Luigi Amerio x B. Segre (Eds.)
Sistemi dinamici e teoremi ergodici Lectures given at the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, June 2-11, 1960
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
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ISBN 978-3-642-10943-0 e-ISBN: 978-3-642-10945-4 DOI:10.1007/978-3-642-10945-4 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2011 st Reprint of the 1 ed. C.I.M.E., Florence, 1960 With kind permission of C.I.M.E.
Printed on acid-free paper
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CENTRO INTERNATIONALE MATEMATICO ESTIVO (C.I.M.E)
Reprint of the 1st ed.- Varenna, Italy, June 2-11, 1960
SISTEMI DINAMICI E TEOREMI ERGODICI
P.R. Halmos:
Entropy in ergodic theory ....................................................
1
E. Hopf:
Some topics of ergodic theory ............................................. 47
J. Massera:
Les equations différentielles linéaires dans les espaces de Banach ........................................................ 113
L. Amerio:
Funzioni quasi-periodiche astratte e problemi di propagazione .................................................................... 133
L. Markus:
Sistemi dinamici con stabilità strutturale ............................. 149
G. Prodi:
Teoremi ergodici per le equazioni della idrodinamica ........................................................................ 159
A. N. Feldzamen:
The Alexandra Ionescu Tulcea proof of Mcmillan’s theorem ....................................................... 179
CENTRO INTERNATIONALE MATEMATICO ESTIVO (C.I.M.E)
PAUL R. HALMOS
ENTROPY IN ERGODIC THEORY
ROMA - Istituto Matematico dell' Università - 1960
1
- I -
P.R.Halmos
CONTENTS PREFACE CHAPTER I. CONDITIONAL EXPECTATION .Section 1. Definition Section 2. Examples Seotion 3; Algebraio properties Seotion 4. Dominated ccnvergence Section 5. Conditional probability Section 6. Jensen's inequality Seotion
7. Transformations
Seotion 8. Lattioe properties Seotion 9. Finite fields Seotion 10. The martingale theorem CHAPTER II. INFORMATION Seotion 11. Motivation Seotion 12. Definition Seotion 13. Transformations Seotion 14. Information zero. Seotion 15. Addit i vity Section 16. Finite additivity Seotion 17. Convergenoe Seotion 18. McMillan's theorem CHAPTER III. ENTROPY Seotion 19. Definition Seotion 20. Transformations Seotion 21. Entropy zero Section 22. Conoavity
3
- II -
P.R.H'llmos
Seotion 23. Addi ti vity Seotion 24. Finite additivity Seotion 25. Oonvergenoe CHAPTER IV. APPLIOATION Seotion 26. Relative entropy Seotion 27. Elementary.properties Seotion 28. Strong ,monotony Seotion 29. Algebraio properties Seotion 30. Entropy Seotion 31. Generated fields Seotion 32. Examples
4
- 1 -
P.R.Halmos PREFACE Shannon's theory of information appeared on th,mathematical soene in 1948; in 1958 Kolmogorov applied the new subject to solve some relatively old problems of ergodic theory. Neither the general theory nor its speoial applioation is as well known among mathematioians as they both deserve to be; the reason, .probably, is faulty oommunioation. Most extant expositions .of inflormation theory are designed to make the subjeot palatable to non,mathematioians, with the result that they are full of words like "souroe" and "alphabet". Suoh words are presumed to be an aid to intuition; for the serious student, however, who is anxious to get at the root of the matter, they are more .likely to be oonfusing than helpful. As for the recent munioa~on
~godic
applioation of the theory,the com-
trouble there is that so far the work of Kolmogorov and
his .sohool exists in Doklady abstracts only, in Russian .only. The purpose ·of these notes is to present a stop-gap exposition of some ·of the general theory and some of its applioations. While a few of the proofs may appear slightly
different from the oorrespon-
ding ones in the literature, no claim is
m~de
for the novelty of
the results. As a prerequisite) ' some familiarity with the ideas of the general theory of measure is assumed; Halmos's leasure
theory (1950) is an adequate referenoe. Chapter I begins. with. ,elatively well known faots about conditional expeptations; for the benefit of the reader who does not · know this technioal.prohabilistioooncept, several standard proofs are reproduced. Standard referenoe: Doob,
~tochastic
processes
(1953). A speoial oase of the martingale convergence theorem is proved by what is essentially L6vy's original method (Thdorie de
5
-
2 -
P.R.Halmos
~1addition des variabLes aLlatoires (1937)). The reader who kn ows the martingale theorem can skip the whole chapter, except possi b ly Section 9, and, in partioular"
equation (9.1).
Chapter II motivateeanddefines infor.mation. Standar.d reference: Khinohin, MathematicaL foundations of information theory
(1957). The ,mor.e reoentbook of Feinstein) Foundations of informa-
tion theory (1958}, ,is quite ,teohnioal, but highly recommended. The .chapter ends with Ii. pnoof :MoMillan f s theonem Gmean ,convergenoe}; the reader who knows that theorem oan skip the chapter after looking at .it .just long enough to absorb the notation. Almost everywhereoonvergence probably holds. ' A recent paper :by :Breiman (Ann. Math. Stat. 28 (1957) 809-811) asserts ,it, but that paper has an error j at the time ,these lines ,were written the correction has not appeara,p. yet. In any ·oase, for the ergodic application not even mean oonvergence is neoessary; all th,t is needed . isthe convergence of the integrals,whioh ,is easy to prove direotly. Chapter iIII studies entnopy (average amount of .infollmation)
j
.all .the faots .hene ane dir.eot :oonsequenoes .of the definitions, via the 'maohinery ,built up .in .the first two chapters. Chapter ,IV ,contains ,the application to ergodic .theory . In general terms, the idea is ,that information theory suggests a new invariant (entropy)ofmeasure~presenving transfonmations. The new invar.iant ,is shanp enough :to distinguish ,between some hitherto indistinguishable transformations (e.g., the 2-shift and the 3.shift). The ,original idea ,of using this invariant is due to K01mogorov ' (Do~lady 119 fi958) 861-864 and 124 (1959) 754-755). An improved ,version of the definition is, given .by. Sinai
(Doklady
124 (1959) 768-771l, who also oomputesthe entropy of ergod i c
6
- 3 P.R.Halmos automorphisms of the torus. The new invariant is in some resp ec ts not so sharp as older ones. Thus for instanoe Rokhlin (Doklady 124 (1959)
9BQ~983)
asserts that all translations (in oompaot a-
belian groups) have the same entropy (namely zero); he also begins the study of the oonneotion between entropy and speotrum. Muoh remains to be done along all these lines.
7
- 4 -
P.R.Hal mos
CHAPTER I. CONDITIONAL EXPECTATION SECTION 1. DEFINITION .We :shall work, throughout what follows, with a fixed probability space
(X, ~, . pl.
~
Here X is a non-empty set, a prpbabili ty measure on
is a field of subsets of X, and P is
-8 ., The
an abbreviation for "oolleotion
word "field" in these notes is of sets olosed under the forma-
tion of oomplements and countabLe unions". A probability measure ,on a fieid .of subsets of X is a measure P suoh that p(X) = .1 Suppose that
S
is a subfield of ~
and f is an integrable
real funotion an X. If Q(C)
for eaoh C in
& ,
= JC
f dP
then Q is a signed measure on (;
, absolute l y
ooptinuous with respeotto P (all, rather, with respeot to the restriation of
P to
~ ).
