The numbers A; arc known as ratio coefficients of the construction since they determine the rate of decreasing the sizes of basic sets We stress that basic sets of a Moran geometric construction at the same step may not be disjoint (they may intersect along boundaries, compare to Condition (CG3)). Therefore, the limit set F may not be a Cantor-like set and the coding map x may not be injective (but it is still surjective). Moran considered only geometric constructions modeled by the full shift. His remarkable observ-ation was that an "optimal" cover, which can be used to obtain the exact values of the Hausdorff dimension of the limit set (we call it a Moran cover) is completely determined by the symbolic representation of the geometric construction. In particular, regardless of the placement of basic sets, the Hausdorff dimension of the limit set is completely determined by the ratio coefficients. Namely, Moran discovered that it is a unique root of the equation

wh:tch is a particular case of Bowen's equation corresponding to the full shift. Moran constructions modeled by subshifts of finite type or sofic systems were studied in (MW]. In [PWl], Pesin and Weiss demonstrated that Moran's approach can be greatly extended to a much more general class of Moran-like geometric constructions with stationary (constant) ratio coefficients. They discovered that the only property of basic sets that matters in constructing Moran covers, is

Dimension of Cantor-like Sets and Symbolic Dynamics the following. there exist closed balls B;0 in of radii rio is a constant) with disjoint interiors such that

121 'n

=

0

IJj=O >.; 0 > 0 3

(

The topology and geometry of basic sets of these constructions may be quite complicated (for example, they may not be connected and their boundary may be fractal). Moreover, the basic sets at step n of the construction need not be geometrically similar to the basic sets at step n - 1 (see Condition (CM3) in the definition of Moran geometric constructions). Another important feature is that basic sets ~;0 in at step n of the construction may intersect each other (although the interiors of the balls Bio ;. must be disjoint). In [PWI], Pesin and WeiSs also showed that a slight modification of Moran's approach allows one to deal with geometric constructions modeled by an arbitrary symbolic dynamical system (Q,u). The main tool of study, as explained above, is the thermodynamic formalism described in Chapter 4 and Appendix II. In particular, regardless of the placement of basic sets the Hausdorff dimension and lower and upper box dimensions of the limit set F coincide and the common value is a unique root of Bowen's equation PQ(S
= 0,

where .; 0 for any w = (i 0 i 1 ... ) E Q Moreover, the push forward to F by the coding map of an equilibrium measure J-t on Q corresponding to the function sr.p is an invariant measure m of full dimension, i.e., dimH F = dimH m (see discussion in Section 5). It is worth emphasizing that if J-t


122

Chapter 5

condition (CG3)) by Moran-like geometric constructions can be fruitfully used to some extent We call a geometric construction regular if it admits such an apprmomation. We stress that basic sets of a regular geometric construction need not resemble balls in any geometrical sense. Roughly speaking the regularity of a geometric construction means that its basic sets can be effectively replaced by balls such that the new geometric construction admits a Moran cover. In Section 14 we give a formal mathematical definition of the notion of regularity and study some properties of regular constructions. In particular, we obtain a general lower bound for the Hausdorff dimension of the limit set of the construction (see Theorem 14.1). One can further use the approach, based on an effective replacement of the basic set of a geometric construction (CG1-CG2) by disJoint balls, to obtain a general upper bound for the upper box dimension of its limit set (see Theorem 14.5). In general, it may be extremely difficult to find an effective approximation of a given geometric construction, for instance, inscribed balls often fail to produce a reasonable approximation (see Example 14.1) A more sophisticated approach to the description of geometric constructions with complicated geometry of basic sets is based upon the study of the induced map G on the limit set F generated by the symbolic dynamical system. We notice that if basic sets of a geometric construction are disjoint the coding map is injective. Therefore, the shift <7 induces a map G on the limit set F given by G = x o q o x- 1 . The dynamics of G bears the complete information on the symbolic representation of the construction (via the map qiQ) as well as on its geometry (via. the coding map x, i.e., the embedding of the symbolic dynamics into lR"' by x). It seems qmte plausible that those characteristics of the dynamics of G that are connected to the instability of its trajectories are relevant for computing the Hausdorff dimension and box dimension of F. This can be clearly seen when G is an expanding map. In Section 15 we introduce appropriate characteristics of instability and demonstrate how to use them in estimating the Hausdorff dimension and box dimension of the limit set. In dimension theory there is a popular class of geometric constructions, known as self-similar geometric constructions, which are effected by a finite collection of similarity maps. This means that the basic sets ~io in are given by ~io in = h; 0 0 h;, 0 • • • 0 h;n (D), where ht. .. , hp: D --+ D are conformal affine maps, i.e., they satisfy the property: dist(h.(x), h;(y)) = >.; dist(x, y) for any x, y E D (where D is the unit ball in lR.m) with fixed 0 < >.; < 1 (for detailed description of self-similar constructions and related results see [Fl] where further references can also be found). These constructions are a. very special type of Moran geometric constructions (CM1-CM5) where not only the sizes of basic sets but also the gaps between them are strongly controlled. A more general class of geometric constructions is formed by geometric constructions with contraction maps where the maps h1 are hi-Lipschitz

Dimension of Cantor-like Sets and Symbolic Dynamics

123

contraction maps. This means that for any x,y ED, _d;dist(x,y):::; dist(h;(x),h;(y)):::; :\.dist(x,y), where 0 <

.d; :::; X; < 1. These constructions are regular

(see Section 14).

13. Moran-like Geometric Constructions with Stationary (Constant) Ratio Coefficients We begin with a geometric construction with the simplest geometry of basic is a compact sets modeled by a symbolic dynamical system (Q, u), where Q c shift invariant set. We assume that basic sets are closed and satisfy· (CPWl) Ll; 0 ini C Ll;0 ; . for j = 1, ... ,p;

Et

(CPW2) balls of radii (CPW3)

!1.;0 ; . C Ll'O r.'O ;,. and 1';0

in

C

B;0

in,

where

!1.;0

n

f.;o

in

;,.

and

B;0

in

are closed

in;

= K1

II A;J' j=O

n

f; 0 in = Kz

II A;i,

(13.1)

j=O

where 0 < Aj < 1 for j = 1, ... ,p, Kt > 0, and K2 > 0 are constants; (CPW4) intll.;o in nintll.;o J= = 0 for any (io . . in) =f: Uo •... ,jm) and m?: n. This class of geometric constructions was introduced by Pesin and Weiss in [PWl]. Basic sets of these constructions are essentially balls, although their topology and geometry may be quite complicated. Furthermore, basic sets Ll; 0 in at step n of the construction may intersect each other. The numbers .>.; are called ratio coefficients. They are fixed and do not depend on the basic sets. Self-similar constructions (see below) or more general Moran geometric constructions (CM1-CM5) (see the introduction to this chapter) are particular examples of geometric constructions (CPW1-CPW4). The geometric simplicity of the geometric constructions {CPW1-GPW4) will allow us to illustrate better the role of symbolic dynamics. Given a p-tuple A = (>.1> ... , Ap) such that 0 < A; < 1, there exists a uniquely defined nonnegative numberS>. such that

(we remind the reader that PQ denotes the topological pressure on Q with respect to the shift u; see Section 11 and Appendix II). Denote by x: Q --t F the coding map (see definition in the introduction to this chapter). Note that it is Holder continuous. To see this let w1 = (ioit inj . ) and w2 = (ioiJ ... i,.k ... ) be two points in Q with j =f: k. We have n

llx{wl)- x(w2)11 5

II Ai; :::; (1~ >.;)" 5 ((Jn)'":::; Cdi3(wl, w2)'"'. j=O _p

Chapter 5

124

Et

where C > 0, 0 < a < 1 are constants and d13 is the metric in (see (A2 18) in Appendix II) This implies that any Holder continuous function on F pulls back by x to a HOlder contmuous function on Q. Notice that, in general, the coding map X is not invertible (since basic sets at the same level of the construction may intersect each other) and even if it is invertible it is not necessarily Holder continuous (this depends on the placement of the basic sets, i.e , the gaps between them). Let 1-'>. be an equilibrium measure for the function (ioi 1 ... ) H B>.log >.; 0 on Q, and m>. the push forward measure to F under x (i.e., m>.(Z) = ll.>.(x- 1 (Z)) for any Borel set Z C F). We describe a special cover of the limit set F for a symbolic geometric construction (CPW1-CPW4) whlch allows one to build an opttmal cover to be used to compute the Hausdorff dimension and box dimension of F. We call this cover a Moran cover. Let >. = (>.h ... ,>.,) be a vector of numbers with 0 < >.; < 1, i = 1, ... , p. Fix 0 < r < 1. Given a point w = (~ 0 i 1 .. ) E Q, let n(w) = n(w, r, >.)denote the unique positive integer such that (13.2)

It is easy to see that n(w) ~ oo as r ~ 0 uniformly in w. Fix wE Q and consider the cylinder set C; 0 inl"'l C Q. We have that w E C; 0 . 1n 1.,.,. FUrthermore, if w' E C; 0 ;n 1..,, and n(w') ~ n(w) then Let C(w) be the largest cylinder set containing w with the property that C(w) = Cio in<"'"' for some w" E C(w) and Cia ;n 1.,.,, c C(w) for any w' E C(w). The sets C(w) corresponding to different w E Q either coincide or are disjoint. We denote these sets by C(j), j = 1, .. . ,Nr. There exist points Wj E Q such that c0l = C 00 ;n 1.... 1 • These sets form a disjoint cover of Q which we denote by 11r = ilr,Q(>.). The sets ~(j) = x(C(j)), j = 1, ... , Nr are not necessarily disjoint and comprise a cover ofF (which we will denote by the same symbolilr if it does not cause any confusion). We have that ~(i) = ~io ;n<~;l for some Xj E F. Given a subset R c Q (not necessarily invariant) one can repeat the above arguments to construct a Moran cover of R which we denote by ilr,R(>.). It consists of sets c 0, there exists a number M > 0 which does not depend on x, r, and>. and satisfies the following property: the number of basic sets~ (j) in a Moran cover Ur,Q(>.) that have non-empty intersection with the ball B(x, r) 'IS bounded from above by M. We call M a Moran multiplicity factor. Moreover, given a subset R c Q, a point x E F, and a number r > 0, the number of basic sets ~(;) in a Moran cover ilr,R(>.) that have non-empty intersection with the ball B ( x, r) is bounded from above by M. )

Dimension of Cantor-like Sets and Symbolic Dynamics

125

Theorem 13.1. [PW1] Let F be the limit set for a geometnc construction (CPW1-CPW4) modeled by a symbolic dynamical system (Q,u). Then (1) dimH F = dimBF (2) dimH ffi>. = S>.; (3)

= dimBF == S>.;

Proof. Set s = S>. and d = dimH F. We first show that s ~ d. Fix e > 0. By the definition of Hausdorff dimension there exists a number r > 0 and a cover of F by balls Bt, l = 1, 2, ... of radius r~ ~ r such that (13.3)

For every l > 0 consider a Moran cover .U.., of F and choose those basic sets from the cover that intersect Be. Denote them by ~~l),.. , ~~m(l)). Note that ~~j) = ~; 0 • in
Kz

II A;. ~ !:io

in(l,j)

~rio


~ Czrt,

(13.4)

k=O

where Gz > 0 is a constant independent of land j. The property of the Moran cover implies that m(£) ~ M, where M > 0 is a Moran multiplicity factor (which is independent of l) The sets {~~l, j = 1, .. , m(l), l = 1, 2, ... } comprise a cover Q ofF, and the corresponding cylinder sets c}'l = c,D in(l j) comprise a co~·er of Q. By (13.3) and (13.4)

Given a number N > 0, chooser so small that n(l,j) 2: N for alll and j. We now have that for any n > 0 and N > n,

M(Q,O,cp,Un,N) ~

L 6~;lEg

exp

Chapter 5

126

where M(Q,O,tp,Un,N) is defined by (A2.19) (see Appendix II) with

tp(w) = (d +e) log A;0 (and a= 0). This implies that Pq((d +e) logA;0 ) $0.

Hence, by Theorem A2.5 (see Appendix II), 8 $ d+e. Since this inequality holds for all e we conclude that 8 $ d. Denoted= dimBF. We now show that d $ s. Fixe> 0. By the definition of the upper box dimension (see Section 6) there exists a number r = r(e) > 0 such that N(F,r) 2': r•-il (recall that N(F,r) is the least number of balls of radius r needed to cover the set F). Consider a Moran cover U,. of Q by basic where · Let ~(il = x(C(;l) = ~·IO ··'n.(:x;)' j = 1 . N.. · sets C(i) =C.1-0 -ln(wJ) r x; = x(wj) Note that this cover need not be optimal, i.e., Nr ;::: N(F,r). By (13.2) there exists A > 1 such that for j = 1,.. , Nr, '

1

'

n(w;)+l

< ~ A-

IJ

,x.,, < - r

k=O

and hence

A

l

C2log- -1$ n(w;) $ C3log- + 1, r r that n(w;) can take on at implies This constants. are 0 > C and 0 > C where 2 3 C2log : + 2 possible values. most B ~r C3log We now think of having Nr balls and B baskets. Then there exists a basket containing at least 1ft balls. This implies that there exists a positive integer N E [C2log ~ - 1, C3log + 1] such that

1-

1

card {j: n(w;)

= N} 2:

N(F r) Br 2: - B ' 2:

N

r•-J A,

C3logr

where card denotes the cardinality of the corresponding set. If r is sufficiently small we obtain that card {j : n(w;) = N}

2'-_

r 2•-d.

Consider an arbitrary cover g of Q by cylinder sets C;0

;N.

It follows that

Dimension of Cantor-like Sets and Symbolic Dynamics

127

where C4 > 0 is a constant. We now have that for any n > 0 and N > n,

where R( Q. 0, cp, U,., N) is defined by (A2.19') (see Appendix II) with a = 0 and

cp(w)

= (d- 2c-) log.-\io·

By Theorem 11.5 this implies that

and hence d- 2£ :.::; s (see Theorem A2.5 in Appendix II). Since this inequality holds for all £ we conclude that d :.::; s. This completes the proof of the first statement. In order to prove the second statement we need only to establish that s :.::; dimn m;.. Assume first that the measure J-L>. is a Gibbs measure corresponding to the function slog >.io. By (A2.20) (see Appendix II) there exist positive constants D 1 and D2 such that for J = 1, ... , Nr (13.5) Consider the open Euclidean ball B(x, r) of radius r centered at a point x. Let N(x,r) denote the number of sets t._(i) that have non-empty intersection with B(x, r). It follows from the property of the Moran cover that N(x, r) :.::; M, where M is a Moran multiplicity factor. By {13.5) and (13.2) we obtain that for every x and every r > 0, N(z,r)

m;.(B(x,r)):.::;

L

j~l

N(z,r)

m;.(t..Ul):.::;

L

j~l

n(:tj)

D2

II >.,.

8

k~

N(x,r) n(zJ)+l

:'S:Cs

L II

i=l

>..,.•:o::;c5N(x,r)r":o::;C5 Mr",

(136)

k=O

where Cs > 0 is a constant. It follows that the measure m>. satisfies the uniform mas.'! distribution principle (see Section 7) and hence dimn m;. ~ s. We turn to the general case when J-L>. is just an equilibrium measure By definition (13.7) h,_., (uiQ) +.~~log >.iodJ-L;. = 0,

Chapter 5

128

where hp(crjQ) ~£his the measure-theoretic entropy. Let us first assume that tt>. is ergodic. Fix c: > 0. It follows from the Shannon-McMillan-Breiman theorem that for J.£>.-almost every w E Q one can find N 1 (w) > 0 such that for any n ~ Nt(w), I'>.(C, 0 ... (w)) ~ exp( -(h- c:)n), (13.8) where C; 0 ;,. ( w) is the cy Iinder set containing w. If the measure IL>. is ergodic it follows from the Birkhoff ergodic theorem, applied to the function slog >.;0 , that for 11>. -almost every w E Q there exists N 2 (w) such that for any n;::: N 2 (w), (13 9)

Combining (13.7), (13 8), and (13.9) we obtain that for J.'>.-almost every w E Q and n;::: max{N1(w),N2(w)}, n

P.>.(C;0

i,.(w))

~

II >.;

3

n •

exp(2en) ~II >.;i s-o.,

j=O

j=O

where a= 2ef mini log(l/>.J) > 0. This implies that for Jt>.-almost every wE Q and any n;::: max{N1(w),N2(w)}, n

J.L>.(C; 0 ;,(w)) ~II>.,:-"'.

(13.10)

j=O

If J.'>. is not ergodic, then (13.10) is still valid and can be shown by decomposing J.L>. into its ergodic components. Given£> 0, denote by Qt ={wE Q: N 1 (w) ~ l and N 2 (w) ~ £}. It is easy to see that Qe C Qt+l and Q = U~ 1 Qt (mod 0). Thus, there exists lo > 0 such that J.L>.(Qe) > 0 if l;::: io. Let us choose l;::: io Given 0 < r < 1, consider a Moran cover i.I,.,Q 1 of the set Qe. It consists of sets c£0), j = 1, .. . ,Nr,l for which there exist points Wj E Q such that cyJ = C·ZQ 'Zn(w.i) · Set A l
m>.(B(x, r)

n x(Qe)) ::;

J=l

~

N n(xj)

2.: m>..(A~j)) ~ 2.: II >.;. •-<> J=l k=O

K2N(x, r, l)r•-or ::0 K2Mr"-"'.

Since m>.(x(Qt)) > 0 by the Borel Density Lemma (see Appendix V) for m>..almost every x E x(Qt) there exists a number r 0 = r 0 (x) such that for every 0 < r ~ r 0 we have m>.(B(x,r))::; 2m>.{B(x,r)nx(Qt)).

Dimension of Cantor-like Sets and Symbolic Dynamics

This implies that for any l > £0 and

!lmJx)

= lim logm~(B(x, r)) r-+0

log r

m~-almost

129

every x E x(Qt),

~ lim logm~(B(x, r) n x(Qt)) ~ r->0 log r

8

-'

_a

Since the sets Qt are nested and exhaust the set Q (mod 0) we obtain that !lm.x (x) ~ s~ -a for m~-almost every x E F. This implies that dimn m~ ~ 8~ -a. Since a can be arbitrarily small this proves that dimn F ~ dimn m-' ~ 8~. Note that J.l.\ is an equilibrium measure corresponding to the function s-' log A;0 • Therefore,

•

and the third statement follows.

In view of Statement 2 of Theorem 13.1 the measure m-' is an invariant measure of full Caratheodory dimension (see {54)). We call it simply measure of full dimension. The next statement provides an upper estimate for the number s-'.

Theorem 13.2. Let F be the limit set for a geometnc construction {CPW1CPW4) modeled by a symbolic dynamical system (Q,a). Then (1) [PWl) < hQ(a) 8 ~ - - log Amax ' where Amax =max{.>.~: : 1 ~ k ~ p} and hQ(a) is the topolo!llcal entropy of a on Q; equality occurs if A; = A fori = 1, ... ,p; in particular, if hq(a) = 0, then

(2) [Fu] if>.,=>. fori= 1, ... ,p, then

. F d!IJlH

F = -d. F = 8~ = -hQ(a), . = d' _!!l!B IIDB - 1ogl\

Proof. It follows from Statement 3 of Theorem 13.1 that s~

h,..,(aiQ)

= - J log A; dJ.L.\ 0

hq(u)

< - -log Amax .

The case of equality is obvious. If .>.; = A. for i = 1, ... , p, then J.l-' is a measure of maximal entropy (since the function . is constant). Thus, h~-', (u IQ) = hQ(u). This proves the desired results. •

Chapter 5

130

b

a Figure 2.

SIERPINSKI GASKETS

a) A Simple Construction, b) A Markov Construction. We consider two special cases of symbolic geometric constructions - simple geometric constructions and Markov geometric constructions, specified by the full shift and a subshift of finite type respectively. Examples are shown on Figure 2. This is the well-known SierpiDski gasket - the limit set for a Moran geometric construction (CM1-CM5) on the plane with >. 1 = >.2 = >. 3 < ~ and p = 3. The case of a simple construction is shown on Figure 2a, and thE' case of a Markov

:::::::c~i:nro:::~:: a2:: ;::r:o:~::::;:r::r::::~r~:sisw(hfe for)n~s 1 1 1 We consider the particular case of a subsh!ft of finite type (~1.u). Given p numbers 0 < At. ... , Ap < 1, we define a p x p diagonal matrix Mt(>.) diag(>. 1t, · , >./). Let p(B) denote the spectral radius of the matrix B.

Theorem 13.3. (1) Let F be the limit set for a geometnc con.~truction (CPW1-CPW4) modeled by a subshift of finite type (~1,u). Then dimH P

= dim8 F = dimBF = S).,

=

where S>. tS the unique root of the equation p(AMt(>.)) 1. (2) Let F be the limit set for a ge.ometnc c.onstruction (CPW1-CPW4) modeled by the full shift. Then

dimH F

= dim 8 F = dimBF = 8>.,

where S>. is the unique root of the equation

Proof. The desired result follows from Theorem 13.1 and Theorem A2.8 (see • Appendix II).

Dunension of Cantor-like Sets and Symbolic Dynamics

131

For instance, for the Sierpinski gMkets shown on Figures 2a and 2b with we have respectively that S>.. = log3(log2)- 1 and S>.. =log ¥(log2)- 1 . A more delicate question in dimension theory is whether the Hausdorff measure of the limit set at dimension is finite. In general, the answer is negative. Below we will show that it is positive provided the mea.'lure It>.. is a Gibbs measure (see Condition (A2.20) in Appendix II). If a geometric construction is Markov then the measure It>.. is Gibbs provided the transfer matrix is transitive, i.e., the subshift is topologically mixing (the c!l.'le of an arbitrary transfer matrix can be reduced, in a sense, to the case of transitive transfer matrix; see Appendix II). A more general class of geometric constructions, for which P,>.. is still a Gibbs measure, includes geometric constructions modeled by mixing sofic systems or by mixing subshifts satisfying specification property (see the definition in [KH]). The latter form a much broader class of geometric constructions than the class of geometric constructions modeled by subshifts of finite type.

>. =

!

Theorem 13.4. [PW1] Let F be the limit set for a geometnc construction {CPW1-CPW4) modeled by a symbolic dynamical system (Q,u). Assume that the measure m>.. is a Gibbs measure then (1) the measure /1->.. satisfies the uniform mass distnbution pnnciple; (2) the Hausdorff measure mH(·, s,.) is equivalent to the measure m,.; in particular, 0 < mH(F,s>..) < oo; (3) 4m,..(x) = dm~(x) = S>. for every x E F; (4) dimH(F n U) = S>. for any open set U which has non-empty intersection with F.

Remark. If p, is an arbitrary Gibbs measure on Q, then by Theorem 15.4 below its push forward me!l.'lure m = X•/1- is exact dimensional (see definition of exact dimensional me!l.'lures in Section 7) and the pointwise dimension of m is constant almost everywhere (and is equal to h~-'(uiQ)/ fqlog>.i 0 dJ.t).

Proof of the theorem. Since It>. is a Gibbs measure one can repeat arguments

ill the proof of Theorem 13.1 (see (13.5) and (13.6)) and conclude that it satisfies the uniform mass distribution principle. This proves the first statement. Moreover, (13.6) implies that S>. ~ 4m~ (x) for every x E F. It follows that B>.. ~ dimH F and mH(F, S>.) > 0. We now prove that m,.(·) ~ const x mH(·,s>.)· Given 6 > 0 and a Borel subset Z c F, there exists e > 0 and a cover of Z by balls Bk of radius rk ~ e satisfying k

It follows from (13.6) that

m,..(Z) ~

E m,.(Bk) ~ G1ME (rk)'~ ~ G1MmH(F, B>.) + G1M6, k

k

where G1 > 0 is a constant. Since tJ is chosen arbitrarily this implies that m>.(Z) ~ G1MmH(F, S>.)

Chapter 5

132

We now show that mH(·, 8>.) :::; const x m~() Let Z c F be a. closed subset. Given 5 > 0, there exists c > 0 such that for any cover U of Z by open sets whose diameter :::; e we have

mH(Z,8~):::; L(diamU)"~ +5.

(13.11)

ueu

Note that one can choose a cover U of Z by basic sets b. {k) diam.b,.(k) :::; c and m>.(b.(k)):::; m>.(Z) + .t.eu

L

= 6;0

;,. k 1 1

satisfying

o.

We can apply (13.11) to this cover U and obtain using (13.1) and (13.5) that

L

mH(Z, 8>.):::;

(diamt.{kl)•>

.t,.!klEU

:::; K2D} 1

L

n

n(k)

L

+ 8:::; K2

Ai; ·~

+5

.t.C•>eu j=O

m.>,(.b.(k))

+ o 5 K2D1 1 m>.(Z) + (K2Di 1 + 1)!5.

.t.<•>eu

Since 5 is chosen arbitrarily this implies the second statement. Fix 0 < r < 1. For each wE Q choose n(w) according to (13.2). It follows from (13.1) that b.;. in(w)+l c B(x, K2r), where X= x(w). By virtue of (13.5) for all wE Q, n(w)+l

m>.(B(x,K2r)) ~ m>.(.b.; 0

;.

1.>+,)

~ D1

fl

>..::

~ C2r"',

k=O

where c2 > 0 is a constant It follows that for all -d

m, (X) =

X E

F,

. log m,x(B(x, r)) 11m l 5 S,x.

r--+0

og r

This implies the third statement. We now prove the last statement. It follows from the second statement that mH(F n .b.;. in' 8>.) > o. Thus, dimH(F n b..., in) ~ S,x. Now, let u be any open set with F n U =F 0. If x = x(i0 i 1 ... ) E F n U and n > 0 is sufficiently large then b.; 0 in C U. Therefore,

This completes the proof of the theorem.

•

Dimension of Cantor-like Sets and Symbolic Dynamics

133

Self-similar Constructions There is a special class of geometric constructions of type (CG1-CG2) which are most studied in the literature (see for example, (Fl]) -self-similar geometric constructions. They are geometric constructions with basic sets Aio . ;. given as follows: where h 1 , •.• , hp: D -t D are conformal affine maps (i.e., maps that satisfy dist(h;(x),h.(y)) = >.;dist(x,y) for any x,y ED, where Dis the unit ball in lim). Here 0 < ..\; < 1 are ratio coefficients. These geometric constructions can be modeled by an arbitrary symbolic dynamical system (Q, o'). Clearly, self-similar constructions are a particular case of Moran geometric constructions with stationary ratio coefficients (CPW1-CPW 4). Therefore, by Theorem 13.1, the Hausdorff dimension and lower and upper box dimensions of the limit set of a self-similar construction coincide. The common value is the unique root of Bowen's equation PQ(s.; 0 for w = (ioi1 ... ) (if a self-similar construction 18 modeled by a subshift of finite type or the full shift the numbers can be computed as stated in Theorem 13.3). If we further assume that basic sets at the same level of a self-similar construction do not overlap (i.e., their interiors are disjoint) then the geometric construction is a Moran geometric construction (CM1-GM5). Moreover, if basic sets at the same level are disjoint the coding map is a homeomorphism. Thus, the induced map G on the limit set of the self-similar construction (which in this case is a smooth map) provides a smooth realization of the subshift (Q, 0'). We also remark that if the ratio coefficients of the maps h. are equal (say, to a number >.) then the coding map is an isometry between the limit set of the geometric construction and Q {endowed with the metric d>.-'; see (A2.18) in Appendix II). Thus, it preserves the Hausdorff dimension and box dimension which can be computed by Statement 2 of Theorem 13.2 (compare to Theorem A2.9 in Appendix II).

14. Regular Geometric Constructions In this section we follow Pesin and Weiss [PWl) and introduce a class of geometric constructions that admit approximations by Moran-like geometric constructions with stationary ratio coefficients. This allows us to obtain effective lower bounds for the Hausdorff dimension of the limit sets. We will control the geometry of basic sets by numbers 'YI, ... , 'Yp such that one can replace the basic sets A; 0 in by balls of radius Dj= 1 'Yi;. In some cases these balls coincide with the largest balls that can be inscribed in the basic sets. However, this is not always the case and below we present an example where the "optimal" numbers 'Yb ... , 'Yp are completely independent of the radii of the largest inscribed balls (see Example 14.1 below). Consider a geometric construction (CG1-CG2) modeled by a symbolic dynamical system (Q, 0') (see the introduction to this chapter; it is worth emphasizing that we do not require the separation condition (CG3)). Given 0 < r < 1 and a vector of numbers 'Y = ('Yl> ••. , 'Yp), 0 < 'Yi < 1, i = 1, ... ,p, consider a.

134

Chapter 5

Moran cover i4 = ilr('Y) = {~(j)} of the limit set F constructed in Section 13. Given an open Euclidean ball B(x, r) of radius r centered at x, denote by R(x, r) the number of sets ~(j) that have non-empty intersection with B(x,r). We call a vector 1 estimating if R(x, r) ::; constant (14.1) uniformly in x and r. We call a symbolic geometric construction (CGI-CG2) regular if it admits an estimating vector If 'Y = ('Yt, ... , 'Yp) is an estimating vector for a regular geometric construction, then any vector 1' = (1'1> ... , "Yp) for which')'; 2:: .:y., i = 1, ... ,pis also estimating. We provide an example of a regular geometric construction on the plane that illustrates how the choice of the estimating vector can be made.

A.

'

::

~ 1

I

:

I

I

------vr---: :_:

' t-'

I I

::

\ I

I

I I

I

~ ------

o~----~+-r4-+.---------1Hr;~~r----;

-A.

---- ------- _'D~-. I 1

I 1

I 1

1 I

I

I I

I I

I I

I I

1 1

I

I I

I I

I I

I

I

I

I

I I

-I~----~---~---~-1----·L-----~---~---~--·--1.&------~ Figure 3. A REGULAR GEOMETRIC CoNSTRUCTION. Example 14.1 (PWI) Let 'Yt.'Y2,'Y3, and ,\ be any numbers in (0, 1). Given a number i = 1,2,3 consider a simple geometric construction (CG1-CG2) on the interval [0, 1) x {i- 2} with 2n basic sets of size 'Yin at step n. We denote this construction by CG(-y;). Since the 2n intervals at step n in each of these constructions are clearly ordered we may refer to the ith subinterval at step n, 1 ::; i ::; 2n of these constructions. Consider the 2n polygons in (0, 1] x [-1, 1) having six vertices which consist of the two endpoints of the ith subinterval at step n for all three constructions. We define the 2n basic sets at step n by intersecting these 2" polygons with the rectangle [0, 1] x (-,\", ,\n). This produces a simple geometric construction (CG1-CG2) on the plane See Figure 3.

Dimension of Cantor-hke Sets and Symbolic Dynamics

135

It is easy to sec that the limit set F of this geometric construction coincides with the limit set of the construction CG(-y2 ). Hence, by Theorem 13.3, dimH F = _ 1;:,~~2 and does not depend on 'Yt. 1 3 , or A. Let us choose numbers 'Yt, 12, 'Y3, and A such that 'Y2 < 'Yt = 'Y3 < A and 'Y2 < kn or 'Y2 < >.13 One can see that the inscribed and circumscribed balls of the basic sets at step n have radii wluch are bounded from below and above by cnr and c2>.n, respectively, where Ct and C2 are positive constants which are independent of n. Thus, these balls cannot be used to determine the Hausdorff dimension of the limit set. • Consider the positive number s-y such that Pq(s-ylog'Y; 0 ) = 0, where Pq denotes the topological pressure with respect to the shift u on Q (see Section 11) Let P.-r denote an equilibrium measure for the function (ioit ... ) >-+ s"''log'Y.., on Q, and let m"'' be the push forward measure ou F under the coding map x (i.e., m"''(Z) = p."''(x- 1 (Z)) for any Borel set Z c F). The following result provides a lower bound for the Hausdorff dimension of the limit set. Its proof is quite similar to the proof of Statement 2 of Theorem 13.1

Theorem 14.1. [PW1] Let F be the l1mit set for a regular symbolic geometnc construction. Then dimH F :::: s"'' for any estimating vector 1 Hence, dimH F :::: sup 8"'1, where the supremum is taken over all estimating vectors 'Y. In the case when the measure J.t. 1 is Gibbs one can strengthen Theorem 14.1 and prove a statement that is similar to Theorem 13.4.

Theorem 14.2. [PWl] Let F be the limit set of a regular symbolic geometnc construction and 'Y an estimating vector. Assume that the measure fJ."'( is a Gibbs measure Then

(1) the measure m"'' satisfies the uniform mass distnbution principle; (2) 0 < mH(F,s"''); moreover, m"''(Z) :0:::: CmH(Z,s"'') for any measurable set Z C F, where C > 0 is a constant; (3) s"'' :0:::: 4m~(x) for every x E F; (4) dimH{FnU):::: s"'' > 0 for any open set U which has non-empty intersection with F.

The second statement of Theorem 14.2 is non-trivial only when s 1 = dimH F. Otherwise, mH(F, s-y) = oo. If 8-y < 8 = dimH F, then the s-Hausdorff measure may be zero or infinite. Theorem 14.2 holds for simple geometric constructions or Markov geometric constructions with transitive transfer matrix In [Barl], Barreira gave sufficient conditions for a geometric construction to be regular. Roughly speaking, it requires that the basic sets rontatn sufficiently large open balls. We begin with geometric constructions on the line.

Theorem 14.3. Assume that each basic set 6.; 0

in of a symbolac geometnc construction {CG1-CG2} on the line contains an interval 1;0 in of length 0 < .>."" in < 1 such that 1;0 in n Iio in = 0 for any (io .. in) # (jo ... in). Assume also that there extSts 0 < 'Y < 1 such that

136

Chapter 5

where the minimum is taken over all Q-admissible n-tuples (i 0 .•. in)· Then the geometnc construction is regular with the estimating vector (re-•, ... , -ye-•) for any e > 0.

Proof. Given e > 0, we have A.., ,n > (~re-")n for every (i 0 i 1 .. ) E Q and any sufficiently large n. Given r > 0, one can find a unique number n = n(r) > 0 such that For any interval I of length r there exist at most two basic sets of length 2: ('Ye-•)'• intersecting I. Therefore, for every point x in the limit set the number R(x, r) in the definition of regular geometric constructions (see (14.1)) we obtain that R(x,r)::; 2 for all sufficiently small r. Hence, the construction is regular with the estimating vector (-ye-•, .. . , -ye-•). • We now formulate a criterion of regularity for a geometric construction in Rm with m > 1 Fix a point x E F and a number A. > 0. Given n > 0, consider two basic sets 6., 0 •n and 6.;0 in intersecting the ball B(x, A.n). Denote by a(6.; 0 in, fl; 0 in) the minimum angle of spherical sectors centered at x which contain both 6.; 0 in and 6.,0 Jn. Let an (x, A.) be the minimum of all the angles a( 6.;0 in, 6-;o ;,J.

Theorem 14.4. Assume that each basic set of a symbolic geometnc comtntction (CG1-CG2} in IR"' with m > 1 contains a ball B; 0 in C ilia in of radius An wzth 0 0 such that an(x, A.) 2: liN' for all x E F and n 2: 1 and that B; 0 in nB, 0 Jn = 0 for any (io ... in)# (jo . . jn)· Then the geometnc construction is r·egular with the estimating vector (A., .•. , A.).

Proof. By elementary geometry there exists a universal constant C = C(li) such that the maximum number of sets 6.; 0 ;n intersecting a ball B(A.n) does not exceed C. This proves the result. • ConsidPr a simple regular geometric construction inlR"" with the limit set F. It follows from Theorem 14.2 that the Hausdorff dimension of any open set U which has non-empty intersection with F satisfies dimH(F n U) 2: s for some .~ > 0. The following example shows that the converse statement may not be true.

Example 14.2. [Earl] For each s E (0, 1), there exists a geometric construction (CG1-CG2} on the line modeled by the full shift (~t, a) such that (1) dimH(F n U)

= s for any open set U with F n U #

0;

(2) the construction is non-regular. Proof. Define the function m· Q

m (w) =

. .

m (totl-··

-t N U { +oo} by

) = { +oo,

w=O

least j E N with ii = 1, w

#

0.

137

Dimension of Cantor-like Sets and Symbolic Dynamics 0

+t

1---------1110

\

!.. 1(4)

~

A

_ A(2)

1----t Ll 01 -

Figure 4. A

NON-REGULAR GEOMETRIC CONSTRUCTION.

Define also the numbt>rs A; 0

Tij= 0 A;

A;o in =

Ll

;,.

3

by

n
(j+2),

{ ( TI;=o ).i; (J + 2)) x (TI~=i+l A;, (l + 2)) ,

where£=m(io ... in0 ... ),Ao(j)=j,and). 1 (j)= ( 1-(j)• )

n ;::: l 1/s

. Noticethat {14.2)

for each j > 0. We consider basic sets spaced as shown on Figure 4. They have the following property: if D.>o. ;. =[an, bn] then an E D.io ;.o and bn E .1.;0 • in 1· Define intervals ,1.0) = .1.;0 ; 1 , where (io .. i;) = (0 .. 01). Inside each D,!i) we have a sub-construction modeled by CEt, a) with rates Ao Olio ;nf Ao 01 = TI~= 0 A;,(j + 2). Therefore, the Hausdorff dimension ofF n D,(i) is equal to s, where s is the unique root of equation (14.2), with j + 2 instead of j (sec Theorem 13.3). Hence, dimn(F n D,Ul) = s. Since F = {0} u U;>o(F n D,CJ)) it follows that dimH F = supi>O dimH{F n AUl) = s. Now, ifF n U # 0, there exists x = x(i 0 i 1 •.. ) E (F n U) \ {0}, and n > 0 such that A; 0 ;,. cU. Hence, .~ = dimn F ;::: dimn(F n U) ;::: dimn(F n A 00 .;,. ) = s This proves the first statement. Consider a vector b1.'Y2). For each n > 0, set rn = 2/{n -1) 1• Select now the Smallest positive integer k that > Tn and n;:~ 'YiJ ::; Tn for 1 some (i 0 i 1 . ) E Et Set 'Y = minb1.'Y2} and observe that 'Yk+l::; rn Hence, k ;::: logrn/ log')'-1 Therefore, for kn = log rn/ log')'-1 we get R(O, rn) 2: 2kn-n {where R(O,rn) il:i defined in {14.1)). We also have

SUCh

k n

-n> log2-log(n-l)l _ 1 _n log 'Y

n;=O ')';

~ n--+oo

logn!

---n -log')'

~

n--+oo

logn! -log')'

Chapter 5

138 Therefore, there exists D > 0 such that kn - n obtain R(O,rn)

~

2Dlognl

~

= n!Dlog2 >

D log n! for all n > 0, and we

(

r:

Dlog2 )

As the sequence 1·n decreases monotonically to 0 and (-n, 12) is arbitrary we proved that the construction is non-regular. • If a geometric construction is regular one can effectively replace its basic sets by balls to obtain a lower bound for the Hausdorff dimension of its limit set (see Theorem 14.1). We further exploit this approach and show that, under some mild assumption, a geometric construction (CG1-CG2) (whose basic sets are, in general, arbitrary close subsets and are possibly intersecting) can be effectively compared with a geometric construction whose basic sets are disjoint balls.

Theorem 14.5. Let F be the limit set of a qeometric construction (CGJ-CG2} modeled by a symbolic dynamical system (Q, u). Assume that there e:nst numbers A1 , ...• Ap, 0 < Ai < 1 such that for any admissible n-tuple (i 0 ..• in), ii = 1, ... ,p, we have n

diamD.to

in

:$ C

IJ Aii,

j=O

(14.3)

where C > 0 is a constant. Then (1) there e:nsts a self-similar geometnc construction modeled by a symbolic dynamical system (Q, a) satisfying: a) its basic sets B; 0 '" are disjoint balls, b) diamB; 0 in = 2C f17=o A;,, and c) there i.~ a Lipschitz continuous map 'lj; from its limit set F onto F; (2) dimBF :$ S:>...

Proof. Consider a self-similar geometric construction with ratio coefficients A1 , ... , Ap modeled by a symbohc dynamical system (Q, a) whose basic sets B'o in are disjoint balls of radii A;j. Denote its basic set by F. Let x: Q -+ F and Q -+ F be the coding maps. Consider the map 'lj; = xox- 1 : F-+ F We shall show that 'lj; is a (locally) Lipschitz continuoub map. Choose x, y E F 1{x) = (ioi1 .. ), 1 with p(x, y) e: We have that (y) = (joj1 ... ), and io = jo,. 'in = jn, in+l # jn+l for some n > 0. Therefore, llx- Yli ~ cl ITJ'=o A;j, where C1 > 0 is a constant. One can also see that 'lj;(x),'lj;(y) E .6.;0 in· Hence, A;r Since the map 'lj; is onto this proves the by (14 3}, li¢(x)- 'lj;(y) II c first statement. The second ~tatement follows immedi.ttely from the first one. •

c n;=O

x:

s

x-

x-

s n;=O

Geometric Constructions with Ellipsis We describe a special class of regular geometric constructions We say that a geometric construction (CG 1-CG3) in 1R2 is a construction with ellipsis (see Figure 5) if each basic set A; 0 in is an ellipse with axes }t/2 and X'/2, for some 0 <~<X< 1 (we stress that we require the separation condition (CG3)). Such constructions were studied by Barreira in [Barl].

Dimension of Cantor-like Sets and Symbolic Dynamics

139

Theorem 14.6. Let F be the limit set of a geometric construction with ellipsts. Then the construction i.s regular vnth the estimating vector (A,>..), where A is any number in the interval {O,t,.2(J.). Proof. For each >.. E (0, I) define the function 9>.= (0, 1) x (0, 1) -+ R by

[(X/t,.2)tt,.]tf2, >..<X Y>.(t,.,X)= (X/t,.)t, >..=X { t,.t-t, >.. >X where t =:log },_flog A. Consider an ellipse En with axes };<>n /2 and X"n /2, where a= log },_flog X. We assume that it is located outside the ball B(O,t,.n), is tangent to this ball at a. point, and that the major axis of the ellipse points towards 0. Denote by 2/3{n) the smallest angular sector centered at 0 that contains En· The desired result follows immediately from Theorem 14.4 and the following lemma.

Lemma. For each A E (0, X), there exists a number C > 0 such that tanf3(n) ~ Cg>.(t,., X)-n as n -+ oo. In particular, f3(n) decreases exponentially with the mte Y>.(t,.,X)- 1 when },.2/X < .>.<X and is uniformly bounded away from 0 if

o < .>. s. !//X.

Figure 5. A CONSTRUCTION WITH ELLIPSIS. Proof of the lemma. Consider an orthogonal coordinate system centered at 0 with the x-axis directed along the major axis of En. If m(n) = tanf3(n) is the slope of a line starting at 0 and tangent to En, the points of tangency are solutions of the equation

140

Chapter 5

provided that the discriminant of this equation is zero:

Set b01 =}:,_f)(>. One can see that 4 m(n)2 following cases:

= b~"'n/[b~(b~ + 1)].

We <'onsider the

>. > ->.; then ba > 1 and 2 m(n) ......, (b1 2alba: 2 ) n/2 = (6.a- 1 ) n = 9A(6., ->.) -n ; >.=A; then b, = 1 and 2v'2m(n}......, (b1"'t =g>.(~,Xfn; 12 (c) >. <X; then ba < 1 and 2 m(n) ....., (bl 2n/ba)" = 9>.(6., Xf". It is easy to check that if ~2/ X< >.<X, we have 9A(6., :\) > 1 and ifO < >.::;~//X, we have 9>.(6., A)::; 1. Since tanx rv x as x ~ 0 the desired result follows. • (a) (b)

15. Moran-like Geometric Constructions with Non-stationary Ratio Coefficients In this section we study Moran-like geometric constructions with ratio coefficients at step n depending on all the previous steps. This class of geometric constructions was introduced by Barreira in [Bar2]. Consider a geometric construction modeled by a symbolic dynamical system (Q, u) We assume that uiQ is topologically mixing and the following conditions hold: (CBl} ~io i,.j C ~io ;,. for j = 1, ... ,p; (CB2) where !1.; 0 ;ft and Bio ;~ are closed balls with radii K1r; 0 ;,. and K2r; 0 ,;,. and 0 < K1 S K2; (CB3) intl1.io i,. n intl1.jo im = 0 for any (io ... in) =F Uo ... jn) and m 2: n. The class of geometric constructions (CB1-CB3) is quite broad and includes geometric constructions (CPW1-CPW4) (see Section 13}, geometric constructions with contraction maps (see below}, and more general geometric constructions with quasi-conformal expanding induced maps (see Theorem 15.5 below). The study of geometric constructions (CB1-CB3} is based upon the non-additive version of the thermodynamic formalism (see Section 12 and Appendix II). We dt>fine the sequence of functions cp = {
Pq(scp) = 0,

CPq(scp) = 0

(15.2)

and to show that they have unique roots. We shall use these equations to produce sharp estinlcJ.tes and sometimes obtain exact values of the Hausdorff dimension

Dimension of Cantor-like Sets and Symbolic Dynamics

141

and box dimension of the limit set of a geometric construction (CB1-CB3). We need the following two assumptions on the radii of the basic sets The first assumption is the following: there ex-est L1 > 1 and L2 > 0 such that for any (ioi 1 ... ) E Q and n 2:: 0, (15 3) The role of Condition (15.3) is the following. First of all, one can easily derive from (15.3) that for any (ioil ... ) E Q and any n 2': 1, (15.4) where a and {3 are positive constants, i.e., the radii of basic sets decay exponentially. Therefore, the sequence offunctions {rpn} (see (15.1)) satisfies Condition (A2.22) (see Appendix II). By Theorem A2.6 Bowen's equations (15.2) have respectively unique roots §. and s that satisfy 0 ::=; §. ::=; s < oo. Furthermore, Condition (15.3) implies that for every n 2:: 0, 'Pn +log L2 ::=; 'Pn+l

Thus, Condition (12.5) holds. At last, given 0 < r < 1, one can construct, using Condition (15.3), a Moran cover U,. of the limit set F. It consists of basic sets .t.,(J) = .6.;0 •n<-;> such that: (a) x 1 E F; r;0 ;,.<•;> > r and r 00 ;,.<•Jl+' ::=; r; {b) the corresponding cylinder sets c C Q are disjoint. Moran covers have the following crucial property: given a point x E F and a number r > 0 the number of basic sets .t..io ;•<•j> = x(C; 0 '•<•j>) in a Moran cover U,. that have non-empty intersection with the ball B(x, r) is bounded from above by a number M, which is independent of x and r (a Moran multiplicity factor). Repeating arguments in the proof of Statement 1 of Theorem 13 1 one can prove the following result Theorem 15.1. [Bar2] Let F be the limit set of a geometnc construction (CB1C!J3) modeled by a symbolic dynam~cal system (Q, cr). Assume that the numbers 7'; 0 .;. satisfy Condition {15.8}. Then §. ~

dimH F :::; dim8 F ::=; dimBF

~

s.

The second assumption we need is the following: the sequence cp is sub-additive, i.e.,
Chapter 5

142

Theorem 15.2. (Bar2] Let F be the limit set of a geometnc construction (CB1CB3) modeled by a symbolic dynamical system (Q,u). Assume that the numbers rio. i. satisfy Conditions (15.3) and (15.5). Then

where s is the unique root of Bowen's equations PQ(s
-

== CPQ(s
=

1 lim -log

n-+CX)n

(>o in) q e.dmis8iblo

The assumption that the sequence

a > 0 and choose 8 > 0 such that 8 < !(b- a). Let mi be a sequence of positive integers. Denote

nk(a, b)= b

2.: i::;k

mi

j is odd

+

2.:

mi,

j:=;k

j is even

We define inductively the sequence mi Set m1 = 1 and choose mk such that lnk(a,b)/nk- bl < 8/k, if k is odd, and lnk(a,b)/nk -ai < 8/k, if k is even. We introduce now the sequence of functions cpn: l:j -+ lR by b(n-

n~:)

'Pn(w) = -nk(a, b)- { ( ) a n - nk

if k is odd if k is even

where k is the largest positive integer such that n" < n. Note that def

cpn ( w) = const = >.n

(15 6)

and 'Pn, > nkcpo This implies that

.n. Let F be the lmut set. Clearly, N(F, e>-n) $ 2n On the other hand, given an interval of length exp>.n there exist at most two basic sets which intersect it and at least one, if

Dimension of Cantor-like Sets and Symbolic Dynamics

143

the interval intersects F. Therefore, 2N(F,e)\•)?: zn. Since An+t- A-n?: -b by Theorem 6.4 we have d'

..ill!B

F _ l' -

liD

n----+oo

--,-- F d lffiD

logN(F,e)l•) - I lim~_ log2 -A - og 2 X _ _ , b , n

n~oo

......--- logN(F,e)l•) 1lffi ,

= n-+oo

-An

= 1og 2 X

/\n.

lim

n

n~oo

--.----An

log2 = --. a

By Theorem 15.1 we obtain that dimH F?: §.,where!!. is the unique root of the first equation in (15.2). Since An ?: -bn for all n > 0 we conclude that log2 -b- =

.

§_

.

-.-

log2 = dimH F == d!mBF < dimRF = a'

We need only to show that

s = log2/a. Observe that for every

s E lR,

1 ""' 1 SAn, -log ~ exp ( s sup 'Pn ) = -log(2n exp(s>.n)) = log2 + n (>o in) C;o '• n n

where C'O ;. are cylinder sets. By definition CPE+(srp) = log2 +slim An • n-too n

A.nh

= -n~c(a, b).

Therefore,

?: log2- slim n~c(a, b) ?: log2- sa k->oo

nJ..

Moreover, we have that -bn ~An ~ -an and thus, CPE+(srp) == log2- sa. 2 This implies that 8 = log2/a. •

Pointwise Dimension of Measures on Limit Sets of Moran-like Geometric Constructions Let v be a Borel probability measure on the limit set F of a geometric construction (CB1-CB3} modeled by a symbolic dynamical system (Q,u). We formulate a criterion that allows one to estimate the lower and upper pointwise dimensions of v. Given x E F, set

. f . log v{A; 0 .. i.) _ x ) =m , d( 11m n->oo logjA; 0 .i.l -( X ) =Ill . f -liD . log v(A;0 .i.) d 1 , n-too log IA'O in I where IA;0 ; . I denotes the diameter of the basic set A; 0 taken over all w == (ioit ... ) E Q such that x = x(w).

;.

and the mfimum is

Theorem 15.3. Assume that a geometric construction (CB1-CB3} satisfies Conditzon (15.3). Then (1) tfv(x) ~ d(x) for all x E F; (2) .d_(x) ~ fl.,(x) for v-almost all x E F; -

dcl

-

= d(x) == d(x) for v-almost every x E F, then !l,(x) == d,(x) == d(x) for v-almost every x E F.

(3) if 4(x)

Chapter 5

144

= x. Given r > 0, choose n = n(r,w) such that IA..,.;nl
Proof. Fix x E F and choose w = (ioi 1 ... ) E Q such that x(w)

We also haver ::; £,1A; 0

;.

I, where L 2 is the constant in {15.3). It follows that

-d ( ) _ ...,..---Iogv(B(x,r)) < -. logv(A;0 ;J < -. logv(A; 0 ;m) 1tm .,x-un 1nn . 1 1 r-+0 logr - n=n(r,w)-+oolog(r,IA; ;.I)- m-too logiA; 0 iml 0 Since this is true for every w = (i 0 i 1 ... ) E Q with x(w) = x the first statement follows. We now prove the second statement. Given a > 0 and C > 0, define Qa,O Set Fa,O

= {w = (ioit ... ) E Q : v(A;

= x(Qa,c).

;J::; C)A; 0

0

If w = (ioit ... ) E Qa,C and x(w)

;J'

for all n? 0}.

=X

then

fl(x) = inf lim logv(A;o in) ? a. n-too log [A; 0 in I Moreover, the set Fa = Uc>oFa,c coincides with the set of points for which d.(x) ? a. We will show that 4.,(x) ? a for almost every x E Fa· This will imply Statement 2. Indeed, if fl(x) > !l.,(x) on a set of positive measure then there exists a such that !l(x) >a> 4.,(x) on a set of positive measure. Fix x E Fa,c and r > 0. Consider a Moran cover 11,. of the set Fa,C and choose those basic sets in the cover that have non-empty intersection with the ball B(x, r) By the property of the Moran cover there are points xi E Fa,c, j = 1, ... , M (where M is a Moran multiplicity factor which is independent of x and r) and basic sets A,(;) such that Xj E .6,0) = A; 0 ·'·<•J>' diamA,(i) ::; r, and M

B(x,r)nFa,C

c

U
It follows that

M

v(B(x, r)

n Fa ,c)::;

L

v(A,U)

n F).

j=l

Assume that v(Fa,c) > 0. By the Borel Density Lemma (see Appendix V) for v-almost Pvery x E Fa,C there exists a number ro = ro(x) such that for every 0 < r ::; r 0 we have v(B(x, r)

n F)::; 2v(B(x, r) n Fa,c)

This implies that for v-almost every x E Fa,c,

. logv(B(x,r)) > . logi:;: 1 v(A.
dv() X = J1 rn

-

r-tO

log r

-

r->0

log r

logi:;: 1 C[A;0 '•<•;>Ia > '·mlogi:;: 1 Ctr"' ll =a, log r - r->0 logr where Ct > 0 is a constant. The last statement is a direct consequence of the preceding statements. .

> hm -

r->0

•

Dimension of Cantor-like Sets and Symbolic Dynamics

145

Let v be a Borel probability measure on the limit set F of a geometric construction (CB1-CB3). Even if the pointwise dimension dv(x) of v exists almost everywhere it may not be constant and may essentially depend on x. AB Example 25.2 shows, the pointwise dimension may not be constant even if v is a Gibbs measure. This phenomenon is caused by the non-stationarity of geometric constructions (CB1-cB3). The situation is different for geometric constructions (CPW1-CPW4) where ratio coefficients do not depend on the step of the construction. Theorem 15.4. Let F be the limit set of a geometnc construction {CPW1CPW4) modeled by a symbolic dynamical system (Q, a) and fl. an ergodic measure on Q. Let also m be the push forward measure of p, to F. Then m is exact dimensional {see Section 7} and for p,-almost every w = (ioit ... ) E Q we have 4m(x) where

X=

-

h,.(qJQ)

= dm(x) = _ J.Q Iog A·•o d1-' ,

X(W).

Proof. Since p. is ergodic by the Birkhoff ergodic theorem applied to the function w >-+ logA"' (where w = (ioit .. ) E Q) we have that for (l-almost every w the following linut exists: lim -1

L: log A,. =1log A; n

0

dJ.L.

Q

n->oon k=O

Exploiting again the fact that p. is an ergodic measure, by the ShannonMcMillan-Breiman theorem we obtain that for J.L-almost every w = (i 0 i 1 .•. ) E Q,

It follows from Condition (CPW3) that for J.L-almost every w = (ioi 1 ... ) E Q, lim

.!. log diam A;

n->oo n

0

= lim

i n

n->oo

t

.!.n k=O log A;. =

llog A; 0 dp, Q

The desired result follows now from Theorem 15.3.

•

Geometric Constructions with Quasi-conformal Induced Map Let F be the limit set of a geometric construction (CG1-CG3) in Rm modeled by a subshift (Q, 0'). Since we require the separation condition (CG3) the coding map x: Q -+ F is a homeomorphism and the induced map G: F -+ F is well defined by G = X o 0' o x- 1 We have the following commutative diagram

Chapter 5

146

It is easy to see that G is a continuous endomorphism onto F. By the result of Parry [Pa] it is a local homeomorphism if and only if the subshift is a subshift of finite type, i.e. Q = ~1, where A is a transfer matrix. From now on we consider tllis case and assume that A is transitive, i.e., the shift is topologically mixing (see Appendix II). The induced map encodes information about the sizes, shapes, and placement of the basic sets of the geometric construction and hence can be used to control the geometry of the construction. In order to illustrate this let us fix a number k > 0. For each w = {ioil· . ) E ~1 and n 2:: 0, we define numbers A(w n)

- '

= A (w ~

'

"X(w n) =X (w n) =sup { ,

k

an(y)ll} llx- Yll ' IIGn(x)- G"(y)ll} llx- !Ill ,

n) = inf { IIG"(x) -

,

(15.7)

where the infimum and the supremum are taken over all distinct x, y E F n ~; 0 in+•. It may happen that ~(w, n) = 0 or "X(w, n) = oo for all sufficiently large n. In the case 0 < .d(w,n) ~ X(w,n) < oo, consider the limits

.d(x)

1 = nlim -log~(w,n), ...... oo n

-

A(x)

1 = n-+!Xl lim -logA{w,n), n

where x = x(w). Notice that by the multiplicative ergodic theorem (see for example [KH]), the limits exist for almost all x with respect to any Borel Ginvariant measure p, on F provided that

If the map G is smooth the numbers ~(x) and X(x) coincide with the largest and smallest Lyapunov exponents of Gat x (see definition of Lyapunov exponents in Section 26). When G is continuous these numbers can serve as a substitution for the Lyapunov exponents (see [Ki]). We consider the case when the trajectories of the induced map are strongly unstable. More precisely, we call the induced map expanding if there exist constants b 2:: a > 1 and ro > 0, such that for each x E F and 0 < r < ro we have B(G(x),ar) C G(B(x,r)) C B(G(x),br). (15.9) Note that if the induced map G is expanding then it is (locally) hi-Lipschitz. Furthermore, if the induced map G is expanding then the placement of basic sets of the geometric construction cannot be arbitrary (sec Theorem 15.5 below). We now specify the choice of the number k in (15 7). Namely, we assume that k is so large that (15.10) diam ~io in+• ~ ro.

Dimension of Cantor-like Sets and Symbolic Dynamics

147

We say that an expanding induced map G is quasi-conformal if there exist numbers C > 0 and k > 0 (satisfying (15.10)) such that for each w E E1 and n ~ 0, X(w, n) :s; C ~(w, n). (15.11) As the following example shows geometry of constructions (CG1-CG3) with expanding quasi-conformal induced maps (i.e , the placement of basic sets and their "sizes") is sufficiently "rigid".

Theorem 15.5. Let F be the limit set of a geometnc construction (CG1-CG3} in IRm modeled by a subshift of finite type (E!, u). Assume that the induced map G is quasi-conformal and that the basic sets on the first step of the construction have non-empty interwrs. Then (1) the construction satisfies Conditions (CB1-CB3}, i.e., it is a Moran-like geometnc construction with non-stationary ratio coefficients; moreover, it also satisfies Conditions (15.3} and (15.5}; (2) dimy F = dimBF == dimBF = s, where s ib a unique number satisfying lim !.tog n-+oon

(diam(Fn~; 0

""' L..J

(io

;.))'

=0.

i.)

E1-admissible

Proof. We outline the proof of the theorem. Smce the basic sets on the first step of the construction have non-empty interiors we observe that each basic set ~io ;,. of the geometric construction satisfies

!J..io

in C

~io

.in C B;o

in'

where !1.;0 ; . and B; 0 ; . are closed balls with radii [.0 there exist numbers Ct > 0 and C2 > 0 such that

c1 - - < rX(w, n) - -•o

;.

and 1';0

; ••

Moreover,

< r·•o '•· < - c2 -- ~(w, n) ·

'• -

Since the induced map G is quasi-conformal this implies that the geometric construction satisfies Condition (CB2). Clearly, Conditions (CB1) and (CB3) hold and thus, the construction is a Moran-like geometric construction with nonstationary ratio coefficients. By straightforward calculations one can show that given w = (i 0 i 1 ... ) E E1 and n,m ~ 1, ~(w, n + m) ~ ~(w, n) x ~(un(w), m), and similarly, X(w, n

+ m) s

X(w, n)

X

X(an(w), m).

Since the induced map G is quasi-conformal the above inequalities imply Condition (15.5). It follows from (15 9) and (15 10) that a :s; ~(w, 1)

:s; X(w, 1) :s; b.

Thus, the first inequality in (15.3) holds. Similar arguments show that the second inequality also holds. This implies the first statement The second statement follows from the first one and Theorem 15.2. •

148

Chapter 5

One can build a geometric construction (CPW1-CPW4) for which the induced map on the limit set is not expanding: whether it is expanding depend on the placement of basic sets on each step of the construction (see below). We present now more sophisticated examples which illustrate properties of induced maps. Example 15.2. [Bar2] There exists a geometnc construction {CG1-CG9} on the line modeled by the full shift (Et, u) such that:

(1) each basic set ~;0 ; , is a closed interval; (2) there e:nsts a point w E Ef such that ~(w, n)

= 0 and A(w, n) = oo for each n;::: 1; hence, the induced map G is not expanding.

Proof. Consider a geometric construction on the line modeled by the full shift on 3 symbols for which ~;0 ; , = [ajn• bjn] for each (io ... in) = (0 ... 0} and j = 0, 1, 2. One can choose the basic sets such that: (a) for each n ;::: 0 the points bon, a 1n, b1n, and b2n lie in the limit set F; (b) the difference a 1, - bon is e-a(n+l) for n even and e-b(n+l) for n odd; (c) the difference a2,.. - btn is e-a(n+l) for [n/2] even and e-b(n+l) for [n/2] odd. Here a and b are positive distinct constants (see Figure 6a where a = log 5, b = log6, and the intervals ~; 0 ;,. are of length 5-n). This implies that ~(w, n) = 0 and A(w, n) = oo for the point (ioi 1 .•• ) = (00 . . ) and all n;::: 1. • The following example shows that there are geometric constructions (CG 1CG3} for which the induced map may be expanding but not quasi-conformal. It also illustrates that in this case Theorem 15.5 may fail. Example 15.3. [Bar2) There extsts a geometnc constructwn (CGI-CG3} on the line modeled by the full shift (Et, u) such that

(1) each basic set ~; 0 ; , is a closed interval of length depending only on n; (2) the induced map G is expanding but not quasi-conformal; (3) dimy F = dim 8 F < dimnF.

Proof. (See Figure 6b.) Let An be numbers defined by (15.6). Consider a geol!letric construction (CG1-<::G3} on the line modeled by the full shift on 2 symbols with the basic sets ~; 0 • in to be closed intervals of length exp An· Given a basic set ~; 0 .. in, we require that the sets ~io inO and ~; 0 ;, 1 be attached respectively to the left and right end-points of the interval ~io ..;, This guarantees that for each x, y E F n ~'<> ;" such that x # y, the ratio IJGnx- Gnyll/llx- Yll is of the form aAm-n + '1;"'00 k ·eAm-n+> ·"-'j=l 3. eArn

00 + "-'3=1 '1;"' k3·eArn+;

for some m ;::: 0, and ki taking on one of the values {-2, -1, 0, 1, 2} for each sequence~ kn are admissible. Since

j ;::: 0. Notice that not all

149

Dimension of Cantor-like Sets and Symbolic Dynamics we obtain

-

e>..,._,.

1+2E~ 1 co.J

A(w, n):::; sup->.- x 1 2 z:::oo -aJ :::; m;:::o e "' i=l e

A(w n)

-

'

e>.,._,. - e>.m

> inf - - x -

m>O

ch"(l+c"') 1- 3e-a

< oo,

1- 2 E~ 1 e-aj e-a"(I- 3e-a) J. > >0 1 + 2 L..,J=l ~ e-aJ 1 + e-a ' 00

for all b ~ a > log 3. If a is sufficiently large we have

It follows that a :::; ~(w, 1) :::; X(w, 1) :::; j3, where a and j3 are some positive constants. Therefore, for a sufficiently small r 0 and any x, y E F we have that allx- vii :::; IIG(x)- G(y)ll :::; J3llx- Yll provided llx- vii :::; ro. Since G is a local homeomorphism one can easily derive from here that G is expanding. Notice that by the construction of the numbers An, we have that sup (Am - Am-n) m;::o for any n

~

= -an,

inf (Am - Am-n) m;::o

= -bn

(15.12)

0. Hence,

This proves that the induced map G is not quasi-conformal. One can see that 2n-l :::; N(F, e>.,.) :::; 2" (recall that N(F, r) is the smallest number of balls of radius r needed to cover F). Therefore, (15.12} implies that .

d --.!illB

F- log2 - -b-'

-d. F 1mB

log2 =--. a

Notice that our construction is a geometric construction (CB1-CB3) with basic sets satisfying (15.3). By Theorem 15.1 we have that dimn F ~ s, where 8 is the uruque root of the equation PE+(stp) = 0. Since A,.~ -bn for all n ~ 0 we • conclude that 8 ~ log 2/b. Hence, dimn F = dim 8 F = log 2/b. •

Chapter 5

150 0

(

J

ao

a!

(J\

\ a2 1 5

5

ao2

aoo ao1

H H 1--t

1

1

36

25

a 0

+(

\+

)! ,an

a 00 .............. +( }

I

_1 { 4

4

} + +{ b Figure 6. GEOMETRIC CONSTRUCTIONS WITH INDUCED MAPS: a) Non-expanding, b) Non-conformal. Geometric Constructions with Contraction Maps There is a special class of geometric constructions (CGl-cG2) that are wellknown in the literature (see for example, (FI]) - constructions with contraction maps. Their basic sets .a.,0 • ;,. are given as follows

.a.;0 . in = h., 0 h1

1 0 • • · 0

h;, (D).

Here D is the unit ball in Rm and hi. . .. , hp: D --+ Dare hi-Lipschitz contraction maps, i.e., for any x, y E D,

A dist(x, y)

~ dist(h;(x), h;(y)) ::;

X; dist(x, y),

where 0 < 41 ::; X;< 1. They arc modeled by a subshift (Q, rr). It is easy to see that these geometric constructions are Moran-like geometric constructions with non-stationary ratio coefficients of type (CB1-CB3) and hence can be treated accordingly.

Dimension of Cantor-hke Sets and Symbolic Dynanncs

151

If we require the separation condition (CG3) then the coding map is a homeomorphism and we can consider the induced map G on the limit set F of the construction which acts as follows G(x) = h,(x)- 1 for each x E F n ~. and i = 1, .. , p. Hence, G is expanding, i.e., it satisfies (15.9). As we mentioned above G is a local homeomorphism if and only if Q is a topological Markov chain, i.e., Q = E1 for some transfer matrix A (which we assume to be transitive). One <".an also verify that for each w = (ioi1 ... ) E E1 and n > 0, n

II ~.j j=O

:;

n

~(w, n) :::; X(w, n) :::;

II x.j. j=O

By Theorem 15.2 the Hausdorff dimension and lower and upper box dimensions of the limit set of a construction with contraction maps admit the following estimates: d.:::; dimH F :::; dimBF :::; dimBF :::; d, where d. and d are unique roots of Bowen's equations PE+(d
A

-

Notice that, in general, d. :::; !!. and d ~ s, where §. and 8 are roots of Bowen's equations (15.2) (see Theorem 15.1). The inequalities can be strict (see [Bar2]}. This illustrates that one can obtain more refined estimates of the Hausdorff dimension and upper box dimension using the numbers ~(w, n) and X(w, n) than using the Lipschitz constants of the contraction maps (i.e., the numbers ~i and X;). If ~ = X, for any i = 1, ... , p then the the contraction maps are affine and conformal The induced map G is quasi-conformal (and in fact, is smooth and conformal; see Section 20) and the Hausdorff dimension and lower and upper box dimensions of the limit set coincide. The common value is given by Theorem 13.3. In particular, if the geometric construction is simple (i.e., is modeled by the full shift) then s is the unique root of the equation

Geometric Constructions Associated with Schottky Groups

A Kleinian group is a discrete subgroup of the group of all linear fractional of the complex plane C with the determinant ad transformations z >-7 az+±db cz be = 1 or -1. A linear fractional transformation g is said to be hyperbolic if tr 2g = (a+ d) 2 > 4 and loxodromic if tr 2 g E C\ [0, 4] A (classical) Schottky group r is a Kleinian group with finitely many generators 911 ..• gp, p ~ I which act in the following way: there exist 2p disjoint circles 'Yl, 'Yi, . . , 'Yp, 'Y~ bounding a 2p-connectcd region D for which 9; (D) n D = 0 and gi('Yi) = 'Yj for j = 1, ... ,p. The group r is known to be free and purely loxodromic, i.e., all non-trivial elements of r have either hyperbolic or Joxodromic type (see [Mas], [Kr]).

C h a p te r

5

152

IN G T O SPOND CORRE . N IO T ATORS UC GENER ONSTR T R IC C THREE E H M IT O E W A G GROUP e E C T IO N E if th A REFL

s ta b il iz e r C z t in z such o at a p rhood U nuously ~eighbo ti n a (r) of o s c Q a is h t d e id to a c t r is finite, a n d z for g E r z· T h e s a s is r p ontinuity T h e grou (z) :: z} of z in z a n d g(U•.) "" U. n o f disc io g re g e factor r : j th r. = {g .)E nrU . "" 0 for all g Edisr continuously is aclallSecdhottky group th e n th e ts ic th a t g(U et is a class hich r a c e li m it s th a t if r E C at w wed a s th eral, a re p o in ts z p r I t is known ie . v p e s u b n n e u ca fg gen o f th e g ro closed surface o k y g ro u p onformal b u t, in d) a c al S c h o tt (one-side is ic a re f ss a / y la h b c f) ic ( d a h n le w r e ed s fo d p ll o t a a e c m s m T h e li m it c o n s tr u c ti o n w it h is c o n s tr u c ti o n is S c h o tt k y g ro u p is t th e h a c j T th -r tn . e :: m ng ow "YJ :::: for a g e o ine nor c o n tr a c ti ne c a n sh up, ial case ff "" 3). O h a reflection g ro r th e spec p In re . e e h n e it h e r a p e w it ty th w 7 e o d it e re n te il u fi of a l wh ssocia e Fig subshift r o u p (se ic c o n s tr u c ti o n a long main diagon it s e t o f th e g n o ti c etr m sa th e r e fl e th e geom fe r m a tr ix h a s O m a p G on th e li s n e ra ti n g maps, g e ti n g a n d th e tr a n th a t th e induced xpanding. e ac a re c o n tr equal to L N o te th conformal a n d s ic S e t s o o re a m jo in t B a s is is D n h o entries it ti ) w c o n s tr u c 13) m o d 1-CPW4 geometric e Section e s (CPW (s n ) o f th e 4 ti c W u sic s e ts o W 1 -C P ic C o n s tr ti o n (C P a t th e ba .. • ) E Ejj:, c th u tr e s G e o m e tr m n u o i1 etric c We ass h w = (i 0 er a geom (~1, a). t for eac a We consid ift o f finite ty p e th fy ri ve subsh O n e can n eled by a disjoint. re a n o ti c u tr .> .i r s IJ n o c ) S K2 n 5 X(w, n ) J=O n , n>O, (w A 5 th e in1 K1 for which a d e r) . ) 4 W P re 1 -C j= O e for th e n (C P W n s tr u c ti o as a n easy exercis om th e above o c ic tr e fr ows t a. geom ave this (t h is foll . is th e n u m b e r in c o n s tr u c expanding (we le nformal o S> -c One c a n re t si e ions o a h n u w q it ts x dimens = S>,, a p G is duced m x p a n d in g .1)). T h u s ,! ! = s wer a n d u p p e r b o e is it if 3 lo However, a n d C o n d it io n (1 f dimension a n d es dorf · .) S> inequaliti .1 (i.e., th e Haus l to 13 a re equa T h e o re m s e t coincide a n d it o f th e lim

. F ig u r e 7

J1.>.i

Dimension of Cantor-like Sets and Symbolic Dynamics

153

16. Geometric Constructions with Rectangles; Non-coincidence of Box Dimension and Hausdorff Dimension of Sets

A crucial feature of Moran-like geometric constructions with stationary or non-stationary ratio coefficients is that they are, so to speak, isotrop~c, i.e., the ratio coefficients do not depend on the directions in the space. This is a very strong requirement and that is why the placement of the basic sets may be fairly arbitrary. A simple example of anisotropic geometric constructions is provided by constructions with rectangles. As in the case of Moran-like geometric constructions one still has a complete control over the sizes and shapes of the basic sets but needs two collections of ratio coefficients to control length and width of rectangles on the nth step. In this section we consider the simplest case when the ratio coefficients are constant and do not depend on the step of the construction. Even so, we will see that the Hausdorff and box dimensions of the limit set may depend on the placement of the basic sets and may not agree. This can happen even if a geometric construction is "most close" to a self-similar construction, i.e., it is given by finitely many affine maps (the so-called general Sierpinski carpets; discussed later in Section 16). In this section we present examples which illustrate how the equality between the Hausdorff dimension and box dimension can be de:,troyed. These examples are also a source for understanding some "pathological" properties of the pointwise dimension (see Section 25). A surprising phenomenon is that the Hausdorff dimension of the lirmt set for constructions with rectangles may also depend on some delicate number-theoretic properties of ratio coefficients corresponding to different directions. Geometric Constructions with Rectangles We call a symbolic geometric construction (CG1-CG3) on the plane modeled by a symbolic dynamical system (Q 1 u) a construction with rectangles if there exist 2p numbers~; and A;, i = 11 • • • ,p, 0 <~;~A;< 1 such that the basic set ~io ;" is a rectangle with the largest side not exceeding K 1 1 A;; and the smallest side not less than K 2 Ilj= 1 ~; where K 1 and K 2 are positive constants. See Figure 8. We stress that we require the separation condition (CG3). Lets~ and sx be the umque roots of Bowen's equations PQ(S~log~; 0 ) = 0 and PQ ( sy; log Aio) = 0 respectively and ml!. and mx be the push forward measures by the coding map of equilibrium measures J.Lli and 11->: corresponding to the functions (ioil· .. ) t-+ S},_log~io and (ioil ... ) t-+ sxlogAio·

IT;=

1

Theorem 16.1. Let F be the limit set for a symbolic geometnc construction unth rectangles. Then (1) s~::; dimH F ~ dim 8 F ~ dimaF::; sy;; (2) If the measures J.L}!. and P.>; on Q are Gibbs, then s~ ~ d.m,~,. (x) and

dm:;;(x)::; sx.

Cha.pter 5

154

is reg ula r wi th cti on wi th rec tan gle tru ns co ic etr om s 14.1, 14.2, th e ge lows from Th eo rem Pr oo f. Notice th at • ). Th e res ult fol '~P ... , (~ r 1 the es tim ati ng ve cto trec cti on s wi th an d 14.5. ge om etr ic co ns tru of n tio rip X sc de a X; d gave at ::\; ,1 an In [B arl ], Ba rre ira me for sim pli cit y th su as d an ch oa pr his ap angles. We follow 1. 1 [0,27r]/{0,27r} so me 0 < ,1 < X< for ,p l,. = i te by /'n{w) E § =: no for de 0 > n d an We first •) E Q sides of len gth Given w = (ioi 1 •• • ,n, i.e., the 6., of es rat e. 0 al sid st nti ge ne e lon th an ex po th e direction of th ly when n --l- oo wi rm ifo un es erg nv co show th at 'Yn(w)

=

=

xn.

. Pr op os iti on 16 .1

lim 'Yn(w). 7( w) ~f n-> oo re ex ist s the lim it the Q, wE ch ea r (1) Fo Q, E w all for t tha > 0 such (2) There exists C MX)". h( w )- 'Yn(w)l < C(

Fi gu re 8. A

NS TR UC TIO GE OM ET RI C CO

(16.1)

AN GL ES . N WI TH RE CT

placeth e ''worstn possible We first consider ir of pa Q a E ) 9): . . re i gu (io 1 ns tru cti on (see Fi Pr oo f. Fix w = co the of p gest ste lon h on th e nt ches the two me nt of rectangles tan gle 6.; 0 ;,.+ 1 tou rec a of s ce rti ve sed dia me tri ca lly oppo 6." ' ;,. . an d 6. 10 '"+' (see st sides of 6.;., ,,. sides of a rectangle ge lon the n ee tw the equation be Le t Bn be the angle 9,. does not depend on wa nd satisfies at th e tic e no

Figure 9). W

Dimension of Cantor-like Sets and Symbolic Dynamics

155

Setting a = >..;~ we obtain that sin On satisfies the fol1owing equation: (an+l sin11n- 1/~) = 1 -- sin 2 On. 2

The formal roots of this equation are

. (}

SJD n

= 1

1 (1xa

+a

2(n+l)

n+l

-

±

1 - ;2

+ a2(n+l))

.

Since cos 8,. > 0 we have to choose the root with minus in front of the square root. The discriminant of this equation is non-negative if and only if lA., in I = [~2(n+l) + )..2(n+l)) 1/Z < _,i". This takes place if n is sufficiently large, i.e., IA;0 ;~+•! < }/'. One can also see that sin8n is asymptotically equivalent to (1/~ -1) a-(n+ll as n--+ oo. Since~< 1 there exist C 1 > 0 and C2 > 0 such that C1a-n < sin Bn < C 2 a-n for all sufficiently large n. One can now choose C 1 and C 2 to obtain in addition that C1a-" 0 is sufficiently large and n > m ?: no one has C1a-(m·H) [

1-a-t

1

_ a-(n-m)] <

..[!-. L.J

k=m+l

8 "'

<

C2a-(m+1)

[l _ a-(n-mJ].

1-a- 1

(16.2)

We now consider a general placement ofthe basic sets. Given wE Q, there exist angles Bk(w) (for each k > 0) with !B~c(w)i :::; 8~c such that 'Yn(w) = L;~=l(jk· Therefore, if n > m ?: no, using (16.2) we have

This shows th<\t the sequence (-y,.(w)),eN converges and satisfies (16.1).

.6, . . lO"'

'n

Figure 9.

THE "WORST" DISPOSITION OF RECTANGLES.

•

Chapter 5

156

Proposition 16.2. The vector field 7 o x- 1 : F

-t

S 1 is uniformly continuous.

Proof. Let w = (i 0 i 1 •.. ) E Q. Omsider a sequence Wn -t w as n -too. There exists a sequence of numbers mn -too as n -too such that x(w) E A;0 in for each n > 0 and k 2:: m ... Therefore, 1n(wk) = 1n(w) for n E N and k 2:: ffin. Using (16.1) we conclude that

i-y(wk) - ')'(w)i~i'Y(Wk) - 'Yn(wk) i+i'Yn(Wk) - 'Yn(w)i + i'Yn(w)- ')'(w)j< 2C(5.j},)-n

(16.3)

for n sufficiently large and k ;::: m ... Given e > 0, let us choose n > 0 such that 2C(5.j},)-n <e. Then I'Y(wk)- ')'(w)l < e fork 2:: m ... Since F is compact this gives the desired result. • Propositions 16.1 and 16.2 show that the basic sets of a symbolic geometric construction with rectangles cannot be spaced arbitrarily, but arc forced to be oriented to approach a continuous vector field. In [Bar 1), Barreira showed that the problem of computing the Hausdorff dimension and box dimension of the limit sets for geometric constructions with rectangles can be reduced to study constructions of a special kind where rectangles are aligned, i.e., the sides of length >:' for all n 2: 0 are parallel (see Figure 10). We present here two of his results in this direction. The proofs are based on Propositions 16.1 and 16.2 and are ormtted.

I I

I

t·

I

Figure 10. A CONSTRUCTION WITH ALIGNED RECTANGLES. Theorem 16.2. Given a geometnc construction with rectangles, modeled by a symbolic dynamical system (Q, a), there ensts a geometnc construction with altgned rectangles, modeled by (Q, a), such that dim8 F = dim 8 F and dimBF = dimBF, where F and F are the limit sets of the two constructions. We say that a geometric construction has exponentially large gaps if there exists o > 0 such that for each (i 0 i 1 •.. ) E Q and each n EN we have that

Dimension of Cantor-bke Sets and Symbolic Dynamics

157

dist(.6.; 0 ini• .6.io ink) 2': 8f!n whenever j =/= k and (io ... inj) and (io ... ink) are Q-admissible. With this extra hypothesis we can study the Hausdorff dimension of a geometric construction with rectangles by comparing it with an appropriate geometric construction with aligned rectangles. Theorem 16.3. Given a geometnc construction with rectangles and exponentially large gaps, modeled by a symbolic dynamical system (Q, u), there extsts a geometnc construction with aligned rectangles and exponentially large gaps, modeled by (Q, u), such t~at the map f: F_--+ F ts Lipschitz with Lipschitz inverse and dimH F = dimH F, where F and F are the limit sets. Non-coincidence of Hausdorff Dimension and Box Dimension We provide several special examples of geometric constructions with rectangles that illustrate some interesting phenomena and reveal the non-trivial structure of these constructions and the crucial difference between them and Moran-like geometric constructions with stationary ratio coefficients. We consider a geometric construction with rectangles in R2 which is generated by finitely many affine maps /!, ... , /p: S --+ S, where S = [0, 1] x [0, 1] is the unit square. Let F be the limit set. Each map !. can be written a.q fi(x, y) = T;(x, y) + b;, where T; is a linear contraction and b; is a two-vector. In [F2] (see also [F4]), Falconer proved that for almost all (b1 .•. bp) E R2P (in the sense of2p-dimensional Lebesgue measure), the Hausdorff dimension and the lower and upper box dimensions ofF coincide and the common value is completely determined by T1 , ... , Tp provided that liT; II < ~· On the other hand, as we mentioned above, the Hausdorff and box dimensions of the limit set for geometric constructions with rectangles may depend on the placement of the basic sets. They may not agree even in the case when the construction is generated by finitely many affine maps. The corresponding example is described by McMullen (see [Mu]; in [LG), Lalley and Gatzouras considered a more general version of his construction). Given integers e 2': m 2': 2 choose a set A conslSting of pairs of integers (i,j) wrth 0 ::; i < l and 0 ::; j < m. Denote by a the cardinality of A (clearly a ::; mn). Let J~c, k = 1, ... , a be affine maps given in the following way: if k enumerates the element (i,j) E A then fk(S) = S;;, where S = [0, 1) x [0, 1] and S;; = [~, ~] x [;;, ~] The affine maps fk generate a simple geometric construction with basic sets .6.io in = /; 0 o · · · o /;n (S) being (rn x m-n)rectangles. We allow some of the basic sets .6. 1, ... , .6.., on the first step of the construction to intersect each other by either a common vertex or a common edge (see Figure 11 where l = m = 4 and a = 6). The geometric description of this construction is the following: starting from the (l x m)-grid of the unit square S choose rectangles corresponding to (i, j) E A, then repeat this procedure in each chosen rectangle and so on. The limit set F for this construction is

F= {

(f~:. n=D

and is known as a general Sierphiski carpet (see [Mu]).

Chapter 5

158

Example 16.1. (1) dimH F = logm(l::j:~ t~og,m) ~ s, where t; is the number of those i for which (i, j) E A. (2) dimBF = dimBF = lo~ r + logf(~), where r is the number of those j for which (i,j) E A for some i. 1

Remarks. (1) For the geometric construction shown on Figure 11 we have t 0 = 1, t 1 = 2, t2 = 2, t3 = 1, and r = 4. (2) The Hausdorff and box dimensions of the limit set F agree if: a) l = m; b) the constants t; take on only one value other then zero. In the first case the construction is conformal self-similar and the common value for dimensions is logm a. Proof. Let X· r:;t -t F be the coding map given as follows:

~ .i::._) ~mn '

n=O

where kn = (in, in)· This map provides a symbolic representation of the limit set F by the (one-sided) shift on a symbols It is surjective but may fail to be injective because some points in F may have more than one representation. Define numbers bk, k = 0, ... , a- 1 in the following way. If k = (i,j) E A then bk is the number ofi' such that (i',j) EA. Note that by the definition of the numbers, 8 _

a-1 ""blog, m-1

m-L.,k

.

k=O

We define the Bernoulli measure on E;t by assigning probabilities blog,m-1

Pk = -'""---::--

m•

to each symbol k = 0, ... , a - 1. In other words, if Cko kn is a cylinder set then tt( Oko kn) = O~=O Pk,. Note that l::~=l Pk = 1. Let A be the measure on F which is the push forward of the measure J.l. (i.e., .\(A) = JJ-(x- 1 (A)) for any Borel subset A c F) We will show that dimH .\ ~ ,q Set q = [nlogt m] $ n. Given (n + 1)-tuple (ko ... k,.) with kt = (idt) E A, consider the set Rko kn C F of all points (x, y) for which 00

X= L.,fj• t=O

00

.,

""Zt

y=

.,

L~t' t=O

where i~ = it for t = 0, ... , q and J; = }t for t = 0, . , n The set Rko kn is "almost" a ball of radius m-n. More precisely, there are constants 0 1 > 0 and 02 > 0 independent of (ko ... kn) and a point (x,y) E Rko kn such that (16.4)

Dimension of Cantor-like Sets and Symbolic Dynamics

159

We will show that the sets R~oo ... kn comprise an "optimal" cover of J which we use to compute the Hausdorff dimension of the measure >... Note that each set R~oo. ,." can be decomposed into finitely many basic sets Llko kok~+> k~ n F, where k~ = (i~, jD and J~ = jt fort= q + 1, ... , n Moreover, the number of such sets is equal to b"<+' b"•+• · · ·b"". Define now the function r/Jn on Et by

It is easy to see that this function is constant on the cylinder Cko "". We denote

the common value by 4>ko . kn . Lemma 1. For any (k0 ••• kn) we have

Proof of the lemma. Let us pick an (n +I)-tuple (ko, k1 ,

,

kn) and set

Note that this sequence is independent of the choice of (k 0 ,k 1 , ... ,kn)· This implies that

As we have mentioned the number of basic sets Llko . "•"~+'. ~.:;. is equal to Therefore,

b"•+• b"•+• · · · bk".

'(R /\

)ko

""

-

SoSJ .. , Sn Jogt m SoSl · • • Sq

m

-•n -

-

("' 'l'ko

This completes the proof of the lemma

•

We now describe the "gaps" between the sets Ll~o0 "•"~+• ~o;. n F inside the set Rko "" in terms of the behavior of the functions ¢.,. over n.

Lemma 2. (1) limn-+ooif>n(w);::: 1 for all wE Et. (2) ¢.,. -+ 1 as n -+ oo J.t-almost everywhere. Proof of the lemma. Define the functions Un and hn on E;t by

Chapter 5

160

2

4

3

Figure 11. A GENERAL SIERPINSKI CARPET WITH l = m A= {(0, 1), (1, 0), (1, 4), (2, 2), (2, 3), (3, 0)}.

= 4 AND

Then ¢>,. (w) = h,.(w) x g,.(w)log, m We claim that the functions g,. and h,. satisfy the following properties wluch immediately imply the desired result: 1) hn(w) --t 1 as n --too for all w. Indeed, 1 :5 bk :5 l for all k and hence

The expression in the right-hand side goes to 1 as n --t oo since q

[nlogtml

n

n

logt m - - = logt m -

--t 0

as n --too. St

2) lim.. ..... oo9n (w} ~ 1 for all w E E;!". Since each bk ~ 1 we have that · b~c,) t ~ 1. Therefore,

~r (b~c0 b~c,

-

-

Sn

lim.. ..... oo9n(w) =limn-too-~ 1 Sq

since this holds for any positive sequence

St

bounded away from zero.

Dimension of Cantor-like Sets and Symbolic Dynamics

161

3) gn(w) --+ 1 ~£-almost everywhere as n --+ oo. From the definition of 1£ it is clear that the functions (kok 1 ... ) ~ Pkn, n = 0, 1, 2, . are independent, identically distributed random variables with respect to I'· Hence, the sequence (pkoPk 1 • • 'Pkn)lfn, n = 0, 1, 2, ... converges for almost every (kokt ... ) E by Kolmogorov's strong law of large numbers. Note that by the definition of Pk

r:;;

and hence Yn --+

1~-t-almost

everywhere.

It immediately follows from Lemma 1, the second statement of Lemma 2, and (16.4) that d.>. ((x, y)) ~ s for A-almost every point (x, y) E F. Thus, s ~ dimH F. We now show that d.ImH F ~ s by constructing an efficient cover of F. Fixe> 0 and consider the collection 14. of those of sets Rko k,. for which r/Jko kn ~ m-•. These sets are disjoint and by Lemma 1 satisfy

Therefore, the number of such sets is bounded by mC•+•}n as ).(F) = 1 Note that any point (x, y) E F is covered by sets Rko k,. E 14. for infinitely many n since limn-too4>n((x, y)) ~ 1 > m-• (see Statement 1 of Lemma 2). Therefore, 'R.(N) = Un;::N'Rn is a cover ofF for any choice of N. Let us choose N so large that

It follows that

L R•o

"" E'R.(N)

(diantRko kJ(s+ 2 ~) =

L card'R.n m<s+ 2e)n ~ L m-en< e n;=:N

n;=:N

(here card denotes the cardinality of the corresponding set). This implies that dimH F ~ s + 2e and the desired result follows since c can be chosen arbitrarily small. In order to compute the box dimension of the limit set F let us choose n > 0 and set rn = m-n. Consider the finite cover ofF by sets Rko k,. and let Nn be the number of elements in this cover. It is easy to see that

where Ct > 0 and C2 > 0 are constants independent of n (recall that N(F, r) the least number of balls of radius r needed to cover the set F). Note that N,. is precisely the number of ways to choose sequences (it), t = 1, ... , q and (jt), t = 1, ... , n (recall that q = [n logt m]) such that a) (it,Jt) E A fort= 1, .. ,q, b) (i~, jt) E A for t = q + 1, ... , n and some choice of i~

ts

Chapter 5

162

It follows that N., = aqrn-q = (~)qrn (recall that a is the cardinality of A and r is the number of j such that (i, j) E A for some i). Therefore, di F ___ffiB

= n-+00lim lolg Nn ogr.,

F d lffiB = = --,--

= logm r

. logN(F,rn) 1liD

n-+oo

-

log r n

+ logm ~r n-+oo lim !!.. = logm r + logm ~. n r

This completes the proof of the statement.

•

Following Pesin and Weiss [PW1] we construct a more sophisticated example than in the previous section, which illustrates that all three characteristics - the Hausdorff dimension, the lower and upper box dimensions - may be distinct.

Example 16.2. There extsts a geometric construction with rectangles in the unite square S C JR2 , modeled by the full shift on two symbols (Et, u), for which _d 1 = _d2 = _d, X1 = "X2 =X, 0 < .d <X< ~, and the limit set F satisfies . log2 d1mHF= - -,, - 1og.
.

log2 - 1og.
d1m 8 F = 'Y--,,

-,-- F dIIDB

log2 -log A

=----,

where 1 E ( 1, a) is an arbitrary number and a = log M log X. Moreover, the induced map G an F is Holder continuous.