The Radon-Nikodymtheorem implies the exis-
.t e nce of an integrable funotion flf , measurable 13
S '
=J
fit dP C The funotion fit is uniquely determined (t o Q(C)
for eaoh C in
, such that
within a set of measure zero); its dependenoe an f and ~
i s in-
dioated by writing
The funotion E{r/!!) is oalled "the oonditional expectation of f with respeot to ~
". It is worth while to repeat the aharac-
teristic properties of oonditional expeotatibn; they are that (1. 1)
E(f/~)
is measurable ~
9
- 5 P.R.Halmos and (1. 2)
I C E(f/~)
=I
dP
C
f dP
for each C in ~ . SECTION 2. EXAMPLES. I f that is
(;=,(;
t;
is the hrgest subfield of
6 '
,then r i tseU satisfies (1.1) and (1. 2), so tltat E(f/~ )
= f .
This result has a trivial generalization: since f always satisfies (1. 2)
(fc
f dP =
Ic
f dP), it follcws that if the field
that f is measurable
13 ,
(2. 1)
E (f/ ~) = f
I;
is such
then
To look at the other extreme, let 2 be the smallest sub field of
~ , thH is the field whose only non-empty member is X . Since the only functions .measurable 2 are constants, and since the only constant (in the role of E(f/S )) that satisfies (1.2) is
Ic
f dP,
it follows that (2. 2)
E(r/2)
The constant
I
C
=I
r dP .
f dP is sometimes called the absolute (as oppose d
toconditionall expectation of
r , and, in that case, it is denc-
ted by E(f). Here is an illuminating example. Suppose that X is the unit square, with the collection of Borel sets in the role of
4
and
Lebesgue measure in the role of P . We Bay that a set in ~ is "vertical" in case its intersection with each vertical line L in the plane is either empty or else equal to X n L . The collection
~ of all vertical sets in ~
is a subfield of ~
10
. A function f
~
6
~
P.R.H!!.lmos i:sme!isur'lble
1$
i f and only if it does not depend on its second
(vertioal) argument; it follows easily that if f i~ integr!!.ble, .then E(f/~
SECTION 3.
AL~EBRAIC
lex, y) =
J
f(x, u) du .
·PROPERTIES. 'Conditional expectation is a
generalized integral and in one form or another it has all the
pro~
perties of an integral. Thus, for instance, (3.1)
E(l/~)
= 1,
W!,0l'e this equation, as well as '111 other asserted equations and inequalities involving oonditional expeotations, holds almost everywhere. (To prove (3.1), apply
(2~1).)
If f and g are integrable
funotions and if a and .b are oonstants, then (3.2)
"-ECaf+bg/~) =aECf/~) +bECg/(:;)
' (Proof: if C is in ~, then the integrals over C of the two sides
»
of (3.2) are equal to eaoh other). I f f:;: 0 , then ~(f/~) ~
(Proof,: i f C=
J
C
O.
E(f/!)(x) < Ot,then C is in
.h
~
and
f dP = 0; this implies that P(C) = 0 ). It is a consequence of
(3.3) that
(Proof: both If I
-
f ~ 0
and
1r:1
+ f ; 0, and therefore, by
(3.2) and (3.3), both E(-f/~) . ~ Eclfl/~) and E(f/~) ~ E(lfl /~) ) . Conditional expeotations also have the following multipli c!!.ti ve property: if f is integrable and if g is bounded !!ond measurabl e
11
- 7P.R.Ha.1mes
~, then (3. ·5) Sinoe the right side. of (3.5) is measurable ~ , the thing to prove is that (3.6)
J
c
E(f/b)gdP=J
C
fgdP
for eaoh C in ~. In oase g is the oharacteristio funotion of a
I; ,
set in
(3.6) follows from the defining equation 0.2) for
oonditiona1 expeotations. This implies that (3.6) holds whenever g is a finite linear oombination of suoh oharaoteristio funotiens J and henoe, by approximation, that (3.6) holds whenever gis a .bounded funotionmeasurab1e ~. SECTION 4. DOMINATED CONVERGENCE. The .usua1 .limit theorems for integrals also have their analogues for oonditional expeataHons. Thus i f f, g, and fn are integrable funotions, if Ifni ~ g and f
~
n
f almost everywhere,. then E(f It;)
(4.1)
-+ E(fl
n
G)
almost everywhere and, also, in the .mean. For, the. proof) . write.
g
n
= sup. (I f n - fl, If
1I.1.1
-fl,
If
n.+2
-fl,.·.. );
observe that the sequence{g } tends monotone1y to 0 almost every/l
where and, that g n ~2g, .. It follows that the sequenoe{E:(g . ' . n It
)}
is monotone decreasing and, therefore, has a limit h almost everywhere. Sinoe (4. 2)
Jh dP ~ J E (g/ ~,) dP = J gn dP ,
12
- 8 -
P.R.Halmos and since
J gn
dP'" 0, this implies that E(g/~) ~ 0
almost every-
where. Since, finally, (4.3)
the proof of almost everywhere convergenoe is complete. Mean oonvergence is implied by the inequality (4.4)
and the Lebesgue dominated convergence theorem. SECTION 5. CONDITIONAL PROBABILITY. If A is a measurable set (that is A is in ~ ) and i f f
= 0 (A)
(where c(A) is the charaoteristic function of A), E(f/t;)
= P(A/~)
~e
write
•
The function P(AI ~) is called lithe conditional probability of A with respeot to ~ ". The oharacteristic properties of conditional probability are that p('V ~)
is measurable ~
a,nd
J
o
for each 0 in
G. If
(5. 1)
P(A/t',) dP ::: peA
n 0)
A is in ~, then
l' (AI ~) =
0
(A) ,
and, in any oase, p(A/2) = P(A). For this reason the constant peA) is sometimes called the absolu-
13
- 9 P.R.Halmos te (as opposed to oonditional) probability of A. The oonverse of the oonolusion (5.1) is true and , sometimes useful. The assertion is that i f P(A/G) is the oharacteristic function of some set, say B, then A is in (and therefore B
= A).
To prove this, note that
JC and therefore peA
n
t;
dP
o(B)
= PCB
C)
= p(An
C),
n C) for eaoh C in ~ .' Since p(AI ~)
is measurable ~, the set B itself belongs to ~ . It is therefore permissible to put C
=B
and to put C
=X
- Bi it follpws that B
c:
A
and AC B (almost), so that B = A (almost). Just as oonditional expectation has the properties of an integral,
oondit~onal
probability has the properties of a probability
measure. Thus if A is
measurable set, then
Ii
and if ,{An} is a disjoint sequence of measurable sets with union A,
then P.(AI
e)
= L
n
peA II:':,)
n
SECTION 6. JENSEN'S INEQUALITY. A useful analytic property of integration is known as Jensen's inequality, which we now proceed to state and prove in its generalized (conditional) form. A real-valued funotion F defined on an interval of the real line is oalled convex if F (ps + qt) < pFeil)
+ qF(t)
whenever sand t are in the domain of F and p and q are non-negative
14
- 10 P.R.H'l.lmos numbers with sum 1. It follows immediately, by induotion, that if t l ,· •• ,t n are in the domain of F and Pi'" "P n are non-negative numbers with sum 1, then
(6. 1) Suppose now that F is a oontinuous oonvex funotion whose domain is a finite subinterval of
[0,00), suppose that g is a mea-
surable funotion on X whose range is (almost) included in the domain of F, and suppose that ~ is an arbitrary subfield of
-0
Jensen a inequality asserts that under these conditions F(E(gl ~ )) ~ E(F(g) / ~)
(6. 2)
almost everywher.e. Sinoe
is the limit of an increasing sequence
~
of simple funotions, and sinoe F is continuous, it is suffioient, to prove (6.2) in oase
where the summation field of
0 . If
extends over
the atoills of some finite sub-
g has this form, then
and
Sinoe E(F(g)/~) :; 2
A
p(A/~)F(t ), the inequality (6.2) is in
this oase a special oase of In the extreme oase,
A (6.1).
i::; -b ,
the conditional form of Jensen's
inequality reduces to a triviality (F(g) ~ F(g)); in the other extreme oase,
~:; 2, it becomes the olassioal absolute Jensen's
inequality
15
- 11 P.R.H~1mos
F(I SECTION
7.
g dP) S
I F~g)
dP .
TRANSFORMATIONS. Later we shall need to know the
effeot of measure-preserving transformations on oonditional expectations and probabilities. Suppose therefore that T is a measurepreserving transformation on X; this means that if A is measurable, ,
-1
then T
A is measurable and peT
-1
A) = peA) .
(For present purposes T need not be invertible). If
~ is a sUbfieU of ~ , then T-~~ -1
is the colleotion (field) of all sets of the form T
C with C in
~; if f is a funotion on X, then fT is the oomposite of f and T.
The basic ohange-of-variables result is that if f is integrable, then
Ic
= I. T--1 C
f dP
fT dP
for eaoh measurable set C. If, in partioular, C is in ~, then
I
-1
T C
E(fT/T
=
I c- f
=
I
-1
-1
dP
T C
~) dP =
=I
C
I
-1
T C
fT dP
\ E(f/~) dP
E(fl t;)T
1
dP . -1
Since both E(fT/T- ~) and E(fl ~)T are measurable T lows that
(7. 1) Since o(A)T
E(ft/T
= 0 (T
-1
-1
~)= E(fl ~)T .
A), this implies that
(7. 2) 16
~, it fol-
- 12 P.R.H'l lrr,os SECTION 8. LATTICE PROPERTIES. The
item of in te rest i s th e
ne~t
dependenoe of E(f/~) on ~ . The colleotion of all subfields of-6 has a reasonable amount of struoture;
it is partially ordered ( by
inolusion), and, in faot, i t is a oomplete lattice.
(The infimum of
two subfields ~ and ~ is just their interseotion, and their supremum
13 v ~
is the field they generate;
similar assertions hold
for the infimum and supremum of any family of subfields). It migh t therefore be hoped that the dependenoe of E(f! ~) on
t,
ex h i b i t s so -
me algebraioally pleasant behavior, such as monotony or addi t i v ity , Nothing like this is true. If,
for instance,
13
~ are subfields of ~ with 33.:; ~ ,
and
then the best that oan be said is that E(E(f/~)!03)
(8. 1)
= E(f/1l!)
and JIE(f / J3)1 dP ~ JIE(f/e,)1 dP.
(8.2)
To prove (8.1) observe that if B is in ~, then B is in
G,
and therefore
=JB
To prove (8.2), write B = B is in ~,
h:
ECf/i;)dP
E(f/~)(x) > O} . Since, c learly,
it follows that
J
B
E(f/13) dP
=
J
B
E(f/~) dP <
JB
IECf/~)1 dP
and
-J X-B In oase f
E(f/"n.) dP = N
= o(A),
-J X-B
ECf/~) dP ~
-
JX-B ImCf / r; ) 1
the equation (8 . 1) becomes
17
dP .
- 13 P.R.Halmos
35) = p(A/3J)
E(P(A/~)/
(8.3 )
•
SECTION 9. FINITE FIELDb~ Some insight can be gained by studying the finite sub fields of ~ mic. This means that
13
. A finite subfield ~ of
0
is ato-
has a finite number of (neoessarily disjoint)
atoms whose union is (almost) equal to X; an atom is a set of positive measure in ~
that inoludes no subset of striotly smaller posi-
tive measure in 13
• A function is measurable 'TD if and only if
it
is a oonst ant on eaoh atom of ~ . It follows easily that if f is integrable, then on eaoh atom B of
35
A oonvenient way of expressing this faot is to write c(B)
= LB P(B) f B
E(f/1D )
f
dP ,
where the summation extends over all the atoms of ~. In case f
= c (A),
this becomes P(A!~ )
13
Suppose next that It follows,
B
7!>vt
where B is an atom of
7b
13\1~
P(B)
~v~
is a finite sub field of
t;
and C is an atom of
-5
0
such that
13
G and
n
C,
More generally, if
is finite, then every e-
is obtained as follows : for each atom B of
let CB be an element of
I.,
are the non-empty sets of the form B
and ~ are sub fields of
lement of
P(A n B)
and ~ are two fini te subfields of
of course, that
and the atoms of
n
= L c(B)
J'j
form the union
ded over all atoms of ~ . Something useful can be said about conditional expectations and probabilities in this case: the assertion
18
- 14 -
P. R.H"llmo s is that if f is integrable, then on eaoh atom B of E(f/7Jvt) :;
1
13
E(c(B)f/~
p(B/~)
).
Equivalently (9. 1)
:; ~B'
c(B) P(B/t,)
E( c ( B) f! ~ ) ,
where the summation extends over all the atoms of ~ f :;
C
. I n case
(A), this beoomes P (AI '12. V ~)
(9. 2)
'/J
:;
~
B
C
(B)
p(AnB/~)
P( BI ~ )
=
To prove (9.1), observe that both sides are measurable ~v~ it is therefore suffici~nt to prove that whenever B is an atom of
1j
and C is in ~ , then the integrals of the two sides of (9.1 )
over B
n e are equal. The proof is the following straightforward
computation:
Ie
c(B)
E(c(B)fl
C)
P(B/~)
:;
Ie
E(c(B)f!~) dP
:;
Ie
o(B)f dP
:;
IB (J
C
E(fl
[by (3.5)J
- IB () ~v t
C
)
f dP
dP
SECTION 10. THE MARTINGALE THEOREM. The only thing alon g t hese lines that remains to be discussed is the continuity of in
0 .
A typical and useful result is that if {t;} n
sing sequence of subfields of ~ and if
e is
(that is,
en C ~
the subfield they generate, then
( 10. 1)
19
E(f l ~ )
i.e an increant1
for · all nl,
- 15 P.R.Halmos almost everywhere and in the mean. This is a non-trivial assertion; it is a speoial oase of Doobls justly oelebrated martingale oonvergenoe theorem. The speoial oase l'
= o(A)
is all that we shall need;
the assertion for that oase is that P ( AI ~
(10.2)
n
) ~
P (AI
t )
almost everywhere. We shall give the proof for a statement of intermed1ate generality: we shall prove (10.1) in oase l' is bounded (and, of oourse, measurable). Mean oonvergenoe is then a oonsequenoe of the Lebesgue bounded oonvergenoe theorem. Assume for a moment that it is already known that E(gl ~ ) n
~
g
almost everywhere whenever g is measurable
~ (and bounded). Note
that in this oase E(g/~) = g • If l' is an arbitrary bounded measurable funotion, write g = E(f/~ ). It follows that g is bounded and measurable ~ Sinoe, by (8.1), E(g/e)
n
= E(f/t') n
,
the desired result (10.1) is proved. Conolusion: it is suffioien t to prove (10.3). If g is bounded and measurahle
t; , then it is the limit al-
most everywhere of a uniformly oonvergent sequenoe {gk} of simple funotions measurable
t; . Sinoe, by (3.4),
it follows that
20
- 16 P.R.H"Ilmos
IE(g/~) - gl <
(10.4)
n
The first and third terms on the right side of (10.4) tend to zer o uniformly as k
'"'*
IX!