Proof. Let n 0 = 0 and for k = 0, 1, 2, . , let nk+l = [ank] and f3k = 2(-r-ar)nsk+t. In order to describe the nth step of the construction we use the basic types of spacings: vertical stacking (A) and horizontal staking (B). See Figure 12. (1) We start with two horizontally stacked rectangles. During steps n 3 k < n ~ nJk+I we use (B). {2) We begin with 2"»+• rectangles. Choose f3k percent of these rectangles arbitrarily and pa.mt them blue, paint the others green. During steps nak+l < n ~ n3k+2> we use (B) in all blue rectangles and usc (A) in all green rectangles. (3) During steps n 3 1<+2 < n ~ n3k+3, we use (A) in all blue rectangles and use (B) in all green rectangles. {4) Repaint all 2"•k+s rectangles white. Repeat steps 1 through 4. The collection of rectangles at the nth step of the construction contains 2" rectangles each with vertical and horizontal sides; the size in the vertical direction is }," and the size in the horizontal direction is X". Any two subrectangles at step n + 1 that are contained in the same rectangle at step n are stacked either horizontally or vertically and the distance between them is chosen to be at least !J:,". The projections of any two distinct rectangles at step n onto the two coordinate axes either coincide or are disjoint. Each point x E X can be coded by a one-sided infinite sequence of two symbols. The induced map G on the limit set F is easily seen to be Holder continuous with HOlder exponent log "X/ log _d.

163

Dimension of Cantor-like Sets and Symbolic Dynamics

I t.'

I

I. .1

> _..!__ ')..n

3 -

I

I -n

-n

a

b

).

).

Figure 12. a) Vertical Stacking, b) Horizontal Stacking. We now compute the Hausdorff dimension and the lower and upper box dimensions of the limit set F. a) Calculation of Hausdorff Dimension.

Given e > 0, choose k > 0 such that An••+~ :5 e. Consider the cover ofF consisting of green rectangles for n = nak+l and blue rectangles for n = nak+2· Consider a green rectangle ~ 10 in•>+' . Dy construction the intersection A = ~; 0 in•>+• n F is contained in 2n••+•-naut small green rectangles corresponding to n = n 3 k+2. These rectangles are vertically aligned and have size 2 +. = xf<>na>+d+ :5 constAn"+' the (1 - f3k)2n••+• An•>+• X X'"+'. Since green rectangles in the construction of F arc each contained in a green square of size An•>+•. Now consider a blue rectangle ~; 0 in••+•. By our construction the intersection B = ~... in••+• n F is contained in 2n•>+a-na•+' small blue rectangles corresponding ton= n 3 k+3· They are vertically aligned and have size ~nsHa x -,r'••+a. Since -,r'ao+a :::; constAn••+• the f3k2n 3 •+'2n 3 '+ 2 -n3 '+' = {J~.,2n••+> blue rectangles in the construction of F are each contained in a blue square of size Ana>+• . The collection of green and blue squares comprises a cover g = {U;} ofF for which

r ..

E (diam U;)" :5 const ( (1 - fik)2nak+l (J2~nau• )" + fJk2n•>+• (J2Ans>+•)•) . U;EQ

The right-hand side of this inequality tends to 0 as k --+ oo if s > .!~~:~.' This implies that dimH F :5 ~~!g\. On the other hand, by Theorem 16.1, we know that dimH F ;::: .!~!:4 . Therefore, we conclude that log2 . ,. dtmnF= - 1og!l We now proceed with the box dimension. b) Calculation of Lower and Upper Box Dimensions.

Chapter 5

164

Choose e > 0. We wish to compute explicitly the number N(F, e) (the least number of ball of radius e needed to cover F). There exists a unique integer .,-n+l .,..,. n > 0 such that ~ < e ~ ~ . Denote by

A,= logN(~e). -log>.. We consider the following three cases: Case 1: n 3 ~;: ~ n < n 3 k+1· One can easily see that N(F,e) = 2" and hence,

A,= log2-· -log~

Case 2: na1:+1 ~ n < na/:+2· We have N(F,e) = Nblue(F,e) + Ngreen(F,e), where Nbtue(F,e) and Ngreen(F,e) are the numbers of e-balls in the optimal cover that have non-empty intersection with respectively blue and green rectangles at step n. It follows N(F,e) = f3~:2" + (1- f3~:)2"•w.

One can see that for all sufficiently large k (for which fJ1: ~ ~),

N(F, e)~ 2 ( 2(-y-n)nsHt2""a>+t+n-nsH2

= 2 (2 and

7

"••+'2"-"••+• + 2" 3 •+ 1 )

N(F,e) 2:: ~ (2(-y-n)ns>+t2"'"••+t+n-n 3Ho +2"•>+•) =

~ (2'Y1'••+'2"-"••+• + 2""+'). 2

One can easily check that the following inequality holds

provided nak+I < n < nak+2· This implies that lim A,.> "(log2 = "(log2 - - log~ -a log >..

n-+oo

Moreover, if n

+ 2ns•+t)

= na1:+1 then lim An= log2_. - log >..

n-+oo

Case 3: n 3k+2

~ n

< nak+3· We have

Dimension of Cantor-like Sets and Symbolic Dynamics

165

It is easy to see that for sufficiently large k (for which f3k ~ ~),

and N(F, e) Since

zn + 2'Yna•+•

;:::

~

( zh-cr)ns•+•z"'n3HI

;::: 2n

li ....ill

n3k+2

=

~ (zn + 2'Yn3k+l) .

we obtain that

n-+oo

provided

+ zn)

log2 A n;::: -1--,. -

< n < n3k+3· Moreover, if n

og~

= n3k+2 then

lim An= logz__ -log A

n-+oo

2 It follows that dimBF;::: - 101og & ,. Combining this with Theorem 16.1, we conclude ...

~~!s2>.. It also follows that dim8 F 2: "f _!~:~. As we have seen d" F ="'-log~. ~ • li A nsk+t =I ~ ab ove, k-+~ -log~· Thus, ...!!!1B

that dimBF

=

We consider another example of a simple geometric construction with rectangles in the plane generated by two affine maps which was introduced by Pollicott and Weiss (see [PoW]). It illustrates that the Hausdorff dimension of the limit set may depend on delicate number-theoretic properties of ratio coefficients while the box dimension is much more robust. Example 16.3 We begin with two disjoint rectangles ~lt ~ 2 C I in the unit square I= [0, 1] x [0, 1] given by ~; = [a;, a;+ A2] x J;, i = 1, 2, where 0 < a 1 ~ a2-< 1 and Jt, Jz are disjoint intervals in the vertical axis of the same length A1. We assume that 0 < At ~ A2 < 1 and At < ~- Consider the two affine maps h;: I-+ ~;, i = 1, 2 that contract the unit square by At in the vertical direction and by A2 in the horizontal direction. See Figure 13. These maps generate a simple self-similar geometric construction with rectangles in the plane with basic sets at step n

Let 'Irk, k = 1, 2 denote the projections of the unit square I onto the vertical side, for k = 1, and onto the horizontal side, for k = 2. Obviously, 7rt o h;(x, y) = a;+ Azx for any (x, y) E I and i = 1, 2. Hence, for any basic set ~; 0 in the left endpoint of the interval 7rt(~io ;J is given by n-t

7rt o h.o o h;1 o · ··oh.n(O,O) = L(at +ik+l(a2- at)) A~. k=O

Chapter 5

166

Taking the limit when n

11"2(F) =

-t

oo yields

{f)a1 +

i1c+l (az

-a I))

A~ :

0,

(i ih ... )

k=O

={ 1 ~\2 +df>k+tA~:

E{0, l}N}

(io,ih···)E{0,1}N}

(16.5)

k=O

c [

1~\ ' 1:2AJ = J,

a2-

where d = a1. Note that if A1 = A2 then the construction is a Moran simple geometric construction (CM1-CM5). It then follows from Theorem 13.1 that . F d lmH

.J:

=~

F

2

F log , • = -d. 1mB = - 1og"'2

(16.6)

There is a very special- degenerate- case when 11" 1 (~ 1 ) = 11" 1 (~ 2 ) (i.e., the rectangle ~ 1 lies directly above the rectangle ~ 2 ) It is easy to check that in this case (16.6) still holds. We now compute the Hausdorff dimension and lower and upper box dimensions of the limit set F of the construction assuming that At < A2 and d = a2- a1 > 0.

a) Calculation of Box Dimension. Lemma 1.

(1) dim8 F (2)

= dimBF ~f climB F. _..!2&1.. dimBF =

log>.,

{ - log ¥;/log At

i/O< A2:::; ~. if~ :::; ).2 < 1.

!·

Proof. We first consider the case 0 < A2 :::; By virtue of (16.5) the set 7rz(F) is aflinely equivalent to the standard Cantor set

after scaling by d = disjoint rectangles

a2 -

a1

> 0 and translating by

~·2 =

1 ~~. .

Consider two new

[a21 -dAz ~] - ).2 ' 1 - A2

><

J2

"

167

Dimension of Cantor-like Sets and Symbolic Dynanncs

Figure 13. A

SELF-SIMILAR CONSTRUCTION WITH RECTANGLES.

It is easy to see that these rectangles have d'ISjoint projections into the horizontal line and ~i C ~;, i = 1, 2 We consider now the simple self-similar geometric sub-construction using the rectangles ~i and ~2 and the affine maps h1 and h2 Obviously, the basic sets of this new construction at step n, i.e., ~ia ;,. , satisfy ~io in C ~io in Hence, the limit set p• C F. This implies that dim 8 F ~ dim 8 F* = - 1~{2 • On the other hand, it follows from Theorem 16.1

-~. that dimBF < log>.2 We turn now to the case ~ ~ ..\ 2 < 1. Given n, consider the cover of the limit ::;et F by squares with sides oflength >.]' such that each rectangle ~ia i,. is covered by [ ~] + 1 squares aligned in a row. One can see from (16.5) that the projection in) contains an interval of length IJII1r1(~; 0 Hence, the proportion of the squares in the cover of ~io ~"'· in n F is at least

1r1(Fn~io

in)l in

= IJI>.~ = 1~1,.

required to cover

This implies that

(recall that N(F, r) is the least number of balls of radius r needed to cover F). • This implies the desired result b) Calculation of Hausdorff Dimension.

Lemma 2. If 0 < ..\2 <

!

then dimH F = dimB F = - ~~~s{•.

168

Chapter 5

Proof. First note that dimH(1r2(F)) ~ dimHF ~ dimnF. As we saw in the proof of Lemma 1 the set 11'2 (F) is the limit set for a simple Moran geometric construction (CM1-CM5) on [0, 1] with 2n basic sets at step n of equal length A~ 1 !.~.,. Hence, dimH(11"1 (F)) = -~~~s;,. The result now follows from Theorem 16.1 and Lemma 1. •

We turn to the case ~ ~ A2 < 1. Following Pollicott and Weiss [PoW], we call a real number f3 E [0, l] a GE-number (after Garsia-Erd<Ss) if there exists a constant C > 0 such that for all x E [0, +oo) card { (io, .. ,in-1) E {0, 1}n :

~ i,fY E [x, x + f3n)} ~ C (2f3t.

The following properties of GE-numbers are known (see [PoW]): (1) no number 0 < f3 < is aGE-number; (2) there exists a non-GE-number that is bigger than (for example, the Golden mean, i.e., the positive root of the equation 1+~ =;b); moreover, the reciprocal of any Pisot-Vijayarghavan number (a root of an algebraic equation whose all conjugates have moduli less than one) is a non-GE number [S]; (3) if {1 is the reciprocal of a root of 2 then it is a GE-number; (4) almost all numbers on the interval(~, 1) are GE-numbers (So]; (5) if < f3 < 1, then f3 is a GE-number if and only if for all sequences p E ITo {0, 1} and any d > 0 there exists K > 0 such that Nn(p) ~ K(2f3)n-m for any 0 < m < n, where

!

!

i,

Nn(p)

= {(im+J .. . in) E {O,l}n-m:

t

dl

(pe- it)f3t

I< f3n}.

(16.7)

l=m+1

Lemma 3. If~< A2
dimB F = dimH F =-log

c~2 ) flog .>'1.

Proof. Given T > 0 choose n ;::: 0 such that n = r·~o~7] + 1 and consider a Moran cover U,. = U,.(~) constructed in Section 13 with ~ = (A2, ..\2) It is easy to see that this cover consists of all rectangles at step n. Given x E F, we first compute the number N(x, r) of those rectangles at step n that intersect the ball B(x,r). Choose m = [nlog..\ 2 /logA!] Clearly Af x ..\2. Assume that x E 6;0 in and consider an asymptotic square S(x) of dimensions Af x A~ that prolongates the rectangle 6; 0 in· Let Nn(x) be the number of those rectangles at step n that intersect the square S(x). Obviously, we have N(x, r) $ C1Nn(x), where C1 > 0 is a constant. We now establish an upper estimate for the number Nn(x). First we observe that if a rectangle 6;0 in intersects the

Dimension of Cantor-like Sets and Symbolic Dynamics

169

square S(x) then io = io, ... , im = im· We note now that the left endpoint of Llr.o in is ~7=0 (al +d it)>..~ and the left endpoint of Ll; 0 imim+> in is ~;~ 0 (a1 + die)>..~+ ~7=m+l(al +die)>..~. Hence, for these two rectangles to lie in the same asymptotic square S(x), we should have that

d

I

t

l=m+l

I

(it- h)>../· < Azn.

(16.8}

It follows from property (5} ofGE-numbers (see (16.7}} that Nn(x) :-:; K(2>..2)n-m. Set SJ,. = -log(-¥,a-)/log>..l. Since the function SJ,.log,:l;0 = s~!og..\2 is constant the corresponding equilibrium measure m>.. is the measure of maximal entropy for the full shift and hence for any basic set Ll~0 in we have that m~(Ll; 0 in)= 2-n. Using (13.5) and following (13.6) we obtain by direct calculation that

where K 2 > 0 and K 3 > 0 are constants. The desired result follows now from the uniform mass distribution principle. • c) Non-coincidence of the Hausdorff Dimension and Box Dimension.

We illustrate that the number-theoretic property (16. 7) that we have used is not just an artifact of the proof. Consider A1 = ~· In this very spec1al case the limit set F reduces to the graph of a Weierstrass-like function (modulo a

countable set). The dimension of such graphs were studied by several authors. In particular, in [PU], Przytycki and Urbanski showed that if >.2 is the reciprocal of a PV number then there exist certain configumtions (i.e., a choice of numbers • a 1 and a 2 ) such that dimn F < dimB F.

Chapter 6

Multifractal Formalism

Let I be a dynamical system acting in a domain U C lR"" and Z an invariant set. We saw in Chapter 2 that the Hausdorff dimension and box dimension of Z yield information about the geometric (and somehow topological) structure of Z. This information, in fact, may not capture any dynamics. For example, if Z is a periodic orbit then the Hausdorff and box dimensions of Z are zero regardless to whether the orbit is stable, unstable, or neutral. In order to obtain relevant information about dynamics one should consider not only the geometry of the set Z but also the distnbution of points on Z under f In other words one should be interested in how often a given point x E Z visits a fixed subset Y c Z under f. If 1J is an /-invariant Borel ergodic measure for which ~J(Y) > 0, then for a typical point x E Z the average number of visits is equal to fL(Y). Thus, the orbit distribution is completely determined by the measure f-L· On the other hand, the measure f-L is completely specified by the distribution of a typical orbit. This fact is widely used in the numerical study of dynamical systems where orbit distributions can easily be generated by a computer. These distributions are, in general, non-uniform and have a clearly visible fine-scaled interwoven structure of hot and cold spots, i.e., regions where the frequency of visitations is either much greater than average or much less than average respectively (see [GOY] for more details; see Figure 14). For dynamical systems possessing strange attractors the computer picture of hot and cold spots reflects the distribution of typical orbits associated with special invariant measures. The latter are naturally interconnected with the geometry of Z and can be used to describe relations between the dynamics on Z and the geometric structure of Z. These measures are often called natural measures. If the strange attractor is hyperbolic, the corresponding natural measure is the well-known Sinai-Ruelle-Bowen measure. (Definition and properties of these measures can be found in [KH].) The distribution of hot and cold spots varies with the scale: if a small piece of the invariant set is magnified another picture of hot and cold spots can be seen. In order to obtain a quantitative description of the behavior of hot and cold spots with the scale let us consider a dynamical system I acting on a hypercube K in lRm and a cover of K by a uruform grid of mesh size r. Let p, be an average number of visits of a "typical" orbit to a given box Bi of a grid, i.e., Pi= ~J(Bi), where J.1. is a natural measure. The collection of numbers {p,} determine the distribution of hot and cold spots corresponding to the scale level r. Define scaling exponents CXi by p, = r"''. In the seminal paper [HJKPS], the authors suggested using the limit distribution of numbers a; when r ---+ 0 as a quantitative 170

171

Multifractal Formalism

characteristic of the distribution of hot and cold spots. It is intimately related to the multifractal structure of X (see [M2]).

. 0

..

..

cold spot

hotspot

(J)·

0

cold spot Figure 14.

AN ORBIT DISTRIBUTION.

The general concept of rnultifractality can be formulated as follows (see a more detailed description in Appendix IV). One can say that the set X has the multifractal structure if it admits a decomposition - called multifractal decomposition - into subsets that are homogeneous in a sense. Of course, tlus is not a rigorous mathematical definition and it depends on how the homogeneity is interpreted. For example, if h: X --+ R is a function then the decomposition of X into level sets Xa = {x E X : h(x) =a} can be viewed as multifractal with the sets Xo being homogeneous. There is a multifractal decomposition of X associated with hot and cold spots of orbit distributions (i.e., the decomposition that is induced by the invariant measure ~-t):

Here X a is the set of points where the pointwise dimension takes on the value a and X - the irregular part - is the set of points with no pointwise dimension. This decomposition can be characterized by the dimension spectrum for pointwise dimensions of the measure f-t or /,.(a)-spectrum (for dimensions), where /,.(a)= dinlH Xa. The /p.(a)-spectrum provides a description of the fine-scale geometry of the set X (more precisely, the part of it where the measure f-t JS concentrated) whose constituent components are the sets X 0 (see Section 18). The straightforward calculation of the f,. (a )-spectrum is difficult and one should relate it to observable properties of the invariant measure 1-'· For example, one can use various dimension spectra as most accessible characteristics of orbit

172

Chapter 6

distribution. Among them let us mention the Renyi spectrum for dimensions introduced by Tel [Tel]. It is defined as follows: 1

.

R qP. ( ) = - - I1l l

q - 1 r-+oo

logL:;~ 1 p,(Bi)q , log r

where N = N(r) is the total number of boxes B, of the grid with p.(Bi) > 0 (provided the limit exists; see Section 17 and discussion in (V1]). In (GHP], Grassberger, Hentschel, and Procaccia suggested computing correlations between q-tuples of points in the orbit distribution for q = 2, 3, . . . (see also [G, GP, HPJ). This led to characteristics known as correlation dimensions of order q (see Section 17). These characteristics proved to be experimentally the most accessible and offered a substantial advantage over the other characteristics of dimension type that were used in numerical study of dynamical systems with chaotic behavior. In Section 17 we give a rigorous mathematical substantiation of the fact that the correlation dimension of order q is completely determined by the function 0 and thus, depends only on the metric on X and the invariant ergodic measure IJ., but not on the dynamical system itself - the unexpected phenomenon noticed first by Hentschel and Procaccia in (HPJ. They also introduced a family of characteristics depending on a real parameter q ::::0: 0 (except q = 1) of which the correlation dimensions of order q are special cases for integers q :;: -: 2. It is now known as the H P-spectrum for dimensions (after Hentschel and Procaccia). The formal definition and a detailed discussion is given in Section 18. For natural measures the HP-spectrum for dimensions can be defined in the following way.

H P. ( ) 1 r q p. = q - 1 r~

login£{ g

E

~J.(B(xi,r))q}

B(x,,r)EQ

log r

where Q is a finite or countable covering of the support of p, by balls of radius e: (provided the limit exists). A group of physicists in the paper [HJKPS] presented a heuristic argument based on the analogy with statistical mechanics to show that the HP-spectrum for dimensions and the Renyi spectrum for dimensions coincide and that the latter (multiplied by the factor 1 - q) and the f~t(a)-spectrum for dimensions form a Legendre transform pair (see Sections 18, 19, and 21; see also Figures 17a and 17b in Chapter 7). Roughly speaking their idea goes as follows (see arguments in [CLP]). Given a grid of size r, consider the partition function N

N

Z(q,r) = Lfl(Bi)q = Ee-qE,, i=l

i=l

where q is the "inverse temperature" and Ei 18 the "energy" of the element Bi of the grid (the sum is taken over those elements of the grid Bi for which 11-(B,) is positive). The "free energy" of p, is defined (when it exists) by F(q) =-lim logZ(q,r). r-+0 logr

Multifractal Formalism

173

The analogy with statistical mechanics is then used to relate the Legendre transform ofF (i.e., the function t >-+ infq[qt-F(q)]) to the distribution of the numbers IJ.(B;), i.e., to f~'(a). Once the Legendre transform relation between the two dimension spectra is established, one can compute the delicate and seemingly intractable !~'(a)­ spectrum through the H P-spectrum, since the latter is completely determined by the statistics of a typical trajectory. In order to convert the above argument into a rigorous mathematical proof one must first establish that the HP-spectrum and the !~'(a)-spectrum for dimensions are smooth and strictly convex on some intervals (in q and in a respectively). This seems amazing since a prion one expects the functions fl'(a) and HPp(q) to be only measurable. Furthermore, it is not at all clear whether, even if the measure JJ is exact dimensional (i.e., d~'(x) exists and is constant, say d, almost everywhere) the pointwise dimension attains any values besides d. In Section 18 we will show that the Hentschel-Procaccia spectrum and the Renyi spectrum for dimensions coincide for any Borel finite measure on JR.m. This allows us to set up the concept of a (complete) multifractal analysis of dynamical systems as a collection of results which establish smoothness and convexity of these spectra as well as the !~'(a)-spectrum for dimensions and the Legendre transform relation between them. Although the multifractal analysis was developed by physicists and applied mathematicians as a tool in numerical study of dynamical systems its significance has not been quite understood. In Section 18 we will demonstrate that dimension spectra - whose study constitutes the multifractal analysis - are Caratheodory dimension characteristics (i.e., they can be introduced within the general Caratheodory construction described in Chapter 1). This alone justifies the study of dimension spectra as part of dimension theory. This also opens new perspectives for the multifractal analysis one can classify dynamical systems up to an isomorphism that preserves dimension spectra. Such an isomorphism keeps track on stochastic properties of the dynamical system (specified by the ii!Variant measure) as well as dimension properties (of invariant measures or sets) and thus, may be of great importance in describing chaotic dynamical systems (see Appendix IV for more details). In Chapter 7 we effect a complete multifractal analysis of Gibbs measures for smooth conformal expanding maps and axiom A-diffeomorphisms on surfaces. As a part of this analysis we show that any equilibrium measure on a conformal repeller corresponding to a Holder continuous function is diametrically regular (see Proposition 21.4). The same result holds for equilibrium measures on conformal locally maximal hyperbolic sets (see Proposition 24.1). In Section 19 of this chapter we consider another class of examples which includes Gibbs measures on limit sets for a large class of geometric constructions of type (CG1-CG3). Our analysis is based on the dynamical properties of the induced map on the limit set generated by the shift map on the symbolic space. Our main assumption is that this map, being continuous, is expanding and conformal in a weak sense. Whether the induced map satisfies these properties strongly

Chapter 6

174

depends on the symbolic dynamics and its embedding into Euclidean space, i.e., the gaps between the basic sets. Examples include self-similar geometric constructions and geometric constructions associated with some (classical) Schottky groups (see Section 13), as well as geometric constructions effected by a sequence of similarity maps whose ratio coefficients admit some asymptotic estimates. The multifractal analysis of these "expanding and conformal" geometric constructions is intimately related to the analysis for smooth conformal expanding maps (see Section 21).

17. Correlation Dimension One can obtain information about a dynamical system that is related to invariant ergodic measures by observing individual trajectories. In [GHPJ, Grassberger, Hentschel, and Procaccia introduced the notion of correlation dimension in an attempt to produce a characteristic of dynamics that captures information on the global behavior of "typical" trajectories by observing a single one (see also [G]). This trajectory should be typical with respect to an invariant measure. The formal definition of the correlation dimension is as follows. Let (X, p) be a complete separable metric space with metric p and f: X-+ X a continuous map. Given x EX, n > 0, and r > 0, we define the correlation sum as C(x,n,r)

=~card {(i,j). p(fi(x),Jj(x))::; rand 0::; i,J::; n}, n

(17.1)

where card A denotes the cardinality of the set. A. Given a point x E X, wP call the quantities

( ) li lim ax=m r~O n-+oo

-

logC(x,n,r) log(1 /r ) ,

_( ) li-.-- li log C(x, n, r) a x = m m ----::-7-:-'-;--:--'r-+0 n-->oo

log(l/r)

(17.2)

the lower and upper correlation dimensions at the point x respectively. We shall first discuss the existence of the limit as n -+ oo. Let p. be an !-invariant Borel probabillty measure on X. Consider the function


L

p.(D(x,r)) dp.(x),

where D (x, r) is the closed ball of radius r centered at x. It is clearly nondecreasing in r and hence may have only a finite or countable set of discontinuity points. Thus, it 1S right-continuous; it is continuous at r if and only if p.(S(x, r)) = 0 for p.-almost every x EX (S(x, r) is the sphere of radius r centered at x).

Theorem 17.1. [P3, PT] Assume that p. is ergodic. Then there exfsts a set Z C X of full measure such that for any e > 0, R > 0, and any x E Z one can ftnd a positive integer N

= N(x, e, R) for which the inequality

Multifractal Formalism

175

holds for every n ~ N and 0 < 1' ~ R. In other words, C(x, n, r) tends to 0, there exists a set Xr c X of full measure such that for any x E Xro lim C(x,n,r) =
(17 3)

n->00

Given a point x E X, a measurable set A C X, and integers n ~ 0, m ~ 0 denote by N(x, A, n, m) the number of points fi(x), -m ~ i ~ n for which Ji(x) E A. Let us fix a countable collection of closed balls Dk = D(yk, rk), k ~ 0 forming a basis of the topology in X. We can assume that JL(8Dk) = 0 for all k ~ 0. Since I' is ergodic there exists a set Y C X of full measure such that for any x E Y and k ~ 0,

1'

n!..~

N(x,Dk,n,m) n +m

= JL

(D) k •

It follows that given x E Y, e ~ 0, and k ~ 0, there exists n{x,s, k) such that for any n ~ 0, m ~ 0, and n+m ~ n(x,s,k), (17.4) Let us fix r > 0, e: > 0, and x E Y and choose sequences xk,l, such that xk,l -t x, Xk,2 -t x, rk, 1 -t r, rk,2 -t r, and

rk,l

and xk,2• rk,z

This imphes that (17 5) Moreover, since J.~-(8Dk) = 0 for all k, one can find K = K(e:) such that for any k ~ K(s) and i = 1,2, (17.6) It follows from (17.4), (17.5), and (17.6) that for any k 2: K(e) and n with n + m ~ n(x,e:, k),

~

N(x, D(x, r), n, m) < N(x, D(xk,2, rk,2), n, m) n+m n+m ~ J1-(D(xk,2• rk.2)) + e: ~ J.~-(D(x, r)) + 2e:, N(x, D(x, r), n,m) > N(x, D(xk,lt rk,2),n, m) n+m n+m ~ ~-t{D(xk,t. rk.d) - e 2: JL(D(x, r))- 2e.

0, m 2: 0

176

Chapter 6

Thus, for any x E Y and c > 0 there exists n 1 = n 1 (x,e:) 2: 0 such that for any n 2: 0, m 2: 0 with n + m 2: n1,

N(x,D(x,r),n,m) -J,£(D(x,r))'::::; Ze.

(17.7)

n+m

Since p. is ergodic and p.(D(x, r)) is a bounded Borel function over x E X we have by virtue of the' Birkhoff ergodic theorem that for J,£-almost every x E X, n-1

lim n--+oo

!_ Ltt(D(i(x),r)) = n

i=O

1

p.(D(y,r))dJ,£(y) = cp+(r).

X

This implies that for any e > 0 there exist a measurable set xJ~j c Y with p.(XJ.~) 2: 1 - e and a number n 2 = n 2(e) such that for any x E x$.~ and (17.8)

Moreover, for JL-almost every

X

E

xJ.~, (1)

lim n-+oo

N ( x, Xr,• , n, 0) _ (X{ll) -P. r<. n ,

Therefore, there exists a set X~~ c x;~ with p.(X;~j) 2: I'(X$~) - e 2: 1 - 2e: and a number n 3 = n3(e:) such that for any x E x;~J and n 2: n3, 1 cx,Xr,e~n,O) (2) ;;,N -1 I 5 2c.

(17.9)

1

Let us fix X E x$~2 and n:::: n(e) 0::::; < n(e:). Denote

n.,

= max{nl> n2, n3}·

Write n == tn(e)

+ n(

with

N; = N(fi(x), D(f;(x), r), n- i, i), i 2: 0. One can rewrite the expression for C(x, n, r) (see (17.1)) in the following form:

C(x,n,r)=-1 n

Ln ~=N· 1 (E n n i=O

N· ~+

ln

L 2n~+ N L 3n~N) ,

where the first sum is taken over all i, 0 ::::; i ::::; tn( e) with Ji (x) E X$~), the second sum over all i such that tn(e) + 1 ::::; i 5 n, and the third sum over all other i (i.e., those i, 0 ::::; i ::::; tn(e) for which Ji(x) ¢ x,\~2). Since N; ::::; n + 1 this implies that

(17.10)

177

Multifractal Formalism if n is sufficiently large. It follows froiYl. (17.9) that

IE

-n

3

2 () 1 N < -(nn ) O)) -< 2e-. N(x t X r,E' ---.! n - n

(I7.11)

Let us now estimate the first sum It follows from (17 8) that

Ni

I

I-n "L....-1' --
r,

5

I+c: 1 N; I5 11-n "L....-' 1 ---"'Jl.(D(f'(x),r)) n L....n .

n-l

i=D

. I L....- 1 n- p(D(f'(x),r)) nI"'IN;

(I7 I2)

. . +e, +-nIE3 f.t(D(!'(x),r)) +-nIE2 f.'(D(J'(x),r)) where the first, second, and third sums are taken over i as above. Therefore, by (17.7) and {17.9)-(I7.11), we find -1 1n

L

1

Ni -


n

n,n + 2e + e 5 6e. +c r>I 5 2e +-

This inequality along with (I710), (1711), and (17.12), imply (I7.3) for all X E x$~}' and hence for any X E Xr = u.>oX$~}. Obviously, Xr is a set of full measure. We now show the uniform convergence of C(x,n,r) to t,p+(r) over ron an interval 0 < r 5 R. Let us fix e > 0. Since 0 such that i 0, C(x, n, rt) ~ C(x, n, r) ~ C(x, n, r2)

o

If n is sufficiently large we have also that

C(x,n,rt) ~
+ c ~
These mequalities rmply that IC(x, n, r)-

~ R if

n is •

We now extend the notion of the correlation dimension by considering correlations of higher order. Namely, given x E X, n > 0, r > 0, and an integer q ~ 2, we define the correlation sum of order q (specified by the points {Ji(x)}, i = 0, 1. ... , n) by

Cq(x, n, r)

= _..!:._ nq

card { (i 1 ... iq) E {0, 1, ... , n}q: p(ii(x), fik(x)) ~ r for any 0 ~ j,k ~ q}.

Chapter 6

178 The quantities a (x) = _l_lim lim logCq(x,n,r), q- lr-+0 n-+oo log(1/r)

=q

_ ( ) x

a

q

. logCq(x,n,r) 1 -. =- 1un 1an ---"':---''7-:-';--;'---'q- 1 r-+0 n-+oo log(l/r)

are called respectively the lower and upper correlation dimensions of order q at the point x or lower and upper q-correlation dimensions at x.

The existence of the limit as n -+ oo is guaranteed by a more general version of Theorem 17.1 which we shall state now. For any r > 0 and any integer q ~ 2 consider the function

rp~(r) =

L

p,(D(x, r))q-l dp,(x),

where p, is an /-invariant Borel probability measure on X (recall that D(x, r) is the closed ball of radius r centered at x). The function tpt(r) is non-decreasing and hence may have only a finite or countable set of discontinuity points. It is clearly right-continuous. The following statement is proved by Pesin and Tempelman in [PT] and is an extension of Theorem 17.1 to correlation dinlensions of higher order. Theorem 17.2. If p, is an ergodic mMsure then for any integer q ~ 2 there extsts a set Z c X of full mrosure such that for any € > 0, R > 0, and any x E Zone can find N = N(x,e:,R) for which

with arbitmry n

~

N and 0

< r S R.

Remarks. (1) We stress that the lower and upper q-correlation dimensions do not depend either on the dynamical system f or on the point x for p,-almost every x (provided p, is ergodic). Instead, they are completely specified by the measure p,. This allows us to introduce the notion of q-correlation dimension for any finite Borel measure p, on a complete separable metric space X (see [PT]). Let Z c X be a Borel subset of pm.itive measure. Define the lower and upper q-correlation dimensions of the measure p, on Z for q = 2, 3, . . . by Corq(p,, Z)

=

_!__ dimqZ,

q- 1

Corq(p,, Z)

1-

= --dimqZ. q-1

For q = 2 these formulae define the lower and upper correlation dimensions of the measure p, on Z. We also define the (q, H)-correlation dimension of the measure p, on Z for q = 2, 3, ... by

Corq,H(JJ., Z)

= q ~ 1 dimq Z.

Multifractal Formalism

179

Consider the function cpq(r) = cpq(X,r) defined by (8.7). We stress that in (8.7) we use open balls ofradius r centered at points in X. A simple argument (which we leave to the reader) shows that

. logcpt(r) . logcpq(r) 11m = 1rm -:-=---7-'---'--7-

r-to

;::;a log(1/r)'

log(1/r)

-.-log cpt(r) -.-logcpq(r) hm =lim . log(1/r) r-+0 log(1/r)

r-+0

If p, is an invariant ergodic measure for a dynamical system follows that for J.t-almost every x E X, ~(x)

= ~(J.!, X),

liq(x)

f

acting on X it

= Corq(J.!, X)

(X can be also replaced by any set of full measure). These relations expose the "dimensional" nature of the notions of lower and upper q-correlation dimensions: they coincide respectively (up to the constant q~ 1 ) with the upper and lowerq-box dimensions of the space X for q = 2, 3, . . . (or any other set of full measure).

(2) Example 8.1 shows that the lower and upper q-correlation dimensions of p, may not coincide: namely, for any integer q 2: 2 there exists a Borel finite measure p. on I= [0, 1] for which

(and J.t is absolutely continuous with respect to the Lebesgue measure on I). One can check that J.t is an invariant measure for a continuous map f on I, where f(x) = J.!([O,x)). Thus, with respect to this map (17.13) for p.-almost. every :r E I Furthermore, using methods in [K) one can construct a diffeomorphism of a compact surface preserving an absolutely continuous Bernoulli measure with non-zero Lyapunov exponents for which (17.13) holds. (3) We describe a more general setup for introducing the lower and upper q-corrclation dimensions. Namely, given x,y EX, n > 0, r > 0, and an integer q 2: 2, we define the correlation sum of order q (specified by the points {Ji(x)} and {f;(y)}, i = 0, 1, ... , n) by Cq(x,y,n,r) =_.!__card {(i 1 .. . iq) E {0, 1, . . ,n}q:

nq

p(J 1i (x),ji•(y)) ::; r for any 0::; j, k ::; q}. Consider the direct product space Y = X x X. We call the quantities

_ .

.

1!Ill 1!ffi !!q (X, Y) - r-+0 n-+oo

JogGq(x,y,n,r) 1 1og (l/ r )

_ ( ) -. . JogCq(x,y,n,r) 11m 11m aq x, y = r-+0 n-+oo log ( 1I r )

the lower and upper q-correlation dimensions at the point (x,y) E Y. One can prove the following statement: let J.! be an f-invanant ergodic Borel

Chapter 6

180 probab-Jity measure and v = 1-1 x 1-'-i then for any integer q (x, y) E Y the limit 1un Cq(x,y,n,r) = cp:(X,r).

~ 2

and v-almost every

n-+oo

e:nsts.

(4} The functions cpq(X, r), q = 2,3, ... admit the following interpretation. Consider the direct product space (Yq, pq, vq), where Yq =X

X .•. X

X,

~

Vq

= ,_____...... 11- X •.. X 11-

q times

q times

=

= 2:::%"' 1 p(xk, Yk) for x = (xt. ... , xq), y (Yt. ... , yq)· Let A. { (x, ... , x) E Yq: x E X} be the diagonal. Then for any r > 0,

and pq(x, jj)

=

cpq(X, r) = vq(U(A., r)), where U(A., r) is the r-neighborhood of A.. (5) Let 1-1 be a Borel finite measure on am with bounded support. Following Sauer and Yorke [SY] we describe another approach to the notion of correlation dimension of p based on the potential theoretic method (see Section 7). Define the quantity (17.14) D(~J-) = sup{ s : I.(/1-) < oo }, where I.(JL) is the s-energy of p, (see (7.5)). The "correlation drmension" D(p) is interpreted as the supremum of those s for which the measure 11- has finite s-energy. Given a subset Z C Rm, one can compute the Hausdorff dimension of Z via the quantity D(p,). Namely, by the potential principle (see (7 6}), dimH Z

= sup{D(~J-) : ~-t(Z) = 1}.

(17.15)

"' A slight modification of the argument by Sauer and Yorke in (SY] shows that

where X is the support of JL. Indeed, set D = Cor2 (J-~, X). By the definition of the lower correlation dimension for any e > 0 there exists r 0 > 0 such that for any 0 < r ~ ro, cp2(X, r) S rD-•. Given 0 S s < D, set e = (D- s)/2 > 0. We have that 00

1 o

dcp2(X, r} = (" dcp2(X, r)

r•

Jo

+

r•

r "' < - n_!!_ e + constant <

1"" ro 00.

dcp2(X, r)

r•

Multifra.ctal Formahsm

181

This implies that ls(l-') is finite. On the other hand, let 0 < D < s. Set e = (s- D)/2 > 0. By the definition of the lower correlation dimension there exists a sequence of numbers rn > 0 such that

Given n > 0, one can find a number m cp2(X, rn)/2. We have that

= m(n)

>

n such that cp2(X, rm)

<

Since rn can be chosen arbitrarily small there is no upper bound for l 8 (J.'). Sauer and Yorke also studied the problem whether the quantity D(p,) is preserved under a smooth map. In particular, they proved the following statement: assume that D(l-') ::;: p; then there exists a residual subset A c C 1 (Rm, JRP) (the space of smooth maps from JRrn to RP) such that the equality D(g.p.) = D(p.) holds for every g EA. In view of (17.15) a similar result holds for the Hausdorff dimension of a subset Z c lRm . On the other hand, it fails in the case of box dimension: there exists a compact subset Z c lRm with ~Z = dimBZ ~r D :5 p such that dimB g(Z) < D for every g E G 1 (1R"', JR.!'). (6) Let (Y, v) be a Lebesgue space (vis a probability measure) and f: Y -t Y a measurable transformation. Let also (X, p) be a metric space and h: Y -t X a Borel function called observable. In practice, !-orbits may not be accessible but the values hn = h(r (y)) for some y E Y can often be computed. One can analyze the data {hn, n = 1, 2, ... } in the following way. For q = 2, 3, ... we define the h-correlation sum of order q (specified by the data {hn, n = 1, 2,.. }) by

Cq( {hn}, r)

= _.!._ card {(it .. . iq) E {0, 1, ... , nF: nq p(h,,,h;k) :5 r for any 0:5 j,k::;: q}.

The quantities ~ ({h n

})

_ . . logGq({hn},r) 11m 11m ( ) , r->O n->oo log 1/r

-

a q

({hn}) =lim lim logGq({hn},r) r->0 n->oo log(l/r)

are called respectively the lower and upper q-correlation dimensions specified by the data { hn, n = 0, 1, 2, ... } . One can prove the following statement: For any q = 2, 3, ... and v-almost every y E Y the limit lim Cq( {h.,}, r) n-+oo

e:nsts where p, = h.v.

=

Jxf p,(D(x, r))diL(x)

Chapter 6

182

We consider the case when Y c JR1" is a compact subset, f is a C 1diffeomorphism of an open domain U c !Rm into !Rm for which Y is an invariant set, and h: U --+ R. is a smooth function. In [T1], Takens developed a method of reconstructing the Hausdorff dimension and box dimensions of Y using the data. hn = h(r(y)) for some y E Y. Given an integer p 2: 1, define the delay eoordinate map Gf,h.,p: Y-+ JRP by

Gf,h,p(x) = (h(x),h(J(x)), .. . ,h(JP- 1 (x))). Takens proved that for a residual set of C 1- diffeomorphisms of Y and a residual set of functions h the delay coordinate map G f,h.,p with p = 2m is a C 1- embedding. In particular, the Hausdorff dimension and lower and upper box dimensions of Y are preserved under G f,h,p· Sauer and Yorke [SY] proved a similar result involving the quantity D(J.t) (see (17.14)): assume that D(p) < p (pis an integer) and that f has at most countably many penodic points; then D((G f,h,p)oJL) = D(J.t) for a residual set of junctions h.

18. Dimension Spectra: Hentschel-Proeaceia, R.enyi, and /(a)-Spectra; Information Dimension Hentschel-Procaccia Spectrum for Dimensions Let J.£ be a Borel finite measure on !Rm and X the support of f-t. We introduce the HP-spectrum for dimensions (after Hentschel and Procaccia; see [HPJ), specified by the measure J.', as the one-parameter family of pairs of quantities (HPq(J.t),HPq(J.£)), q > 1, where

HP (J.£) = _1_lim logfx !J(B(x,r))q-I dJL(x), q- 1 r-->0 log(1/r)

==-q

-

( )

HPq 1-'

1

- . logfxJL(B(x,r))q-l dJL(x)

= q -1!~

log(1/r)

·

It follows from (8.10) that -

1-

HPq(lt) = --dimqX. q-1

{18.1)

Thus, the HP-spectrum for dimensions is a one-parameter family of characteristics of dimension type. for every q > 1 the quantities HPq(f£) and HPq(/-l) coincide (up to a normalizing factor q_: 1 ) with the lower and upper q-box dimensions of the set X respectively. Equalities (18.1) allow us to rewrite the definition of the H P-spectrum for dimensions in the following way: using (8.5) we obtain that

HP ( ) = _1_ltm log.:lq,..,(X,r) /-1 q- 1r-+0 log(1/r) '

==-q

HP (p) q

= _1_lim log.:lq,..,(X,r), q - 1 r-->0

log( 1/r)

Multifractal Formalism

183

where 1 > 1 is an arbitrary number and 6.q,'Y(X, r) is defined by {8.6). As Remark 1 in Section 17 shows for q = 2,3, ... the values HPq(P.) and HPq(P.) coincide with the lower and upper correlation dimensions of X, i.e, ~(JL, X) and Corq(JL, X). Following Pesin and Tempelman (PT] we introduce the modified H Pspectrum for dimensions specified by the measure JL as a one-parameter family of pairs of quantities (HPM q(JL), HPMq(p.)), q > 1, where 1 . lim!ogfzJL(B(x,r))q-l dp.(x) HPM ( p.) = - - 1 m 1 sup q - 1 o->0 z r->0 log(1/r) ' HPM ( ) 1 r rlogfzt-t(B(x,r))q-ld p.(x) q J1. = q- 1o~ s~p r~ log(1/r)

==-==q

and the supremum is taken over all sets Z c X with ft(Z) One can derive from Theorem 8.4 that for any q > 1, --

HPMq(p.)

~

1-6.

= -q-1 di1nqp.. 1

Thus, the modified HP-spectrum for dimensions is a one-parameter family of Caratheodory dimension characteristics specified by the measure 1-l· It follows from Theorem 9.2 that for any q > 1,

In particular, if the measure JL satisfies d_,.(x)

= dp(x) = d"'(x) then

(compare to (9.4)) . • The modified H P-spectrum for dimensions is completely specified by the equivalence class of JL. This was shown in (PT]. We present the corresponding result omitting the proof. Theorem 18.1. Let 11-1 and t-t2 be Borel measures on X. If these measures are equivalent then

It follows from the definitions that for q > 1

As Example 8.1 shows there exists a Borel finite measure p. on [O,p] for some > 0 such that

p

Chapter 6

184

for all 1 < q ::; Q, where Q > 0 is a given number (the reader can easily check .2=.!. that this holds provided /3 > a • ) .

R.enyi Spectrum for Dimensions Let U c am be an open domain. A finite or countable partition ~ = { 0;, i ?: 1} of U is called a (/3, r)-grid for some 0 < /3 < 1, r > 0 iffor any i?: 0 one can find a point x; E U such that

B(x;, {Jr)

c

C;

c B(x;, r).

The Euclidean spacer admits (/3, r)-grids for every 0 < /3 < 1 and r > 0. Let p, be a Borel finite measure on Rm and X the support of p,. We assume that X is contained in an open bounded domain U c am. Given numbers q ?: 0 and /3 > 0, we set .

D diDq

X

. log Aq(X, r) = r1liD , .... o log(1/r)

--.-- X -. logAq(X,r) DliDq = 1liD , r--+0 log(l/r)

where

Aq(X, r)

= inf { ~

2: p,(O;)q}

C,E~

and the infimum is taken over all {/3, r )-grids in U (compare to (8.6) ). We wish to compare the quantities DimqX, DimqX with the quantities dimqX, dimqX. The following result obtained by Guysinsky and Yaskolko [GY] establishes the coincidence of these quantities for q > 1 and demonstrates that one can lL~e grids instead of covers in the definition of q-dimension.