Conolusion: i t is suffioient to prove (10.3)
•
for simple funotions measurable ~ , and henoe for the oharacteristio funotions of sets in ~ . The desired result has th us been reduoed to (10.5) almost everywhere, whenever A is in ~ Let € and
S be
arbitrary positive numbers less than 1. Since
the union of the fields
~n is dense in
B in that uniOn, say B is in
t; ,
there exists a set
~ , suoh that peA + B) < " 0/ 2 k
(where the plus eign denotes Boolean sum: A + B = (A - B) U (B - A)) Write
and write {Fn} for the sequence obtained by disjointing the sequenc e {B
n
Dn}; that is, F1 =
Bn
D1 , etc.
~ n whenever n > k . It follows that i f
Observe that F n is in n ~ k , then P(F
n
- A) = f
and henoe that if F =
Fn
U n~k
P(F -
[1 - peAl F , then n
A) ~ P(F)€ 21
c; n ) 1
dP > P(F ) € n
J
- 17 P.R.Halmos This implies that P(B - A) ~ P(F)
< cS!2 , it follows that I f x is in B - F,
€
Sinoe P(B - p.) ~ P(B + A) <
•
P(F) < 0/2 then P(A!~ )(x) ~ 1 -
n
This oan be expressed as follows: if n ~ k
€
whenever n ~ k
then P(A/ ~n) > 1 -
throughout B, exoept possibly in a set of measure lesa than S oall that
€o/2
+ Z/2 < 0 • Sinoe
€
j
€
re-
is arbitrarily small, it fol-
lows that P(A/ ~ ) oonverges to 1 throughout A, exoept possibly n
in a set of me~sure lees than 0
sinoe 8 is arbitrarily small, it
follows that p(A/~ n ) oonverges to 1 almost everywhere in A . This .. result applied to X - A in plaoe of A shows that p(A/l3 n ) oonverges to o(A) almost everywhere.
22
- 18 P.R.H~lmos
CHAPTER II. INFORMATION
SECTION 11. MOTIVATION. What is a reasonable numerioal measure of the amount of information oonveyed by a statement? How muoh information, for instanoe, do we get about a point x of X when we are told that x belongs to a subset A of X? It seems reasonable to require that the answer should depend on the size of A (that is, on peA)) and on nothing else. In other words, the answer should be expressed in terms of a function F on the unit interval; the amount of information oonveyed by A shall be F(P(A)). Two further reasonable requirements are that the funotion F be non-negative and oontinuous. Two experiments (or, alternatively, two statements) do not neoessarily yield more information than one. If, however, two experiments are independent of each other, then it is reasonable to expeot that the amount of information obtainable from the two together is the sum of the two separate amounts of information. If we perform two independent experiments, and if the result of the first tells us that x is in A and the result of the seoond tells us that x is in B , then we know that x belongs to a set (namely An B) of ~e~ sure P(A]P(B) (by independenoe). The reqUirement of additivity implies, therefore, that the funotion F should satisfy the funotional equation F(st)
= F(s)
+ F(t)
throughout its domain, It is well known, and it is easy to prove, that the oonditiDDF imposed on F uniquely oharaoterize F in the interval (0,11
23
J
to wi-
- 19 -
P.R.Halmos thin a multiplioative oonstant. Indeed, two induotions (one for the numerator and one for the denominator) show that F(t r ) never r is a positive rational number. Continuity result whenever r is a F (e -,r) = r F ( e
-1
positl~e
= rF(t)
yi~lds
real number. put t
1 - ~ e
whe-
the same to get
) , or, in other words,
Conolusion: exoept for a constant factor, F(t)
=-
log t; the a-
mount of information oonveyed by the assertion that x is in A is satisfaotorily measured by - log peA).
SECTION 12. DEFINITION. As the preoeding discussion indicates, the mathematioal model of an experiment with a finite number of possible outoomes is a finite partition of X into measurable sets, or equivalently, and from the point of view of the intended applications more elegantly, a finite sub field of ~
The amount of in-
formation oonveyed by one of the possible outcomes of an. experiment is a number depending on that outoome; in other words, the· amount of information is a funotion of outoomes (associated with some experiments). These oonsiderations motivate the following definition. It
a
is any finite subfield of '"
assooiated with atom A of
a.
a is
, the information funotion I(a)
the funotion whose value at eaoh point of e!l.ch
is - log peA); equivalently
(12.1)
I(
a) =
~A
0
(A) log peA) .
Conditional information is a natural and useful generalization of this conoeptj it is obtained by using conditional probabilities instead of absolute probabilities. Explicitly, i f o'and t) are subfields of
.-6 ,
with
a
finite, then, by definition,
24
- 20 -
P.R.Halmos (12.2)
Observe that I( a!~) is always non-negative. The oonneotion between oonditional information and absolute information is that ( 12.3)
I (
see (5.2). The funotion
a12),
a.)
= I(
j
Qv~, a'nci, in
I(al!;) is measurable
partioular, I( a}is measurable
a.
SECTION 13. TRANSFORMATIONS. If T is a measure-preserving
tr~ns-
t'ormation on X , then
the proof is a straightforward applioation
(7.2). It follows in
~f
particular, that
SECTION 141. InFORMATION ZERO .. (14.1)
I(
rra. c t
Q.! ~) = 0
The proof 1ei &&Sedo'll ',65. U and
1(2)
=0
then
.
th& ~.C()nv:.ent,i.Qn
log t = 0 . It follows in particular, with (14.2)
,
0..=
tha.t i f t = 0, t he n 2 and
~ = 2, that
.
These equations are in harmony with the intuitive meaning of information. Thus, for example, .cU.~) Ilxpresses that i f before some partioular
experim~nt
experiment oan veyed by the
is perfor,ed r mere
posslb~r
outoo~e
is '
reveal,
. *~~~ ! ~h,
oertaJnly ~ z~ro.
25
IS
already known than the
a.ount.gf information oon-
- 21 P.R.Halmos The converse of the conclusion (14.1) is true; the assertion is that i f I(
(i/f;) = 0 almost everywhere, then
(14.3) Indeed, i f I( (l/~) = 0 almost everywhere, then c(A) log P(A/~) = 0 almost everywhere for eaoh atom A of
a . This
implies that P(A/
t,)
is a oharaoteristio funotion and henoe (by (5.3)) that A is in (;
.
SECTION 15. ADDITIVITY. Conditional probability (of a set with respeot to a field) is algebraioally well-behaved as a function of its first argument and analytioally well-behaved as a funotion of its second argument. These faots are refleoted by the behavior of oonditional information. If
!l,
~ and
G
a
are fields suoh that
and ~ are finite,
then I(
a.>'7;) /~)
= I( 'd/~)
The left side of (15.1) is symmetrio in
+ I(Q./ ~vt;
a,
and
) .