Theorem 18.2. If p. is a Borel finite measure on Rm with a compact support X then for any q > 1,

Let~= {C;, i?: 1} be a ({J,r)-grid. There are points {x,} such that B(x,,{Jr) C C; C B(x,,r). Therefore,

Proof.

::; 2: j

p.(B(x, 2r))q-l dp. =

i~l C;

This implies that Aq(X, r) ::; cpq(2r).

j p.(B(x, X

2r))q-l dJJ = cpq(2r).

Multifractal Formalism

185

We now prove the opposite inequality. Consider again a (.8, r)-grid {C;, 1} and choose points {x;} such that B(x;,,Br) C C; C B(x;,r). We have

i;:::

cpq(r)

= j ft(B(x,r))q-ld!J. =

~

L

j J.t(B(x,r))q-ldJL

L i~l

X

J

c,

~L

JL(B(x;,2r))q-ld1J

~1~

J.t(B(x;,2r))q.

~1

Therefore,

cpq(r) ~

L tt(B(x;, 2r})Q. i~l

Let us fix a ball B(x;, 2r) and consider the set D; = {x; : Cj n B(x;, 2r) :j= 0}. Note that D; C B(x;,3r) and that any two points x;.,xh ED; are at least 2.Br apart (recall that balls B(x., .Br) are disjoint). This implies that there exists N == N(.B) such that cardD; ~ N (we stress that N does not depend on r). This implies that any ball B(x;, 2r) can be covered by at most N clements C;~ of the grid. We will exploit the following well-known inequality: for any N > 0 and q > 1 there exists K = K(N, q) > 0 such that for any collection of numbers {a;, 0::; a; 1, i = 1, ... ,N},

s

We have that for any i

~

1, N

J.t(B(x;,2rW ~ K,Ltt(C;.)q. k=l

Now let us fix an element C; of the grid and consider the set A;== {x; : B(x;, 2r)n C;.:j= 0}. Again, one can see that A; C B(x;, 3r) and tha.t any two points Xj,, xh E A; are at least 2f3r apart. Therefore, we conclude that cardA; ~ N and hPnce cpq(r) S NKLp(C;)q t<:l

for any (,B,r)-grid. This implies that NKAq(X,r) ~ rpq(r) and completes the proof of the theorem. • We introduce the Renyi spectrum for dimensions (see [Tel]), specified by the measure ~-'• as a one-parameter family of pairs of quantities {Jlq(tt), 1{1 (~-t)), for q ~ 0, q :j= 1, where _ 1 . X_ 1 limlogAq(X,r) R (J.l ) - - - 0 liD --q- 1 ===q q- 1 ;:::;o log(1/r) '

=q

-R ( ) _ q

1 - . X_ 1-1 - - - 0 lffiq -

q- 1

1

--

q- 1

limlogAq(X,r) log(1j1·) ·

r-tO

Chapter 6

186 It follows from Theorem 18.2 that for any q > 1,

(18.3) Information Dimension Let~ be a finite partition of X. We define the entropy of the with respect to p, by

where Ce is an element of the partition

e.

partition~

Given a number r > 0, we set

H,..(r) = inf {H,..(O: diam~ :-:::; r}, ~

where diame = maxdiamC~ We define the lower and upper information dimensions of p, by

.

H,..(r)

-( )

I (p, ) = rhm ( / ), I ... o 1og 1 r

- . H,..(r)

/-L = r-+0 lim 1og (1/ r ) .

(18.4)

Obviously, l(p,) :-: :; f(p,). Proposition 18.1. (Y2) (1) f(p,) :"S; dimB/-L· (2) If !4(x) ;:::: d for p,-almost every x then l(p,);:::: d. Proof. Fix o > 0, r > 0 and let B 11 ••. ,BN, N = N(o, r) be r-balls that cover a set of measure;:::: 1- o. Set B1 = B 1 and fork= 2, 3, ... , N, k-l

Bk = Bk \

U B;.

i=l

The sets Bk arc disjoint and have diameter :-: :; r. Therefore, one can find measurable sets Ui, j = 1, ... , M(r) such that together with Bk they comprise a finite measurable partition~ of X of diameter 5 r. We recall the following inequality: if numbers 0 < t :-: :; 1 and 0 < Pk :-:::; 1, k = 1, ... , s are such that E~=l Pk = t then s

- :~:::>k logpk 5

-t logt + t logs.

k=l

It follows that

H,..(r) :-: :; H,..(O :-: :; log N(o, r) - o logo+ clog M(r)

187

M ultifractal Formalism

Therefore, -I() -r-logN(6,r) -:--6log6+6logM(r) -d. >:d-,-- X 1Im fJ. :::; r-+0 liD l l og (1/ r ) :::; lffiB/.L + u Ims . og (l/ r ) + r-+0 Since dimsX is finite the first statement follows by letting {J--+ 0. In order to prove the second statement choose a < d and {J > 0. There exists a set Z C X with 1-1(Z) 2: 1- {Janda number r 0 > 0 such that Jt(B(x, 1·)) ::=; r" for any 0 < r :::; r 0 and x E Z Fix r ::; r 0 and consider a finite partition E;, = {C1 , ... , Cn} of diameter::; r. Let B 1 be the collection of sets C; E E;, that have non-empty intersection with Z and B2 the collection of other sets C; E £;,. We have that J.t(Uc,EB 2 C,) ::; 8. Each C; E B1 is contained in a ball B(x;, r) centered at some point x; E C; and therefore, has /.£-measure::=; r". Thus, 1-28 H,.(E;,) 2: "'"" L /.L(Ci)log /.L(C;) 2: -"-(-r"'log r"')

= (1- 28)alog(1/r).

r

C,EB,

It follows that H,.(E;,) > (1- 28)a. log(1/r) -

•

This implies the second statement.

As an immediate consequence of Proposition 18.1 and Theorem 7.1 we obtain the following claim.

Proposition 18.2. Let /.L be a finite Borel measure on R"'. Assume that /.L is exact dimensional (see Section 7). Then l(fJ.) = l(J.£) = dimH J.l. It is conjectured that for "good" measures

l(J.t)

= q-+l,q>l lim !1q(J.L)

J(p,) =

lim Rq(J.L)

q-+l,q>l

=

lim HPq(fl),

q-H,q>l

= q-+l,q>l lim HP9 (~-t)

Below we will show that this conjecture holds for equilibrium measures corresponding to Holder continuous functions on conformal repellers for smooth expanding maps (see Section 21) and on basic sets for "conformal" axiom A diffeomorphisms (see Section 24).

/(a)-Spectrum for Dimensions We now introduce another dimension spectrum in order to describe the distribution of values of pointwise dimension of a measure. Let fJ. be a Borel finite measure on R"' and X the support of J.l· There is the multifractal decomposition of X associated with the pointwise dimension of f.t:

Chapter 6

188 where

X= {x EX: fl,.(x) < d"(x)} is the irregular part and the level sets Xa are defined by -

def

Xa = {x EX: .r!.,.(x) = d,.(x) = d,.(x) =a}.

(18.5)

In order to characterize this multifractal decomposition quantitatively we introduce the dimension spectrum for pointwise dimensions of the measure 1-' or fl'(a)-spectrum (for dimensions) by

We first consider those values a for which f.t(Xa) > 0. Theorem 18.3. If 1-'(Xa) > 0 then dimH Xa = a. Proof. The statement immediately follows from Theorems 71 and 7.2.

•

If ~t(Xa) = 0 the set Xa may still have positive Hausdorff dimension and hence be "observable" from a physical point of view. Thus, the !,.(a)-spectrum can be used to describe the fine-scale structure of the measure It provided the function d"(x) exists on an "observable" set of points in X. An invariant ergodic measure f.t supported on a repeller of a smooth expanding map (see definition in Section 20) or on a hyperbolic set of an axiom A diffeomorphism (see definition in Section 22) can be shown to be exact dimensional (see Theorems 21.2 and 24.2). This means that the function dl'(x) exists and is constant on a set of full measure. Therefore, f.t(Xa) = 0 for all values a hut one. A pnon, it is not clear at all why the sets X a are not empty for all values a except this special one. Even if these sets are not empty, it seems unclear from a point of view of the classical measure theory that the function f I' (a) would behave "nicely" and provide meaningful information about the measure I.J. Furthermore, Barreira and Schmeling [BS] showed that for any equilibrium measure (corresponding to a HOlder continuous function) for a smooth expanding map or for an axiom A diffeomorphism the set X IS obseroable; moreover, it has full Hausdorff dimension (see Appendix IV for details; some very special cases should be excluded). Despite this it was conjectured in [HJKPS] that for "good" measures, which are invariant under dynamical systems, the function f"(a) is correctly defined on some interval [a~, a2], real analytic and convex. This wa.s supported by a computer simulation of some dynamical systems: a typical "computer made" graph of the function f"(a) is drawn on Figure 17b in Chapter 7. Furthermore, a strong connection between the HP-spectrum and /"(a)spectrum was discovered to support an important role that these spectra play in numerical study of dynamical systems: heuristic arguments (based on an analogy with statistical mechanics; see introduction to this chapter) show that the HP-spectrum for dimensions (multiplied by q- 1} and the /"(a)-spectrum for dimensions form a Legendre transform pair.

Multifractal Formalism

189

The rigorous mathematical study of the above multifractal decomposition and f 14 (a)-spectrum is based upon constructing a one-parameter family of equilibrium measures Va, a E [a1. a2] which are supported on X 0 (i.e., li0 (Xa) = 1) and have full dimension (i.e., dimE v0 = dimH X,.). We will construct such families of measures for multifractal decomposition generated by equilibrium measures invariant under conformal dynamical systems of hyperbolic type (see Chapter 7) We will also reveal extremely complicated "bizarre" structure of this multifractal decomposition: each set X 0 is everywhere dense in X and so is the set X. This also shows that the Hausdorff dimension in the definition of the f 14 (a)-spectrum cannot be replaced by the box dimension. Finally, we will observe that the !,..(a)-spectrum is complete, i.e., for any a outside the interval [at, a2], the set X a is empty.

19. Multifractal Analysis of Gibbs Measures on Limit Sets of Geometric Constructions

In this section we will show how to compute the Hentschel-Procaccia spectrum, Renyi spectrum, and !(a)-spectrum for dimensions of equilibrium measures supported on the limit sets of some geometric constructions (CG1-CG3) in JRm (see Section 13). We will also undertake a complete multifractal analysis of these measures. Continuous Expanding Maps Let X C lR"' be a compact set. We say that a continuous map f: X -+ X is expanding iff is a local homeomorphism and f satisfies Condition (15.9), i.e., there exist constants b :2: a > 1 and ro > 0 such that

B(f(x),ar)

C

J(B(x,r))

C

B(J(x),br)

(19.1)

for every x E X and 0 < r < ro. Without loss of generality we may assume that for any x E X the map I rem:ricted to the ball B(x, ro) is a homeomorphism. Note that a continuous expanding expanding map is (locally) bi-Lipsch.ttz. We recall that a Markov partition for an expanding map f is a finite cover R = {Rb ... , Rp} of X by elements (called rectangles) such that: (a) each rectangle R, is the closure of its interior intR,; (b) intR, n intRi = 0 unless i = j; (c) each f(R,) is a union of rectangles Rj. An expanding map has Markov partitions of arbitrary small diameters. Let us fix a Markov partition R. It generates a symbolic model of the map I by a subshift of finite type (:E1, u), where A = (a;;) is the transfer matrix of the Markov partition, namely, a;j = 1 if intR, n f- 1 (intRj) f= 0 and a;j = 0 otherwise. This gives the coding map x: :E1 -+ X such that

x(w) =

nh(R,n), for n~O

w = (ioil ... )

(19.2)

190

Chapter 6

(where his an appropriate branch of 1-n) and the following diagram ~1~E1

x~x is commutative. Under the coding map the cylinder sets Cio .in to the basic sets in X generated by the Markov partition

R.o in

= R.o n r 1 (R.l n r

1

(· ..

n

r

1

C

E1 correspond

(R,J ... )).

(19 3)

The map x is Holder continuous and injective on the set of points whose trajectories never hit the boundary of any element of the Markov partition. The pullback by x of any Holder continuous function on X is a Holder continuous function on E1. Let f be a continuous expanding map of a compact set X c R"'. We obtain effective estimates for the Hausdorff dimension and box dimension of X following the approach suggested by Barreirain (Bar2]. Let n = {R1 , .•. ,Rp} be a Markov partition of X of a small diameter c (which should be less than the number r 0 in (19.1)) and (E1, o-) the symbolic representation of X by a subshift of finite type generated by n. The Markov partition allows one to view X as the basic set for a geometric construction (modeled by the subshift of finite type) whose basic sets are defined by (19.3) and whose induced map is f. We will extend the approach, developed in Section 15 for expanding induced maps, to arbitrary continuous expanding maps, and we will apply the non-additive version of thermodynamic formalism to compute the Hausdorff dimension and box dimension of X. Fix a number k > 0 For each w = (ioi 1 . ) E E1 and n 2: 1 define numbers A (w n) ~ '

X (w k

= inf { lir(x) - r(y)li} llx-yjj

n) =sup { llr(x)'

rCY)II}

llx- Yll

'

(19.4)

'

where the infimum and the supremum are taken over all distinct x, y E X n R,0 '"+> (compare to (15.7)). Note that since f is expanding Condition (15.8) holds. Define two sequences of functions on E1 5f(k)

= {i,kl(w) = -logAk(w,n)},

~(k) = {~~kl(w) = -log_&(w,n)}. (19.5)

Consider Bowen's equations (19.6)

One can show that for any sufficiently large k, any wE ~1. and n 2: 1, an:::; _&(w, n):::; Ak(w, n) :::; bn (see the proof of Theorem 15 5). Thus, Condition (A2.23) (see Appendix II) holds. By Theorem A2 6 this ensures the existence of unique roots of Bowen's equations (19.6). We denote these roots by §.(k) and s-(k) respectively. Clearly, lk) :::; s
191

Multifractal Formalism

Proposition 19.1. [Bar2] Assume that

I

is topologically mixing. Then

We say that a continuous expanding map I is quasi-conformal if there exist numbers C > 0 and k > 0 such that for each w E E~ and n ~ 0 Condition (15.11) holds. The following result is similar to Theorem 15.5. Proposition 19.2. [Bar2] Assume that f is topologically mrn-ng. Then for any open set U C X and all sufficiently large k,

and s is a unique number satisfying (diam(X nR; 0 i.))' = 0.

lim .!.log

n-+oon

(io

in) E!·a.dmissibJe

Weakly-conformal Maps We say that a map f of a compact set X is weakly-conformal if there exist a Holder continuous function a(x) with Ja(x)J > 1 on X and positive constants C1, 0 2 , and ro such that for any two points x, y E X and any integer n ~ 0 we have: if p(fk(x), fk(y)) :::; r0 for all k = 0, 1, ... n then

n n

Ctp(x,y)

Ja(f"(x))J- 1

:::;

p(f"(x), J"(y))

k=O

n

(19.7)

n

:::; C2p(x, y)

Ja(J"(x))l-

1

.

k=O

Obviously, a weakly-conformal map is continuous, expanding, and quasi-conformal and Proposition 19 2 applies We now discuss the diametrically regular property of equilibrium measures corresponding to Holder continuous functions for weakly-conformal maps (see Condition (8.15)). If f were a smooth map this property would follow from Proposition 21.4. The proof in the general case is a modification of arguments in the proof of this proposition and is omitted. Proposition 19.3. Let f be a weakly-conformal map of a compact set X and


192

Chapter 6

Multifractal Analysis of Equilibrium Measures for Weakly-conformal Maps From now on we assume that the map f is topologically mixing. Let r.p be a Holder continuous function on X and v = v"' an equilibrium measure corresponding to cp. Note that since f is topologically mixing the measure is unique and is ergodic (in fact it is a Bernoulli measure). Define the function 1/J such that log 1/J = r.p - Px (r.p). Clearly '1/J is a Holder continuous function on X such that Px(log1/J) = 0 and vis a uruque equilibrium measure for log,P. We denote by m a unique equilibrium measure corresponding to the function x t-t -slog Ja(x)J on X, where sis the unique root of Bowen's equation

v"'

Px( -slogJa(x)J) = 0. Define the one-parameter family of functions cpq, q E ( -oo, oo) on X by

r.pq(x) = -T(q)logJa(x)J +qlog'I/J(x), where T(q) is chosen such that Px(r.pq) = 0. One can show that for every q E IR there exists only one number T(q) with the above property. It is obvious that the functions 'Pq are Holder continuous on X. The following statement effects the complete multifractal analysis of equilibrium measures corresponding to Holder continuous functions for weaklyconformal maps. Its proof uses the diametrically regular property of these measures (see Proposition 19.3) and is a slight modification of arguments in the proof of Theorem 21.1 where we consider the case of smooth expanding conformal maps.

Theorem 19.1. [PW2] Let f be a topologically miXing weakly-conformal map on X. Then for any Holder continuous function r.p on X we have (1) the pointwise dimension d.,(x) exists for v-almost every x E X and d.,(x)

=

fx log,P(x) dv(x) fx log Ja(x)J dv(x)'

(2) the function T(q) is real analytic for all q E R; T(O) = dimH F and T(1) = 0; T'(q)::; 0 andT"(q);::: 0 (see Figure 17a in Chapter 7}; (3) the function a(q) = -T'(q) takes on values in an interval [at.aa], where 0::; 0:1 = o:(oo)::; a2 = o:(-oo) < oo; moreover, fv(o:(q)) = T(q) + qa(q) (see Figure 17b in Chapter 7); (4) for any q E IR there e:nsts a unique equilibrmm measure vq supported on the set Xa(q)• i.e., vq(Xcr(q)) = 1 {the sets Xcr are defined by {18.5) with respect to the measure v) and d.,.(x) = d.,.(x) = T(q) + qo:(q) for Vq·almost every x E Xa(q)i (5) if v f= m then the functions f,_(o:) and T(q) are stnctly convex and form a Legendre transform pair (see AppendiX V};

193

Multifractal Formalism

(6) the v-measure of any open ball centered at points in X is positive and for any q E JR. we have

. log infgr 2::BEQ v(B)q I T(q) = - hm ogr r-->0 r

where the infimum is taken over all finite covers mdms r In particular, for q > 1,

,

9r of X by open balls of

T(q) - - = HPq(v) = HPq(v) = Eq(v) = Rq(v). 1-q

The following statement is an immediate corollary of Theorem 19.1 (see Statement 1). Proposition 19.4. Any equilibnttm measure corresponding to a Holder continuous function for a topologically mmng weakly-conformal map f on X is exact dimensional. In fact, this result holds for an arbitrary (not necessarily equilibrium) ergodic measure for a weakly-conformal map. The proof is quite similar to the proof of Theorem 21.3 (which deals with the smooth case).

Induced Maps for Geometric Construction s (CG1-CG3) : Multifractal Analysis of Equilibrium Measures Consider a geometric construction (CG1-CG3) in IRm (see Section 13) and assume that it is modeled by a transitive subshift of finite type (:E1, u). Let F be the limit set. Since we require the separation condition (CG3) the cod.ing map x is a 1 homeomorphism and we can consider the induced map G = x o u o x- on the we constructions geometric the limit set F. It is a local homeomorphism since a builds one if Moreover, (:E1,u). type finite of subshift a by modeled consider are geometric construction modeled by an arbitrary symbolic system (Q, u) with the expanding induced map on the limit set then uiQ must be topologically conjugate to a subshift of finite type. This follows from a result of Parry [Pa]. Therefore, the induced map G is expanding if and only if it satisfies Condition (15.9). Note that the placement of basic sets of a geometric construction, whose induced map is expanding, cannot be arbitrary and must satisfy the following special property: there are constants C1 > 0 and C2 > 0 such that for every x E F and any r > 0 there exists n > 0 and basic sets C,(l), ... , c,(m) (with m = m(x, r)) for which B(x,C1r)nFc

U

L\(!:lnFcB( x,C2 r)nF.

l:S;k~m

We assume that the induced map is weakly-conformal. Note that in this case it is also quasi-conformal (see Cond.ition (15.11)). We conjecture that one can

Chapter 6

194

build a geometnc construction modeled by the full shift with disjoint basic sets such that the induced map is quasi-conformal but not weakly-conformal Theorem 19.1 can be used to effect the complete multifractal analysis of equilibrium measures, corresponding to HOlder continuous functions, supported on the limit sets of geometric constructions (CG1-CG3) with weakly-conformal induced map. We apply this result to a special class of self-similar geometric constructions (see Section 13). Recall that this means that the basic sets ~.0 • >n are given by where h11 ... ,l~,p:D---+ Dare conformal affine maps, i.e., llhi(x)- ht(Y)II = >.illx- yl! with 0 < >.; < 1 and x, y ED (the unit ball in lRm ). Assuming that the basic sets~ •• i = 1, .. ,pare disjoint, one can easily see that the induced map G on the limit set F is weakly-conformal (with a(x) = >.;;; 1, where x(x) = (ioi1 ... )). Thus, Theorem 19.1 applies. For Bernoulli measures this result was obtained by several authors who used various methods (see, for example, [CM], [EM], [F1], (0], (Ri]) We describe a more general cla::~s of geometric constructions to which Theorem 19.1 applies Namely, consider geometric constructions build up by p sequences of hi-Lipschitz contraction maps h~n): D---+ D such that

and for any x, y ED,

4ln) dist(x, y)::::; dist(h~n)(x), h~n}(y)) ::::; x~n) dist(x, y), where 0 < 4lnl : : ; X~n) < 1 (see [PW2]). We assume that the following asymptotic estimates hold: there exist 0 <

>., < 1 such that

(19.8) One can check that the induced map G is weakly-conformal (with a(x) = >.i;; 1 , where x(x) = (i 0 i 1 ... )) and Theorem 19.1 applies Example 25.2 shows that there exists a geometric construction produced by three sequences of hi-Lipschitz contraction maps which do not satisfy the asymptotic estimates (19.8). Although the ba::~ic sets at each step of the construction are disjoint it does not admit the multifractal analysis described by Theorem 19.1 (otherwise, by Proposition 19 3 any equilibrium measure corresponding to a Holder continuous function on the llmlt set F of this construction would be exact dimensional which is false for the geometric construction described in this example). It is still an open problem in dimension theory whether one can effect the complete multifractal analysis of Gibbs measures supported on the hmit set of a Moran geometric construction (CM1-CM5) modeled by a transitive subshift

Multifractal Formalism

195

of finite type (for some results in this direction see [PW2, LN]). Notice that by Theorem 15.4, the pointwise dimension of such a measure exists (and is constant) almost everywhere. Remarks. (1) One can show that if 11 = m.>. then T(q) = (1 - q)s (thus, T(q) is a linear function) and f.,(o:) is the ¢-function (i.e., f.,(8) = 8 and f,.(o:) = 0 for all a 1- s; see Remark 1 in Section 21; thls case was studied by Lopes [Lo]). (2) The graphs offunctions T(q) and f(o:) are shown on Figures 17a and 17b in Chapter 7. Note that the function f(a) is defined on the interval (o: 11 o: 2 ], where a1 =- lim T'(q), az =- lim T'(q). q4+oo

q-4-oo

It attains its maximal value 8 at o: = o:(O). Furthermore, f(a(l)) = a(l) is the common value of the lower and upper information dimensions of v (see Section 21) and /'(o:(l)) = 1. We also note that the /(a)-spectrum is complete, i.e., for any o: outside the interval [o:1, 0:2] the corresponding set X a is empty (see Section 21). (3) Consider again a self-similar geometric construction modeled by the full shift u and assume that v is the Bernoulli measure defined by the vector (Pl , ... , Pr), where 0 < Pk < 1 and 2:~= 1 Pk = 1. It is easily seen from Theorem A2 8 (see Appendix II) that PEt {log(.Afo(q)P~0 l) is equivalent to

r

:L "'r(q)Pk = 1. k=l

=0

Chapter 7

Dimension of Sets and Measures Invariant under Hyperbolic Systems

In this Chapter we study the Hausdorff dimension and box dimension of sets invariant under smooth dynamical systems of hyperbolic type. This includes repellers for expanding maps and basic sets for Axiom A diffeomorphisms. The reader who is not quite familiar with these notions, can find all the necessary definitions and brief description of basic results relevant to our study in this chapter. For more complete information we refer the reader to (KH]. We recover two major results in the area: Ruelle's formula for the Hausdorff dimension of conformal repellers (see Theorem 20.1) and Manning and McCluskey's formula for the Hausdorff dimension of two-dimensional locally maximal hyperbolic sets (see Theorem 22.2). Our approach differs from the original ones and is a manifestation of our general Caratheodory construction (see Chapter 1) and the dimension interpretation of the thermodynamic formalism: it systematically exploits the "dimension" definition of the topological pressure described in Chapter 4 and Appendix II. This unifies and simplifies proofs and reveals a non-trivial relation between the topological pressure of some special functions on the invariant set and the Hausdorff dimension of this set. Furthermore, we use Markov partitions to lay down a deep analogy between conformal repellers (as well as two-dimensional locally maximal hyperbolic sets) and the limit sets for Moran-like geometric constructions We then apply methods developed in Chapter 5 (see Theorem 13.1} to study dimension of these invariant sets. In particular, this allows us to strengthen the results of Ruelle and of Manning and McCluskey by including the box dimension of repellers and hyperbolic sets into consideration. We also cover the case of multidimensional conformal hyperbolic sets. The crucial feature of the dynamics, to which our methods can be applied, is its conformality. In the non-conformal case (multidimensional non-conformal repellers and hyperbolic sets) the approach, based upon the non-additive version of the thermodynamic formalism, allows us to obtain sharp dimension estimates. We stress that in this case the Hausdorff dimension and box dimension may not agree. We describe the most famous examples of repellers (including hyperbolic Julia sets, repellers for one-dimensional Markov maps, and limit sets for reflection groups; see Section 20) and hyperbolic sets (including Smale horseshoes and Smale--Williams solenoids, see Section 23). We also provide a brief exposition of

196

Dimension of Sets and Measures Invariant under Hyperbohc Systems

197

recent results on the Hausdorff dimenaion of a class of three-dimensional solenoids by Bothe and on the Hausdorff dimenaion and box dimension of attractors for generalized baker's transformations by Alexander and Yorke, by Falconer, and by Simon (see Section 23). These results illustrate some new methods of study that have been recently developed, as well as reveal some obstructions in studying the Hausdorff dimenaion in non-conformal and multidirnenaional cases. A significant part of the chapter is devoted to the recent innovation in the dimension theory of dynamical systems - the multifractal analysis of equilibrium measures (corresponding to Holder continuous functiona) supported on conformal repellers (see Theorem 211) or two-dimensional locally maximal hyperbolic sets (see Theorem 24.1; in fact, we cover more general multidimensional conformal hyperbolic sets). The first rigorous multifractal analysis of measures invariant under smooth dynamical systems with hyperbolic behavior was carried out by Collet, Lebowitz, and Porzio in [CLP] for a special class of measures invariant under some one-dimensional Markov maps. Lopes [Lo] studied the measure of maximal entropy for a hyperbolic Julia set. Pesin and Weiss [PW2] effected a complete multifractal analysis of equilibrium measures for conformal repellers and conformal Axiom A diffeomorphisms. In this chapter we follow their approach. Simpelaere [Si] used another approach, which is based on large deviation theory, to effect a multifractal analysis of equilibrium measures for Axiom A surface d.iffeomorphisms. There are two main by-products of our multifractal analysis of equilibrium measures on conformal repellers and hyperbolic sets. The first one is that these measures are exact dimensional (see Theorems 21.3 and 24.2; in Chapter 8 we extend these results to arbitrary hyperbolic measures). The second one is the complete description of the dimenaion spectrum for Lyapunov exponents for expanding maps on conformal repellers (see Theorem 21.4) and diffeomorphisms on locally maximal conformal hyperbolic sets (see Theorem 24.3 and also Appendix IV). This spectrum provides important additional information on the deviation of Lyapunov exponents from the mean value given by the Multiplicative Ergodic Theorem.

20. Hausdorff Dimension and Box Dimension of Conformal Repellers for Smooth Expanding Maps Repellers for Smooth Expanding Maps Let M be a smooth Riemannian manifold and f: M -4 M a Cl+"-map. Let J be a compact subset of M such that f(J) = J We say that f is expanding on J and J is a repeller if (a) there exist C > 0 and>.> 1 such that lldf;,'vll 2: CN'IIvll for all x E J, v E T.,M, and n 2: 1 (with respect to a Riemannian metric on M); (b) there exists an open neighborhood V of J (called a basin) such that J = {x E V : r(x) E V for all n 2: 0} Obviously, f is a local homeomorphism, i.e., there exists ro > 0 such that for every x E J the map JIB(x, r 0 ) is a homeomorphism onto its image. Thus, f is expanding as a continuous map (see Section 19 and Condition (19.1)).

198

Chapter 7

We recall some facts about expanding maps. A point x E M is called non-wandermg if for each neighborhood U of x there exists n 2 1 such that r(U) n U =/= 0. We denote by fl(f) the set of all non-wandering points of f. It is a closed /-invariant set. The Spectral Decomposition Theorem claims that the set fl(f) can be decomposed into finitely many disjoint closed /-invariant subsets, fl(J) = h U · · · U Jm, such that f I J; is topologically transitive. Moreover, for each i there exist a number n; and a set A; c J; such that the sets f"'(A.) are disjoint for 0:::; k < n;, their union is the set J;, fn•(A;) =A;, and the map I A; is topologically mixing (see [KH] for more details). From now on we will assume that the map f is topologically mixing. This is just a technical assumption that will allow us to simplify proofs. In view of the Spectral Decomposition Theorem our results can be easily extended to the general case (with some obvious modifications). An expanding map f has Markov partitions of arbitrarily small diameter (see definition of the Markov partition in Section 19; see also [R2)). Let R = {Rll ... , R,} be a Markov partition for f. It generates a symbolic model of the repeller by a subshift of finite type (I:~,
r·

xl

lx

J ~ J is commutative. We remind the reader that the map

x is Holder continuous and injective on the set of points whose trajectories never hit the boundary of any element of the Markov partition. We need the followmg well-known estimates of the Jacobian of C 1 +a expandmg maps. Proposition 20.1. (1) Let '1/J be a Holder continuous function on J such that '1/J(x) 2 c > 0 There enst positive constants L 1 = L 1 ('1f;) and L 2 = L 2 ('1/J) such that for any (n + 1)-tuple (io ... in) and any x, y E Rio in,

rr n

Ll :::;

'1/J(!k(x))

k=O '1/J(f"(y))

:::; L2.

{2) Let '1/J be a Holder continuous function on J such that '1/J(x) 2 c > 0. There enst positive constants L 1 = L 1 ('1f;) and L2 = L2('1/J) such that for any n > 0, any branch h of and any points x E J, y E h(B(x, ro))

rn,

Statement 1 holds. (3) There exist positive constants L 1 and L 2 such that for any (n +I)-tuple (io .. in) and any x,y E R 00 in'

< IJacr(x)l Ll- IJacfn(y)l :::; L2, where Jacfn denotes the Jacobian of fn.

Dimension of Sets and Measures Invariant under Hyperbolic Systems

Proof. Let {3 > 0 be the Holder exponent and C1 Then

ll'I/J(f:(x)) 5 ll (1 + k=O (y)) By the expanding property we find that k=O '1/J(f

199

> 0 the Holder constant of '1/J.

d

lfk(x)- fk(y)I.B).

l!k(x)- !k(y)l 5 C2>.k-nlr(x)- r(y)l, where Cz > 0 is a constant. Therefore,

fi'I/JCJ:(x)) 5

k=O '1/J(f

(y))

n

(1+CICf(>.")k).

k=O

This implies the upper bound. The lower bound follows by interchanging :E andy. This completes the proof of the first statement. The second statement can be proved in a similar fashion The third statement follows by applying the first one to the function t/J(x) = Jac f(x) which is Holder continuous since f is of class

•

c~+a.

A Markov partition n = {Rb ... ,Rp} allows one to set up a complete analogy between limit sets of geometric constructions (CGI-CG2) and repellers of expanding maps by considering the sets R.o in as basic sets. Namely,

J=

n u R.o n::O:O {io

int

in)

where the union is taken over all admissible (n + 1)-tuples (io ... in). By the Markov property every basic set R.o. ;n = h(Rin) n R; 0 for some branch h of ~-n+l,

A smooth map f: M -7 M is called conformal if for each x E X we have df., = a(x) Isom,, where Isom, denotes an isometry of T.,M and a(x) is a scalar. A smooth conformal map f is expanding if la(x)l > 1 for every point x E M. The repeller J for a conformal expanding map is called a conformal repeller. Note that a smooth conformal expanding map is weakly conformal (as a continuous expanding map; see Condition 19.7). The converse is not true in general: there exists a C 00 -map which is not conformal in the above sense but is weakly-conformal (see [Bar2]). Conformal repellers can be viewed as limit sets for Moran-like geometric constructions with non-stationary ratio coefficients since the basic sets R.o in satisfy Condition (B2). Proposition 20.2. (1) Each basic set Rio ;. contains a ball of radius r., 0 in and is contained in a ball of mdi'IJ.S 1';0 in· (2) There eX?-st positive constants K 1 and K2 such that for every basic set R; 0 • in and every X E R.o. in , n

K1fl la(Ji(x))l-l 5 !:; 0 i=O

n

in

5 7'; 0 in 5 K2fl la(!l(x))l- 1. j=O

(20.1)

200

Chapter 7

Proof. Since f is conformal and expanding on J we have for every x

E J,

n

lld/;'11

= ll ia(!i(x))i = IJacr(x)l. )=0

This fact and the third statement of Proposition 20.1 imply that for every x E

Rzo

1n'

. = d1amR,n

(maxyER.n IJach(y)l) I ( nc ))I IJach(/"(x))J Jach f x ~

c1 J=O Tin I ( i( ))1-1 af x ,

where h is some branch of 1-n and C 1 > 0 is a constant. Since each Rj is the closure of its interior we have diamRio in (x) ~ diamR;" .

= diam.R.n

X

min 1Jdh11 11 = diam.R." x min IJac h(y) I

yERin.

yER,n.

(min11eR,n IJach(y)l) n IJach(tn(x))J IJach(f (x))J ~

Tin c2 j=O

j

Ja(f (x))i

-1

'

where C 2 > 0 is a constant. This completes the proof of Proposition 20.2. • We use the analogy with geometric constructions to define a Moran cover of the conformal repeller. It allows us to build up an "optimal" cover for computing the Hausdorff dimension and lower and upper box dimensions of the repeller. Given r > 0 and a point w E ~1. let n(w) denote the unique positive integer such that n(w)

n(w)+l

k=O

k=O

n Ja(x(uk(w)))l-1 > r, n la(x(uk(w)))i-l ~ r.

(20 2)

It is easy to see that n(w) -t oo as r -t 0 uniformly in w. Fix w E ~1 and consider the cylinder set Cio ;n<w> C ~1. We have that wE C;0 ;n<wJ" Furthermore, if w' E C; 0 •• in<w> and n(w') ::; n(w) then

Let C(w) be the largest cylinder set containing w with the property that C(w) = C; 0 ;n<..,"> for some w" E C(w) and C;0 ;n<w'> C C(w) for any w' E G'(w). The sets C(w} corresponding to different wE~~ either coincide or are disjoint We denote these sets by cW' J = 1, ... ' Nr· There exist points Wj E ~1 such that C0l = C;0 • ;"<"'i >. These sets comprise a disjoint cover of ~1 which we denote by ll,. and call a Moran cover. The sets R(J) = x(C(il), j = 1, . . , Nr may overlap along their boundaries. They comprise a cover of J (which we will

Dimension of Sets and Measures Invariant under Hyperbolic Systems

201

denote by the same symbolU,. if it does not cause any confusion). We have that Xj E J. Let Q C E! be a (not necessarily invariant) subset. One can repeat the above arguments to construct a Moran cover of the set Q. It consists of cylinder , Nr for which there exist points Wj E Q such that C(j) = sets c and the intersection c 0, the number of basic sets R(i) in a Moran cover 14 that have non-empty intersection with the ball B(x, r) is bounded from above by a number At!, which is independent of x and r. Analogously to Moran-like geometric constructions (CPWI-CPW4) we call this number a Moran multiplicity factor. In order to explain this property of a Moran cover let f = max{diam.R. : i = 1, ... ,p}. Since the sets R. are the closure of their interiors there exists a number 0 < r 1 :::; r such that each R. contains a ball of radius r 1 . By Proposition 20.2 each basic set R(i) in the Moran cover contains a ball of radius Cr, where C > 0 is a constant independent of r and J This implies the desired property of the Moran cover. R(i) = ~ in(•;> for some

Examples of Conformal Expanding Maps (1) Rational Maps [CG]. Let R: C -t C be a rational map of degree ~ 2, where Cdenotes the Riemann sphere. The map R, being holomorphic, is clearly conformal. The Julia set J of R is the closure of the set of repelling periodic points of R (recall that a periodic point p of period m is repellmg if J ( Rm )'(p) J > 1). One says that R is a hyperbolic rational map (and that J is a hyperbolic Julia set) if the map R is expanding on J (i e., it satisfies Conditions (1)-(2) in the definition of smooth expanding map with respect to the spherical metric on C). It is known that the map z t-+ z 2 + c is hyperbolic provided lei < (see Figure 15). It is conjectured that there is a dense set of hyperbolic quadratic maps. (2) One-Dimensional Markov Maps [RaJ. Assume that there eXIsts a finite family of disjoint closed intervals Jr, [z,. Ip c I and a map f: Ui I; -t I such that

l

(a) for every j, there is a subset P ~ P(j) of indices with /(I;)== UkEPh (mod 0); (b) for every x E Ujintl;, the derivative off exists and satisfies I f'(x) I ~ ar for some fixed ar > 0; (c) there exists A > 1 and no > 0 sucli that if fm(x) E U; intij, for all 0 :S m:::; no- 1 then J(/" 0 )'(x)J ~A. Let J = {x E I : r(x) E U~=I I; for all n E N}. The set J is a repeller for the map f. It is conformal because the domain off is one-dimensional (see Figure 16).

202

Chapter 7

a

b

Figure 15. THE

BOUNDARIES OF THESE "BLACK SPOTS" ARE JULIA SETS FOR THE POLYNOMIAL z 2 c WITH

a) c = -

fo- + ~i and b) c = -fo -

+

~i.

0

Figure 16. A

ONE-DIMENSIONAL MARKOV MAP.

(3) Conformal Toral Endomorphisms. This is a map of multidimensional torus defined by a diagonal matrix (k, ... , k), where k is an integer and JkJ > 1.

Dimension of Sets and Measures Invariant under Hyperbolic Systems

203

(4) Induced Maps. Let h1, ... ,hp:D --t D be conformal affine maps of the unit ball D in !Rm. Assume that the sets h;(D) are disjoint. Consider the self-similar construction generated by these maps and modeled by the full shift (or a subshift of finite type; see Section 13) Define the map G on U; inth;(D) by G(x) = hi 1 (x) if x E inth;(D). Clearly, G is a smooth conformal expanding map and the repeller for G is the limit set of the geometric construction (i.e., G is a smooth extension of the induced map). Similar result holds for the induced map on the limit set of the geometric construction generated by reftection groups (see Section 15). Hausdorff Dimension and Box Dimension of Conformal Repellers

We compute the Hausdorff dimension and box dimension of a conformal repeller. Let f: M --t M be a Cl+"'-conformal expanding map with a conformal repeller J. Let m be a unique equilibrium measure corresponding to the Holder continuous function -slogja(x)l on M, where sis the unique root of Bowen's equation P;( -slog Ia/) = 0 (see Appendix II). Theorem 20.1. (1) dimH J = dim 8 J

= dimsJ = s; moreover s= J;logia(x)idm(x)"

(2) The a-Hausdorff measure of J is positive and finite; moreover, it is equivalent to the measure m. (3) s = dimH m, in other words, the measw-e m is an invariant measure of full Caratheodory dimension {see Section 5).

In [R2], Ruelle proved that the Hausdorff dimension of the conformal repeller J of a Cl+"-map is given by the root of Bowen's equation PJ( -slog Ia I) = 0. He also showed that the a-Hausdorff measure of J is positive and finite. In [F4], Falconer showed that the Hausdorff and box dimensions of J coincide. One can derive Theorem 20.1 from a more general result for continuous weaklyconformal expanding maps (see [Bar2] and Section 19). In particular, this shows that Theorem 20.1 holds for C 1-conformal expanding maps. This result was also established by Gatzouras and Peres [GaP]; see also Takens [T2] for a particular case. We provide here an independent and straightforward proof of Theorem 20.1 which is similar to the proof of Theorem 13.1 (where we dealt with limit sets of geometric constructions (CPW1--CPW4)) • Proof of the theorem. Set d = dimH J. We first show that s:::; d. FixE> 0. By the definition of the Hausdorff dimension there exists a number r > 0 and a cover of J by balls Bt, l = 1, 2, ... of radius rt :::; r such that

(20.3)

204

Chapter 7

Let 'R. = {Rl> ... ,Rp} be a Markov partition for f. For every f > 0 consider a Moran cover U,., of J and choose those basic sets from the cover that intersect Note that R(J) Bt. Denote them by Rt(I) , ... , R(m(t)) l . t = R ;0 ;,.<, ;) £or some (io ... in(t,i))· By Proposition 20.2 and (20.2) it follows that for every x E RY),

n

n{l,j)

K1

ja(fk(x))l- 1 $ !:;0 ;,.< 1 Jl $

7'; 0

i,.<•.n $ Gtrt,

(20.4)

k=O

where C1 > 0 is a constant independent of f. and j. By the property of the Moran cover we conclude that m(f.) $ M, where M > 0 is a Moran multiplicity factor (which is independent oft). The sets {R~il, j = 1, ... , m(f.), e =I, 2, ... } comprise a cover g of J, and the corresponding cylinder sets c~> = C;0 in(l,i) c. comprise a cover of E~. By (20.3) and (20.4) we have

Given a number N > 0, chooser so small that n(l,j) ;,:: N for all£ and j. We now have that for any n > 0 and N > n,

where M(E~,O,v>,Un,N) is defined by (A219) (see Appendix II) with a= 0 and v>(w):::::: -(d +c) log ia(x(w))j. This implies that PE~(-(d+c-)logjaoxl) $0,

where PE+A is the topological pressures on E~. Hence, by Theorem A2.5 (see Appendix II), s $ d +c-. Since this inequality holds for all e we conclude that s$d Denoted= dimsJ. We now proceed with the upper estimate and show that d $ s. Fix t: > 0. By the definition of the upper box dimension (see Section 6) there exists a number r = 7'(t:) > 0 such that N(J, r);,:: re-J (recall that N(J, r) JS the least number of balls of radius r needed to cover the set J).

205

Dimension of Sets and Measures Invariant under Hyperbolic Systems Let 'R = { Rt. . .. , Rr} be a Markov partition for

f and U,. a Moran cover of

J by basic sets Rf.Jl = R;0 inc•;>, j = 1, ... , Nr. The sets CUl = x- 1 (R(il) = C; 0 inc"' > (where w; -= x- 1 (x;)) comprise a Moran cover of :E:A (recall that X is the c~ding map generated by the Markov partition). Repeating arguments in the proof of Theorem 13.1 one can show that there exist A > 0 and a positive integer N such that for any sufficiently small r, card {j : n(w;)

Consider an arbitrary cover

2::

g of I:~

= N};::: r 2•-ci.

by cylinder sets Cio

It follows that

'N

I:

j n(j)=N

where C 2 > 0 is a constant. We now have that for any n > 0 and N > n,

R(:E~,O,cp,Un,N)= I: C;o

I: C, 0

exp(

'NEg

sup

wEC;o

;NEQ

N

sup

iN

"f.

cp(a-"'(w)))

k=O

-

II iaukcx>>rd+Z• 2:: c2,

.,ERI'l k=O

where R(E1, 0, cp,Un., N) is defined by (A2.19') (see Appendix II) with a = 0 and cp(w) = -(d- 2e:) log la(x(w))l By Theorem 11.5 this implies that CP J ( -(d- 2e:) log Ia I) = PJ( -(d- 2e:) Jog Ia I) = PE~ ( -(d- 2e:) Jog Ia 0

xi) 2:: 0

and hence d- 2c ::; s. Since this inequality holds for all e: we conclude that d::; s. The last equality in Statement 1 follows from the variational principle (see Theorem A2.1 in Appendix II). This completes the proof of the first statement. We now prove the other two statements. Note that PJ( -slog Ia I)= hm(f)- s llog la(x)l dm(x) = 0,

where hm(f) is the measure-theoretic entropy ofm. One can use formulae (21.20) and {21.21) below to conclude that dm{x) = s form-almost every x E J The third statement follows now from Theorem 7.1. However, we present here a simple and straightforward proof of the third statement. Having in mind that the measure m is an equilibrium measure and

206

Chapter 7

PJ (-slog Jal) = 0, there exists a constant 0 3 any basic set ~ '· (x)

> 0 such that for any x

E M and

n

m(R.. '· (x)) $ 03

IT Ja(f"'xW"

(20.5)

k=O

(see Condition (A2.20) in Appendix II). Given r > 0, consider a Moran cover U,. = {RO)} of J constructed from a Markov partition R for f. It follows from the property of the Moran cover and (20.5) that m(B(x, r)) $

E

m(R!?l) $ M03 r 8 •

R(jlEU,

Thus, m satisfies the uniform mass distribution principle (see Section 7) and hence dimH m 2: s. The above arguments also imply the second statement. •

Remark. By slight modification of arguments in the proof of Theorem 20.1 one can strengthen the first statement of this theorem and prove the following result (see (Bar2]; compare to Statement 4 of Theorem 13.4): gwen an open set U C M

such that U n J f.

0

we have

dimH(U n J)

= dim8 (U n J) = dimn(U n J) = 8

(recall that 8 is the unique root of Bowen's equation PJ( -slog Jal) = 0). Analysing the proof of Theorem 20.1 one can also obtain a lower bound for the Hausdorff dimension as well as an upper bound of the upper box dimension of any set Z c J (which need not be invariant or compact). Namely,

where s and s are unique roots of Bowen's equations Pz (-slog Jai) = 0 and OPz( -slog JaJ) = 0 (existence and uniqueness of these roots are guaranteed by Theorem A2.5 in Appendix II). One can apply Theorem 20.1 to conformal expanding maps described in Examples 1-4 above. We leave it as an exercise to the reader to show that: 1) iff is a linear one-dimensional Markov map with the repeller J then

where s is the unique root of the equation Cl -•

(here c;

+ · · · +Cp-s

= 1

> 1 is the slope of f on the interval I;,

j = 1, ... , p).

Dimension of Sets and Measures Invariant under Hyperbohc Systems

207

2) iff is the induced map generated by conformal affine maps hl> . .. , hp: D -+ D (D is the unit ball in R"' ), h1(x) = >..ix + a1, then the number s (that is the common value of the Hausdorff dimension and lower and upper box dimensions) is the unique root of the equation

>.r" + ... + >.p' = 1. One can also use Theorem 20.1 to compute dimension of hyperbolic Julia sets of rational maps. We present two additional results (without proofs) that provide more detailed information on the Hausdorff dimension of Julia sets. The first result is due to Ruelle [R2] and deals with the two-parameter family of rational maps z >-+ zq- p Let Jq,p be the corresponding Julia set. Proposition 20.3. If Jq,p is hyperbolic then 2

dimH Jq P = 1 + - 1PI ' 4 1og q

+

(terms of order > 2 in p).

Consider a family {RA : >. E A} of rational maps. Given>., let J..\ be the corresponding Julia set The family R>. is said to be J-stable at >. 0 E A if there exists a continuous map h: A' x J-'o -+ C such that A' is a neighborhood of >. 0 in A and h(>., ·) is a conjugacy from (J>. 0 , R>. 0 ) to (J>., R>.) satisfying h(>.o, ·) = idiJ.x 0 • The second result was obtained by Shishikura [Shi] Consider the oneparameter family of complex quadratic polynomials R>.(z) = z2 + >.. The set 8M = {>.. E C : R..\ is not J -stable} is known to be the boundary of a set M c C called the Mandelbrot set. Proposition 20.4. There e:nsts a residual subset A C 8M such that dimH J >. = 2 for any >. E A. Estimates of Hausdorff Dimension and Box Dimension of Repellers: Non-conformal Case Let J be a repeller for an expanding Cl+"'-map f. If f is not conformal the Hausdorff dimension and the lower and upper box dimension of J may not coincide. An example is given by the induced map on the limit set of the selfsimilar geometric construction, described in Section 16.1 (or Section 16.3), which is smooth expanding but is not conformal. Nevertheless, a slight modification of the above approach, which is relied upon the thermodynamic formalism (and its non-additive version), can be still used to establish effective dimension estimates (i.e., estimates that can not be improved). We follow Barreira [Bar2]. Consider two HOlder continuous functions cp_ and cp on J

cp_(x) Let f_ and

= -log

lldxfll,

cp(x) =log ll(dxf)- 1 11.

t be the unique roots of Bowen's equations

(20.6)

Chapter 7

208

Let R = {R 1 , ... , Rp} be a Markov partition of J of a small diameter and f is a continuous expanding map we can consider two sequences of functions cp(k) and ~p(k) onE~ defined by (19.5). One can verify that given e > 0, there exi;i;s k 2:: 0 such that for every w E E~ and n ;::: 1 we have (E~, u) the symbolic representation of J by a subshift of finite type. Since

-ne +

n

n

j=O

j=O

L 1£.(/i(x)) ~ r£.kl(w) ~ -q;~kl(w) ~ ne + LW(Ji(x)),

where x ~ x(w) E J. This implies that t ~ flkJ ~ s
We describe another method which allows one to obtain even sharper dimension estimates. Consider two sequences of functions on J defined as follows: ~ = {y~)x) =-log lld.,rll},

Note that there exists a constant C

'P = {Wn(x) =log li(d,r)- 1 11}.

(20.8)

> 1 such that

for any x E J and n 2:: 1, where K = max{l[d.,/11: x E J}. We wish to apply the non-additive version of thermodynamic formalism and find roots of Bowen's equations (20.9) In order to do this we ought to establish Property (12.1). Given 15 E (0, 1], we call the map f 15-bu.nched if for every x E J we have (20.10)

Proposition 20.5. [Bar2] Iff is a Cl+"'-expanding 15-bunched map (for some 8 > 0} then the sequences of functions cp and cp satisfy Condition (12.1}. Moreover, there exists e ;::: 0 such that for e;ry w E E~,

for all sufficiently large n and some k ;::: 1 {where x = x(w)).

In view of Proposition 19.1 this imphes dimension estimates for the repeller J, (20.11)

Dnnension of Sets and Measures Invariant under Hyperbolic Systems

209

where§. and 8 are unique roots of Bowen's equations {20.9) (one can show that under the assumptions of Proposition 20.5 Bowen's equations (20.9) have unique roots). The lower and upper estimates in (20.7) and (20.11) cannot be improved. Note that n n

L~(Ji(x)) ~ ~(x) ~ "fPn(x) ~ L"fP(Ji(x)) i=O

j=O

for every x = x(w) E J and n ;::: 1. These inequalities rmply that if I is an 15-bunched Cl+"-expanding map then t ~ §. ~ s ~ t. If f is conformal it is easy-to see that t = §. = s = t. If f is not conformal the numbers §. and s may provide sharper estimates then the numbers~ and t. Indeed, in (Bar2], Barreira constructed an example of a !-bunched C 00 -expanding map of a compact manifold for which t < §. = 8 < t. This map is not conformal but can be shown to be weakly-conformal (as a continuous expanding map; see (19.7)). 21. Multifractal Analysis of Gibbs Measures for Smooth Conformal Expanding Maps We undertake the complete multifractal analysis of Gibbs measures for smooth conformal expanding maps. Let J be a conformal repeller for a Cl+<>_ conformal expanding map I· M ~ M of a compact smooth Riemannian manifold M. We assume that f is topologically mixing. The general case can be reduced to this one (with obvious modifications) using the Spectral Decomposition Theorem.

Thermodynamic Description of the Dimension Spectrum Let 'R = {Rl> ... , Rp} be a Markov partition for f and X the corresponding coding map from E1 to J (see Section 20). Consider a Holder continuous function cp on J. The pull back by x of cp is a Holder continuous function rp on E1, i.e., rp(w) = cp(x(w)) for wE Et Since f is topologically mixing (and so is the shift u) an equilibrium measure corresponding to this function, 11- = J-L.p, is the unique Gibbs measure for u (see Appendix II). Its push forward is a measure on J which is a unique equilibrium measure corresponding to cp. We denote it by 11 = 11v>. Define the function '1/J on J such that log '1/J = cp - PJ (cp) Clearly '1/J is a Holder contrnuous function such that PJ(Iog '1/J) = 0 and 11 is a unique equilibrium measure for log'I/J. By the variational principle (see Theorem A2.1 in Appendix II) we obtain that

1log'I/J(x)diJ.(x) = h (f) 11

= h,_.(u).

Define the one parameter family of functions cpq, q E ( -oo, oo) on J by

cpq(x) = -T(q) log ia(x)i + q log'I/J(x), where T(q) is chosen in such a way that PJ(cpq) = 0. It is obvious that the functions cpq are Holder continuous. Clearly, T(O) = dimH J = s.

Chapter 7

210

The function T(q) can also be described in terms of symbolic representation of the repeller by a subshift of finite type (E1,u). Namely, let i:i, ~~and cpq be the pull back by the coding map x of the functions a, 'if;, and <{lq respectively. Therefore, the function T(q) satisfies Note that log~= ip- PE+(ip). A

cpq(w) = -T(q) log lii(w)l + q log {;(w) with Pr.+ (ipq) = 0. This simple observation allows us to work with the funcA tion T(q) using either the dynamical system (J, f) or the underlying symbolic dynamical system (EA, u). We first study some basic properties of the function T(q).

Proposition 21.1. The function T(q) is real analytic for all q E R. Proof. Consider the function c: R2 -+ C"'(EA, R) defined by c(r, q) = -r log Iii!+ q log 1f;. This function is clearly real analytic. It is known that the pressure P = Pr.+ IS a real analytic function on the space of Holder continuous functions A (see for example, [Rlj), i.e the function P(r,q) = P:r;+(c(r,q)) is real analytic A with respect to r and q. The desired result follows immediately from the Implicit Function Theorem once we verify the non-degeneracy hypothesis. The latter is 8P(c(r, q))

8r

I

# O.

(ro,qo)

In order to compute the partial derivative we use the well-known formula for the derivative of the pressure (see [Rl)): given two Holder continuous functions h1 and h2 on E1, we have

d 1 P(hl dc;: e=O

+ c::h2)

=

r

jE!

h2 dp,,."

(21.1)

where J.tn 1 denotes the Gibbs measure for the function h 1 . Applying this formula with h1 = -ro log Iii!+ Qo log{;, h2 =log liil, and c:: = ro- r we obtain that -LQ __ { 1 l~ld 8P(c(r,q))l og a f.tro,qo r ' ar E! (ro,qo)

1!

where J.tr0 ,q0 is the Gibbs measure for c(ro, Qo).

(21.2) •

We show that the function T(q) is strictly decreasing by computing its first derivative. Fix q E lR and let P,q denote the Gibbs measure corresponding to the function cpq. We also denote the push forward of /1q to J by vq· The measure vq is a unique equilibrium measure corresponding to the function <{lq· Given q E R., let us set

The following statement establishes monotonicity of the function T(q).

Dimension of Sets and Measures Invariant under Hyperbolic Systems

211

Proposition 21.2. For all q we have a(q) = -T'(q). In particular, T'(q) < 0 for all q. Proof. Since P(
= oP(c(q, r-)) + 8q

oP(c(q, r) or

I

T'(q) = O.

r=T(q)

Hence, T'(q)=-oP(c(q,r))l oq ~=T(q)

)-l

(oP(c(q,r))l or

r>=T(q)

In order to compute the partial derivative OP(~~q,r} we apply (21.1) with h1 =-ro log /ii/

+ Qo log~,

h 2 = log ,b, and e = qo - q This results in

a

8P(c(r, q)) q

I (ro,qo)

Using (21.2) and assuming that ro

-- -1

:E;;

1og .i. d '!' J.tro,qo

_J__

..,..

0•

= T(q0 ) we conclude that

/r;+ log(,b(w)} dp.q0 T'(qo) =- J A I 1-( )ld E~ og a w J.tqo

= -a(qo).

•

The lemma follows.

We show now that the function T(q) is convex by computing its second derivative. We recall that by m we denote the measure of full dimension (see Theorem 20.1), i.e., a unique equilibrium measure corresponding to the Holder continuous function -slog la(x)l on J, where s is the unique root of Bowen's equation PJ( -slog lal) = 0.

Proposition 21.3. The function T(q) is convex, i.e., T"(q) convex if and only if v =J m (see Figure 17b}.

~

0. It is stn.ctly

Pt-oof. Differentiating twice the equation P(cpq) == 0 we obtain that 2

T'(q)2 (Q P(c~q,r)l) "( )

T q =-

8r

+ 2T'(q) (O'P(c(q,r))) + (O'P~c~q,r))) tJqiJr

(8Pbr">)

q

'

where the partial derivatives are evaluated at (q, r) = (q, T(q)). We use the explicit formula for the second derivative of pressure for the shift map on E~ obtained by Ruelle in [Rl]. Namely,

Chapter 7

212 where Qh is the bilinear form defined for ht, h2 E G" (E,;t, R) by

and /1-h is the Gibbs measure for the potential h. Ruelle also showed that Qh(g,g) 2: 0 for all functions g E G"'(E1,JR) and that Qh(g,g) > 0 if and only if g is not cohomologous to a constant function (see definition of cohomologous functions in Appendix V). Applying the second derivative formula we obtain that -

()2

82 r P(-(ro +et +c2)logliil +qlog'ljl), 8 2 = Qh(log liil, log liil).

or 2 P(-rlogliil +qlog'ljl) =

Arguing similarly we find that 82 Bq 2 P(-rlogliil + qlog'ljl) = Qh(log'ljl, log¢),

82 q rP(-rlog liil 8 8

-

-

+ qlog'ljl) = Qh(log'ljl,

logliil).

This implies that T"(q)

= Qq(logti}- T'(q) log Iii!,

logti}- T'(q) log !iii)

JE+ log liild~tq A

It follows that T"(q) 2: 0 for any q Moreover, T"(q) > 0 for some q if the function logti}(w) -T'(q) log lii(w)! is not cohomologous to a constant function. The latter can be assured provided that the functions log'¢(w) and - T' (q) log Iii(w) I are not cohomologous. On the other hand, if they are cohomologous for some q then

Hence, -T'(q)

= s.

This implies that v

= m.

•

Diametrical Regularity of Equilibrium Measures An important ingredient of the multifractal analysis of equilibrium measures is the remarkable fact that these measures are diametrically regular (see Condition (8.5)) as the following statement shows.

Proposition 21.4. Let cp be a Holder continuous function on a conformal repeller J. Then any equilibrium measure for tp with r-espect to f is diametrically regular.

Proof. Let v = v'f' be an equilibrium measure for cp. Choose a Markov partition R of J. Given a number r > 0, consider a Moran cover .U,. of J. Fix a point x E J

213

Dimension of Sets and Measures Invariant under Hyperbolic Systems

and choose those elements R(l), ... , R(m) from the Moran cover that intersect the ball B(x, 2r). We recall the following properties of the Moran cover: for every j = l, ... ,m, (1) R(i) = Ri, in<~;l, where Xj E J is a point; (2) diam RU> :<;; r; (3) m :<;; K, where K is a constant independent of x and r. There is an clement R(t) of the Moran cover that contains x. We have that m

c

R(t) C B(x,r) C B(x,2r)

U R. j=1

Define the function 'lj; such that log 'if;= cp- P(cp). Clearly, 'lj; is a Holder continuous function on X such that P(Iog'lj;) = 0 and vis an equilibrium measure for log 'if;. By (A2.20) (see Appendix II) we also have that for any j = 1, ... ,m,

In view of the second statement of Proposition 20.1, if the diameter of the Moran cover does not exceed ro, then for any j = 1, ... , m,

nk-o

))

n(o:.)-1.1,(/k(x

C <

"'


t

I - n~~~)-11/J(!k(x;)) -

2

It now follows that m

v(B(x, 2r)) :<;;

j=l

j=l

n

n(z,)-1

:<;; KC2D2

n

m n(z;)-1

L v(RU)) :<;; D2 L

1/J(/k(x;))

k=O

D D 2 2 1/J(J'•(xt)) :<;; KC2 D v(R
k=O

I

1

•

This completes the proof.

Multifractal Analysis of Equilibrium Measures We state the result that establishes the multifractal analysis for equilibrium measures (corresponding to Holder continuous functions) supported on repellers of smooth conformal expanding maps. One can apply this result to conformal expanding maps listed in the previous section (i.e., hyperbolic rational maps, one-dimensional Markov maps, etc.). For a ~ 0 consider the sets

la

= {x E J: dv(x) =a}

(compare to (18.5)) and the fv(a)-spectrum for dimensions f,(a) = dimH Jo..

Chapter 7

214

Theorem 21.1. (PW2) (1) The pointwise dimension d.,(x) extsts for v-almost every x E J and _ JE11ogi);(w)dtt(w) _ d.,(x)- - JE+ log lii(w)l dtt(w) - A

h11 (!)

JJ log la(x)l dv(x) ·

(2) The function f.,( a) is defined on the interval [al> a2] which is the range of the function a(q) (i.e., 0 ::=:; a1 ::=:; a2 < oo, Cl'l = a(oo) and Cl'z = Cl'( -oo));

=

this function is real analytic and fv(a(q)) T(q)+qa(q) (see Figure 17b). (3) For any q E lR we have that vq(Jo(q)) = 1 and 4,._(x) =d.,. (x) = T(q) + qCl'(q) for Vq-almost every x E Jo(q). (4) If v f. m (i.e., v is not the measure of full dimension) then the functions fv(a) and T(q) are stnctly convex and form a Legendre transform pair (see Appendix V). (5) The v-measure of any open ball centered at points in J is positive and for any q E lR we have

where the infimum is taken over all finite covers 9r of J by open balls of radius r. In particular, for every q > 1, T(q) 1-q

- - = HPq(v)

= -HPq(v) = !lq(v) =

Rq(v).

Proof. We begin with the following lemma.

Lemma 1. There exist constants C 1 > 0 and C2 > 0 such that for every q E lR and for all basic sets R; 0 .in, (21.4)

Proof of the lemma. Note that p. and IJ.q are Gibbs measures corresponding to the Holder continuous functions log!); and ij?q = -T(q) log Iii I + q log!); whose topological pressure is zero Note also that m is a unique equilibrium measure corresponding to the Holder continuous function -slog Ia I whose topological pressure is zero. It now follows from (A2.20) (see Appendix II) that the ratios

(where x E R.o •J are bounded from below and above by constants independent of n. The desired result follows. •

Dimension of Sets and Measures Invariant under Hyperbolic Systems

215

Given 0 < r < 1, consider a Moran cover 1.1,. of the repeller J by basic sets R~i) = R; 0 ;., •. 1 with radius approximately equal to r (see Section 20). Let J N(x, r) denote the number of sets that have non-empty intersection with a. given ball B(x, r) centered at x of radius r. By the property of the Moran cover N (x, r) :$ M uniformly in x and r, where M is a Moran multiplicity factor (recall that M does not depend on x and r). It follows from (20.1) and (20.2) that there exist positive numbers C 3 and C4 such that for every RJ?l E iJ..,

mil

(21.5)

Since 11,. is a disjoint cover of J we have

L

Vq(R!.i>)

= 1.

R!f>eu, Hence, summing (21.4) over the elements of the cover U,., we obtain that there exist positive constants C5 and C6 such that Cs :$

I:

rT(q)

v(R~i))q :$ C5.

R~i)EU,

Taking logarithm and dividing by log r yields for all q E R, (")q

log~R<J>eu v(R/) - lim r ' r-+0 logr

= T(q).

(21.6)

Given 0 < r < 1 and a point wE ~1, consider the unique positive integer n(w) = n(w, r) defined by (20.2). For a number o 2: 0 denote by ia the ''symbolic" level set, i.e., the set of points w E ~1 for which the limit n(w,r)-1

r

r~

2::

_

log¢(uk(w))

k=o

(21.7)

n(w,r)-1

2::

loglii(uk(w))l-1

k=O

exists and is equal to o. Define the "symbolic" spectrum for dimensions by (21.8)

Lemma 2. For every q E lR we have (1) vq(x(ia(qJ)) = 1; (2) dv. (x) = T(q) +qa(q) for vq-almost all x E x(Ja(q)) and dv0 (x) :$ T(q) qo(q) for all X E x(Ja(qj); (3) dimH x(fa(q)) = T(q) + qo(q).

+

216

Chapter 7

Proof of the lemma. Consider the functions w >-+log lii(w)l and w >-+ logtfr(w). Since l-'q is ergodic the Birkhoff ergodic theorem yields that

f:

lim n-+oo

logtfr(uk(w)) :=o L: log lii(uk(w))l-1

=:

a(q)

k=O

for JJq-almost every w E E1. This implies the first statement. It follows that for any e > 0 and every wE la(q) there exists r(w) such that for any r :::; r(w), n(w,r)-1

L:

a(q)- c: :::;

_

log1ji(uk(w))

:S a(q) +e.

k)=O

n(w,r -1

L:

(21.9)

log lii(uk(w))l-1

k=O

lJ·

Given l > 0, denote by Qt = {wE Jc.(q) : r(w) :S It is easy to see that Qt c Qt+l and Ja(qJ = U~ 1 Qt Thus, there exists l 0 > 0 such that 17 (Qe) > 0 9 if l ~ t 0 • Let us choose t ~ lo. Fix r, 0 < r < 1 and consider a Moran cover .U,.,q, of the set Qt (see Section 20). It consists of cylinder sets Ofl, j = 1, ... , Nr,t for which there exist points Wj E Ql such that C~i) = 0, 0 .. ;,<w;J. If r is sufficiently small we have n(w;) ~ l for all j. Since JJq is a Gibbs measure we obtain by (A2.20) (see Appendix II) that for every w = (ioi1 ... ) E E1 and n > 0, (21.10) where 0 7 and Cs are positive constants. It follows from (21.9), (21.10), and (20.2) that for all n ~ land any x E x(Qt), M

v9 (B(x, r) n x(Qt)) :::; 2:>9 (0~il) j=l

M n(w;)-1

:::; Cs

L IT j=l

lii(uk(w))I-T(q)tfr(uk(w))9

k=O

M n(w,)-1

:::; Cs

L IT j=l

ia(uk(w))I-T(q)-q(a(q)-e):::;

CgrT(q)+q(a(q) -.:),

k=O

where 0 9 > 0 is a constant and M is a Moran multiplicity factor (we stress that both 0 9 and M do not depend on l and j). Since v (x(Qt)) > 0, by the Borel 9

Dimension of Sets and Measures Invariant under Hyperbolic Systems

217

Density Lemma (see Appendix V), for llq-almost every x E x(Qt) there exists a number ro = ro(x) such that for every 0 < r::::; r 0 we have

vq(B(x, r)) S 2vq(B(x, r) n x(Qt)). This implies that for any l > £0 and almost every x E x(Qt),

= x d () •

. logvq(B(x,r)) . Iogvq(B(x,r)nx(Qt)) > 11m -"'---'!...:~:-'--"---~~~ log r - r-+0 log r ;?: T(q) + q(a(q) -e).

= l1m

r-+0

Since sets Qt are nested and exhaust the set Q we obtain that fb-.(x);?: T(q) + q(a(q) -e) for Vq-almost every x E Jcx(q)· Since e is arbitrary this implies that 411.(x) ;?: T(q) + qa(q) for vq-almost every x E x(Jcx(q))· By Theorem 7.1 we obtain that dimH x(Jcx(q)) ;?: T(q) + qa(q). Fix 0 < r < 1. For each w = (ioi1 ... ) E Qt choose n{w) = n(w, r) according to (20.2). It follows that R.o .i,.{..,) c B(x, r), where X = x(w). By virtue of (21.9) and {21 10) for all wE Qt,

n(w)

;?: Cr

IJ la(qk(w)))I-T(q)-q(a{q)+•);?: CrrT(q)+q(a(q)+e).

k=O

It follows that for all

X

E

x(Qt),

- ( ) - . logvq(B(x,r)) d..,. x = hm ::::; T(q) + q(a(q) +e) r-+0 1ogr Since e is arbitrary this implies that d11• (x) S T(q) +qa(q) for every x E x(Jo:(q))· Therefore, dv. (x) (x) T(q) + qa(q) for llq-almost every X E x(Ja(q)) and the second statement of the lemma follows. Moreover, by Theorem 7.2, we obtain that dimH x(Jcx(q))::::; T(q)+qa(q). This implies the last statement and completes the proof of the lemma. •

=a... =

It immediately follows from (21.6) that T(l) = 0 and thus, J.L = J.Ll· The first statement of the theorem now follows from Lemma 2 and the following lemma.

Lemma 3. Jcx = x(Jcx)· Proof of the lemma. Fix 0 < r < 1. For each w = (i 0 i 1 ... ) E Qt choose n = n(w, r) according to (20.2). It follows that B(y, C10r) C

Rio ;,.

C B(x, r),

218

Chapter 7

where x = x(w), y E R.., ..in is a point, and Cw > 0 is a constant. Since the measure v is diametrically regular (see Proposition 21.4) we obtain

v(R. 0

in)~ ~

v(B(x,r) ) ~ v(B(y, 2r)) Cuv(B(y , Cwr)) ~ v(R..,

;J,

where C 11 > 0 is a wnstant. Note that J1. is a Gibbs measure corresponding to the Holder continuous function log (f whose topological pressure is zero. Note also that m is a unique equilibrium measure corresponding to the Holder continuo us function -slog [aJ whose topological pressure IS zero. It now follows from (A2.20) (see Appendix II) that the ratios

(where x E R, 0 in) are bounded from below and above by constants independ ent of n. This implies that the two limits n(w,r)-1

lim Iogv(B(x ,r)) and Jim r-+0 logr

r-+0

2:':

~

log¢(uk( w))

k=O

n(w,r)-1

2:

log [ii(uk(w))l-1

k=O

exist simultaneously, and the proof of the lemma follows. • Applying Lemma 3 we conclude that dimH .J"'(q) = T(q) + qa(q), and Stat~ ments 1-5 follow. We now prove the final statemen t of the theorem. Given r > 0, consider a Moran cover .U,. = {R!.i>} of J. There are positive constants C and C 13 12 independent of r such that for every j one can find a point Xj E RJ!> satisfying (21.11) Since the measure vis diametrically regular (see Proposition 21.4) it follows from (21.11) that for every q E R, (21.12) j

j

j

where C 14 > 0 is a constant independent of j and r. Let 9r be a cover of the repeller J by balls B (y;, r). For each j ~ 0 there exists B(y;,,r) E 9r such that B(y;;,r) nR~j) =I= 0. Consider the new cover of J by the balls B; = B(Yii, 2C1ar). By (21.11) each basic set is contained in at least one element of the new cover. Since the measure v is diametrically regular we obtain by (21.11) that for any q E lR,

RJil

L Rtee.

v(R~i))q ~ C1s11(Bk)q,

(21.13)

Dimension of Sets and Measures Invariant under Hyperbolic Systems

219

where 0 15 > 0 is a constant. Exploiting again the fact that the measure v is diametrically regular we conclude using (21.13) that for any q E R,

L

v(R~)q ~ Cls

R~1 )EUr

L v(B~.:)q ~

c16

k

L

v(B)q,

BEQr

where 016 > 0 is a constant. Statement 5 of the theorem follows immediately • from (21.6), (21 12), and (21.13).

Remarks. (1) Assume that v = m is the measure of full dimension. This implies that the functions -slog Ia o xi and log 1/J are cohomologous to each other (see Appendix V). Since PE+(-T(q)Ioglaoxl +qlog'I/J) = 0 it follows that T(q) = 1\ (1- q)s (thus, T(q) is a linear function). Since the pointwise dimension of m is equal to s everywhere in J we have that fv(s) = s and fv(a) = 0 for all a# s.

T(q)

slope= -a(oo)

~

\

',-z__ ' slope= -a(-co)

Figure 17a. "TYPICAL" GRAPH OF THE FUNCTION T(q).

(2) Assume that v :j:. m. There is an interesting manifestation of the multifractal analysis which immediately follows from Statement 4 of Theorem 21.1: for any e > 0 there ex~sts a distinct (singular) equdibnum measure p. on J such that IdimH p.- dimH vi ~ e. In fact, using analyticity of the topological pressure and Statement 1 of Theorem 21.1 (see also (21.20) and (21.22} below) one can show that (we leave detail arguments to the reader): the Hav.sdorff dimension

Chapter 7

220 tangent to the curve at a( 1), with slope= I

~

f(CX)

s = dimH(support Jl)

f(a(l))

0

a (I)

a(oo)

a(O)

a(-oo) I

I

q>O

:q
I

Figure 17b. "TYPICAL" GRAPH OF THE FUNCTION fv(a). of an equilibrium measure on a conformal repeller (corresponding to a Holder continuous potential) depends continuously on the potentiaL (3) Assume that v # m. The function f,(a) is defined on the interval [at. a2] where a1 = - lim T'(q), a2 = - lim T'(q). q-+-oo

q-++oo

We have that 0 < a 1 < a 2 < oo. Moreover, f,(a) < s = dimH J for any a # a(O) and f,(a(O)) = T(O) = s (since the function fv(a) is strictly convex a(O) is the only value where this function attains its maximum s; see Figure 17b). Notice that a(O) <sand v0 = m. Differentiating the equality fv(a(q)) = T(q) + qa(q) with respect to q we obtain that Z,Jv(a(q)) = q for every q E JR. This implies that lim dd fv(a) 0!4-0:t

= -oo.

{21.14)

Q

Furthermore, fafv(a(l)) = 1 Since T{l) = 0 we have that fv(a{l)) = a{l). Taking into consideration that the function /,(a) IS strictly convex we conclude that the equation fv(a) = a has the unique root a(l). Moreover, the function j,(a) is tangent to the line of slope 1 at a(l) (sec Figure 17b). Notice that v1 = v and 0! = a(l) is the only number for which v(Ja) = 1. (4) Given a number q0 # 0, consider the measure Vqo· We claim that its dimension spectrum can be computed by the following formula:

Indeed, since

P(rp~~o)

fv 90 (a(q)) = f,(a(qqo)).

(21.15)

= 0 we have that log1/;q = rpq

and hence for every q E lR,

0

0

(rpllo)q(x) = -Tqo(q)logla(x)l +qlog1/;q0 (x) = -(T~~o(q) + T(qo)q) logja(x)l + qq0 log1/;(x).

Dimension of Sets and Measures Invariant under Hyperbolic Systems Since P((rpq0 )q) = 0 we obtain that Tq0 (q) + T(qo)q Differentiating over q yields that

aq0 (q) = -T~ (q) = a(qqo)qo

221

= T(qqo) for every q E JR.

+ T(qo)

and the desired formula follows. In particular, we conclude that for every q E JR., q 'I= 0,

{f.... (a( +oo)),J.... (a( -oo))} = {f,.,(a( +oo)),J.,(a( -oo)) }. (5) Assume that vq is the measure of maximal entropy for some q 'I= 0. This implies that the function rpq is cohomologous to 0, and hence the functions T(q) log Ia I and q log 1/J are cohomologous. Therefore, the measure v must be a unique equilibrium measure corresponding to the Holder continuous function tloglal, where t:::;; ~ (6) In (Si], Simpelaere obtained a variational description of the function T(q): namely, _. f hp(f)- q J1 log'I/Jdp (21.16) T( q ) - I l l J1 1ogadp '

where the infimum IS taken over all Borel ergodic measures p on J. It follows from Statements 1 and 3 of Theorem 21.1 and {21.3) that the infimum is attained at p:::;; v 9 . Completeness of the j.,(a)-Spectrum Schmeling (Sch] showed that the /.,(a)-spectrum is complete, i.e., for any number a outside the interval [a 1 ,a2 ] the set Ja is empty. More precisely, the following result holds. Theorem 21.2. We have that a 1 :::;;

and hence fv(a)

=0

inf 4,(x),

~EJ

if and only if a¢

a2:::;;

supd.,(x) xEJ

[a~,

a2].

Proof. Using {21.16) we find that for every q 2: 0,

J1 !og1/Jdvq

J1 !og1/Jdvq

.

_

qat ~ -q JJ Iog Ia Idvq ~ dimH v- q f Iog Ia Idv - T(q) 1

9

. ( . J1 log 1/J dp ) . . J1 log 1/J dp =mf dtmHp-qDioglaJdp ~dtmHJ+qmf- J,IogJajdp' where the infimum is taken over all Borel ergodic measures p on J. Dividing by q and letting q -+ oo yields that

.

f 1 1og1/Jdp

al ~ +mf- J,logjajdp'

(21.17)

222

Chapter 7

Recall that a 1 = liiilq-+oo a(q). Therefore, in view of Theorem 21.1, a 1 ~ inf,.EJ dv (x). Assume that this inequality is strict One can find e > 0, x E X, and a sequence of numbers nk such that -1

nk

- ~log1j>(x)

n~r

(

~logla(x)l

)

:::; a1- e.

nk-1

Let p be an accumulation measure of the sequence of measures Pk = ,:

E

{;!'("')

• j=O

(here 8y denotes the unit mass concentrated at the point y). We have that

Passing to the limit as k ---t oo yields that

which contradicts (21.17). This implies the first equality. The second one can be proved in a similar fashion. • Schmeling also proved that for any T1, r2 E (0, s] there exists an equilibrium measure von J corresponding to a Holder continuous function for which fv(al) = r 1 and f.,(a 2 ) = r2. However, "generically" f.,(ad = j.,(a2) = 0; more precisely, this holds for equilibrium measures whose potential functions belong to an open and dense set in the space of Holder continuous functions on J - the phenomenon noticed previously in physical literature (most of the graphs of the function J.,(a) that one can find there end up on the x-axis) Note that in the latter case the measures v<>(-oo) and Va(oo) have zero measure-theoretic entropy. We have the following multifractal decomposition of a conformal repeller J associated with the pointwise dimension of an equilibrium measure supported on J and corresponding to a Holder contrnuous function: (21.18)

Here J., is the set of points where the pointwise dimension takes on the value a. Each set Ja is everywhere dense in J and supports a measure v"' (i.e., va(Ja) = 1), which is a unique equilibrium measure corresponding to the HOlder continuous function 't'q(x) = -T(q)logia(x)i + qlog1j>(x), where q is chosen such that a= a(q). The set j - the irregular part of the multifractal decomposition consists of points with no pointwise dimension. We will see in Appendix IV that j =f:. 0; moreover, it is everywhere dense in J and carries full Hausdorff dimension (i.e., dimH j = dimH J) and full topological entropy (i.e., hJ(f)hJ(/)).

Dimension of Sets and Measures Invariant under Hyperbolic Systems

223

Pointwise Dimension of Measures on Conformal Repellers Given a point x E J, we define the Lyapunov exponent at x by

>.(x)= lim logJJd/~11 >0 n-+oo n

(21.19)

(provided the limit exists, see Section 26 for more details). If vis an !-invariant measure then by the Birkhoff ergodic theorem, the above limit exists v-almost everywhere, and if v is ergodic then it is constant almost everywhere We denote the corresponding value by Av > 0. Let v be an equilibrium measure corresponding to a Holder continuous function on J. It follows from Statement 1 of Theorem 21.1 that for v-almost every X E

J, - h,(f) d v (X ) '

(21.20)

>..,

where h,(J) is the measure-theoretic entropy off and

>.,

=

1 ElogJa(Jk(x))J n limn-toon k=O

=

1

logJa(x)Jdv(x).

(21.21)

J

F\1rthermore, let G., be the set of all forward generic points of v (i.e., points for which the Birkhoff ergodic theorem holds for any continuous function on X; see Appendix II). It follows from Lemmas 2 and 3 in the proof of Theorem 21.1 (applied to the measure v 1 = v) that for every x E G,,

In view of Theorems 7.1 and 7.2, this unpiles that

hv(J) = d"imHV = d"imH G v·

~

(21.22)

We extend (21.20) and (21.22) to any Borel ergodic measure on J of positive measure-theoretic entropy which is not necessarily an equilibrium measure.

Theorem 21.3. For any Borel ergodic measure v of positive measure-theoretic entropy supported on a conformal repeller J we have (I) the equality {21.20} holds for almost every x E J; (2) h';.~f) = dimH v = dimH G, = dim 8 G, = dimsG,... Proof. Set d,.. = h-;.SD. We first show that d,(x) ;::: d,.. The proof is a slight modification of the proof of Statement 2 of Theorem 13.1. Consider a Markov partition = { R" ... ' Rp} for I and the corresponding symbolic model (E1, u) (see Section 20). Let 11- be the pullback to E~ of the measure v by the coding map X

n

Chapter 7

224

Fix c > 0. It follows from the Shannon-McMillan-Breiman theorem that for JL-almost every wE E_i one can find N1(w) > 0 such that for any n ~ N 1 (w),

JL(C; 0

;,.

{w)) ::; exp( -(h- c)n),

{21.23)

where Cio ;,.(w) is the cylinder set containing wand h = h,.(O") is the measuretheoretic entropy of the shift u. It follows from the Birkhoff ergodic theorem, applied to the function log la(x)l, that for v-almost every x EM there exiSts N 2 (x) such that for any n ~ N 2 (x),

1 M

n

1 n log la(x)ldv::; -log la(Ji(x))l n i=l

+€

(21.24)

In order to prove the desired lower bound for d.,(x) it remains only to use (21.23) and (21.24) and to repeat readily the argument in the proof of Statement 2 of Theorem 13.1. We now prove the opposite inequality. Fix 0 < r < 1 By (20.1) it follows that R, 0 ;,.<,.l+l c B(x,C1r), where C1 > 0 is a constant Therefore,

where C2 > 0 is a constant. By virtue of (20.1) we obtain for all x E J,

-d ( ) _ 1. logv(B(x, r)) < d liD l _ v 11 X r-+0 ogr

+ 2c.

Since c: can be arbitrarily small this proves that 4... (x) ::; d., In order to prove the second statement we note that dv = dimH v ::; dimH 0 11 • On the other hand, by (A2.16) (see Appendix II), we conclude that 0 =h.,(!)JJ d...>...,dv = Po"(-dv>...,), i.e., d., is the root of Bowen's equation. Repeating arguments in the proof of Theorem 20.1 (applied to the set Gv instead of J) we obtain that dimBGv ::; d.,. This completes the proof of the theorem. • Some results, similar to Theorem 21.3 for measures supported on conformal repellers of holomorphic maps, were obtained in [Ma2, PUZ]. Since the measure-theoretic entropy is a semi-continuous function we obtain as an immediate corollary of Theorem 21.3 and (21.21) that. the Hausdorff dimension of a Borel ergodic measure v on a conformal repeller is a semicontinuous function of 11.

Information Dimension We compute the information dimension of a Gibbs measure 11 on a conformal repeller J. Applying Theorem 21.1 and taking into account that the function T(q) is differentiable we obtain that the limit lim T(q) 1- q

q-+1

Dimension of Sets and Measures Invariant under Hyperbolic Systems

225

exists and is equal to -T'(l) = a:(l). As we know the latter coincides with the Hausdorff dimension of v (see (21.22) and Remark 3 above) This implies that

fv(a:(l))

= a:(l) = -T'(l) = I(v) = l(v),

where l(v) and 1(v) are the lower and upper information dimensions of v (see Section 18). We note that Statement 5 of Theorem 21.1 allows one to extend the Hentschel-Procaccia spectrum and Renyi spectrum for dimensions for any q f= 1 Moreover, the above argument makes it possible to define these spectra even for q = 1 (as being equal to a:(l)). Dimension Spectrum for Lyapunov Exponents We consider the multifractal decomposition of the repeller J associated with the Lyapunov exponent >.(x) (see (21.19)

(21.25)

where

L = {x E J: the limit in

(21.19) does not exist}

is the irregular part and LfJ = {x E J: the limit in (21.19) exists and >.(x) =

{3}.

If v is an ergodic measure for f we obtain that >.(x) = >., for v-almost every x E J. Thus, the set L>." f= 0. Moreover, if v is an equilibrium measure corresponding to a Holder continuous function the set L>.u is everywhere dense (si.nce in this case the support of v is the set J). We note that if the set LfJ is not empty then it supports an ergodic measure VfJ for which .>.,~ = {3 (indeed, for every x E Lp the sequence of measures ~;;;;~ &J•(x) has an accumulation measure whose ergodic components satisfy the above property). There are several fundamental questions related to the above decomposition, for example, (1) Are there points x for which the limit in (21.19} does not exist, i.e., Lf= 0? (2) How large is the range of values of .>.(x)? (3) Is there any number {3 for which any ergodic meMure v with v(L{3) > 0 is not an equilibnum measure?

k

In order to characterize the above multifractal decomposition quantitatively, we introduce the dimension spectrum for Lyapunov exponents of f by

£(.8) = dimn Lp.

Chapter 7

226

This definition is inspired by the work of Eckmann and Procaccia [EP). In [We), Weiss derived the complete study of the Lyapunov dJmension spectrum for conformal expanding maps by establishing its relation to the f vmo.x (a )-spectrum, where Vma.x is the measure of maximal entropy. Notice that the measure of maximal entropy is a unique equilibrium measure corresponding to the function cp = 0, and hence t/J =constant= exp(-hJ(J)), where h;(f) is the topological entropy of f on J. Therefore, for every x E L13,

This implies the following result.

Theorem 21.4. [We) Let J be a conformal repeller for a smooth expanding map f. Then (1) If Vmax f. m (m is the measure of full dimension) then the dimension spectrum for Lyapunov exponents

£({3) =

fvmax

CJ~f))

is a real analytic strtctly convex function on an open interval (fh, f32] containing the point f3 = hJ(f)fs. (2) If Vmax = m then the dimension spectrum for Lyapunov exponents is a delta function, i.e.,

l(f3)

= { s, 0,

for f3 for f3

= h;(f)/ s

f.

h;(f)/s

As immediate consequences of Theorem 21.4 we obtain that if Vmax f. m then the range of the function A.(x) contains an open interval, and hence the Lyapunov exponent attains uncountably many distinct values On the contrary, if the Lyapunov exponent A.(x) attains only count ably many values, then Vmax = m. There is an interesting application of this result to rational maps. In [Z), Zdunik proved that in the case of rational maps the coincidence llmax = m implies that the map must be of the form z -+ z±n. Therefore, we obtain the following rigidity theorem for rational maps.

Theorem 21.5. If the Lyapunov exponent of a rational map with a hyperbolic Julia set attains only countably many values, then the map must be of the form z-+ z±n. We can now answer the above questions. Namely, (1) the set L is not empty and has full Hausdorff dimension (see Appendu IV; compare to {21.18}); (2) the range of values of A.(x) is an interval ({31> f3z] and for any f3 outside this interval the set L13 is empty (i.e., the spectrum is complete); (3) for any {3 E (f3t, {32} there exists an equuibnum measure v corresponding to a Holder continuous function for which v(L(J) = 1.

Dimension of Sets and Measures Invariant under Hyperbolic Systems

227

22. Hausdorff Dimension and Box Dimension of Basic Sets for Axiom A Diffeomorphisms

Axiom A Diffeomorphisms In this section we study the Hausdorff dimension and box dimension of sets invariant under smooth dynamical sy:;tems with strong hyperbolic behavior. Let M be a smooth finite-dimensional Riemannian manifold and f: M --+ M a Cl+"'-diffeomorphism (i.e., f is a Cl+"-invertible map whose inverse is of class Cl+"). A compact /-invariant set A C M is said to be hyperbolic if there exist a continuous splitting of the tangent bundle TAM = E(•) ® E(") and constants C > 0 and 0 < ). < 1 such that for every x E A (1) dfE(•l(x) = E(•l(J(x)), dfE(ul(x) = E('-'l(J(x)); (2) for all n

~

0

lldrvll S C).nllvll lidf-nvll S C).nllvJI

if v E E<•>(x). if v E E("l(x).

The subspaces E(•l(x) and E<"l(x) are called stable and unstable subspaces at x respectively and they depend HOlder continuously on x. It is well-known (see, for example, [KH]) that for every x E A one can construct stable and unstable local manifolds, W1~~(x) and W1~"j{x). They have the following properties:

Wl~2(x), X E W 1'.,"}(x), (4) T.,W1':2(x) = E<•>(x), T,W1~"}(x)

(3)

X

E

= E<"l(x);

(5) f(W1~~(x)) c W1~~(f(x)), f- 1 (W1~"J(x)) c W1~'::Cf- 1 (x)); (6) there exist K > 0 and 0 < p, < 1 such that for every n ~ 0, (22.1)

and

where p is the distance in M induced by the Riemannian metric; (7) there exist£> 0 such that the intersection W1~l(x)nB(x,c:) (respectively, W1<;1(x) nB(x,E)) consists of all points in B(x,E:) that satisfy (22.1) (respectively, (22.2)) A hyperbolic set A is called locally maximal if there exists a neighborhood U of A such that for any closed /-invariant subset A' C U we have A' C A. In this case A= r(u).

n

-oo

The set A is locally maximal if and only if the following property holds:

Chapter 7

228

(8) there exists 8 > 0 such that for all x, y E A with p(x, y) s o the set W1~2(x) n W1~"j(y) consists of a single point z E A, which we denote by z = [x, y]; moreover, the map

[·,-). {(x,y) E A

X

A: p(x,y) S 8} -t A

(22.3)

is continuous. We define stable and unstable global manifolds at x E A by

wC•>(x)

= U rn(W1~~(fn(x))),

wCu>(x)

= U r(W1~"j(f-n(x))).

n~O

n~O

They can be characterized as follows:

as n -too},

w<sl(x) ={yEA: p(r(y),r(x)) -t 0

wCul(x) ={yEA: p(f-n(y),rn(x)) -t 0

as n-+ oo}.