'tJJ , whereas the
r i ght side is apparently not. _This is no oause for alarm;
(15.1)
i s merely one of two equations, whioh between them describe the who-
~= 2, then (15.1) becomes
le (symmetrio) truth. If (15.2) if
I(
rtv'i'J)
= I(lJ) + I(Q../1j)
QC ~ , then (15.1) beoomes
If, in partioular, (15.4)
(;. = I (
Q
a , then V
13 /Ov)=
I (
26
J3! Q)
- 22 P.R.H'l.lmos
CtcTj ,
If, finally,
then
this follows from (15.1) and the faot that I(~/a..V~
) is non-
negative. The proof of (15.1) is the following oomputation: I(
Qv1:J1 ~) = - rArB o(A)o(B) log P(A n BI ~) A) 0 ( B) log ( P ( Bib
= -
~ AIB
0 (
=
~A
A) IB log P (B I ~)
0 (
-
)
P(AfiB/G)
IA 0( B)
ptA (f BI t,) P(BI ~)
SECTION 16. FINITE ADDITIVITY. The following assertion is a
13
useful oorollary of (15.3). If
I(
V._n
L-1
'l1
·(Ji)=~
The proof is induotive. For n I(
13 ) l
= I(
are finite
~o = 2,and n= 1, 2, 3, ... ), then
fields (with (16.1)
1'J 1 , •.. ,1O n
0,
131 12).
n
k=1
I('t>IV
= 1,
k
k-1
i=O
1).) 1
the assertion reduoes to
For the induotion step, use (15.3) as foI-
lows:
27
- 23 P.R.Halmos n+1
I(
n
Vi=l 131 ) = I( V1=1 'l3 i =
I(
V1:1
V 1)n+1)
lj1) + I( ijn+1/
V 1:1
1'0 1 )
An important speoial case of (16.1) is obtained as follows. Suppose that
a,
,6
is a f1nite subfield of
and that T is a mea-
sure-preserving transformation on X • For each positive integer n ,
13 i
"rite
= T-(n-i),., IN
( i = 1, ... ,n, ) and apply (6 1 . 1). The re-
sult is
I
n ( Vi=1
T-(n-i) .... ) = I(T-(n-1)".) vc. I.N n
+ ~k =2
>1
for n
I (T
-(n-k)
a
k-1
I
V i= 1
T
-(n-i)h IoU
)
. This can be rewritten in the form I(V n - 1 T-i(t) = I(T- Cn -l)Q..) i=O +
I(
n-1
V 1 =0
T
Ln -1 I ( T- ( n - k - 1)
-1
k=l
av)
aI V
k T - ( n - 1) i=1
a ).
a )Tn-1
=I(
+ ~n -1 I ( (L I k=l
V
k T- ( k - 1+ 1) i=l
a.. } Tn - k -1
or, equivalently, (16. 2)
n-1 I(
V 1=0
= I( ~)T
+
n-1 };k=l
n-1
I ( Q, /
V k T-iet... )Tn - k - 1 i=1
whenever n > 1. [The preoeding oomputation made use of the faot that
28
- 24 -
P.R.Halmos
for any oolleotion
E.,
To prove this, observe that sinoe T
-1
(V 6 )
V here
of sets;
'
GC Vt
denotes the generated field. so that T- 1 G C
-1
f...
C
V (T -1 C).
Sinoe the oolleotion of those sets A for whioh
V(T- 1 G )
V G );
is a field, it follows that
For the reverse inolusion observe that T
to
T- 1 (
T
-1
A
belongs
is a field, it follows that
and the proof is oomplete. If, in partioular, ~ and
t;
!lore fields, then put
G = ~U t.
and apply (16.3); the result is that
) =
-1
T
a., v
[The similar faots about infinite suprema are proved tha same way] .
SECTION 17. CONVERGENCE. If { ~n} of sub fields of {) , and if (17.1)
I(a/~n)
~
~= I (
is an inoreasing sequenc e
then Vn I:; n ,
(1;
t, )
almost everywhere. The proof of this assertion is immediate from the definition (12.2) and the oonvergenoe theorem (10.2). For some purposes oonvergenoe in the mean is more useful than almost everywhere oonvergenoe. Convergenoe in the mean is olosely oonneoted with uniform integrability. Reoall that a se-
29
- 25 P.R.H'l.lmos quen0e {fn} of measurable funotions on X is uniformly integrable if
tends to 0 , as t
~
Ol
,
uniformly in n
The faots are these. If
f is an integrable funotion and if {f } is a sequenoe of integran
ble funotions suoh that fn ~ f
in the mean, then {fn} is unifor-
mly integrable. If, oonversely, f is a measurable funotion, is a uniformly integrable sequenoe, and if f re, then f is integrable and f
n
~
n
~
{fn}
f almost everywhe-
f in the mean. In view of these
faots, the way to prove that (17.1) may also be interpreted in the sense of mean oonvergenoe is to prove uniform integrability. Note that even the integrability of one I( a.,/~) needs proof; there is no obvious boundedness to invoke. UNTFORM INTEGRABILITY THEOREM. If N is a positive integer, then the oolleotion of all funotions of the form I( (J.,/~) (where
0....
is a finite subfield of ~ with not more than N atoms and ~
is an arbitrary subfield of ~ ) is uniformly integrable. PROOF. Take N, (t, and ~
!HI
desoribed, and let rand s be
positive numbers, r < = s • Write D If A is an atom of
= h:
r ~ I(Q.,/t;)(x) ~ s } .
(L, thsn I(a/~) =
log P(A/~) on A . It
follows that if CA then
AnD =
belongs to
t,
A
n
= h:
r ~ - log P(A/~)(x) ~ s } ,
CA' Sinoe P(A/
G)
is measurlJ.ble ~ , the set CA
for eaoh A . Sinoe on CA log P(A/~) < - r
30
- 26 P.;; .H a llllo s or P(A/~) < e
-1'
it follows that
= Ic < = e
IUl/C) < s on
Since
Sum over all the atoms A of
D
Put
=t
l'
(17.2)
+ n , s
J {x:
I
< e -1' p(C A) =
-1' < dP ~ sP(A fl D) = se
a.. to
get
I( a.!~) dP ~ Nse
=t
_1'
p(A/~) dP
D ,
J n 1«1/0) A D J
A
+ n - 1 (n
((1 I ~ ) (x) ~ t}
= 0,
l'
1, 2, ... ); it follows that
aI ~) 100
I(
<
dP = N
Since the sum in (17.2) tends to 0 as t
~
00
In =0
00
,
the proof of
uniform integrability is oomplete. COROLLARY. If { ~} n
of ~ with
Vn ~n
= ~,
for each finite field
is an increasing sequence of subfield s then I( 0..,1 ~n)
-?
I( Q.,/~)
in the mean
(L.
SECTION 18. MCMILLAN'S THEOREM. The preceding corollary can be applied to the following important situation. Suppose that ~ is a finite subfield of ~ and that T is a measure-preserving tr~ns formation on X . Write
31
- 27 P.R.H'llmos fn
=
-1 r( V n1.=0
T- i
~
t1
)
= 1, 2, 3, ... ,
n
go = I ( Ct. )
V
gn = I( ().. /
11
i=l
n = 1, 2, 3, . . . .