(22.4)

A diffeomorphism f is called an Axiom A diffeomorphism if its nonwandering set n(f) is a locally maximal hyperbolic set (see Section 20 for definition of a non-wandering set). The Spectral Decomposition Theorem claims (see [KH]) that the set n(f) can be decomposed into finitely many disjoint closed /-invariant locally maximal hyperbolic sets, fl(f) = A1 U · · · U Am, such that f I A; is topologically transitive. Moreover, for each i there exiSt a number n; and a set A; C A; such that the sets Jk(A;) are disjoint for 0 s k < n;, their union is the set A;, r•(A;) =A;, and the map r· I A; is topologically mixing. Let A be a locally maximal hyperbolic set of f. From now on we assume that f I A is topologically mixing. This assumption is technical and we require it to simplify arguments iJ?. the proofs The general case can be reduced to this one by using the Spectral Decomposition Theorem. . A non-empty closed set RcA is called a rectangle if diamR s 8 (where 8 is chosen as in (22.3)), R = intR, and [x, y] E R whenever x, y E R. By Property {8) a rectangle R has "direct product structure", i.e., given x E R, there exists a Holder continuous homeomorphism {22.5)

We remind the reader that a finite cover R Markov partition for f if (1) intR. n intR; = 0 unless i = j; (2) for each x E intR; n f- 1 (intR;) we have

= {R 1 , ••. , Rp}

f(W1~1(x) n R;) c W1~2(!(x)) n R;, f(W1~"J(x) n R;) ::> W1~':/(f(x)) n R;.

of A is called a

Dimension of Sets and Measures Invariant under Hyperbolic Systems

229

n

A Markov partition = {RI> ... ,Rp} generates a symbolic model of A by a subshift of finite type (EA, u), where EA is the set of two-sided infinite sequences of integers which are admissible with respect to the transfer matrix of the Markov partition A= (a.;) (i.e., a;1 = 1 if intR; n r 1(intR;) ::/= 0 and a,; = 0 otherwise; see Appendix II). Namely, define

Now we define the coding map

x: EA -t A by

x( ... i_n·· io .. i,. ... )=

nR.-n

in•

n<:O

Note that the diagram

xl

lx

A ~A is commutative. The map x is Holder continuous and injective on the set of points whose trajectories never hit the boundary of any element of the Markov partition. For anypointw = (... i-1ioi1 ... ) E EA and any pointw' = (... i~ 1 iiA .. ) E EA with the same past as w (i.e., ij = ii for any j ::; 0) we have that x(w') E w.~(x)nR(x}, where X= x(w) and R(x) is the element of an Markov partition containing x. Similarly, for any point w11 = ( ... i~ 1 i~i~ ... ) E EA with the same future as w (i.e., i'J = i1 for any j ;::: 0) we have that x(w") E W1~J (x(w)) nR(x). Thus, the set W1~"}(x) n R(x) can be identified via the coding map x with the cylinder C~ in E1 and the set W1~~(x) n R(x) can be identified via the coding ma_p x with the cylinder C;: in EA:. It is well-known that a locally maximal hyperbolic set A admits Markov partitions of arbitrary small diameters (see [KH]). Let t/J be a Holder continuous function on A The following result allows one to compare the values of this function along trajectories of two points lying on the same stable (or unstable) local manifold.

Proposition 22.1. There extst positive constants such that for any x E A and n > 0,

L1

= L 1 (t/J) and

L2 = L2(t/J)

L2 if y E

W1~J(x),

L1 ::; }]t/Jcri(y)) (gt/Jcr;Cx))) -l.::; L2 if y e

w.~';!Cx).

L1 ::; gt/J(fi(y)) (fit/J(f;(x})) - l

Proof. See the proof of Proposition 20.1.

<::;

•

Chapter 7

230

Given x E A, we set

J<•>(x) and J(ul(x) are Jacobians at x in the stable and unstable directions respectively. Proposition 22.1 applies to these functions. Consider a Markov partition 'R- = {R1, ... , Rp} of A and a Holder continuous function cp on A. The pull back by the coding map x of cp is a Holder continuous function cp on EA, i.e., cp(w) = rp(x(w)) for w E EA. Iff is topologically mixing (and so is the shift u) an equilibrium measure corresponding to this function, J1. = Jl..p, is the unique Gibbs measure for u (see Appendix II). Its push forward is a measure on A which is a unique equilibrium measure corresponding to rp. We denote it by v = v'P. Following Appendix II we consider stable and unstable parts of the measure J.t., i.e., the measures Jl.<•> and Jl.(u) Fix a point x E A and choose the element R(x) of the Markov partition containing x (we assume that x E intR(x)). The sets W1~~(y) n R(x), y E R(x) either are disjoint or coincide and hence comprise a measurable partition of R(x) which we denote by {("). Similarly the sets W1~j (y) n R(x), y E R(x) comprise a measurable partition of R(x) which we denote by{<">. Since J1. is a Gibbs measure we have v(R(x)) > 0. For almost every y E R(x) the conditional measure v<•>(y) generated by 11 on wi;l(y) nR(x) (respectively, the conditional measure v<">(y) generated by von W1~"j (y) nR(x)) coincide with the push forward of the measure J.t.<•> (respectively, Jl.(ul). By Proposition A2.2 (see Appendix II) we obtain the following result. Proposition 22.2. There are positive constants A 1 and A2 such that for any Borel sets E E W1~2 (x) n R(x) and FE W1: ' (x) n R(x), Al(v<•l(E) x v<"l(F)) S v(E x F) S A2(v<•>(E) x v<"l(F)) In other words, the measure von R(x) is equivalent to the direct product of measures v<•>(x) and v<">(x) Conformal Axiom A Diffeomorphisms We say that a diffeomorphism f of a locally maximal hyperbolic set A is u-conformal (respectively, s-conformal) if there exists a continuous function a<">(x) (respectively, a<•>(x)) on A such that df I E<">(x) = a<">(x) Isom,. for every x E A (respectively, df I E<•>(x) = a<•>(x) Iwm.,; recall that Isom,. denotes an isometry of E<"l(x) or E(•l(x)). Since the subspaces E("l(x) and E(•l(x) depend Holder continuously on x the functions a(u) (x) and a<•> (:r) are also Holder continuous. Note that Ja<">(x)l > 1 and Ja<•>(x)l < 1 for every x E A. A diffeomorphism f on A is called conformal if it is u.-conformal and s-conformal as well. Consider a Markov partition n = {R 1 , ... ,Rp} of A of a small diameter (which is less than the number c in Property (7)) Given a point x E A, define the sets A<">(x) = W1~"j(x) n R(x), A<•>(x) = wi;i(x) n R(x) (22.7)

Drmension of Sets and Measures Invariant under Hyperbolic Systems

231

(we assume that x E intR(x)). Note that by (22.5) the rectangle R(x) has the direct product structure (defined up to a homeomorphism 0): R(x) = A<"l(x) x A<•l(x). From now on we assume that f is u-conformal. Note that one can view A(u) (x) as the limit set for a geometric construction (CG1-CG2) with R~:) in to be the basic sets. Namely,

A<"l(x)

=

nU n;?:O (io

n(uJ. n w<"l(x) so tn loc '

in)

where the union is taken over all admissible n-tuplcs (i 0 ••. in) with R.o = R(x). Moreover, this geometric construction is a Moran-like geometric construction with non-stationary ratio coefficients since its basic sets satisfy Condition (B2) (see Section 15) as the following statement shows.

Proposition 22.3. Let A be a locally maximal hyperbolic set for a u-conformal cl+o:_diffeomorphism I We have that (1) eachbasicsetR~:) ;.nw,~:>(x) containsaballinW1~:>(x) ojradiusr..;0 i. and is contained in a ball in W1~"j (X) of radius 1';0 in ; (2) There exist positive constants K1 and K2 such that for every basic set (u) d n
n-1

Ktfl la("l(!i(x))l-

1

$[; 1

;,.

$ f; 1 , in$

j=O

K2fl la("l(Ji(x))l-

1

.

j=O

•

Proof. See the proof of Proposition 20.2.

Using the analogy with geometric constructions described above we will construct a Moran cover of the set A<">(x). It is comprised from basic sets and it alli>ws us to build up an optimal cover for computing the Hausdorff dimension and lower and upper box dimensions of A. Let w = (... L1ioi1 ... ) E 'EA is chosen such that x(w) = x We identify the set of points in 'EA having the same past as wwith the cylinder set C:!- c 'E_t •o Givp_n r > 0 and a point w E C:!-, let n(w) denote the unique positive integer •o such that

n

n(w)

la(u) (x(uk(w)))

k=O

n

n(w)+l

1-l > r,

la(u) (x(uk(w)))

1-l :5 r.

(22.8)

k=O

It is easy to see that n(w)---+ oo as r ---+ 0 uniformly in w. Fix w = (ioh . . ) E and consider the cylinder set C; 0 '•l"'> C C;~. We have w E C;0 w' E C; 0 i,.(w) and n(w') ~ n(w) then

in (w)

C:!•o

and if

232

Chapter 7

Let C(w) be the largest cylinder set containing w with the property that C(w) = C; 0 ;,.<..,"l for some w" E C(w) and C; 0 in(w) C C(w) for any w' E C(w). The sets C(w) corresponding to different wE C"!" either coincide or are disjoint. We denote these sets by

c,
•o

= 1, ... , Nr There exist points

Wj

E C"!" such that •o

c,+ which we denote by

. m">. The sets Rj"> = x(Cj">),j = 1, ... , Nr may overlap along their boundaries and comprise a cover of A<">(x) (which we will denote by the same symbol m"> C3(u) = C; 0

;

< ·l. ~

These sets form a disjoint cover of

if it does not cause any confusion). We have that

R]">

~

=

R;0

'"<>;l

for some

Yj E A(u)(x).

Let Q C C"!" be a subset (not necessarily invariant). One can repeat the above arguments"' to construct a Moran cover of the set Q. It consists of cylinder sets cJ"> ,j = 1, ... ,Nr for which there exist points Wj E Q such that cj"> =

Cio ;,.<..,;> and the intersection

cj"> n cfu.> n Q

is empty as soon as i =f. j. We

denote this cover by~~~· Moran covers have the following crucial property Given a point z E A(u) (x) and a sufficiently small r > 0, the number of basic sets R]"> in a Moran cover that have non-empty intersection with the ball B(u)(z, r) is bounded from above by a number M which is independent of z and r. We call this number a Moran multiplicity factor. In order to establish this property let f = max {diam R. : i = 1, ... , p}. Since the sets R. are the closure of their interiors there eXIsts a number 0 < r 1 ~ f such that for every x E A,

m">

The desired property of Moran covers follows from Proposition 22.3.

Hausdorff Dimension and Box Dimension of Locally Maximal Hyperbolic Sets for u-Conformal and s-Conformal Diffeomorphisms Let A be a locally maximal hyperbohc set for a C 1+a.diffeomorphism f. Assume that f is topologically mixing. We denote by ,.(u) a unique equilibrium measure corresponding to the Holder continuous function -t(u) log Ja(u)(x)J on A, where t(y) the conditional measure on w(y)nR(x) generated by ,_(u) as explained above (sec Proposition 22.2; recall that m(u) (y) is the push forward by the coding map of the unstable part of the measure ,.(ul). We now state our main result assuming that f is u-conformal.

Theorem 22.1. Let f beau-conformal Cl+"'-d1Jjeomorphism of a locally maximal hyperbolic set A. Then (1) for any x E A and for any open set U c W("l(x) such that UnA =f. 0, dimH(U n A)= dim 8 (Un A)= dim 8 (U n A)= t<">;

Dimension of Sets and Measures Invariant under Hyperbolic Systems

233

(2) t(u.) _

h,.(u)

(f)

- J,.. log ia
where h,.<•> (!) '-S the mea.mre-theoretic entropy of f with respect to the measure n(u.); (3) the t = dimHm<"l(x) for every x E A, i.e., the measure m<"'>(x) is the measure of full dimension (see Section 5}.

Proof. It is sufficient to prove the theorem assuming that U = A<"'l(y) for some pointy E w 0. By the definition of the Hausdorff dimension there exists a number r > 0 and a cover of A("l(y)nA by balls Bt, i = 1, 2, ... of radius re ::; r such that

e

For every > 0 consider a Moran cover f4~l of A("l(y) n A and choose those basic sets from the cover that intersect Bt. Denote them by Ri1), • • , R}m(t)). Note that R}il = R~:) inct i> for some (io ... in(t,j))· Using Proposition 22.1, the property of the Moran cover, and repeating the argument in the proof of Statement 1 of Theorem 20.1 one can show that

where C1 > 0 is a constant. Given a number N > 0, choose r so small that n(i,j) 2: N for all i and j. We now have that for any n > 0 and N > n,

where M(c.+, 0, cp,Un, N) is defined by (A2.19) (see Appendix II) with a !.() and cp(w) = -(d+c)logla(ul(x(w))l.

=0

234

Chapter 7

This implies that

P0~ (-(d+e:)logJa<">oxl) ::;o. Notice that for any continuous function cp; A--+ lR

Hence,

P,.. ( -(d+e:}logJa<">i)

S0

Therefore, by Theorem A2.5 (see Appendix II), t<"> s d+c. Since this inequality holds for all e: we conclude that t 0. Repeating arguments in the proof of Theorem 20.1 one can show that there exists a positive integer N such that for an arbitrary cover g of C7 by cylmder sets Cia iN' to

s

N

L: G; 0

sup

-

II Ja<">uk<xJ)I-d+2

e

2:: c2,

;NE9 o:ER~j) k=O

where C2 > 0 is a constant. We now have that for any n > 0 and N > n,

R(Ct,O,cp,Un,N)

=

c,o~E9 exp cE~~p 'N ~ cp(ak(w))) L:

N

sup

-

II Ja<">ukcxJ)I-d+2e 2:: c2,

G; 0 , ;NE9 o:ER~;J k=O

where R(Cf, 0, cp,UN, N) is defined by (A2.19') (see Appendix II) with a = 0 •o and cp(w) = -(d- 2c) log Ja<">(x(w))J. This implies that

Notice that for any continuous function cp: A --+ lR

It follows that

PA ( -(d- 2e:) log Ja<">J) 2::

0

and hence d- 2c ::; t<"> Since this inequality holds for all c we conclude that d s t
Dimension of Sets and Measures Invariant under Hyperbohc Systems

235

Since the measure ~~;(u) is an equilibrium measure corresponding to the function -t<"J log ia<"l(x)l on A, whose topological pressure is zero, we obtain in view of the variational principle (see Appendix II) that

TIJ.is implies the second statement. Since the measure 11-(u) is the stable part of the measure ,_(u) (i.e., the projection of ~~;(u) to :E:;ti) for every x E A and any basic set i. containing x we have that

R}:.'

n

m<">(x)(R~:!.i. nw<">(x)) ~ const IJia<"l(Jk(x))l- 1 • k=O

Fix x E A, 1' > 0 and consider a Moran cover of the set A("l(x) by basic sets R
m<">(x)(RUl) ~ const IJia<"l(Jk(xi))l-e<•J. k=O

Therefore, by the property of the Moran cover we find that m<"l(x)(B(u)(x,r)) ~ L:m<">(x)(R(j)) ~ Mrt<•l, j

where M is a Moran multiplicity factor (which does not depend on r). Thus, m<"J (x) satisfies the uniform mass distribution principle (see Section 7) and hence dimHm<"l(x) 2:: t<"J. The opposite inequality follows from the first statement. This also shows that the t<"LHausdorff measure of UnA is positive and finite and is equivalent to the measure m<"l(x)IA(uJ(x) • Let A be a locally maximal hyperbolic set for a C 1 +"'-difft>omorphism f. Assume that f is s-conformal. Denote by ,<•> a unique equilibrium measure corresponding to the Holder continuous function t<•l log ia<•l (x)l on A, where t<•> is the unique root of Bowen's equation PA(tlog ia<•ll) = 0. Consider the system of conditional measures m<•>(y) on w<•l(y) n R(x), y E R(x) generated by ,_(•) (see Proposition 22.2; recall that m<•l(y) is the push forward by the coding map of the unstable part of the measure ,.<•>; see Appendix II). Similarly to Theorem 22.1, one can prove that for any x E A and any open set u c w<•>(x), dimy(U n A) = dim 8 (UnA) Moreover,

= dimn (UnA) = t<•>.

Chapter 7

236

where h,.<•> (f) is the measure-theoretic entropy off with respect to the measure ~~:<•>. In addition, the t<•LHausdorff measure of UnA is positive and finite. We assume now that A is a locally maximal hyperbolic set for a Cl+a_ diffeomorphism f which is both s- and u-conformal. By results of Hasselblatt [Ha] in this case the stable and unstable distributions on A are smooth. Hence, given a point x E A and a rectangle R containing x, the homeomorphism (} in (22.5) is Lipschitz continuous. Therefore, using Theorems 6.5 and 22.1 we compute the Hausdorff dimension and box dimerusion of A.

Theorem 22.2. We have dimn A = dims A = dims A = t<">

+ t<•l,

where t<") and t<•> are unique roots of Bowen's equations

respectively and can be computed by the formulae

(22.9) This result applies and produces a formula for the Hausdorff dimension and box dimension of a basic set of an Axiom A CH"-surface diffeomorphism, which is clearly seen to be both s- and u-conformal. For this case, McCluskey and Manning [MM] proved that dimn A = t<"> + t<•l For diffeomorphisms of class C 2 Takens [T2] showed that the Hausdorff dimension of A coincides with its lower and upper box dimensions. Palis and Viana [PV] extended this result to difl'eomorphisms of class C 1 . Barreira [Bar2] obtained the same result using another approach based on the non-additive version of the thermodynamic formalism. This approach allowed him to extend Theorem 22.2 to "conformal" homeomorphisms possessing locally maximal "topologically hyperbolic" sets. Consider the measures m<">(x) and m<•>(x) for x EA. By (22.9), Proposition 26.1, and Theorem 71, dimn m
d(u)::;

+ d(s),

dimn m<•>(x) = t<•>

+ d(u),

t
In this case, ~~: is the measure of full dimension. Condition (22.10) is a "rigidity" type condition. It holds if and only if the functions -t(x)l and t<•l log Ja<•l(x)l are cohomologous (see [KH] and Appendix V). One can show that this is the case if and only if for any periodic point x E A of period p, p-1

IT a<"l(J"{x))t = 1.

k=O

Dimension of Sets and Measures Invariant under Hyperbolic Systems

237

Estimates of Hausdorff Dimension and Box Dimension of Locally Maximal Hyperbolic Sets: Non-conformal Case Let A be a locally maximal hyperbolic set for a topologically transitive Cl+a_ diffeomorphism f of a compact manifold M. In the multidimensional case the map f may not be either u- or s-conformal and the Hausdorff dimension of the intersection UnA (where U c w<•>(x) is an open set) may not coincide with its lower and upper box dimensions (see example of a multidimensional horseshoe in Section 23). Nevertheless, the above approach, which relied upon the thermodynamic formalism (or its non-additive version), can be still used to establish effective dimension estimates (i.e., estimates that can not be improved). We follow Barreira (Bar2]. Define functions 'f!.(u) and
and functions

'£.(•)

and on A by

Let f(ul(x) and t(u)(x) be the unique roots of Bowen's equations

and t<•>(x) and t<•>(x) be the unique roots of Bowen's equations

Proposition 22.4. For any x E A, f<">(x)::; dimH(W1~':{(x) n A)::; dim8 (W1~"1(x) n A) ::; dimB(WI~':{(x) n A)::; t(u)(x), t<•>(x)::; dimH(W1~2(x) n A)::; dim8 (W1~~(x) n A) ::; drms(W1~~(x) n A)::; t<•>(x).

One can obtain even sharper dimension estimates using another approach. Define two sequences of functions tp(u) and
(22.11) and two other sequences of functions

tp<•>

-n

and
238

Chapter 7

If the derivatives dfiE(u) and dr 1 iE<•> are 8-bunched (see Condition (20.9)), one can show that all four sequences of functions, (22 11) and (22.12), satisfy Condition (12.1). Thus, by Theorem A2.6 (see Appendix II), Bowen's equations CP Al•l(z)(r~(u))

= 0,

PAt•l(z)(rcp(u))

=0

have unique roots r.
have unique roots r.C•l(x) and r<•l(x). We can now state dimension estimates. Proposition 22.5. If the derivatives dfiE(u) and df- 1 IE(•) are 8-bunched for some 8 > 0 then for any x E A,

r.<">(x) ~ dimH(W1~~(x) n A) ~ dim8 (W1<,:](x) n A) ~ dimB(W1~~(x) n A)~ r(x),

r<•>(x) ~ dimH(W1~J(x) n A)~ dim8 (W1~J(x) n A) ~ dimB(W1~1Cx) n A)~ r<•>(x).

23. Hausdorff Dimension of Horseshoes and Solenoids Multidimensional Horseshoes Let Bt C JRk and B2 C JRf be balls and !:::. = Bt x B2 C Rk E11 JRf = Rn. We refer to Rk as "horizontal" and JRf as "vertical" directions in lR". Let also U c lR" be an open set and f: U -+ !Rn a -diffeomorphism, r ~ 1. The set !:::. = B 1 x B 2 c U is called a horseshoe for f if the intersection!:::.' = !:::. n f(!:::.) can be decompo.'led into simply connected closed disjoint sets !:::.' = 1:::. 1 U · · · U t:J.'P such that (1) 7rt(tl.i) = Bt, 7r2(!- 1 (!:::.i)) = B2, where 11"1 :an -+ JRk and 11"2 : Rn -+ JRk are the canonical projections; (2) 11"2lr 1 (!:::.i n (B 1 x 1r(z))) is a bijection onto B 2 for any z E !:::.i; (3) the map df preserves and expands a vertical cone family C 1 (z, a) and the map df- 1 1t:J.' preserves and expands a horizontal cone family C2 (z,a).

cr

The latter means that C 1 (z, a) and C2 (z, a) are cones in lR" at z of angle a around Rt and JRk respectively such that

df(C1 (z,a)) C C1 (j(z),a)

dr and

1

(C2(z,a))

lldfzvll lld/; 1 vll

~

~

c

Allvll Allvll

1

C2{f- (z),a)

if z E!:::.

if z E !:::.'

for any v E C 1 (z, a) for any v E C2(z, a),

Dimension of Sets and Measures Invariant under Hyperbolic Systems

239

where >. > 1 is a constant. One can show that the set A=

n

r(D.')

nEZ

is a locally maximal hyperbolic set with "almost" vertical unstable and "almost" honzontal stable subspaces (see [KH]). In the two-dimensional case the above construction was first introduced by Smale [Sm] and the set A is known as a "Smale horseshoe " We describe the topological structure of A. Note that for every n 2': 0 the set fn(l:!.') consists of pn simply connected close disjoint "almost" vertical components which we denote by 1:1.~:) in. Similarly, for every n 2': 0 the set J-n(l:!.') consists of pn simply connected close disjoint "almost" horizontal components which we denote by !:!.~~ in (see Figure 18). Given a point z E A, denote by 1:1.!:) in (z) and !:!.l:) ;. (z) the vertical and, respectively, horizontal component that contains z (clearly, it is uruquely defined). One can show that (1) for every z E A the set nn>O 1:1.~:)

in (z)

is a smooth £-dimensional unstable

submanifold that is isom~phic to B2; similarly, the set nn;::o !:!.~) ;n (z) is a smooth k-dimensional stable submanifold that is isomorphic to B 1 ; (2) every point z E A can be coded by a two-sided infinite sequence of integers ( ... L 1 ioi 1 ..• ), ii = 1, . .. ,p such that

moreover, the coding map X: A--+ Ep defined by x(x) = ( .. L1ioh .. is bijective and onto. The map x establishes the symbolic representation of the horseshoe by the full shtft on p symbols. There is another description of the horseshoe which is more smtable for studying its dimension. Notice first that the sets

define a geometric construction in 1:!. (modeled by the full shift on 2p symbols) whose limit set is A. Furthermore, consider the sets

They define geometric constructions in B 1 and B 2 respectively (modeled by the full shift on p symbols). We denote these constructions by ca<•l and CG(u). If p(•) and p(u) are their limit sets then A= p(•) x p(u).

Chapter 7

240

......... I

II

I

(s) A(s) A II 12

Figure 18. A

... ....... I

II

I

A(s) A(s)

21

22

SMALE HORSESHOE.

A horseshoe A is called linear if every component D.; is a vertical strip and 1 every component f- 1 (.6.;) is a horizontal strip and the maps h; = flf- (.6.;) are by affine generated arc CG(u) and ca<•> constructions the case, thls In linear. maps g}•l and g~u) respectively given as follows: g}•>(y)

= 11'I(/- 1 (11'1 1 (y) n D.;)),

g~u>(x)

= 11'2(r 1 (11'2 1 (x) n D.;)).

(23.1)

On the other hand, let CG(•) and CG(u) be two geometric constructions in B1 and B 2 modeled by the full shift on p symbols and generated by affine maps g1•l and g}u) respectively. Let also p(•) and p(u) be their limit sets. One can build a linear horseshoe A for a map f such that A= p<•> x p(u) and relations (23.1) hold. If the maps ui•l and g~u) arc conformal then by Theorem 22.2, we obtain that dimH A = dim 8 A =dims A= dimH p(•) + dimH p(u) . On the other hand, using Example 16.1 we conclude that there exist a linear horseshoe in JR4 for which dimH A < dim 8 A = dimaA . We also remark that if the ratio coefficients of the affine maps g~•) and g~u) are equal then the coding map is an isometry between the the horseshoe A and l::P. Thus, it preserves the Hausdorff dimension and box dimension (compare to Theorem A2.9 in Appendix II). Three-Dimensional Solenoids

We follow Bothe [Bot). Let 8 1 be the unit circle and 7) 2 the unit disk in Then v = 8 1 X 'D 2 is a solid torus. The projections 11' : v -+ 8 1 and R 1 Pl, P2 : V -+ 8 X (-1, 1] are defined by 2•

1r(t,x,y)=t,

Pl(t,x,y)=(t,x),

P2(t,x,y=(t,y).

Dimension of Sets and Measures Invariant under Hyperbolic Systems

241

We also denote by V(t) = {t} x 'D 2 = u- 1 (t) (where t E S 1 ). Let '9 be the space of all 0 1-embeddings f : V ~ V of the form

j(t,x, y) = (cp(t), At(t)x + zt(t), .X2 (t)y + z2(t)),

(23 2)

where

are C 1-maps, and

cp IS expanding, i.e. dcp > 1.

dt The last condition implies that the degree (} of


n

f"(V)

n;:::o

is called a solenoid. One can show that A is a hyperbolic attractor for f (see definition in Section 26). The local topological structure of A is described as follows. For each t E S 1 the set A(t) = An V(t) is a Cantor-like set; it is the limit set for a geometric construction in V(t) modeled by the full shift on 0 symbols; its basic sets on the step n are mutually disjoint ellipses V~a ;. (where i; = 1, ... , 9) which comprise the intersection r(V) n V(t) (see Figure 19). Furthermore, for any arc B c S 1 containing t there is a homeomorphism

h: B x A(t)--+ An 1r- 1 (B)

(23.3)

which can be chosen such that for q E B and x E A(t), 1rh(q,x} = q,

h(t,x) = x.

For each X E A(t) the embedding hz = h(·,x} : B --+ depends continuously on x in the 0 1-topology. A map f E

vis

of class cl, and h,

~is

called intnnsically transverse (with respect to the projections c S 1 and any two components Bt, B2 cAn 1r- 1(B) the arcs p;(B1 ) and p1 (B 2 ) are transverse in 8 1 x [-1, 1] at each point of p;(B 1 ) n p;(B2) We denote by~· the set of all intrinsically transverse maps f E ~· For i = 1, 2 we also denote by P1 and P2) if for any arc B

~; =

{f

E ~ : sup A;

. (dcp)

< inf dt

(d
sup

dt

,

where the infimwn and supremum are taken over 0 :::; t :::; 1. Obviously, open in '9. In [Bot], Bothe proved the following statement.

~;

is

Chapter 7

242

Figure 19.

THE CROSS-SECTION OF A SMALE-WILLIAMS SOLENOID FORt= 0.

Proposition 23.1. The set ;r n ~~ is open and dense

in~~.

We now compute the Hausdorff dimension of the cross-sections A(t). Since the map cp of the circle 8 1 is smooth and expanding and the functions Ai are of class 0 1 fori= 1, 2 there exists a unique root of Bowen's equation

(where P8 , is the topological pressure with respect to the map cp). One can show that s; is the unique number of which the functional equation

L

A;(t')8;e(t') = e{t)

t'E•r'(tJ

has a positive continuous solution ~ : 8 1 --+ R (see [Bot]). Proposition 23.2.

(1) For every f E ~ and every t E S 1,

(2} If Sio =max{ Sl, S2} then for every IE ~ion~· and every t E S 1 ' dimH A(t) =

8; 0

•

Dimension of Sets and Measures Invariant under Hyperbolic Systems

243

Note that the stable foliation of the solid torus by disks 'D(t) is obviously smooth. However, the one-dimensional unstable distribution on A is, in general, only Holder continuous. So is the homeomorphism h defined by (23.3). We set

It follows from results of Hasselblatt [Ha] that the homeomorphism h is Holder continuous with the HOlder exponent a satisfying Ba log.6 - >1. log>.

In particular, if we assume that

Blog~ > 1

(23.4)

log>.

then the map his Lipschitz continuous. By Theorem 6.5 and Proposition 23.2 we conclude that under the assumption (23.4), if s, 0 = max{st.8 2 }, then for every f E ~; 0 n;§'*, dimH A= max{ st. s2} + 1. We consider now the special case when
(mod 1), A1 (t) = >.1, A2(t) = .\2, z 1(t) =a cos21rt, z2(t) =a sin27rt,

= 2t

where 1 > a > 0 is a constant. If we assume that 0

< >.1,>.2 < min{~,a}

then the map f {see {23.2)) is a 0 1-embedding. The solenoid A for f is known as the Smale-Williams solenoid (see Figure 19). Assume that >. 1 S .\2 • It is easy to check that f E ~2 and log2 81 S 82 = -log.\ • 2

In [Sim2], Simon showed that, in fact,

f

E ~·. Thus, we obtain that

log2 . \ . dlmH A(t) = - - 1 og "2 Moreover, it is easy to see that the map h is Lipschitz continuous. Therefore, log2 . , +1. d!mHA = - 1og "2

(23.5)

Note that if .\ 1 = >. 2 then for every t E 8 1 the set A(t) is the limit set for a geometric construction (CPW1-CPW4) with basic sets on the nth step to be

Chapter 7

244

balls of radius>.~. By Theorem 13.3 we have that the formula (23.5) holds for any 0 < ..\ 1 = >.2 < min {!,a} Using results in [Sim2] Simon extended this statement and proved that the formula (23.5) holds for any 0 < >. 1 ::; >. 2 <min {!,a}. Baker's Transformations Consider the map T of the squareS= [-1, 1] x [-1, 1) given as follows 2y -I)

ify ~ 0

2y + 1)

ify

<0

where we assume that (23.6) and fori= 1,2, (23.7) Conditions (23.6) and (23.7) assure that T maps S into itself In fact, the set T(S) consists of two right rectangles both of vertical height 2, one of width (>. 11 and the other of width j,\ 2 J. They may or may not overlap. The map Tis called a generalized baker's transformation. In the case >.1 = >.2 = c1 = c2 the map T is the classical baker's transformation. Clearly, T is piecewise linear with the discontinuity set [-1, 1) x {0}. For every z E S the map T is expanding by a factor of 2 in the vertical direction and is contracting by a factor .>. 1 or ..\ 2 in the hori2ontal direction depending on whether z lies in the top part or bottom part of the squareS.

!,

= -t

t.

y~O

y
<0)

T( y ~ 0)

Figure 20. A SKINNY BAKER'S TRANSFORMATION.

The set

is the attractor forT (i.e., it attracts all the trajectories inS). Clearly, the set A is the direct product of the interval [-1, 1] in the y-rurection and a Cantor-like

Dimension of Sets and Measures Invariant under Hyperbolic Systems

245

set Ll in the x-direction. The latter is the limit set for a self-similar geometric construction on [-1, 1] whose basic sets at a step n are not necessarily disjoint. Alexander and York [AY] considered the case >.1 =:: >.2 = /3, c1 = 1 - /3, and c2 = f3 - 1. Following their work we call T a skinny baker's transformation if I><2l < 1 (see Figure 20) and a fat baker's transformation otherwise. We first study the case of a skinny baker's transformation T. Under an appropriate choice of numbers c1 and c2 the two rectangles that comprise T(S) are disjoint. The map Tis clearly seen to be one-to-one. Moreover, basic sets at each step of the geometric construction, mentioned above, are disjoint. Therefore,

where s is the unique root of the equation (23 8)

Hence, dimHA = dimBA

= dimBA = 1 + s.

We now consider the case of a fat baker's transformation T assuming for simplicity that >. 1 = >.2 = /3 > ~~ c1 = 1 - /3, and c2 = /3- 1. Note that T lS non-invertible and area expanding and that the attractor for T is the whole squareS. Alexander and York [AY) proved that for every 1;::: {3 > ~ the map T = Tp possesses the Sinai-Bowen-Ruelle measure P,f3, that is the limit evolution of the Lebesgue measure mesonS, i.e., n-1

11-f:J

= n-+oo lim .!. "(T/).mes. nL..J k=O

The measure #f:J has a characteristic property that its conditional measures on vertical lines (where Tp expands) are equivalent to the linear Lebesgue measure. Let Vf:J be the factor-measure induced by Jlof:J· It has the following "measurearithmetic" interpretation and plays an important role in the old and not yet completely solved problem by Erdos. Let En, n = 0, 1, 2, ... be a sequence of independent random variables, each with the values +1 and -1 with equal probabilities. The measure vp can be shown to be the measure with distribution function of the random variable 00

L

En(l-

/3)/3n.

n=O

In other words, for any interval (a, b) and any integer N let YfJ,N(a, b) be the N-1

proportion of points of the form

2::

fn(l- {3){3n that lie in (a, b), i.e.,

n=O N-1

Vf3,N(a, b)

= 2-N card {

x: x

= ~ fn{l

- {3){3n,

246

Chapter 7

where card( A) denotes the cardinality of the set A. Then vp(a, b) = lim Vf3.N(a, b). N--too

The measure Vf3 is called an infinitely convolved Bernoulli measure. It is easy to see that for /3 < ~' vp has a Cantor distribution and for f3 = the uniform (Lebesgue) distribution For fJ > vp is known to be continuous and always pure, i.e., either absolutely continuous or totally singular (see [JW]). For fJ the nth root of ~' vp is absolutely continuous (and indeed, progressively smoother as n increases; see [W]). Erdos [Er2] proved that for almost all {3 sufficiently close to 1 the measure vp is absolutely continuous, on the contrary, he showed [Er1] that if fJ is the reciprocal of a PV (Pisot-Vijayarghavan) number (i.e., an algebraic integer whose conjugates lie inside the unit circle in the complex plane; an example of a PV number is the Golden mean (-1 + J5) /2; see Section 16), Vf3 is singular. If vp is absolutely continuous then its Hausdorff dimension is 1 and hence the Hausdorff dimension of the Sinai-Bowen-Ruelle measure f.J./3 is 2. Ledrappier and Porzio [LP] considered the case when fJ is the Golden mean (and hence vp is singular). They found an explicit formula for the Hausdorff dimension of vp which implies that it is strictly less than 1 (and hence the Hausdorff dimension of f.l/3 is strictly less than 2). Alexander and York (AY] explicitly computed the information dJmension of J-tf3 provided fJ is the reciprocal of a PV number. They also showed that if (3 is the Golden mean then the information dimension is strictly less than 2.

!

!,

In [F1], Falconer studied the so-called slanting baker's transformation of the square S given as follows if y ~ 0 if y < 0. We assume that (23.6) and (23.7) hold and also fori= 1, 2,

(23.9) This guarantees that T maps S into itself and hence possesses an attractor. Assuming that Tis "skinny", i.e., 1>. 1 1+ 1>.21 < 1, Falconer proved that for almost every (ct. c2 ) the Hausdorff dimension of the attractor for T is the solution s of the equation (23.8). In [Siml], Simon found that if I-Ll f= J-t2 and 0 < l>.1l, l>.2l < IP.1- P.2l then for every pair (ct,c2) satisfying (23.7) and (23.9)) the Hausdorff dimension of the attractor is equal to s and that the result holds true for any sufficiently small perturbation ofT in the first component (which may make T non-linear).

Dimension of Sets and Measures Invariant under Hyperbolic Systems

247

24. Multifractal Analysis of Equilibrium Measures on Basic Sets of Axiom A Diffeomorphisms

We undertake the complete multifractal analysis of Gibbs measures on a locally maximal hyperbolic set A of a Cl+"'-diffeomorphism assuming that f is both s- and u-conformal. We follow the approach suggested by Pesin and Weiss in [PW3]. We assume that f is topologically mixing. The general case can be reduced to this one using the Spectral Decomposition Theorem

Thermodynamic Description of the Dimension Spectrum Let rp be a Holder continuous function on A and v = vcp an equilibrium measure for rp. Define the normalized function 1/J on A by log¢ = cp- PA(cp). Clearly, P(log1/J) = 0 and vis an equilibrium measure for 1/J. Consider the one-parameter family of functions cp~•), q E ( -oo, oo) on A

where r<•l(q) is chosen such that PA(cp~")) = 0. Furthermore, consider the oneparameter family of functions cp~"l, q E ( -oo, oo) on A rp~") (x) = _y(u) (q) log la("l(x)l + q log1f;(x),

where y(ul(q) is chosen such that PA(rp~"))

T(q) = r<•l(q)

= 0.

Finally, let us set

+ r<"l(q)

The functions r<•>(q) and r, and (i(u) be the pull back to EA by the coding map of the functions cp, 1/J, a<•>, and a(u) respectively. Note that log {1 = rp- IT,_. (rp). Therefore, the functions r<•l(q) and r
= r<•>(q) log la<•l(w)l + qlog{l(w), rp~">(w) = -T<">(q) log lii(u) (w)l + q log{l(w), where rp~•) and rp~u) are the pull back toEA by the coding map of the functions rp~•l and cp~u) respectively. We have that PEA(rp~•l) = PEA(rp~")) =

o.

There is also a description of the functions r<•l(q) and r on E:;t by setting

Iog~<·>cw-)

=-lim log Jl(Ci_n ,,) , n-+oo

Jl( C;_n

to)

Chapter 7

248

where wsetting

= (... Ln ... L1io) E

E:4. We also define the function ~(u) onE~ by

log ¢<">(w+)

=-

lim log p.(C;, in), p.{ C;o in)

n-+oo

where w+ = (ioil ... in ... ) E E1. According to Appendix II the above limits exist and the functions log~<•> and logr,b


(see Appendix V). Let p.(•) and p.(u) be the Gibbs measures corresponding to the functions ~(•) and ~(u) respectively. As we saw in Appendix II, these measures are the stable and unstable parts of the Gibbs measure p. corresponding to the function ({! (recall that II = x.p.). In addition,

r

lEA

log-ifi<•>(w-) dtt<•>(w-)

=

r

JE~

log-ifi(w+) dtt(ul(w+) =

= h,.<•) (uiE::l) = hi'
r

lr.A

log 1/i(w) dtt(w)

hi'(u) = hv{f).

Arguing as above one can also find the function a<••> defined on E::i and the function a is strictly cohomologous to the projection of the function a<•> to E::t and loga to E~. Therefore, the function

is strictly cohomologous to the projection of the function rp~•) (w) to E::i and the function r,O~""l(w+) = -T(ul(q) log lii(uul(w+)l + q log~(ul(w+) is strictly cohomologous to the projection of the function r,O~"\w) to E1. This observation allows us to use results in Section 21 to study the function T(q). It follows from Proposition 21.1 that the function T(q) is real analytic. In order to compute its first derivative we define for each q E IR,

a(q) = a<•>(q)

+ a<•>(q),

where 11-~•l and 11-~u) are Gibbs measures corresponding to Holder continuous functions ,;;( .. ) on Eon E+A· .,-q A and ,;;(uu) ..-q It follows from Propositions 21 2 and 21.3 that a(q) = -T'(q) and the function T(q) is convex, i.e., T"(q) ;:::: 0. Moreover, one can show that the function

Dimension of Sets and Measures Invariant under Hyperbolic Systems

249

T(q) is strictly convex if and only if the measure 11jR(x) is not equivalent to the measure m<•>(x) x m<">(x) for every x EM. We recall that, according to Section 22, m<">(x) and m<•>(x)) denote the conditional measures on A<">(x)

and A<•l(x) generated by the measures ,._(u) and ,..<•l on A respectively. The latter are equilibrium measures corresponding to the HOlder continuous functions t
(x)l) = 0 and Pt.(-tlogia<•>(x)l) = 0 respectively. We leave it as an easy exercise to the reader to show that if the measure v!R(x) is equivalent to the measure m<•l(x) x m<">(x) for some x EM then it holds for every x E M (note that f is topologically mixing). The equivalence also means that 11 is the measure of full dimension (see Condition (22.10))

Diametrical Regularity of Equilibrium Measures Our approach to the multifractal analysis of equilibrium measures is based upon the diametrically regular property of these measures (see Condition (8.15)). Proposition 24.1. Let rp be a Holder continuous function on A. Then the equilibnum measure 11 = 11"' for rp is diametncally regular and so are the measure v<•l on A(•) and the measure 11(u) on A(u). Proof. The fact that the measures 11<•l and 11(u) are diametrically regular uses Proposition 22.1 and is a shght modification of arguments in the proof of Proposition 21.4 The diametrical regularity of the measure 11 follows from Proposition 22.2.

•

Multifractal Analysis of Equilibrium Measures We now establish a complete multifracta.l analysis of equilibrium measures (corresponding to Holder continuous functions) supported on locally maximal conformal hyperbolic sets For a ~ 0 consider the sets A{)( defined by

A{)(= {x E A: d,(x) =a} (compare with (18.5)) and the /v(a)-spectrum for dimensions f.,(a)

= dimH A{)(.

Theorem 24.1. [PW3] (1) The pointw'tBe dimension dv(x) extSts for 11-almost every x E A and

dv(x) =

JE log?,Z(w)dp.(w) JEA log?,Z(w) dp.(w) JEA tog ia
= hv (f)

-

c~ >.~ ) '

(24.1)

where h., (f) is the measure-theoretic entropy off and >.;t, >.;;- are poS'Itive and negative values of the Lyapunov exponent of 11 (see definition of the Lyapunov exponent in Section 26}.

250

Chapter 7

(2) The function j,(a) is defined on the interval [a 1, a 2 ], which is the range of

the function a(q) (i.e., 0:::; a 1 :::; a2 < oo, a 1 = a(oo) and a 2 = a(-oo)); this function is real analytic and f.,(a(q)) = T(q)+qcx(q) {see Figure 17b}. (3) For any q E lR we have that vq(Ao(q) n R(x)) = 1 for every x E A and lb-. (x) =d.,. (x) = T(q) + qa(q) for vq-almost every x E Aa(q) n R(x). (4) Ifv is not the measure of full dimension then the functions j..,(a) andT(q) are stnctly convex and form a Legendre transform pair {see Appendl.X V) (5) The v-measure of any open ball centered at points in A is positive and for any q E lR we have . loginfg, EBeg v(B) 9 T (q) = - hm I r , r->0 ogr

where the infimum is taken over all finite covers 9r of A by open balls of radius r. In particular, for every q > 1, T(q) = HPq(v) = HPq(v) =11q(v)

1-q

= Rq(v).

Proof. First we note that T(O)

= r<•l(o) + r
Repeating arguments in the proof of Theorem 21.1 we obtain that (see (21.6))

T

Cl s

• l og "' L...ciJ>eu'•> It C•l(c<1l)q (q) = - lrm 1r ' r-+0 ogr

{24.2)

' . 1og " L...cweulu) It (u)(C(j))q ( ) (q) = -lim T" r , r-+0 logr

where

.u}.•l

and

T(l) = r<•>(1)

ti}."l

are Moran covers of ~A and

+ r<"l(1) = o.

E1 respectively.

In particular,

Fix an element of the Markov partition R(x), where x E intR(x). Note that x(C;0 ). Given 0 < r < 1 and w L1ioi1 ... ), choose n- = n-(w,r) and n+ = n+(w,r) such that

R(x)

=

= (...

0

0

II

Ia<••) (uk(w-)) I > r,

II

Ia<••) (uk(w-)) I :::; r,

k=l-nn+-1

II

{24.3)

n+-1

la(uu)

(uk(w+))

1-1 > r,

k=O

Fix a number a ;::: 0 and let the limit

II

lii(uu)

(ak(w+))

~-1

:::;

r.

k=O

A0

be the set of points w = ( ... i_ 1 i 0 i 1 ••. ) for which

(24.4)

Dimension of Sets and Measures Invariant under Hyperbolic Systems

251

exists and ill equal to a (the "symbolic" level set). Define the "symbolic" spectruro (24.5) Given x E A, we consider the measures v~•) = x.JL~•>rc;: on A"(x) and v~u) X·ft~")IC;~ on A(u)(x) and let Vq = v~•l x v~u) be the measure on R(x).

=

Lemma 1. For every q E lR we have (1) Vq(X(Aa(q))

n R(x))

= 1;

(2) dv.(Y) == T(q) + qa(q) for Vq-almost ally E x(Aa(qJ) n R(x); (3) dv. (y) ~ T(q) + qa(q) for ally E x(Aa(q)) n R(x); (4) dimH x(Ao(qJ) n R(x) = T(q) + qa(q).

Proof of the lemma. Given w = (... L 1ioit ... ) E EA, consider the functions w- 1-t Jog fa<••>(w-)1 and w- 1-t log -J;<•>(w-) on EA and the functions w+ 1-t logfii(uul(w+)[ and w+ 1-t log.J;Cul(w+) on E1. Since the measures f£~8 ) and fJ.~u) are ergodic the Birkhoff ergodic theorem yields that for fJ.~•)_almost every w E EA and fJ.~u)_aJmost every wEE!, 1-n-

L

lim

log~<•l(ak(w-))

k=~

= a<•>(q),

r---J-01-n

I:

log fa<••l(ak(w-))1

k=O

(24.6)

n+-1

L

log-J;Cul(uk(w+))

lim k=O r-+0 n+-1

I:

= -a<">(q). log [a
k=O

This implies the first statement. By (24.6) for any e: > 0 and every wE Ao(q) there exists r(w) such that for any r ~ r(w), 1-n-

L

a<•>(q)- e ~ ~=~ 1

I:

log.J;<•>(uk(w-)) log fa<••>(uk(w- ))I

:;

a<•>(q) + e:,

k=O n+-t

I:

(24.7) log-J;
o(q) +e. I: log ja(uu)(uk(w+)))l k=O

Given l > 0, denote by Qe = {w E Aa.(q) . r(w) :S i} It is easy to see that Qe C Q£+1 and Aa(q) = U~ 1 Qe. Thus, there exists lo > 0 such that fJq(Qe) > 0 if l ~ lo. Let us choose f. ~ lo.

252

Chapter 7

Since J-4~8 ) and 1-'~u) are Gibbs measures we obtain by (A2.20} (see Appendix II) that for every w- = ( ... L 1 i 0 ) E E:;t and w+ = (ioi 1 ••• ) E E~,

C1 -<

,.,~·>cc,_,.+,

;,)

n iii<••l(ak(w- )})1-T<•>(q)yj;(•)(ak(w- )}q

1-n

:5

c 2•

k=O

~-'&u>cc'&Q .• J

c <

n lii(uu)(uk(w+ )) )1-T<•>(q)yj;(u)(uk(w+) )q

1 - n+-1

(24.8}


2•

k=O

where C1 and C2 are positive constants. Repeating argwnents in the proof of Lemma 2 of Theorem 21.1 one can show that for any l > l 0 ,

4.,<•> (z) ~ r<•l(q) + q(a<•l(q) -e) if z E A<•>(x) n x(Qt),

• •

4,<•>(z) ~ r
dv. (y)

~

T(q)

+ q(a(q)- e).

Since sets Qt are nested and exhaust the set Q we obtain that g,_ (y) ~ T(q) + 9 q(a(q) -e) for vq-almost every y E Aa(q)· Since e is arbitrary this implies that 4.-. (y) ~ T(q) + qa(q) for vq-almost every y E x(Aa(q))· The second statement of the lemma follows. By Theorem 7.1 we also obtain that dimn x(Aa(qJ) ~ T(q) + qa(q) Fix 0 < r < 1. It follows from (24.3} that R._,.- •,.+ C B(y,Kr}, where y = x(w) and K > 0 is a constant independent of y and r. By virtue of (24.7} and (24.8} for all w E Qt,

llq(B(y, Kr}} ~ llq(R,_,.- . 'n+}

k=O

It follows that for ally E x(Qt),

d,_9 (y)

= lim log v~(B(y, r)) :5 T(q) r-+0 ogr

+ q(a(q) +e).

Dimension of Sets and Measures Invariant under Hyperbolic Systems

253

Since e is arbitrary this implies that d119 (y) ::; T(q)+qa(q) for every y E x(Aor(q))n R(x). The third statement of the lemma follows Moreover, by Theorem 7.2 we obtain that dimH(x(Aa(q)) n R(x)) ::::; T(q) + qa(q). This completes the proof of • the lemma. We also need the following statement. Lemma 2. A"

= x(Aa)·

Proof of the lemma. Notice that the measures v<•l and v
r-->0

logr

exists simultaneously with the limit (24.4). This completes the proof of the • lemma. Since T(1) = 0 we have that I'IR(x) = 111· The first statement of the theorem now follows from Lemmas 1 and 2 and the following obvious observation:

Note that by Lemma 2, Aa(q) = x(Aa(q))· Hence, dimH Aa(q) = T(q) + qa(q). This implies Statements 2, 3, and 4. Since the measure v is diametrically regular and has local product structure one can use (24.2) to prove Statement 5 by • repeating arguments in the proof of the last statement of Theorem 21.1. Remarks. (1) Assume that v is the measure of full dimension One can show (see Remark 1 in Section 21) that T(q) = (1 - q) dimy A (thus, T(q) is a linear fll,tlction) and that fv(dimH A) = dimy A and fv(a) = 0 for all a f dimy A. (2) (see Remark 2 in Section 21). Assume that v f m. Then for any e > 0 there e:nsts a distinct {singular) equilibrium measure I' on A such that Idimy I' - dimH vi ::::; e One can also show that: the Hausdorff dimension of an equilibrium measure on a A (corresponding to a Holder continuous potential} depends continuously on the potential. (3) The function fv(a) has properties described in Remarks 3 and 4 in Section 21. ( 4) We have the following multifractal decomposition of the hyperbolic set A associated with the pointwise dimension of an equilibrium measure on A corresponding to a Holder continuous function (compare to the case of conformal repellers of smooth expanding maps; see (21.18)): (24.9)

Chapter 7

254

where Ao. is the set of points for which the pointwise dimension takes on the value a and the irregular part A is the set of points with no pointwise dimension. It follows from results in [BS] that A I= 0; moreover, it is everywhere dense in A and dimH A = dimy A. We also have that each set Aa is everywhere dense in A.

Pointwise Dimension of Measures on Hyperbolic Sets Let 11 be an equilibrium measure corresponding to a Holder continuous function on A. According to Statement 1 of Theorem 24.1 for 11-ahnost every x E A, 1

d.,(x) =h.,(!) ( + -

>..,

~) . )."

(24.10)

It follows from a result by Young (Y2] that this fonnula holds for an arbitrary ergodic measure 11 on A which is not necessarily an equilibrium measure (see Proposition 26.3; it is a particular case of Theorem 26.1). We present a straightforward proof of this result based upon the approach which was used in the proof of Theorem 21.3

Theorem 24.2. Let 11 be a Borel ergodic measure on A. Then for v-almost every x E A the equality (24.10) holds. Proof. Set d., = h., (f) (~- >.~). Let R = {R1 , ... , 14,} be a Markov partition for f, (EA, a-) the corresponding symbolic model, and x the corresponding codtng map from EA to A (see Section 20). Also, let J1. be the pullback of 11 by

X·

Fix c > 0. It follows from the Shannon-McMillan-Breiman theorem that for J,£-almost every wE EA one can find N1(w) > 0 such that for any n, m ~ N1(w), J-£(C;_,.. ;,. (w)) :S e:xp( -(h- c)(n + m)),

where C;_m .;,. (w) lS the cylinder set containing wand h = hl'(a-) is the measure-theoretic entropy. It follows from the Birkhoff ergodic theorem, applied to the functions logla<"l(x)l and logla< 8 l(x)l, that for 11-ahnost every x EM there exists N 2 (x) such that for any n, m ~ N2(x),

For wE EA we set N2(w) = N2(x(w)). Given i > 0, denote by Qt = {w E EA : N1(w) :S f and N2(w) :=:; i}. It is easy to see that the sets Qe are nested and exhaust EA. Thus, there exists £0 > 0 such that Jl.(Q£) > 0 if i ~ io. Let us choose i ~ f 0 .

255

Dimension of Sets and Measures Invariant under Hyperbolic Systems

Given 0 < r < 1, consider a Moran cover U,.,q 1 of the set x( Qt)· It consists of sets R~il, j = 1, ... , Nr,t for which there exist points Xj E A such that R~il = ~-m(z 3 ) .. in(2:j) • Consider the open Euclidean ball B(x, r) of radius r centered at a point x. Let N(x, r, l) denote the number of sets R~j) that have non-empty intersection with B(x, r). We have that N(x, r, £) ::; M, where M is a Moran multiplicity factor (which is independent of r and 1). It now follows that N(x,r,l)

v(B(x, r) n x(Qt)) ::;

L

v(Rf)) ::; M exp( -(h- e:)(n(xj)

+ m(xj))).

j<;l

Since v(x(Qt)) > 0 by the Borel Density Lemma (see Appendix V) for v-almost every x E x(Qt) there exists a number ro = r 0 (x) such that for every 0 < r :::; r 0 we have v(B(x,r))::; 2v(B(x,r)nx(Qt)). By virtue of (22 8) this implies that for any

e> fo and v-almost every X E x(Qt),

_ . log v(B(x, r)) log v(B(x, r) n x(Qt)) > d _ d " (x ) - 11m 1rm 2e. > r--+0 logr - r-+0 logr - "

-

Since sets Qt are nested and exhaust the set Q we obtain that 4,(x) ;:::: d,- 2e for v-almost every x E A. Since e can be arbitrarily small this proves that f!v(x) 2: d.,. We now prove the opposite inequality. Fix 0 < r < I. By (22.8) it follows that R._c,.cz)+•) . '·<•)+~ c B(x, C1r), where C1 > 0 is a constant. This implies that

v(B(x1, C1r)) 2: v(R;_<.,.<•)+I)· i.cz)+t) 2: C2 exp( -(h + e)(n(x) + m(x))), where C2 > 0 is a constant. By virtue of (22.8) we obtain for all x E A, -d ( ) _ li logv(B(x,r)) < d v x -....ill _ v 1ogr •-+0

+ 2e.

Since e can be arbitrarily small this proves that f!v(x) :::; d,.

•

Since the measure-theoretic entropy is a semi-continuous function we obtain as an immediate corollary of Theorem 22.2 and (24.10) that: the Hausdorff dimension of a Borel ergodic measure v on A is a semi-continuous function of v. Let v be again an equilibrium measure corresponding to a Holder continuous function on A and G~' the set of all forward generic points of v (i.e., points for which the Birkhoff ergodic theorem holds for any continuous function on X). It follows from Lemma 1 in the proof of Theorem 24.1 (applied to the measure v 1 = v) that for every x E G~', dv(x) =

h~~).

Chapter 7

256 In view of Theorems 7.1 and 7.2 this implies that h., (f)

>.;

(•) ( )) ( . = dJmH G., n wloc X '

h,(f) .>.;!'

)) loc X • = d'lmH (Gv n W(u)(

We extended this result to any Borel ergoruc measure on A which is not necessarily an equilibrium measure.

Theorem 24.3. Let v be a Borel ergodic measure on A. Then for v-almost every x E A, dim 8 (G, n W ~J(x)) = dims(G, n W ~·~(x)),

1 1 h~~) = dimH(Gv n W1~~(x)) = dim8 (G,.. n W1~-:,>(x)) =dims(G., n W1~"}(x)). h~V) =dimH(G, n W1~(x)) =

• Proof readily repeats arguments in the proof of Theorem 21.3. Manning [Mal] proved the part of Theorem 24.2 involving the Hausdorff dimension of the sets G., n wi;2(x) and G,.. n W1~"}(x)

Information Dimension We study the information dimension of the measure v. As in the case of conformal repellers one can show that

j,(a(l)) = a(l)

= -T'(l) = I(v) = J(v) = dimH v,

where l(v) and I(v) are the lower and upper information dimensions of v (see Section 18). We note that Statement 6 of Theorem 24.1 allows us to extend the notion of the Hentschel-Procaccia spectrum and Renyi spectrum for dimensions for any q ;:j:. 1 and the above argument defines these spectra for q = 1.

Dimension Spectrum for Lyapunov Exponents Consider the following multifractal decomposition of the set A associated with positive values of the Lyapunov exponent >.+(x) at points x E A (see Section 26)· (24.11) where

£+ = {x

E

A: the limit in (26.1) does not exist for any v E E("l(x)}

is the irregular part and

Lt =

{x

E

A: .>.+(x)

=

fi}.

257

Dimension of Sets and Measures Invariant under Hyperboltc Systems

If vis an ergodic measure for f we obtain that >.+(x) = >.~u) for v-almost every x E A. Thus, the set L +(•l f 0. Moreover, if v is an equilibrium measure >-v corresponding to a Holder continuous function this set is everywhere dense (since in this case the support of v is the set A). We note that if a set Lp is not empty then it supports an ergodic measure vp for which >.,~ = {:J (indeed, for every x E Lp the sequence of measures ~ 2:;;:~ 8r•(x) has an accumulation measure whose ergodic components satisfy the above property). As in the case of conformal repellers there are several fundamental questions related to the above multrl'ractal decomposition, for example: (1) Are there points x for which the limit in (26.1) does not exist for any

v E E(ul(x), i.e., i+ f 0? (2) How large is the range of values of >.+(x)? (3) Is there any number j3 for which any ergodic measure v Wtth v(L13) > 0 is not an equilibnum measure? We introduce the dimension spectrum for (positive) Lyapunov exponents off by e+(/3) = dimHLt. In [PW3], Pesin and Weiss established the relation between the Lyapunov dimension spectrum and the f,.,.. (n)-spectrum, where Vmax is the measure of maximal entropy. Notice that the measure of maximal entropy is a unique equilibrium measure corresponding to the function tp = 0 and hence '1/J = constant = exp(- hA (f)), where h11. (f) is the topological entropy of f on A. Therefore, for every x E Lt

d

{u) Vmax

(x) = hA(f) {3

(recall that vf::b_ denotes the conditional measure induced by stable manifolds). This implies the following result.

llma.x

on local un-

Theorem 24.4. [PW3]

(1) If vf::b.IR(x) is not equivalent to the measure m
£+({3) = f <•l (hA(f)) {3 "m•;x is a real analytic stnctly convex function on an interval [/31> {:12) containing the point {3 = hA(/)/ dimH(A n W1~~(x)). (2) If vf::b.IR(x) is equivalent to mCu) IR(x) for some x E A then the Lyapunov spectrum ts a delta function, i.e.,

t+(/3) = { dimH A, for (:3 = hA(f)/ dimH(A n w,~"j(x)) for /3 f hA(f)/ dimH(A n W1~"j(x)). 0,

Chapter 7

258

As immediate consequences of this result we obtain that if the measure v~k:IR(x) is not equivalent to the measurem<"l(x) forsomex E A then the mnge of the function .X+ (x) contains an open interval, and hence, the Lyapunov exponent attams uncountably many distinct values. On the contrary, If the Lyapunov exponent >..+(x) attains only countably many values then v~kiR(x) is equivalent to m..+(x) Ul an interval [,81 ,/:12] and for any ,8 outside is empty (i.e., the spectrum is complete). this interval the set (3) For any ,8 E [,81 , ,82 ) there exists an equilibrium measure v corresponding to a Holder continuous function for which v(L13) = 1. Similar statements hold true for dimension spectrum for (negative) Lyapunov exponents off corresponding to negative values of the Lyapunov exponent >..-(x) at points x EA.

Lt

Appendix IV

A General Concept of Multifractal Spectra; Multifractal Rigidity

Multifractal Spectra In Sections 19, 21, and 24 we observed two dimension spectra- the dimension spectrum for pointwise dimensions and the dimension spectrum for Lyapunov exponents. The first one captures information about various dimensions associated with the dynamics (including the Hausdorff dimension, correlation dimension, and information dimension of invariant measures) while the second one yields integrated information on the instability of trajectories. These spectra are examples of so-called multifractal spectra which were introduced by Barreira, Pesin, and Schmeling in [BPS2] in an attempt to obtain a refined quantitative description of various multifractal structures generated by dynamical systems. The formal description follows. Let X be a set, Y c X a subset, and g: Y -7 [-oo, +oo] a function. The level sets

Kg= {x EX: g(x) =a},

-oo

~a~

+oo

are disjoint and produce a multifractal decomposition of X,

X=

u

Kgu.X,

where the set X = X \ Y is called the irregular part. If g is a smooth function on a Euclidean space then the non-empty sets Kg are smooth hypersurfaces except for some critical values a. We will be interested in the case when g is not even a continuous (but Borel) function on a metric space so that Kg may have a very complicated topological structure. Now, let G be a set function, i.e., a real function that is defined on subsets of X. Assume that G(Z1) ~ G(Z2) if Z1 C Z2. We introduce the multifractal spectrum specified by the pair of functions (g, G) (or simply the (g, G)multifractal spectrum) as the function :F· (-oo, +oo]-7IR defined by

F(a)

= G(K~)

The function g generates a special structure on X, called the multifractal structure, and the function :F captures inlportant information about this structure

259

Appendix IV

260

Given a, let

Va

be a probability measure on

Kg.

If

F(a) = inf{G(Z): Z c Kg, va(Z) = 1} then we call v"' a (g, G)-full measure. Constructing a one-parameter family of (g, G)-full probability measures Va seems an effective way of studying multifractal decompositions (see examples of Va-measures below). We consider the case when X is a complete separable metric space. Let f: X --*X be a continuous map. There are two natural set functions on X. The first one is generated by the metric structure on X: Gv(Z) :=: dimH Z,

and the second one is generated by the dynamics on X. GE(Z) :=: hz(f}

Multifractal spectrum generated by the function G v is called the dimension spectrum while multifractal spectrUlll generated by the function GE is called the entropy spectrum. There are also three natural ways to choose the function g. (1) Let J.l be a Borel finite measure on X. Consider the subset Y consisting of all points x E X for which the limit

c

X

d (x) :=:lim log~t(B(x,r)) IJ. r-+0 logr exists (i.e., the lower and upper pointwise dimensions coincide). We set g(x) = d,.(x) ~f gv(x),

X E

Y

This leads to two multifractal spectra Vv = vif> and VE = v~> specified respectively by the pairs of functions (gv,Gv) and (gv,GE)· We call them the multifractal spectra for (pointwise) dimensions (note that Vv is just the /p(a)-spectrum studied before). We stress that these spectra do not depend on the map f. (2) Assume that the measure p, is invariant with respect to f. Consider a finite measurable partition {of X. For every n > 0, we write {n = { V f- 1{ v · · V /-ne, and denote by qn (x) the element of the partition {n that contains the point x. Consider the set Y = Ye c X consisting of all points x E X for which the limit

exists. We call h,.(f,{,x) the local entropy off at the point x (with respect to e). Clearly, y is !-invariant and h,.(f,€,f(x)) = h,.(f,e,x) for every X E Y.

A General Concept of Multifractal Spectra

261

By the Shannon-McMillan -Breiman theorem, tt(Y) = 1. In addition, if generating partition and IL is ergodic then

eis a

for tt-almost all x E X. Set g(x) = h11 (f,e,x) ~t 9E(x),

x E Y.

e.

We emphasize that 9E may depend on We obtain two multifractal spectra t:v = t:ifl and t:E = t:~) specified respectively by the pairs of functions (gE, Gv) and (gE, GE)· These spectra are called multifractal spectra for (local) entropies (see similar approach and discussion in [V2]). These spectra provide integrated information on the deviation of local entropy in the ShannonMcMillan-Breiman theorem from its mean value that is the entropy of the map. (3) Let X be a differentiable manifold and f: X --+ X a C 1-map. Consider the subset Y C X consisting of all points x E X for which the limit .A(x)

= n-++oo lim .!.tog lld,J"II n

exists. The function .>..(x) is measurable and invariant under f. By Kingman's sub-additive ergodic theorem, if J1. is an /-invariant Borel probability measure, then J.t(Y) = 1. We set g(x) = .A(x) ~ gL(x),

xEY

This produces two multifractal spectra Cv and CE specified respectively by the pairs of functions (g L, GD) and (gL, G E). These spectra are called multifractal spectra for Lyapunov exponents. It is worth emphasizing that these spectra do,.not depend on the measure IL· If J1. is ergodic the function .>..(x) is constant almost everywhere. Let .>. 11 denote its value. Multifractal spectra for Lyapunov exponents provide integrated information on the deviation of Lyapunov exponent in Kingman's sub-additive ergodic theorem from its mean value Ap.. We consider some examples. First, we analyze four multifractal spectra Vv, VE, C:v, and t:E for a subshift of finite type (E~, u) Denote by 'P the class of finite partitions of E~ into disjoint cylinder sets (not necessarily all at the same level). Clearly, each E 'P is a generating partition. We use it to define the spectra for entropies cv and t:E.

e

The following theorem establishes the relations between the multifractal spectra for dimensions and entropies.

Theorem A4.1. [BPS2] For every a E IR, we have t:E(a)

= t:v(a) log.B = VE(Oi/log .B) = 'Dv(ajlog .B) log .B

(A4.1)

262

Appendix IV

{recall that {3 > 1 is the C()ejfic1ent in the dp metric on E1; see (A2.18} in Appendz:z: II). In particular, the common value is independent of the partition { E P. Moreover, the multifractal decompositions of the spectra CE, &v, DE, and Dv coincide, i.e., the families of level sets /or these four spectra are equal up to the parametenzations given by (A-/.1).

Proof. Notice that there exist constants C 1 > 0 and C2 > 0 such that Cta-n ::; diamf.n(x) ::; C2a-n

e

for every E P, X E E1, and n ~ 1. If the pointwise dimension dt<(x) or the local entropy hl'(f,t;,x) exists for some x E E~ then h,..(f,t;, x) = lim _.!_logp({n(x)) =log a lim n-+oo n n-+oo

l~~p({n~x~) !am{n X

=log a· dl'(x).

This shows that d,.. (x) exists at a point x if and only if h,.. (f, t;, x) exists. One can also see that hz(f) = dimH Z ·log a for any subset Z c The desired result follows. •

E!

One can now obtain the complete description of all four spectra using Theorem 19.1 (note that the map a is obviously a continuous weakly-conformal expanding map). We observe that these spectra are complete (see Section 21). We describe four multifractal spectra Dv, CE, Cv, and LE for equilibrium measures supported on conformal repellers of smooth expanding maps. Let J be a repeller of a conformal Ol+"'-expanding map f. Consider a Markov partition { of J It is clear that { is a generating partition. The same holds true for any partition of J by rectangles obtained from {(not necessarily all at the same level) and corresponding to disjoint cylinder sets in We denote the class of such partitions by Pt. It is easy to check that x; 1 € E P for every partition t; E P,. Let


E!.

'{JD,q = -Tv(q) log Ia I+ qlog'f/1,

'PE,p = -Ts(p)

+ plog'f/1,

(A4.2)

where the numbers Tv(q) and TE(p) are chosen such that

P(t.pD,q)

= P('PE,p) = 0.

Clearly, TE(p) = P(plog'f/1). As in Section 21 one can show that the functions Tv(q) and TE(P) are real analytic decreasing and convex. Set

The complete description of the spectra Dv and £v is given by Theorems 21.1 and 21 4 and the corresponding multifractal decompositions are (21.18) and (21.25) respectively. Since the coding map preserves the entropy the complete description of the ~pectrum eE follows immediately from Theorem A4.1. More precisely, the following statement holds.

263

A General Concept of Multifractal Spectra

Theorem A4.2. (1) There e:ruts a set S c X with p,(S) = 1 such that for every partition E 'PJ and every X E S, the local entropy of fl. at X exists, does not depend on X and and

e

e,

gs(x) = h,.(f,e,x) = -llog'ifJdft.

(2) The function TE is real analytic, and satisfies T[;(p) :s; 0 and T~(p) ;?: 0 for every p E R. We have TE(O) = hJ(f) and TE(l) = 0. (3) The domain of the function a >-t &s(o:) is a closed interval in [0, +oo) and coincides with the range of the function o:s(p). For every p E IR, we have &E(o:E(P)) = Ts(p)

+ po:E(p).

(4) If fl. is not the measure of maximal entropy then &E and TE are analytic strictly convex functions, and hence (&s, TE) form a Legendre transfer pair with respect to the vanables a., q. (5) If p, is the measure of max~mal entropy then &E is the delta function

( )-{ho

eE a -

ifa.=h if a. =1= h ·

We now give a complete description of the spectrum CE.

Theorem A4.3. [BPS2] For every a. E R we have

where m is the measure of full dimension {see Theorem 20.1}

Proof. Since m is the Gibbs measure corresponding to -slog ial (where s = - dimn J), by Proposition 20.2, for every x = x( i 0 i 1 .•. ) E J we have

Therefore, for every X E J n KgL and hm(f,e,x)

and hence

X

= n-+oo lim _.!.logm(R. n

0

eE 'P,, )

;

=slim _.!.logdiamR;,.

n

E K:f:. This implies that .CE(o:)

n-+oo

n

= e1m) (a.dimn J).

i

=so:

•

Appendix IV

264

Let us notice that the Gibbs measures corresponding to the functions 'PD,q 'PE,p (see (A4.2)) are (g, G)-full measures of the corresponding spectra. It is an open problem to obtain a complete description of the spectra 'DE and &v for equilibrium measures on conformal repellers of smooth expanding maps. and

Irregular Parts of Multifractal Decompositions Consider the set

I= {x E J: lim

n-+oo

.!.s .. g(x) n

does not exist for some g E G(J)}

= J\

U

G,.

I'E!m(J)

where S.. g = :E~:~ go fk (recall that G,. is the set of all forward generic points of the measure p,; see (A2.16); !m(J) is the bet of all Borel ergodic measures on J). This set is called metrically irregular. Note that I has zero measure with respect to any /-invariant measure. In [BS], Barreira and Schmeling demonstrated the surprising phenomenon that the metrically irregular set is observable, i.e., it has full Hausdorff dimension and carries full topological entropy. More precisely, they proved the following result. Theorem A4.4. Let J be a repeller for a Gl+" -conformal expanding map Then dimHI = dimH J and hi(!)= hJ(f).

f.

In the numerical study of dynamical systems one usually collects the data by observing typiCtLl trajectories, i e , trajectories that correspond to points in G" for some invariant measure p,. This is based on the common belief that a computer always picks points "randomly" and thus, reproduce8 only ·'typical" orbits. Theorem A4.4 demonstrates that some fundamental characteristics of dynamical systems can be in fact computed while dealing with non-typical trajectories. Although such trajectories cannot be observed by random scannmg of the phase space they can be obtained by "mixing up" two typical trajectories corresponding to two distinct invariant measures (in the case of the full shift on two symbols the procedure of "mixing up" two typical trajectories is explained in the proof of Proposition A2.1; for general subshifts a more sophisticated procedure is described in [BS]). Metrically irregular sets arc closely related to irregular parts of multifractal decompositions generated by the dimension spectrum for pointwise dimensions (i.e., the 8et J in (2118)) and the dimension spectrum for Lyapunov exponents (i.e., the set i in (21.25)). Barreira and Schmeling showed that these irregular parts are observable, i.e., they have full Hausdorff dimension and carry full topological entropy Theorem A4.5. Let J be a repeller for a Gl+"-conformal expanding map f. (1) Let p, be an equilibrium measure corresponding to a Holder continuous function. Assume that p, is not the measure of full dimenswn. Then dimH J = dimH J and hJCf) = hJ(f). (2) If the measure of full dimension and the measure of maximal entropy are distinct then dimH i = dimH J and hL (f) = hJ (f).

A General Concept of Multifractal Spectra

265

The fact that the set J has positive Hausdorff dimension was first observed by Shereshevsky (Sh]. The proofs of Theorems A4.4 and A4.5 exploit the notion of cp-dimension of invariant measures (see Appendix III). We sketch these proofs. Given a collection of sequences of continuous functions Q(i) =

{g~i): J --+ R+ }nEN

fori = 1, .. , m, we define the corresponding metrically irregular set

I(G< 1l, ... , a<ml) =

{x

E

J: lim g~l(x) < lim n-+oo

n~oo

g~l(x)

for i = 1, ... ,

m}.

We say that a collection of measures p.(i), i = 1, . , k, k ::; 2m distinguishes the collection {G< 1l, ... , a<ml} if for every 1 ::; i ::; m, there exist distinct integers h = it(i), i2 = h(i) E [1, k] and numbers =/=a~!) surh that

a;:>

lim g
n-+oo n

]1

p.3·1 -almost

every x,

lim g
n--+oo n

•

The main result in [BS] claims the following.

Lemma. (1) Let 'if;;, i = 1, ... , m be Holder continuous functions on J. Assume that each '1/Ji is cohomologous neither to the function t = -slog lal (where s = dimH J) nor to the junction t = const. Then for any e > 0 there exists a collection of measures p.(i), i = 1, ... , 2m which satisfies the following conditions: a) it distinguishes the collection of sequences of functions

G(i)-

{

9n(i)- -Sn"I/Ji}

8nt

, 1. -- 1,

.. ,

m,.

nEN

b) where cp is a stnctly positive Holder continuous function and BS'f'(v) (respectively, BS'f'(J)) is the Barreiro-Schmeling dimension of the measure v (respectively, of the repeller J ); see Append1X III. (2) Let p.(i), i = 1, .. . , k be a collection of measures which dtStinguishes a collection of sequences of Junctions {G< 1l, ... , G(mJ}, k ::; 2m; then for every Holder continuous function cp on J we have

Appendix IV

266

We proceed with the proof of Theorem A4.4. Fix e > 0. Since Holder continuous functions are dense in the space of continuous functions we obtain that the set I consists of all points x E J for which there exists a Holder continuous function 'f/1 satisfying

Therefore, I(lll) C I, where Ill= {Sn'f/l}nEN· One can choose a Holder continuous function 'f/1 which is not cohomologous either to the function t = -slog Ia I or to the function t = const. Applying Lemma with t = 1,



=log Ia!, m = 1, and 'f/1 1 = 'f/1 we conclude

Since e 1S arbitrary this yields the statement about the Hausdorff dimension and concludes the proof of Theorem A4.4. We leave the proof of Theorem A4.5 to the reader. Hint: to prove the first statement choose the function 'f/1 such that log 'f/1 = ~ - PJ( ~), where ~ is the potential of the measure 11-; then apply Lemma with t = -slog Ia I, m = 1, 'f/1 1 = 'f/1 and set


Ei

a(x) =

c;

and r/>(x) =log /3;

if x E I;,

for i = 1, 2. For every p, q E R, the functions TE(p) and Tv(q) satisfy the identities and (A4.3)

One can explicitly compute the measures 11; 8 and 117i0 : they are the Bernoulli measures with probabilities e-TE(P) j3 1P, e-TE(P) f32P and c~Tv(q) (31q, c;To (q) ('12 q respectively Let f and be two one-dimensional linear Markov maps of the unit interval as above with conformal repellers J and J respectively. Let also x: r:t --+ J and Et --+ J be the corresponding coding maps. We consider two Bernoulli measures JL and Ji on Et with probabilities /31, /32 and /31, (h respectively, where /31 + /32 = jjl + fh = 1. We also consider the numbers c1, c2 and Ct, c2 wluch are the absolute values of the derivatives of the linear pieces of f and respectively Define the functions a and ¢> on J as well as the functions and '¢;on J as above Recall that an automorphism p of r:t is a homeomorphism p: Et --+ Ei which commutes with the shift map cr. The involution automorphism is defined

1

x:

a

1

268

Appendix IV

by p(i1i2 ... ) = (i\.i~ .. ), where i~ = 2 if in= 1 and i~ = 1 if in = 2 for each integer n :;:: 1. Since x and are invertible one can define a homeomorphism (} J -+ J by (} = xox- 1 . We note that (}of= jo(J on J and hence 8 is a topological conjugacy between fiJ and fJJ. If pis an automorphism of :Et, then the homeomorphism 8' = o p o x- 1 is also a topological conjugacy between fiJ and jjJ, and all topological conjugacies are of this form. We consider the spectrum Vv = v~> specified by the mell8ure p. ll8 wellll8 the spectrum i5v = vgi) specified by the measure 'ji.

x

x

Theorem A4.6. [BPS2] lfVv(a) = i5v(a) for every a and these spectra are not delta functions, then there is a homeomorphism (,· J -+ J such that: (1) (of=

fo(

on J, that is, ( is a topological conjugacy between

fiJ and

fJJ;

(2) the automorphism p of Et satisfying X o ( = p o X is either the identity or the involution automorphism; (3) a= 0 (, i.e., either cl = Ci. and C2 = C2 or cl = C2 and C2 = cl; (4) ¢ = "¢; o (, and p. = 11 o (,.

a

Proof. It is sufficient to prove that the spectrum Vv uniquely determines the numbers f3t. /32, c1, and cz up to a permutation of the indices 1 and 2. By the uniqueness of the Legendre transform the spectrum Vv uniquely determines Tv(q) for every q E JR. Therefore, it remams to show that one can determine the numbers /31 , /3z, c 1 , and c2 uniquely up to a permutation of the indices 1 and 2 from the equation (A4.3). One can verify that the numbers O:± = av(±oo} can be computed by

et± = - lim (Tv(q)fq). q->±oo

We observe that since the spectrum Vv is not a delta function, Tv(q) is not linear and is strictly convex. Hence, a+ < dimH J < tL. Therefore, raising both sides of the equation (A4.3) to the power 1/q and letting q-+ ±oo we obtain max{/31 c1 "'+, f32c2 "'+} = min {/31 c1 "'-, f32c2 "'- } = 1. We ll8sume that f3tct''+ = 1 (the case /32 c2 "'+ = 1 can be treated in a similar way; in this case, pis the involution automorphism). Since a+ < a_ we must have /32c2"'- = 1. Setting q = 0 and q = 1 in the equation (A4.3) we obtain respectively, Ct- dimH J

+ c2- dimn J = 1

Set x = c1- dimn 1 , 'Y =a+/ dimH J can easily derive the equation x"~

and

f3t

+ /32

= 1.

< 1, and b = a_f dimH J > 1. Then one

+ (1- x)b = 1.

We leave it ll8 an easy exercise to the reader to show that this equation has a unique solution x E (0, 1) which uniquely determines the numbers c 1 and c2 and hence also the numbers f3t and /32. •

269

A General Concept of Multifractal Spectra

Remarks. (1) We have shown that for a one-dimensional linear Markov map of the unit interval one can determine the four numbers /31> /32, c1, and c2 using the spectrum T>v. If instead the spectrum CE is used then only the numbers fJ1 and !32 can be recovered: one can show that if /31 ~ 132, then /31 = exp lim (TE(p)/p) p--t+oo

and

lim (TE(P)/p). 132 = exp p-+-oo

In a similar way, using one of the spectra Cv or CE one can determine only the numbers c1 and c2. (2) The multifractal rigidity for two-dimensional horseshoes is demonstrated in [BPS3].

Chapter 8

Relations between Dimension, Entropy, and Lyapunov Exponents

In the previous chapters of the book we have seen that the pointwise dimension is a useful tool in computing the Hausdorff dimension and box dimension of measures and sets. The key idea is to establish whether a measure is exact dimensional (i.e., its lower and upper pointwise dimensions coincide and are constant almost everywhere). If this is the case then by Theorem 7.1, the Hausdorff dimension and lower and upper box dimensions of the measure coincide. This also gives an effective lower bound of the Hausdorff dimension of the set which supports the measure. One can obtain an effective upper bound of the Hausdorff dimension of the set by estimating the lower pointwise dimension at every point of the set and applying Theorem 7.2. In the previous chapters of the book we described several classes of measures, invariant under dynamical systems, which are exact dimensional. We also obtained formulae for their pointwise dimension. These classes include Gibbs measures concentrated on limit sets of geometric constructions (CPW1-CPW4) (see Theorem 15.4), invariant ergodic measures supported on conformal repellcrs (see Theorem 21.3) or two-dimensional locally maximal hyperbolic sets (see Theorem 24.2). In [ER], Eckmann and Ruelle discussed dimension of hyperbolic measures (i.e , measures invariant under diffeomorphisms with non-zero Lyapunov exponents almost everywhere). This led to the problem of whether a hyperbolic ergodic measure is exact dimensional. This problem has later become known as the Eckmann-Ruelle conjecture and has been acknowledged as one of the main problems in the interface of dimension theory and dynamical systems. Its role in the dimension theory of dynamical systems is similar to the role of the Shannon-McMillan-Breiman theorem in ergodic theory. In [Y2], Young showed that hyperbolic measures invariant under surface diffeomorphisms are exact dimensional. Later Ledrappier [L] proved theis result for glc'neral Sinai-Ruelle-Bowen measures In [PYJ, Pesm and Yue extended his approach to hyperbolic measures satisfying the so-called semi-local product structure (see below; this class includes, for example, Gibbs measures on locally maximal hyperbolic sets). Barreira, Pesin, and Schmeling (BPSl] obtained the complete affirmative solution of the Eckmann-Ruelle conjecture which we present in this chapter. We also demonstrate that neither of the assumptions in the Eckmann-Ruelle conjecture can be omitted. Ledrappier and Misiurewicz (LM] constructed an ex270

Relations between Dimension, Entropy, and Lyapunov Exponents

271

ample of a smooth map of a circle preserving an ergodic measure with zero Lyapunov exponent which is not exact dimensional (see Example 25.3 below). In [PWl], Pesin and Weiss presented an example of a Holder homeomorphism whose measure of maximal entropy is not exact dimensional (see Example 25.1 below). Barreira and Schmeling [BS] showed that for "almost" any Gibbs measure on a two-dimensional hyperbolic set the set of points, where the lower and upper pointwise dimensions do not coincide, has full Hausdorff dimension (see Appendix IV). 25. Existence and Non-existence of Pointwise Dimension for Invariant Measures We begin with examples that illustrate some problems of the existence of pointwise dimension. Our first example shows that the pointwise dimension may not exist even for measures of maximal entropy invariant under a continuous map. Example 25.1. [PWI] There exists a geometnc construction with rectangles modeled by the full shift on two symbols (see Section 16} such that the corresponding induced map G on the limit set F of the construction is a Holder continuous endomorphism for which the unique measure of mllXImal entropy m satisfies 4m(x) < dm(x) for almost every x E F. Proof. We begin with the geometric construction presented in Example 16.2. Note that for this construction de£ S

-

=

S>,

-

log2 -logd

= ---1

log2 s = sx= ----· -log.\

_de£

Hence, the functions . = ll>. ~r p.; moreover, Jl. is the measure of maximal entropy for the full shift a (see Appendix II). Consider the induced map G. It is a Holder continuous endomorphism. Obviously, the measure m, which is the push forward of~ to F, is the invariant measure for G of maximal entropy We: describe its lower and upper pointwise dimensions. Lemma. For m-almost every x E F we have

log2

gm(x)=~=-1-''

-

og~

dm{x)

_

log2

= s = ---· -log.\

Proof of the lemma. The fact that dimH F = §. immediately implies that d.m(x) ~ §. for m-almost every x E F. Otherwise there would exist a set A of positive m-measure with 4m(x) ~ §. + E for any x E A. The non-uniform mass distribution principle would then imply that dimH F ~ dimH A ~ !!. + c. The first equality now follows from Statement 2 of Theorem 16.1.

272

Chapter 8

In order to prove the second statement consider rk = );"..+, and denote by .6.k(x) the unique cylinder set .6.; 1 .,. 3 k+l that contains x E F It is easy to see that B(x, rk) n F c .6.,.(x) n F. Since the measure IL is a Gibbs measure the inequalities (13.6) imply that for all x E F,

and hence -d ( ) m

x

> -11m . logm(B(x, rk) n F) _ > s. - k-+oo !ogrk -

The second equality now follows from Statement 2 of Theorem 16.1.

•

As we have mentioned the map G constructed above is a HOlder continuous endomorphism but not a one-to-one map. Using the map G we now construct a Holder continuous homeomorphism G for which the measure of maximal entropy is not exact dimensional. Example 25.1'. [PWl] There e:nsts a Holder continuous homeomorphism of a compact subset in IR4 with positive topolog1ca/ entropy for wh1ch the unique measure of maximal entropy has different lower and upper pointwise dimensions at almost every point; moreover, the homeomorphism is Holder continuously conjugate to the full shift on two symbols.

Proof. Consider the set F = F x F endowed with the metric

and the coding map x:~p--+ F defined by (x,y) = x( ... LJioiJ ... ) where ~P denotes the space of two-sided sequences ( ... i_ 1 i 0 i 1 ... ), ii = 1, ... ,p and 1. It is easy to see that G is a X= x( .. . Lt), y = x(ioi1 ... ). Set G = xoao Holder continuous homeomorphism and that for any (x,y) E F,

x-

where 1r1, 11"2 are the projections defined by 1r 1 (x, y) = x and 1r2(x, y) = y. Moreover, since G is expanding the map G is topologacally hyperbolic. Consider the measure m= jJ. where jJ. is the measure on ~t defined by

x·

~(A.

p,u,k

. )_

tn

,(n-k)§. _ \(n-k)s _

-,.:1

-A

-

2

-(n-k)

•

By virtue of Theorem 13.3 jJ. is invariant under a and hence m is invariant under It is easy to see that m = m x m. It follows that form-almost every (x, y)

G.

It is not difficult to check that the measure the measure of maximal entropy for G.

m (after a proper normalization) is •

Relations between Dimension, Entropy, and Lyapunov Exponents

273

We remark that the Holder exponent of the map G is log X/ log~ (see Example 16.2) and hence can be chosen arbitrarily close to 1 The following example illustrates that the pointWise dimension may exist almost everywhere but the measure, nevertheless, may not be exact-dimensional (see Section 7). We remark that this phenomenon may happen for measures invariant under continuous maps but never occurs for measures invariant under smooth maps (see Theorem 7.3). Example 25.2.

(1) There ex1sts a geometnc construction (CB1-CB3) on [0, 1] modeled by a subshift of finite type and detennined by a sequence of affine maps such that (a) the basic sets at each step of the construction are disjoint; (b) the induced map G on the limit set F is Holder continuous; (c) there exists a G-invartant ergodic measure on F of positive entropy whose pointwise dimension exists almost everywhere but u not constant. (2) There exzsts a Holder homeomorphism of a compact subset in R4 that possesses an invanant ergodic measure with positive entropy whose pointtmse dimension exists almost everywhere but is not constant.

Proof. Our approach follows Pesin and Weiss [PW1] and is a refined version of the approach by Cutler [Cu]. Consider a geometric construction (CB1-CB3) on [0, 1] with non-stationary ratio coefficients modeled by a subshift of finite type. We assume that p = 3 and the ratio coefficients ..\;,n, i = 1, 2, 3 and n = 1, 2, 3, ... depend only on the steps of the constructions and are given by ..\ l,n

-

-). 3,n -

>.2 n '

={

a { f3

-y,

t5,

if n is even ·r . dd , I n IS 0

ifn is even ifn is odd,

where 0 < a :::; {3 < ~. 0 < -y :::; t5 < ~ and at5 transfer matrix A is given by

=f -y/3. We also assume that the

We need the following lemma

= 1, ... ,p, n = 1, 2, ... be sequences of numbers satisfying: 0 < a :::; >.i,n :::; b < 1 and for any n

Lemma. Let {..\s,n}. i

p

L>.i,n =). < 1. i=l

Chapter 8

274

Then there enst a sequence of affine maps {h.,n} and a geometric construction (CB1-CB3} on [0, I] modeled by a given symbolic dynamical system (Q, u) such that (I) each basic set Ll; 0 • in = h; 0 ,n o · · · o h;n,n((O, I]); (2) 6."' inn l!..;o in = 0 if (io .. i,.) f. (jo .. . j,.); (3) the induced map G on the limit set F is Holder continuous.

Proof of the lemma. For each n

= 1,2, ... define affine maps h;,n(x) = >..,nx+

f,;

1]) are at least apart. The result • follows. It is easy to check that Conditions {15.3) and (15 5) hold. One can also see that the following limit exists: a;,n and choose a;,n such that the sets h;,n([O,

log >-•

. 1~ { = n-+oo bm - L..J log .>.; k = n k=l '

def

~ log(ao), )

1

if i = 1, 3 .

.

2 log(/h , 1f ~ = 2.

{25.1)

Let p, be a Gibbs measure on ~1 corresponding to a Holder continuous function and v = x• J.i· Consider the sets A:::; {x E F : x:::; x(i 0i 1 ... ) with i1 = I or 3} and

B = {x E F ; X= x(ioil ... ) with il = 2}. These sets are disjoint and comprise the space They are not invariant under G. Since J.1 is a Gibbs measure and the sets A and B are open we have that v(A) > 0 and v(B) > 0. One can check that for every x E A,

E1.