T-ia.) ,
It follows from (16.2) that
= ~n-1
f
If
gn
~
g
g = I(
n
k=O
n
0../ V
(l)
1=1
= 1, 2, 3, ....
T-iQ,) , then (17.1) implies that
almost everywhere, and the oorollary of the uniform inte-
gn ... g.
grability theorem implies that theorem applied to
g
in the mean. The ergodic
implies that the averages 1
n
n-1 n-k-1 \=0 gT
tend to some invariant integrable funotion
h
both almost every-
where and in the mean. From these faots it is easy to deduoe the 1
following assertion, known as MoMill 'a n' s theorem: -
n
the mean. Indeed,if
II f"
1 n
-. II
II <
II::'n <
=
f
n
-. hI!
n
n-1 Lk =0
II
gk -
~
n
in
= flfl dP, then 1
n-k-1 T - h k=O gk
~n-l
n
n-k-1 n-k-1 n-1 T L' ) II + - gT (gk k.=O
1
f '"7 h
II
+
II
1 n
~
II .;.n
II
Ln - 1 k=O
gT
n..,k-l
- h
II
n-J.
k=O
and the proof of MoMillan's · theorem (for not neoessarily invertible transformations) is oomplete. Presumably MoMillan's theorem oan be sharpen'ed by adding the oonolusion that where.
32
1 n
f
n
~
g almost every-
- 28 -
P.H.H"llmo s
CHAPTER III. ENTROPY
SECTION 19. DEFINITION. What we he,ve been studying so far is the amount of information (absolute and oonditional) furnished by an experiment. An assooiated oonoept of importanoe is the average amount of information so furnished. The best sense of "average" here is "expeoted value", where the expeotation in question may itself
a, e
be absolute or oonditional. Suppose, to be speoifio, that and
5J
are subfields of
~
(L
,with
finite,
and write
the funotion H is oalled entropy, the entropy of nal entropy of
0...
iJ
~ =
the oonditio-
with respeot to ~ , or the mean oond-itional en-
tropy lith respeot to If
0... ,
iJ
of
~
with respeot to
~
.
2 , then the mean oondi tional entropy with respeot to
is a oonstant
H(
(1,/ l;
);
it follows from the elementary proper-
ties of oonditional expeotations that
H(a,./~) =
(19.1)
If
~ = G,
f
I(Q,IC) dP.
the mean oonditional entropy with respeot to
funotion H( 0..,/
~
is a
0L
(19.2) it follows from the elementary properties of oonditional expectations that
f
(19.3) If
H( Ct.)
= H(
H ( ()., / ~) dP =
a../2) and
H( Q,) = H(
H( Q. /
~ )
a,/2), then it follows fr om
33
• 29 -
P.R.Halmos (19.2) that H(a.,) is a oonstant, and it folloll's from (19.3) that
-
the oonstant is H( Q,). Note also that H(Q.) =
H(1) = J
I(a.) dP .
Conditional entropy oan be oomputed direotly from the definition (12.2) of information;
H«llt;) = If, in partioular, (19.6)
G=
- L
2
A
P(A/(;) log p(A/~) •
then
Jr( Il) = - L
A
p(A) log p( A) .
SECTION 20. TRANSFORMATIONS. Yost of the properties of entropies are relatively easy to derive from the oorresponding properties of· information. Thus, for instanoe., serving transformation on
if
T
H,(a/~)T = H(T-1a.IT-1~), and, in partioular,
and
= H('l~
-1
a.)
SECTION 21. ENTROPY ZERO. If H(
0.,1
t)
measure~pre-
X, then, by (13.1) and (7.1),
\
(21.1)
is a
=
0 ,
and, in partioular,
34
a. c ~
I
then
- 30 P.R.H!llmos (21.2) and
H( 2) = of.
0
(14.1) and (14.2). Conversely i f H(a../~)
tly, i f
=0
almost everywhere (or, equiV!llen-
H(a./~) = 0), then I(a,/~) = 0
(19.1 »,
a..C~ (by (14.3».
and therefore
SECTION 22. CONCAVITY. If Q.,
a.
that
ljc
is finite and
almost everywhere (by
~
13
and
~
are fields such
,then
(22.1) The proof is an aHlioation of Jensen's inequality. I f F(t) = t log t whenever t > 0 funotion;
and
F(O)
=0
, then
F
is a oontinuous convex
it follows that
F(E(p(A/~)/15); E(F(P(A/~)lra)
(22.2)
for eaoh atom A of , a,. By (8.3) the left side of (22.2) is equal to F(P(A/'b
»;
this implies that
To get (22.1), ohange sign and integrate; of. The speoial oase
~
(19.5).
= 2 is worthy of note :
(22. 3)
35
- 31 P.R.Halmos
a,
SECTION 23. ADDITIVITY. If
~
, 'loud ~
are fields such
Q., and ~ are finite, then, by (15.1),
that
It follows that
(23.2) and, in partioul!l.r,
Further speoial oases: if
a c~,
then
H«(lv'tJ/t;) =H(~/~) , H(a,v~/0) = ii(1Jllj) ,
H( ct v 1f> 1 <1-)
=
H(lj/Q,)
It follolls from (23.2) and (22.1) that (23.5)
H( a. V ~ 1 ~)
.~
H( (1,1 ~ )
3.nd henoe, in partioular (of.
If
a,c'3
J
+
(23.4)),
then (by (15,5))
and therefore,
H(o../~)
< >= H(~/\.;» 36
.
H( lbl
~)
,
- 32 P.R.Halmos It follows (put
t; =
2) that i f
a,c
,
~
SECTION 24. FINITE ADDITIVITY. If finite fields (with (24.1)
~
H(
o
V
n 1=1
= 2,
and
7j) = ~ i
n
then
~1'"''
1'3 0 '
= 1,2,3, ... ),
H( 10 / Vk-1
n k=l
k
i=O
~ "I,) n
ar e
then
~.) 1
this is an immediate consequenoe of (16.4). If T is a measure-preserving transformation on X , then (16.2) applies; the oonolusion is that (24.2)
-( H
n-l T.- i
V 1:0'
1\ ) IN;:
whenever n > 1 .
SECTION 25. CONVERGENCE. If{ oe of subfields of
~
J
and i f
t. n }
is an inoreasing sequen-
!; = V ~ , n
n
then
(25.1) almost everywhere and in the mean. The proof is jmmediate from the oonvergenoe theorem (10.2) and the equation (19.5); reo~ll t h ~ t sinoe t log t is bounded for t in [0,1], the Lebesgue bounded c on vergenoe theorem is applicable. An immediate oorollary is th at (25.2)
37
- 33 -
P.R.Halmos
CHAPTER IV. APPLICATION
SECTION 26. RELATIVE ENTROPY. If
T
is a measure-preserving
transformation on X , then
ii (tt; V
(26. 1)
k
i=l
) ~ ii (Q,; V
T-i a,.
(Xl
i=l
T
-1
~)
alternatively, the same conclusion can be derived by applying termby~term
integration to McMillan's theorem.
The ,limit in (26.2) is an important one in the theory of measure-preserving transformations . If we write h(Q.;T)=
(26.3)
H(
(1;
V
(Xl
1=1
T ,-1~) ,
then (26.4)
h(
Ct, T)= lim
l. n
ii (
n-1
Vi=O
-i T
~).
We may call h( (1, T) the entropy of Trelative to Q.,
SECTION 27. ELEMENTARY PROPERTIES. I f
tl
and ~ are finite
fields, then, by (23.2) and (26.4), (27.1)
If
ae 'e '
( 27 . 2)
h{
a, V 1b,
T)
~ h(~, T) + h(~, T) .
then h (Q., T) ~ h (
1j,
T) .
If Sand Tare aommutative measure-preserving transformations) then h(S Observe, indeed, that
-1
a, , 39
T)
h(a"T).
- 34 P.R.}blmos 00
V i=l
S
-1
V
(
00
i.=l
it follows that
V
~(S-l~/
00
i.=l
A speoial oase of value is h(T
-1
n
T) = h (
IN,
0.., T)
.
SECTION 28. STRONG MONOTONY. The next result is that if T is invertible, then f27.2) oan be given an infinitely sharper form. If, to be preoise,
Q, and 'If) are finite fields suoh that
V
,., C
IN
+00
l'
T ".,
1.=-00
'I)
then
(28. 1)
h(
eL,
T) ;
h(
1b
T)
I
The proof begins with the observation that k-1
V i=O
T- i Q.., <::
for all positive integers
(28.2)
_
H(
k-1
V
i=O
.
T-1 (1 )
V
k-1 1=0 and
k
< = H(
+
T-iQ.;
n
V
V j =-n-k+1
Tj
1J
It follows that
n
V
+n j j=-n-ktl T
'TV)
He V ~:~
The seoond summand on the right side of (23.2) is dominated by
40
- 35 P.R.Halmo s
=
l-l i=O
H(
aI V
+n
Tj .",.) ,
j=_n'J
and therefore H(
V k-1-i . T Q.,)
<
1=0
+ This implies that 1
( 28. 3)
R(V
k
k-1 T- i (0) ~ i=O
gn .+k
1
k
2n+k
+H«(t
2n+k-1 V j=O
H(
V
+n
j
~c
Sinoe
+00
=-n
T-j'Yl. .
~,;
Tj~)
•
Vj=_OO_
TJ~
, it follows from (25.2) and (-21.2)
that the seoond summand on the right side of (28.3) tends to as
n ....
00
•
Choose
the result is that
n h(
large and t.hen,
(1, T)
for fixed
is dominated by
h(
n, let
0
k ....
00;
~, T_ ) plus an
arbitrarily small positive number. This oompletes the proof of (28.1).
SECTION 29. ALGEBRAIC PROPERTIES ·. . The preceding results give some idea of how
h(
Q"
T) depends on
a,;
lie turn nOli to stu-
dy the lIay it depends on T . The prinoipal result along these nes is that
41
li~
- 36 P.R.Halmos for each positive integer k .
Vjn-1 =0 It
V ~-1
'v =
k -j ~
(T)
V k-1
fA 0/J _-
To prove this, write
T- i
i=O
T-jk
J=O
V
1\
I..(,...
;
then
k-1 T-1a, 1=0
= Vkn-1
T -i(;0
L=O
follows that
.2
= k
kn
HeVkn-l T-i~) i=O
and hence that (29.2) k
Since
, the assertion (29.1) follows from (27.2) (for T ) . If
T
is invertible, then
(29.3)
h(
This follows
Vn-1
-
H{
i=O
Cl,
i~mediately
i
-
T (1 )= H (T
T
-1
)
h(
a..,
T)
•
from the equations n-l ·
V n-l. T- i i=O
Il ) IN
=
H( V n-l
i .=O
T-ia, ) .
Combining (29.1) and (29.3), we obtain
for every integer
k.
SECTION 30. ENTROPY. The entropy h(T) of a measure-preserving transformation
e30. 1)
T
1s defined by h ( T) = sup h e
a..,
T) ,
where the supremum extends over all finite subfields of Since, by (29.2)"
42
A>
- 37 P.R.Halmos k. h (
k h (T ) ,
ct, T)
and since, by (29.1),
it follows that ,
(30.2)
It
= k.h(TJ
h(T)
for every positive integer If
T
is
then, by (29.3),
i~vertible,
h(T
k.
-1
)
= h(T)
,
and therefore j{
h(T ) = Ikl.h(T) for every integer
k.
SECTION 31. GENERATED FIELDS. The following result is an efficient tool for calculating the entropy of some transformations. If T is invertible and i f
f3
is a finite subfield of
then (31.1)
h(T) = h(~, T)
Indeed, (28.1) implies that h(~, T) ~ h(~, T)
for all finite . subfields
~ of..4; 43
, and hence that
~ such that
-' 38 P.R.Ha.l!nos h(T) ~ h(15. T) . The reverse inequality follows from the definition of h(T). A ourious oorollary of the preoeding result is the assertion that i f
T
rj'
is invertible a.nd i f
of ~
is a finite subfield
that
suo~
then h(T) = 0 . Indeed, (31.1) implies that h(T) = h(75, T). The invertibility of
T
implies that
l T - 1 Iv
and therefore, since
v
.00
=
~c. V
that
(l)
i=l
T-ir;;
1=1
T-i ~
. The conolusion now follows from (26.3)
and (21. 2).
SECTION 32. EXAMPLES. We are now in a position to oompute the entropy of some transformations. T
Suppose, to begin with, that se
T-ifL=
and all
n
a.
for all
0.. ,
so that
This implies that
h(
a.,
is the identity. In that can-1 i = II for all rL V 1=0 T1 T) = lim - H( a.) = 0 for n
a-
all (1, and hence that h(T) = 0 . If
T
k
has finite order, i.e., T h(T k ) = 0
sitive integer k , then
is the identity for some po-
by the preoeding paragraph, and
it follows (from (30.2)) that h(T) = 0 Suppose
ne~t
that
X is the circle group and that
rotation, Tx = cx , where
o
T
is a
is not a root of unity. Let ~ be the
field consisting of the half-open top arc and its complement (and,
44
- 39 P.R.H!llmos of oourse, the empty set and X). For suitable values of -i
sult of applying T
i
the re-
to the top aro yields arbitrarily sm!lll !lros,
whioh oan then be rotated around to nearly arbitrary positions. -i
(All rotations needed are by
T
, i : : 0, 1, 2, ... ). It follows
that every half-open aro is approximable arbitrarily olosely by
Vi : O T-i l3 '
sets in \ /
00 V i=O
T- i ".
and henoe that every suoh aro belongs to
) This implies, by ( 31.2, that h ( T ) = O.
'U
Consider now the
~easure
spaoe with the points
0, ... , k-1
X be the
(k :: 1, 2, 3, ••. ) bearing the measures
P
bilateral sequence spaoe based on that spaoe, let produot measure in
X, and let
T
be the bilateral shift on
is the field generated by the sets then
tOO
V1.=,.00
i
T
'fb
Since the fields
=,6
be the usual
.<x:
x
o
=
d,
i
= 0, .. ,k-1,
and therefore, by (§1.1), h(T) = h(
T-i~ , i ::
0, ... ,
n-1
X.
10,
T).
are independent, it fol-
lows that
H(
Vn-1 1=0
T-i'A ) = r()
nH(~)
,
and henoe that
This implies, finally, that h(T) = - ~
k-1 i=O
p
i
log Pl'
If, in particular, Po = ... = Pk-1 =
1 k'
then
h(T) = log k . This
proves that the 2-shift is not oonjugate to the 3-shift.