!

lim log l!!..n(x)l :::; lo ('Yf3) n-+oo n 2 g and for every x E B,

r

n~~

loglt!.n(x)l

n

11

( o)

= 2 og a

·

Since x- 1 (6.n(x)) is a cylinder set the Shannon-McMillan-Breiman theorem implies that for v-abnost every x E F,

_ lim log v(.!ln(x)) = h,_.(e1) > O. n-tOO n Thus, by Theorem 15.3 h (u) 2h,_.(CT) for abnost every x E A d(x) =d(x) = -liffin-+oo(logl~n!z!) = -log(f3'Y) and d( ) _ d( ) _

- x -

x --lim

h,.(u) _ 2h,.(C1) (Iogl6.ft(xl1) - -log(ao)

n-+oo

for abnost every x E B.

n

The first statement follows now from Theorem 15.3. Repeating the arguments in Section 25.1 one can define the HOlder homeomorphism G and show that it possesses an invariant Borel ergodic measure i/ with respect to which d;;(x, y) = 4;;(x, y) d;;(x, y) for i/-almost every (x, y); however, the function d;;(x, y) is not essentially constant. This implies the second statement and completes the construction of the example. •

=

Relations between Dimension, Entropy, and Lyapunov Exponents

275

Remark. The geometric construction described in Example 25.2 is called an asymptotic Moran-like geometric construction since its ratio coefficients admit asymptotic behavior (25.1) It is proved in (PWl] that S>..::; dimn F, where S>. is a unique root of Bowen's equation PE+(slog>.;,) = 0. Example 25.2 illustrates A that the strict inequality can occur. Indeed, one can easily check, using Statement 1 of Theorem 13.3, that s>.. = Jog 2/log(afry6). On the other hand, one can compute, using Theorem 15.2, that . { log 2 log 2 } dtmn F = max -log(}3-y), -log(a6) . The following example demonstrates non-existence of pointwise dimension for smooth maps. It was constructed by Ledrappier and Misiurewicz in [LM]. Example 25.3. For every integer r ~ 1 there exzst a C'" -map f of the interval [0, 1] and f-invanant ergodic measure 1-1. for which !t,.(x) < d11 (x) for almost all points x. Proof. We outline the proof in the case r = 1. Choose numbers A, B, and C such that A > 1 and 0 < B < C < A!t. Let On be a sequence of numbers satisfying 81 = 1 and B::; On+l/On ::; C. Since C < 1 the sequence On decreases towards 0. We define two sequences of points {an}, {bn} of the interval (0, 1] 0 = az <

b:l < a4 < b4 < as < bs < .. <~<~<~<~<~<~<~<~=!

such that

and IMn I

= lan+2- bnl =en- On+! - On+! A'

where L,. and Mn are the intervals with endpoints an, bn and bn, a,.+2 respectively. We have IM,.I >1-CA+l >0 On A and

00

00

n=l

n=l

Therefore, all points with even indices lie to the left of all points with odd indices and there exists a common limit c = lim a.. = llm bn. n--too n-+oo We now define the map f. First we set: f(c) = 1, f(an) = 1 --y,., where In= 9n/An-l, f(bn) = 1- 6n, where 6n =(On- On+t)/A"- 1 .

276

Chapter 8

Since 8,. - 8n+l

An-t

-

8n+l A"

IMnl

= An-t'

we have 6,. > 6,.+1 and thus, 1 = 'Yt > 61 > 62 > · · · -+ 0. Therefore, we can define f as a linear map on each interval L,. with the slope >.,. given as

Note that

ro, 11 = {c} u

(uL. ) (u u

n;:=:l

M .. )

n?;l

and it remains to define f on the intervals M,.. Denote

For n even we set

f (b,. + ~IMnl)

f'

= f(b,.) + ~IM,.i(>.,. +a,.),

(b.,+ ~IMnl) =a,..

One can now define f separately on [b,., b,. + ~IM,.I) and [b,. + ~IM,.I, a,.+2) to obtain a C00 -function on [b,., a,.+2] (see details in [Mi]). For n odd one can define f analogously. This gives a map f on (0,1]. The condition on f to be of classC 1 is We have !Ani= A2-n-+ 0 as n-+ oo. Furthermore,

Thus,

lwnl-+ 0 as n-+ oo and hence

as n-+ oo. Let K,. be the interval [a,.,a,.+ 1] or (an+t 1 a,.]. We need the following properties of sets K,. (see {Mi]).

Relations between Dimension, Entropy, and Lyapunov Exponents

277

Lemma 1. (1) The sequence of sets (Kn)~=l is decreasing; (2) f 2 n-l (Kn) = Kn, f 2n-l (Kn+d = L,.; (3) The sets J'(K,.), i = 1, ... , 2n-l are d'tBjoint; (4) p"-'+'(Kn+I) n J'(Kn+I) = 0 fori= I, ... , zn- 1 ; (5) The map j'- 1 1/(K,.) is linear fori= 1, ... , 2"- 1 . For each i

n-1

= 1, ... , 2n-l we write i = 1 + L: ek(i)2k-l

with ek(i)

= 0 or 1.

k=j

Let Lemma 2. For all n

~

1 and i = 1, ... , 2n-l we have n-1

lfi(Kn)l

= IT (k(ek(i)). k=l

Proof of the lemma. We use induction on n. For n = 1 the result is obvious. Assume that the statement holds for n = m. We shall prove it for n = m + 1. Observe that for i = 1, ... , 2m-l the set /'(Km) is a disjoint union (modulo endpoints) of sets J'(Km+l), t•(Lm) = / 2m-'+i(Km+I), and a remaining gap

Gi,m·

.

Observe also that the map /'- 1 is linear on /(Km) so that the lengths of these intervals are in the same proportions as

1/(Km+l)l IJ(Km)l

=

'Ym+l I'm

1 llm+l

IJ(Lm)l

=A 0::: ' lf(Km)l =

I'm -Om I'm

The induction follows clearly from the above two observations.

Om+l

= 9:;: . •

This lemma gives us information on the lengths of intervals building the at tractor for the map j. The following lemma provides estimates of the lengths of the gaps Gi,m· Denote

Clearly, /3 > 0.

Lemma 3. For all m

~

2 and i

= 1, ... , 2m-l

we have

Proof of the lemma. Note that by linearity of J'- 1 1/(Km) the above inequalities are equivalent to

Chapter 8

278 or

< {3 [1 - ( 1 + ~) Om+l] . Om A Om ity follows from the inequal second The The first inequality holds since A > 1.

~ Om+l <

Om+l

A Om

inequality {3_ 1 ~

which is clearly true since 1/C The attract or A for the map

< ~- A+1 A

- Om+l Om/Om+l·

f

•

is defined as 00 2"- 1

A=

nU

J'(Kn)-

n=t i=l

e. We define a By Lemmas I and 2 A is a Cantor-like set of zero Lebesgue measur ... , 2m-t the 1, = i , f'(Km) set every probab ility measur e J1. on A by assigning to 1-m. The measur e J1. is clearly non-ato mic and invaria nt. 1 =2 measure p.(/ (Km)) , where e,..(x) = Each point x E A admits a symbolic coding x 1-+ w = (en(x)) if i ~ 2"- 1 and 0 = e,(x) set we 2" ~ i ~ 1 t), 0 or 1. Namely, for x E /i(Kn+ 1 . Using this coding map one can show that the measur e J1. en(x) = 1 if i > 2"is ergodic and has zero measur e-theor etic entropy (see [CEJ).

Lemm a 4. For all x E A -n log2 . hm 2::"j=llog (j(e;(x )) ri"(x) = n-+oo

,

-d ( ) _ li-:-

I' X

Proof of the lemma . For x E f'(Kn+ t), i

"'n

-nlog2

( ( )) ·

m n-+ao '--i=llo g(; Ej X

= 1, . . , 2" we set n

1Jn(x)

= 1/'(Kn +l)l = TI<~o(e~;(x))

lo+l Clearly, the interval [x -1'/n(x), X+ c-no. < {3 (see Lemma 2). Fix no such that exists i(k) such that 1 ~ i(k) ~ there n < k each For t)f'(Kn+ s TJn(x)] contain have 2,. and x E f'(lol(Kn+I)- By Lemma 2 we IJ'
and hence if /c

~

n - no then lfi(k>(Kk+l)l ~ f3TJn(x).

By Lemma 3 we obtain that if k ~ n - no then IGi(k),kl ~ hn(x) from A the set Therefore, since the sets G;(k),lo are gaps and hence are disjoint that follows It +l)l(Kn-no fi(n-no [x -17n(x), X+ TJn(x)] n A is contain ed in

Tn ~ JJ.([x- TJ,.(x), x + TJn(x)]) ~ 2-(n-no ). the inequalities The desired result follows immediately from this relation and 0 < BA- 1 < '1/n+l(x) < C < 1. - 17n(x) The lemma is proved.

•

Relations between Dimension, Entropy, and Lyapunov Exponents

279

Note that log(j(O) =log

O·+z T -log A, J

Since ei(x) are independent stationary sequences for p-almost every x E A we have

-1~ -1 1 lim - L....-log(j(c3 (x)) = lim -logOn- -logA. . n-+oon 2

n-+oo n

J=l

Let us choose the sequence On such that B ~ ~ ~ C and lim

.!. log On = B ,

n-+oon

-1

lim -logOn= C. n-.oon

From Lemma 4 it follows that for JK-almost every x E A, log2 log2 -J.I(x) = log(VA/B) < log(VA/C) d

=

d '"'(x).

This completes the construction of the example.

•

26. Dimension of Meamres with Non-zero Lyapunov Exponents; The Eckmann--Ruelle Corijecture

We assume that M is a compact smooth Riemannian p-dimensional manifold and f. M -t M is a Cl+<>.diffeomorphism. We recall some basic notions of the theory of dynamical systems with non-zero Lyapunov exponents (see [KHJ, [M] for more details). Given x E M and v E T:zM, define the Lyapunov exponent of v at x by the formula '( ) -,--- log lldi:vll . (26.1) 1\ x,v = 1lm n-+oo n If x is fixed then the function ,\(x, ·) can take on only finitely many distinct values ,\( 1l(x) > · · · > ,\{ql(x), where q = q(x) and 1 ~ q ~ p. Let k;(x) be the multiplicity of the value ,\('l(x), i = 1, ... , q. The functions q(x), >.<;>(x), and k; (x), i = 1, ... , q are measurable and invariant under f. Let p be a Borel /-invariant measure on M. We will always assume that p is ergodic Then the function q(x) = q is constant JL-almost everywhere and so are the functions >.<•>(x) and k;(x), i = 1, . .. ,q. We denote the corresponding values by ,\~) and k~). A measure J1. is said to be hyperbolic if

Chapter 8

280

for some k, 1 ::; k < q. If J1- is such a measure then for Jl--almost every point x EM there exist stable and unstable subspaces E<•>(x), E("l(x) C T.,M such that

(1) E<•l(x) E11 E<">(x)

= T,M,

dj.,E<•l(x) = E<•>(!(x)), dj.,E<">(x)

=

E(u) (J(x) );

(2) for any n 2': 0, ildf~vii

::; Cl(xhnllvll ildf;nvll::; CI(xhnllvll

if v E E<•>(x), ifv E E<">(x),

where 0 < 1 < 1 is a. constant and Ct (x) > 0 is a measurable function;

{3) L(E<•>(x), E<">(x)) 2': C 2 {x) > 0, where C2 (x) is a measurable function and L denotes the angle between subspaces E<•>(x) and E<">(x); (4) C 1 (tn(x)) ::; C 1 (x)en6, C2 (tn(x)) ~ C2 (x)e-n.S for any n ~ 0, where 8 > 0 is a constant which is sufficiently small compare to 1- I· Denote by At, £ ~ I the set of points x for which C1 (x) ::; £, C 2 (x) ~ l· The sets At are closed, At c At+I• and the set A= Ut>l At is !-invariant and wincides with M up to a set of Jl--measure zero. For any x E A one can construct stable and unstable smooth local manifolds which we denote by W1~(x) and W1~"j(x) respectively. They have the following properties:

(5) x E<">(x);

E

W1~~(x) and x

(6) for any n

~

E

W1~~(x); T.,W1~~(x)

= E(•l(x)

and T.,W1~~(x)

=

0,

p(r(x),r(y))::; C3(x)lnp(x,y) p(J-n(x),rn(y))::; C3(x)'ynp(x,y)

if y E W1~J(x), if y E W1~(x),

where pis the distance in M generated by the Riemannian metric and C 3 (x) > 0 is a measurable function satisfying C3 (r(x))::; C3 (x)e 10n<5 for every n ~ 0;

(7) wi~J(x) and w.~~(x) depend continuously on X EAt in the C 1-topology and have size at least rt > 0; this means that they contain respectively balla B<•l(x,rt) and B<">(x,rt) centered at x of radius rein the intrinsic topology generated by the Riemannian metric;

(8) there exists a function £1 = '1/J(l) < oo, such that for any x, y E Ae, if the intersection B<•>(x,rt) n B("l(y,re) f. 0, then it consists of a single point

z E Ae,; (9) there exists K

> 0 such that for any

x E A1 ,

p<">(y,z) ~ Kp(y,z)

if

y,z E W1~"j(x),

p<•l(y,z) ~ Kp(y,z)

if

y,z E W1~~(x),

Relations between Dimension, Entropy, and Lyapunov Exponents

281

where and p(u) and p<•> arc the intrinsic distances in the local unstable and stable manifolds induced by the Riemannian metric. A closed set II c A is called a rectangle if : a) there exists l > 1 such that IInAe is an open subset of At l (with respect to the induced topology of Ae); moreover, for any x E II there exists y E II nAt such that x E B(s) (y, re) or x E B(u) (y, rt); b) for any x, y E II the intersection s<s>(x, rt) n B("l(y, re) is not empty and consists of a single point z E II (one can see that II c At,). Given a rectangle II and points x, y E II, define the u-holonomy map s<•>(x, rt) n II -+ s<•>(y, rt) n II by H~~J(z) = B<">(z, rt) n n<•>(y, rt). One can similarly define the s-holonomy map H~~~· For l'ach rectangle II such that ~t(II) > 0 let e and e<•> be the measurable partitions of II : e<">(x) = II n B<">(x,rt) and e<•>(x) = II n n<•l(x,rt) (see [KH]}. Let {IL~")}xen and {J.t~•l}.,en be the canonical families of conditional measures associated with the partitions e 0 and J,t-almost every x, y E II the associated holonomy map H~~~ is absolutely continuous with respect to the conditional measures J.'~u) and J4">, i.e., the measure (H~~~)•I-'~u) is absolutely continuous with respect to the measure J4"l (or if the same is true for the holonomy maps HJ~ and the conditional measures ~tf:>, for JL-almost every x, y E II). We say that a measure J.' has the local product structure if for any rectangle II with J.t(II) > 0 and J,t-almost every x, y E II the associated holonomy maps H~~J, H~~~ both are absolutely continuous with respect to the conditional measures 1-'~•), 1-'~•l and J.tf,"l, ~u), respectively. There are two special cases of measures which have local product structure. A hyperbolic measure 1-' is called a Sinai-Ruelle-Bowen measure (or SRB-measure) if for every rectangle II with J.t(II) > 0 and tL-almost every x E II th& conditional measure J.t~u) is absolutely continuous with respect to the Riemannian volume on W 1<;,>(x) SRB-measures have semi-local product structure. Let A be a locally maximal hyperbolic set for a Cl+"-diffeomorphism on a compact smooth manifold M (see definition in Section 22) and IL = /1-

0, define

ntJ:

4")

(26.2) (assuming that the limits exist). One can show that the functions d~>(x) and d~u)(x) are measurable and invariant under f. Since the measure 1-' is ergodic these functions are almost everywhere constant. We denote the corresponding values by d(u) and d(•).

Chapter 8

282

We first consider the cases k = 1 and k = q - 1. The following result is a slightly more general multidimensional version of the result obtained by Young in the two-dimensional case (see [Y2]). Proposition 26.1. For any ergodic hyperbolic measure /-L invariant under a Gl+"'.diffeomorphism the limits {26.2) exist almost everywhere and (1) if k

= 1 then d(u) = ~i

(2) if k

= q- 1 then d(•) = - k,.~:i(~~> . >.,.

k~ "~

The existence of the values d~l(x) and d~")(x) in the general case was established by Ledrappier and Young in [LY]. We recall that smce /-L is ergodic 4,..(x) = const = 4(!-L) and diL(x) = const = d(/-L) almost everywhere (see Section 7). Proposition 26.2. For any ergod1c hyperbolic measure /-L invar~ant under a Cl+<>.diffeomorphism the limits {26.2) eXISt almost everywhere and

(1) d(!-L) :$ d(u) + d(•) i (2) d(u) > hu(f) and d(•) > -~

hu(Jl. - >JJr

In [ER], Eckmann and Ruelle discussed the existence of pointwise dimension for hyperbolic mvariant measures. We summarize this discussion in the following statement, which was proved by Barreira, Pesin, and Schmeling in [BPSl] Theorem 26.1. Let f: M-+ M be a C 1 +0 -diffeomorphism of a compact smooth Riemannian manifold M and /-L a hyperbolic ergodic measure. Then for 1-L-almost anyx EM, Remark. In [Y2], Young proved this theorem in the two-dimensional case. Proposition 26.3. Let f be a C1+ 01 -diffeomorphism of a smooth compact surface M and JL a hyperbolic ergodic measure with Lyapunov exponents A~l) > 0 > ,\~2 ). Then

Proof of the theorem. For the sake of reader's convenience we first consider the special and simpler case of measures with local semi-product structure. The proof in the general case will be given later Since it is technically more complicated it can be omitted in the first reading. We follow [PY]. Without loss of generality we may assume that the holonomy maps H~:~ are absolutely continuous with respect to the measures 1-Li,"l, JL~u) for any rectangle II with !-L(II) > 0 and 1-L-almost every x, y E II We fix such a rectangle II. According to Proposition 26.2 it is sufficient to show that 4(1-') 2:: d(s) + d(u).

Relations between Dimension, Entropy, and Lyapunov Exponents

283

Proposition 26.2 (that states the existence of the limits in {26.2)) implies that there exist a closed set A 1 c II with !k(A1 ) > 0 and a number r 1 > 0 satisfying the following condition: (10) for any 0 < r ~ r1 and any x E A1, (26.3)

It follows from the Borel Density Lemma (see Appendix V) that one can find a closed set A2 c A1 with tt(A2) > 0 and r2 > 0 such that for any 0 < r :0:::: r 2 and any x E A2, tt(B(x, r)) ~ 2p,(B(x, r) n A 1 ). (26.4) Fix a point Xo E A2 for which J.l.~~) (WI~") (xo)

n A2) > 0 and

lim logJ.!.(B(xo,r)) = d(J.I.). log r -

;:::;a

We first study the factor measure ji. induced by the measurable partition e<•l. Denote by 1r(u} the projection of B(x 0 , r) n A 1 into W1~ (x 0 ) given by 71'(u) (y) =

wl~':!Cxo) n wl<,;2(y) For any X E B(xo,r) n Al we also have that 71'(u)IW(u)(x) loc

n Al

= H(•) Jw(u)(x) n AI a:o,x Joe

Without loss of generality we may assume that the holonomy map H~~~z is absolutely continuous with respect to the conditional measures J.l.~~) and J.l.~u). Therefore, the factor measure ji. = 1riu) J.1. is absolutely continuous with respect to J.l.~~>:

where cp(z) ;::: 0 is an L 1-function. Since J.l.~~l(W1~"j(x0 ) n A2 ) > 0, without loss of generality, we may assume that the Ra.don-Nikodym derivative djJ/ dJ.!.~~) is bounded. Therefore, we have for sufficiently small r > 0, (26.5) where C 4 > 0 is a constant. We now apply the Fubini theorem to the measurable partition e<•l and write p.(B(xo,r)) =

(26.6)

If the intersection w1<,;2(z) n B(xo, r) nAt contains a point X then by Condition (9),

284

Chapter 8

a.nd hence in view of (26.3) {26.7} We also have that

Conditions (26.5)- (26.8) imply that for sufficiently small r, IL(B(xo, r) n AI):::;

I

(2Kr)d<·>-•dp(z)

B(:to,r)nw,<;'j(:to)

= (2Kr)t:i<•>-e;J(B(xo,r}

n W1~j(xo))

:::; (2Kr)£1(•>-e;J(B(ul(xo, Kr)} :::; (2Kr}d<•>-•C4 ~t~~l(B(x 0 , Kr)) :::; (2Kr)d<•>-•c4(Kr)d_, = C5 rd''>+d<•>- 2•, where 0 5 > 0 is a constant. Consider a decreasing sequence of positive numbers Pk -+ 0 (k -+ oo) such that IL(B(xo, p~o;}) ~ Pk!l,.(p)-e for all k. We can also assume that Pl < min{r1,r2,r3 }. It follows now from (26.4) that

If k -+ oo this yields Since e is arbitrary this implies the desired result. We now proceed with measures which do not have local (semi-) product structure. We will first establish a crucial property of hyperbolic measures: they have nearly local product structure. This enables us to apply a slight modification of the above approach to obtain the desired result In order to highlight the main idea and avoid some complicated technical constructions in the theory of dynamical systems with non-zero Lyapunov exponents we assume that the map f possesses a locally maximal hyperbolic set A which supports the measure 1-'· Although the set A has direct product structure hyperbolic measures supported on it, in general, do not have local (semi-) product structure. For the general case see [BPS1]. Consider a Markov partition 'R. = {R 1 , ... ,Rp}. In order to simplify notations we set R~(x) = R,k ;1 (x) (the element of the partition vl=kfi'R. that contains x). We point out the followmg properties of the Markov partition. Given 0 < e < 1, there exists a set r c M of measure Jt(r) > 1-e/2, a.n integer

Relations between Dimension, Entropy, and Lyapunov Exponents

285

n 0 ;:: 1, and a number C > 1 such that for every x E r and any integer n ;:: no the following properties hold: (a) for all integers k, l ;:: 1 we have (26.9) JL~•>(R2(x))

c-Ie-k(h-e}

S

S ce-k(h+el,

(26.10)

c-le-l(h-e}

s JL~u)(~(x)} s ce-l(h+•)'

(26.11}

where h = h,_.(j) (the measure-theoretic entropy off with respect top,); (b) e-(d(•>+e}n

S p,~·>cs<•>(x, e-")) S e-(d(•> -•)n,

(26.12) (26.13)

(c) define a to be the integer part of 2{1 +e) max{1/ ..\ 1 , -1/ Ap, 1}; then R:~(x)

R~,.(x) nA<•>(x)

c B(x,e-") c R(x),

(26.14)

c s<•>(x,e-") c R(x) nA<•>(x),

(26.15)

Rg"(x) n A<"'>(x) c s<"'>(x, e-") c R(x) n A<"'>(x),

(26.16)

where the sets A<•l and A(u) are defined by (22.7); (d) define Q,.(x) to be the set of points y E M for which the intersections Rg"(y) n B<">(x, 2e-") and R~.. (y) n s<•>(x, 2e-") are not empty; then (26.17) and Jl!~(y) C Q,.(x) for each y E Q,.(x); (e) there exists a positive constant D = D(f) < 1 such that for every k ;:: 1 and X E r we have JL~(R~(x) n r);:: D,

p,~(R2(x) n r);:: D;

(f) for every x E rand n;:: no we have

Property {26.9) shows that the Shannon-McMillan-Breiman theorem holds with respect to the Markov partition 'R. while properties (26 10} and (26.11) show that "leaf-wise" versions of this theorem hold with respect to the partitions ng and Rh. The inequalities (26.12) and (26.13} are easy consequences of the existence of the stable and unstable pointwise dimensions d{•) and d(u) (see Proposition 26.2). Since the Lyapunov exponents at J.l·almost every point are constant the properties (26.14), (26.15), and (26.16) follow from the choice of the number a indicated above. The inclusions (26.17) are based upon the continuous dependence of stable

Chapter 8

286

and unstable manifolds in the CH"' topology on the base point on A. Property (f) follows from the Markov property. For an arbitrary Cl+"'-diffeomorphism preserving an ergodic hyperbolic measure Ledrappier and Young [LY] constructed a countable measurable partition of M of finite entropy which has properties (26.9)-(26.17). In [BPSl], the authors also showed that this partition satisfies Property (e) and simulates (in a sense) Property (f). They used this partition to obtain a proof in the general case. We follow their approach. It immediately follows from the Borel Density Lemma (see Appendix V) that one can choose an integer n1 ~ no and a set f' c r of measure J.!(f') > 1 - E such that for every n ~ n 1 and x E f', (26.18) 1

J.!~·l(B<•l(x, e-")

n r)

~ A•l(B<•l(x, e-"));

p.~"l(B<">(x, e-")

n r)

~

2 1

J.!i"'(B(u) (x, e-")).

2

(26.19) (26.20)

r

Fix X E and an integer n ~ nt. We consider the following two classes '!'(n) and ~( n) of elements of the partition 'R~~ (we call these elements "rectangles"): 'I(n) = {~~(y) c R(x) · R:~(y) n ~(n)

= {R:~(y) c

r # 0};

R(x): ~,.(y) n f # 0 and Rg"(y) n f' # 0}.

The rectangles in '!'(n) carry all the measure of the set R(x) n r Obviously, the rectangles in '!'(n) that intersect f' belong to ~(n). If these were the only ones in ~(n), the measure p.JR(x)nr would have the local direct product structure at the ~eve!" n and its pointwise dimension could be estimated as above. In the general case, the rectangles in the class ~(n) are obtained from the rectangles in '!'(n) (that intersect f') by "filling in" the gaps in the product structure. See Figure 21 where we show the rectangle R(x) which is partitioned by "small" rectangles R~~(y). The black rectangles comprise the collection 'I(n). By adding the gray rectangles one gets the direct product structure on the level n. We wish to compare the number of rectangles in 'I(n) and \V(n) intersecting a given set. This will allow us to evaluate the deviation of the measure JL from the direct product structure at each level n. Our main observation is that for "typical" points y E f' the number of rectangles from the class 'I(n) intersecting A< 8 l(y) (respectively A(u)(y)) is "asymptotically" the same up to a factor that grows at most subexponentially with n. However, in general, the distribution of these rectangles along A(•l(y) (respectively A(ul(y)) may "shift" when one moves from point to point. This causes a deviation from the direct product structure. We will use a simple combinatorial argument to show that this deviation grows at most subexponentially with n.

288

Chapter 8

Lemma 2. For each y E R(x) n f and integer n 2: n 1 we have

N(n, Qn(Y)) 2: p.(B(y, e-")) · 2Ce-Zan(h+e). Proof of the lemma. It follows from (26.17) and (26.18) that 1

2p.(B(y, e-n)) ::;'; p.(B(y, e-n) n r) ::;'; p.(Q.. (y) n r)

:::; N(n, Q,.(y)) max{p.(R): R E '!'(n) and R n Q.. (y) 1= 0}.

•

The desired inequality follows from (26.9).

Lemma 3. For p.-almost every y that for each n 2: n 2 (y) we have

E

R( x) n f' there is an integer n 2 (y) 2: n 1 such

N(n + 2, Qn+2(y)) ::; j{(•l(n, y, Q.. (y)). j{(u)(n, y, Q.. (y)). 2C2e4a(h+e)e4an"

Proof of the lemma. Since f' c r, by (26.17) and the Borel Density Lemma (applied to the set A= f', see Appendix V), for p.-almost every y E f there is an integer n 2 (y) 2: n 1 such that for all n 2: n 2 (y),

2p(Q,.(y) n r) 2: 2p.(B(y, e-n) n r) 2: p.(B(y, e-n))

2: t-t(B(y,4e-n-z)) 2: JL(Qn+2(Y)).

(26.21)

For any m 2: nz(y), by (26.9) and Property (d), we have

p.(Qm(Y)) =

L

IL(~:;!(z))

2: N(m, Qm(Y)). c-Ie-2am(h+e)_

R~:::(~)CQm(Y)

Similarly, for every n 2: n 2 (y) we obtain

where N .. is the nlliilber of rectangles R~~(z) E '!'(n) that have non-empty intersection with f. Set m = n + 2. The last two inequalities together with (26.21) imply that N(n+ 2,Qn+2(y)) ::;'; Nn · 2C 2e4 a(h+e)+4ane. {26.22} On the other hand, since y E f' the intersections R8n(y) n A(ul(y) n f and n A<•l(y) n f' are non-empty. Consider a rectangle R~~(v) C Qn(Y) that has non-empty intersection with f'. Then the rectangles R~n(v) n .ngn(y) and R~ .. (y) n Rgn(v) belong to ;j(n) and intersect respectively the stable and unstable local manifolds at y. Hence, one can associate to any rectangle R~~(v) c Qn(Y) in '!'(n), that has nonempty intersection with f the pair ofrectangles (R<•J = R~.. (v) nRgn(y),R(u) = R~n(Y) n R 0n(v)) such that R<•>, R(u) E ;j(n), R<•> n AC•l(y) n Q .. (y) 1= 0, and R(u) n AC"l(y) n Q,.(y) 1= 0. Clearly this correspondence is injective. Therefore, R~.. (y)

.N<•>(n, y, Q.. (y)) · fi<">(n, y, Q.. (y)) 2: N ... The desired inequality follows from (26.22).

•

Relations between Dimension, Entropy, and Lyapunov Exponents

Lemma 4. For each x

E

t

289

and integer n 2:; n 1 we have

Proof of the lemma. Since the partition n is countable one can find points y; such Lhat the union of the rectangles Rgn(yi) is the set R(x) and thebe rectangles are mutually disjoint. Without loss of generality we can assume that y; E t whenever Rgn(y;) n t # 0. We have

(26.23)

By Properties (e), (f), and (26.10) we obtain that N<•>(n

J.L~~)(Rgn(y,) nf) (•) () max{J.Lz (Rg~(z)) : z E A • (y;) n R(x) n r D

#

0}

- max{J.L~·>(R~n(z)): z E A<•>(y;) n R(x) n r

#

0}

. uan(y·)) > ,y.,uo ' -

(26.24)

;::: nc-lea.n(h-c).

Similarly, (26.9) implies that N(n R(x))

'

<

J.L(R(x)) < Ce2anh+2ane. - min{p.(R~~(z)): z E R(x) n r} -

(26.25)

We now observe that

(26.26)

Putting (26.23)-(26 26) together we conclude that

Ce 2an(h+e) 2:; N(n,R(x)) 2:; i Rgn (y;)nf';o!0"

2:: .JV(n,x,R(x)) · DC- 1e'm(h-e). This yields .JV(n,x,R(x)) :::; proved in a similar way.

v- 1C 2 ea.n(h+3
The other inequality can be •

Chapter 8

290

We emphasize that the procedure of "filling in" rectangles to obtain the class ~(n) may substantially increase the number of rectangles in neighborhoods of some points. However, the next lemma shows that this procedure does not add too many rectangles at almost every point. In other words the procedure is "locally homogeneous" on a set of "big" measure.

Lemma 5. For p.-almost every y E R(x) n f' we have

Proof of the lemma. By (26.20) for each n?: n1 andy E f',

Since R:~(z) that

C R~n(z)

N<•l(n, y, Qn(Y)) ?:

for every z we obtain by virtue of (26.10) and (26.12)

t4•>

(Qn(Y)) max{t-t~8 l(R~~(z)): z E A<•l(y) n R(x) n r

#

> ~ t4'\B<•l(y,e-")) - 2 max{t-t~')(R~n(z)): z E Al•l(y) n R(x) n r 1

0}

# 0}

e-(d('l+e)n

> - -20 ----,,..--, e-
F={yEf:

lim .N<•l(n,y,Qn(Y))e_7ane>1} N(•l(n,y,Qn(Y)) -

n-+oo

For each y E F there eXIsts an increasing sequence {mj}~ 1 positive integers such that

.N<•l(m;,y,Qm,(Y))?:

>

= {mj(y)}~ 1

of

~N(•l(mj,y,Qm/Y))e7am,e (26.27) _2__e-d(•lmi+amih+5am,e

- 4C

for all j (note that a> 1). We wish to show that p.(F) = 0. Assuming the contrary consider the set F' c F of points y E F for which the following limit exists lim logp.~·>cB<•>(y,r)) = d<•>. logr

r-+0

Relations between Drmension, Entropy, and Lyapunov Exponents

291

Clearly, JJ.(F') = JJ.(F) > 0. Then one can find y E F such that ~-t~•l(F) = p.~•l(F') = p.~•l(F'

n R(y) nA<•l(y)) > 0.

It follows from Frostman's lemma that dim 8 (F' n A(•l(y)) = d(•l.

(26.28)

Consider the collection of balls

8 = { B(z, e-m;(zl) : z E F' n A{•l(y), j = 1, 2, .. }. By the Bcsicovitch Covering Lemma (see Appendix V) one can find a countable subcover C C B of F'nA<•l(y) of arbitrarily small diameter and finite multiplicity p (which depends only on dimM). This means that for any L > 0 one can choose a sequence of points {z; E F' n A<•l(y)}~ 1 and a sequence of integers {ti}~ 1 , where t; E {m;(z;)}~ 1 and t; > L for each i, such that the collection of balls C = {B(z;,e-t'). i = 1, 2, ... } comprises a cover ofF' n A(•l(y) whose multiplicity does not exceed p. We write Q(i) = Qt, (z;). The Hausdorff sum corresponding to this cover is 00

l)diamB)a<•>_, BEC

= :~:::e-t;(a<•>-•J. i=I

By (26.27) we obtain 00

00

~::::e-t;(-e) i=l

: :; L

.zVC•l(t;, Z;, Q(i)). 4ce-at;h-4at;e

i=l 00

:::; 4CLe-aqh-4aqe L q=l

.zVC•l(q,z,,Q(i)).

i t;=q

Since the multiplicity of the subcover C is at most peach set Q(i) appears in the sum 2:::; t; =q .N< 8 l (q, z;, Q (i)) at most p times. Hence, L i

.N<•l(q,z;,Q(i)):::; pN(•l(q,y,R(y)).

t~.=q

It follows from Lemma 4 that 00

L(diamB)a<•>-< :-; 4CLe-aq(h+4elp.zV<•>(q,y,R(y)) BEC

q=l

Since the number L can be chosen arbitrarily large (and so are the numbers t;) it follows that dimH(F' n A(•l(y)) :::; d(s)- e < d(•l. This contradicts {26.28). Hence, ~-t(F) = 0. This yields the first inequality The proof of the second inequality is similar. •

292

Chapter 8

Lemmas 3, 4, and 5 show that the deviation of the distribution of rectangles in 'I' from direct product structure is subexponentially small. The rest of the proof simulates the case of measures which have local (semi-) product structure considered above. By Lemmas 2 and 3 for p-almost every y E R(x)nf' and n;::: n 2 (y) we obtain

p.(B(y, e-"- 2 ))

:s Jlr<•l(n, y, Q.. (y)) x N'<"l(n, y, Q,.(y)) x 4C3e4<>{h+•le-2<>nh+6<>n<.

By Lemma 5 for p-almost every y E R(x)nf' there exists an integer n3(y) ;::: n2(y) such that for all n 2:: n3(y),

N'<•>(n,y,Qn(Y)) < N<•>(n,y,Qn(y))e 7""", ]\r(u)(n,y,Q,.(y)) < N("l(n,y,Q,.(y))e7anc. This implies that p.(B(y, e-"- 2 )) :S N<•>(n, y, Q,.(y)) x X

Applying Lemma 1 we obtain p(B(y, e-"- 2 )) :5: p.~•) (B<•>(y, 4e-")) X

N(u) (n,

y, Q,.(y))

4c3e4a(h+•le-2anh+20an£. X

p.~u) (B{u) (y, 4e-"))

4 cse4a(h+£)e22anc.

This implies that lim log,Jl.(B(y, e-")) > d(•) + d(u)- 22ae: ..~ -n for p.-almost every yEt. Since p(f) > 1-£ and£> 0 is arbitrary we conclude that d= lim logp(B(y,r)) = lim log,Jl.(B(y,e-")) > d(s) +d(u) r-+0 logr ..~ -n for ~t-almost every y E M. This completes the proof. • Let A. be a hyperbolic set for a C1+ 0 -diffeomorphism f on a compact smooth "lllanifold M. Assume that /lA. is topologically transitive. The set A. is called a hyperbolic attractor if there exists an open set U such that A. C U and f(U) c U. Clearly, A. = nn>o r(U). One can show that if A. is a hyperbolic attractor then wl~u; (x) c A. fur every X E A. (see, for example, [KH]). Consider the unique equilibrium measure p = fl."' on A corresponding to the function cp = J(u) (x). It is known that Jl."' is the SRB-measure (see [Bo2]). Obviously, d(u) =dim wi<:JCx) ~f m for every X EA. By Proposition 26 2 and the entropy formula (see (LY]) we have that d(•)

> - hp.(/) -

= - 2:~=1

>.~q)

k~) >.~)

>.~q)

This implies that

. A> d lmH _ dimH J1. > _ m-

"~ L...J=l

k(j) >. (j) ~' 1-1 (q)

>.p.

Appendix V

Some Useful Information

1. Outer Measures (Fe]. Let (X,p) be a complete metric space and m a a-sub-additive outer measure on X, i.e., a set function which satisfies the following properties: (1) m(0) = 0; (2) m(Zl):::; m(Z2) if Z1 c Z2 c X; (3) m(.U Z;) :::; Em(Z;), where Z; C X, i = 0, 1, 2, .. •;:::o

;;:::o

A set E c X is called measurable (with respect tom or simply m-measurable) if for any A c X, m(A) = m(A n E) + m(A \E) The collection 2l of all m-measurable sets can be shown to be a a-field and the restriction of m to 2l to be a a-additive measure (which we will denote by the same symbol m). An outer measure m is called (I) Borel if all Borel sets are m-measurable; (2) metric if m(EUF) = m(E) +m(F) for any positively separated sets E and F (i.e., p(E, F)= inf{p(x, y): x E E, y E F} > 0); (3) regular if for any A C X there exists an m-measurable set E containing A for which m(A) = m(E). One can prove that any metric outer measure is Borel.

2. Borel Density Lemma (Gu]. We state the result known in the general measure theory as the Borel Density Lemma We present it in the form which best suits to our purposes.

Borel Density Lemma. Let A C X be a measurable set of positive measure. Then for p.-almost every x E A, lim p.(B(x, r) n A) = 1. p.(B(x, r))

r->0

Furthermore, tf J.t(A) > 0 then for each o> 0 there is a set t. c A with ~J-(t..) p.(A)- and a number ro > 0 such that for all x E t. and 0 < r < ro, 1 p.(B(x, r) n A) 2: /.L(B(x, r)).

o

2

293

>

Appendix V

294

3. Covering Results [F4], [Fe]. In the general measure theory there is a number of "covering statements" which describe how to obtain an "optimal" cover from a given one. We describe two of them which we use in the book. Consider the Euclidean space lRm endowed with a metric p which is equivalent to the standard metric.

Vitali Covering Lemma. Let A be a collection of balls contained in some bounded region oflRm. Then there is a (finite or countable) disjoint subcollection B = {B;} C A such that

UBcU.B;, BEA

i

where B; is the closed ball concentric with B; and of four times the radius. be a set and r: Z --+ JR+ a Besicovitch Covering Lemma. Let Z c r bounde.d function. Then the cover g = {B(x, r(x)) : x E Z} contains a finite or countable .mbcover of finite multiplicity which depends only on m. As an immediate consequence of the Besicovitch Covering Lemma we obtain that for any set Z c r and e > 0 there exists a cover of Z by balls of radius e of finite multiplicity which depends only on m.

4. Cohomologous Functions [Rl]. Two functions cp 1 and cp 2 on a compact metric space X are called cohomologous if there exist a Holder continuous function "1: X --+ lR and a constant C such that cpl - cp2 = "' - 1J 0 f + c. In this case we write cp 1 "' cp 2 • If the above equality holds with G = 0 the functions are called strictly cohomologous. We recall some well-known properties of cohomologous functions: (1) if cp 1 "'cp2 then for every x EX, I n-1 lim - L[cpt(x)- cp2(x)] = C;

n---too

n

k=O

(2) cp1 ,...., cp2 if and only if equilibrium measures of cp1 and cp2 on X coincide; (3) if the functions cp1 and cp 2 are strictly cohomologous then Px(cp1 ) = Px(cp2)· 5. Legendre Transform (Ar]. We remind the reader of the notion of a Legendre transform pair of functions. Let h be a strictly convex C 2-function on an interval I, i e., h"(x) > 0 for all x E 1. The Legendre transform of h is the differentiable function g of a new variable p defined by (A5.1) g(p) = max(px- h(x)). :tEl

One can show that: 1) g is strictly convex; 2) the Legendre transform is involutive; 3) strictly convex functions h and g form a Legendre transform pair if and only if g(a) = h(q) + qa, where a(q) = -h'(q) and q = g'(a).

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Index

Axiom A diffeomorphlsm, 228 u ( 8)-conformal, 230

coding map, 117, 229, 239 conformal map, 199 repeller, 199 toral endomorphism, 202 j-coordinate, 100 correlation sum, 174, 177, 181 cylinder (cylinder set), 100, 101

baker's transformation classical, 244 fat, 245 generalized, 244 skinny, 245 slanting, 246 basic set, 117, 190 Besicovitch cover, 31 Bowen's equation, 100 box dimension of a measure, 41, 61 of a set, 36, 61 BS-box dimension of a measure, 112 of a set, 111 q-box dimension of a measure, 58 of a set, 52, 62 (q, 1)-box dimension of a measure, 57, 63 of a set, 49, 62

0!-density, 45 dimension box of a measure, 61 of a set, 61 Caratheodory of a measure, 21, 67 of a set, 14, 66 pointwise, 24, 67 correlation at a point, 174, 178, 179 of a measure, 178 specified by the data, 181 Hausdorff of a measure, 41, 61 of a set, 36, 61 information, 186, 224, 256 pointwise, 42, 61, 143, 223, 254 BS-dimension of a measure, 112 of a set, 111 q-dimension of a measure, 58 of a set, 52, 62 (q, 1 )-dimension of a measure, 57, 63 of a set, 49, 62 pointwise, 58

CaratModory capacity of a measure, 22, 67 of a set, 16, 67 dimension of a measure, 21, 67 of a set, 14, 66 dimension spectrum, 32 dimension structure, 12, 66 measure, 13 outer measure, 13 pointwise dimension, 24, 67 chain topological Markov, 101 301

302 entropy capacity topological, 75 with respect to a cover, 77 measure-theoretic, 77 of a partition, 186 topological, 75 with respect to a cover, 77 equilibrium measure, 97 estimating vector, 134 expanding map, 146, 189, 197 expansive homeomorphism, 97 function stable, unstable, 105 geometric construction associated with Schottky group, 151 Markov, 118 Moran, 120 Moran-like asymptotic, 275 with non-stationary ratio coefficients, 121 with stationary (constant) ratio coefficients, 120, 152 regular, 122, 134 self-similar, 122, 133 simple, 118 sofic, 118 symbolic, 118 with contraction maps, 122, 150 with ellipsis, 138 with exponentially large gaps, 156 with quasi-conformal induced map, 145 with rectangles, 153 geometry of a <'onstruction, 118 Gibbs measure, 102, 103 grid, 184 group Kleinian, 151 reflection, 152 Schottky, 151 Hausdorff dimension of a measure, 41, 61 of a set, 36, 61

Index

measure, 36, 61 outer measure, 35, 61 holonomy map, 281 horseshoe,238 linear, 240 hyperbolic attractor, 292 measure, 279 set, 227 locally maximal, 227 induced map, 122, 145, 203 irregular part, 171, 188, 222, 225, 254, 256, 259, 264 Julia set, 201 Kleinian group, 151 length of string, 68 limit set, 118 local entropy, 260 Lyapunov exponent, 223, 279 Mandelbrot set, 207 manifold stable, unstable global, 228 local, 227 map coding, 117,229,239 conformal, 199 conformal affine, 133 expanding, 146, 189, 197 induced, 122, 145, 203 one-dimensional Markov, 201 quasi-conformal, 147, 191 weakly-conformal, 191 Markov partition, 189, 228 mass distribution principle non-uniform, 43 uniform, 43 matrix irreducible, 101 transfer, 101 transitive, 101 measure (g, G)-full, 260 a-Caratheodory outer, 13 a-Hausdorff, 36, 61

Index outer, 35, 61 Caratheodory, 13 diametrically regular, 55, 62, 212 doubling, 55 equilibrium, 97 exact dimensional, 44 Federer, 55 Gibbs, 102, 103 hyperbolic, 279 of full Caratheodory dimension, 33 of full dimension, 129 of maximal entropy, 99 Sinai-Ruelle-Bowen (SRB), 281 (q, /')-measure, 49 Moran cover, 120, 124, 141, 200, 231 multiplicity factor, 124, 141, 201, 232 multifractal analysis, 173, 213, 249 classification, 267 decomposition, 171, 222, 225, 253, 256, 259 rigidity, 266 spectrum, 259 structure, 171, 259 point irregular, 47 regular, 47 potential principle, 44 potential theoretic method, 44 pressure capacity topological, 70 non-additive, 84 topological, 70 non-additive, 84 pressure function, 106 ratio coefficients, 120, 123, 133 rectangle, 189, 228, 281 reflection group, 152 repeller, 197 conformal, 199 scaling exponents, 170 Schottky group, 151 separation condition, 117

303 sequence of functions additive, 85 sub-additive, 85 set basic, 117, 190 hyperbolic, 227 locally maximal, 227 invariant, 100 metrically irregular, 95, 264, 265 a-set, 47 shift one-sided, 100, 101 two-sided, 102 Sierpinski carpet, 157 similarity maps, 122 solenoid, 241 space isotropic, 62 metric Besicovitch, 62 of finite multiplicity, 61 spectrum complete, 189, 195, 221, 226, 258 dimension, 260 Caratheodory, 32 for Lyapunov exponents, 225,257, 258 for pointwise dimensions, 171, 188 entropy, 260 multifractal, 259 for local entropies, 261 for Lyapunov exponents, 261 for pointwise dimensions, 260 Renyi for dimensions, 172, 185 !,..(a)-spectrum, 171, 188 H P-spectrum for dimensions, 172, 182 modified, 183 string, 68 structure local product, 281 loeal semi-product, 281 multifractal, 171, 259 subordinated, 15 C-structure, 12, 66 subshift one-sided, 100

304

of finite type, 101 topologically mixing, 101 topologically transitive, 101 two-sided, 102 of finite type, 102 subspace stable, unstable, 227 symbolic dynamical system, 100 representation, 117 system even, 101 sofic, 101

Index

topological entropy, 75 with respect to a cover, 77 Markov chain, 101 pressure, 70 non-additive, 84 variational principle, 33 for topological entropy, 94 for topological entropy inverse, 83 for topological pressure, 88 non-additive, 99 for topological pressure inverse, 83