45
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
EBERHARD HOPF
SOME TOPICS OF ERGODIC THEORY
ROMA - Istituto Matematico ddll' Universitit - 1960
47
SOME TOPICS OF ERGODIC THEORY by E. HOPF 1)
INTRODUCTION. The first topic discussed in these lectures is a
gen~-
ral ergodio theorem for positive linear transformations of certain general funotion spaoes into themselves. The author conjeotured this theorem sometime ago but it was proved only recently by Chacon and O~nstein.
Itoontains as speoial cases the wellknown individual er-
godl0 theorems in the oase when the linear function-transformation is· induced by a point-transformation of the underlying point-space onto itself. The theorem is presented in two equivalent forms (first and seoond ergodic theorem) the first of which deals mainly with additive set functions whereas the second expresses it in terms of point
f~otions.
There is nc denying that the second form of the
theorem has at .present greater immediate appeal than the first. However, the theorem in that seoond form actually deals with equivalenoe olasses of point-functions (i.e. with set funotions) rather than point-functions themselves, and lor this reason -the first form of the theorem has been included here. We also oonsider .implications of the theorem. We discuss ·1;he oase where the operator is induced by a point-transformation and weoonsider here not only ·one_to-one (classicaloase) but also ·manjl,::,to-onemappings. The .lat.ter case is interesting beoausethe ergodio theorem deals with operators in the first place and beoaus-6, ,in gener.al, a many",to_one mapping may be ----------------~--
1)
The author expresses his appreoiation to his Italian colle'1gues for their invitation and to the Office of Naval Research, Washington, D.C., for their support of this work.
49
- 2 E.Hopf regard~d
as an operator in many different ways. Operators corres-
ponding to the same mapping may have widely differing speotral oharacteristics. A lot of ingenuity has
be~n
applied to the generalization of
the classical ergodic theorems and les,s attention has been gi ven to the ergodic behaviour in specific cases, speoifio mappings and specific differential systems . There are v'arious open problems whioh are undoubtedly of considerable m!lthematic!ll interest. It is true that the solutions of the classioal problem of three bodies behave "in general" (apart from a set of measure zero in phase spaoe) such that at least one of the bodies goes infinitely far away from the oenter of gravity as t
~ 00
?
In the language of ergodic theory, is the general problem of three bodies of dissipative type? In certain limit oases of the proble1!i this can oertainly not be so, for instanoe, in Hill's lunar theory where one mass is zero (moon), anpther (sun) infinitely large and infinitely far aW!ly from the earth whioh is the origin of a speoially ohosen rotating ooordinate system. For oertain values of the moon's energy the manifold of ,oonstant energy in phase space oonsists of two 'disjoint parts one of which has finite phase volume. In this part the phase motion oannot possibly be of dissipative type. It is metrioally transitive in this part or, in other words, does tKe moon "in general" run through all positions of the part with mean
ti~e
of sojourn proportional to the volume of the
portion? While such questions can hardly claim astronomical interest (the time intervals for prediotions of ·ergodio theory to come true are probably so .large .that the Newtonian three body sys t em is no longer a good idealization) they are, mathematioally, justified and .interesting. The reason for that is that the aotual stab i-
50
- 3 E.Hopf ·li ty of the classical .per.iodic motions is doubtful ,in spite of their infinitesimal stability. Such questions are not new. They have been voiced already by K.Schwarzschild and G.D.Birkhoff. They are also very difficult, and we have no intention to discuss them i n det ail in these leotures. However, there is another ' oirole of diff e rential problems ,in whioh the ergodio properties of the solutione are oomparatively easy to determine, namely, the geodetic flow on oomplete surfaoes of negative curvature. These surfaoes furnish many interesting .examples for ergodic theory. We present thi ~
subjeot here essentially in the same simple way in whioh we
had treated it in our original.memoir [1-1]. The main idea there has been oondensed here into a "prinoipal lemma".
1a. THE ERGODIC THEOREM FOR POSITIVE LINEAR OPERATORS .
....
Consider a fixed spaoe X with points x, y, . . . and a fixed .u-field long to
f of subsets A, B, ... of X . X itself is supposed to bef , X! f . Real ... valued point funotions in X are denoted
by small Latin letters, f
= f(x)
, g
= g(x), . . . .
tions oonsideredin this paragraph are
All point funo-
f . . measurable,
either by
assumption ,or by implioation. Real~valued .set funotions in
f
whioh
are finite and u-additive are denoted by small Greek letters, '¢ = ¢(A) , .p = .p(A), . . . . A finite u-additive set funotion 1T in
f
whioh is :> 0 is a fin1 te measure in
f.
By 0 we denote the totality of the finite u-additive set funotions
¢
defined in
r.0
forms a Banaoh spaoe with the norm
II¢I! = total
variation of ¢ in X .
We use the symbol
(1)
¢-<1T,1T?O, 51
- 4 -
E.Hopf in the sense that 1T(A) = 0 , A (;
t ,~
¢(Al =0 (¢ is absolutely
oontinuous with respect to the measure 1T).Throughout this paper
,m.
the left hand side is ned .in
f )
The right is always a measure (defi-
but not necessarily finite (not ·neoessarily
Em). Howe-
ver we require 1T to be a-finite: X is the union ,of 'count ably many sets (
61 )
with 1T <
CD
•
We are mainly ,interested ,in finite a-ad-
ditive set functions ,beoause ,they arise naturally from the applic ations to Markov prooesses (mentioned fallther. )elow) ,but other OIpplications (to infinite' invariant .measures ,of ,point..,transformations) require the oocasional ,use of possibly infinite but a-finite measures J.L. For a given CT-finite measure J.L defined inf.. we make use of the no· tation (2)
¢J.L
= [¢I~e,¢,
¢-(J.Ll
Consider now a linear .operator T whioh aots on ·membersof
,m
and whioh.,..produoes again ,membens 'of 0, ,¢' =T¢ . In view of the most ,import ant applioations it .wouldnotbe .natural to demand of Tthat ,it be defined in all of ·0, but ,of the ,subspaoe ,0'ln whioh it ,is defined .we ,require .that i t ,has the ,following :olosune propert .ies Csee [14], § 2) : (I) ) (I)' is a linear ,spaoe. ,(I) ) (I)'
1 ,¢ ... ¢B' ¢B(A) = ¢(A-B), BEf
is olosed .underoontraotions
2
. (I) ) ¢ €
3
n
m',
¢
n
<.
¢, ¢ finite in
f '
implies that ¢! (Il'. (I) itself has all these propenties and so ,does every sp .. oe (I)
J.L
,where J.L .is a a.,finite measure .in ~ . The latter ,faot refleots well known faots of the .theory of integration if ·it is translated by .means of the isometry between
,m
and·L J.L
stated farther .below. It J.L
is also true that 1T
E (I)' "
1T.?
0 ~
(Il C
1T
(I)'
•
(see [14], § 3). Of the .operatorT we demand that
52
- 5 E.Hopf T1 ) T0 'C OJ' • T ) Tis li n ear. T ) Tis po s it.~i v e
3
2·
rp ~ 0 ~ T¢ ~ 0 • T4) II T¢ II ~ II ¢ II, ¢ E OJ' •
The last condition means simply that liT 1I ~ 1 . In order that a positive linear operator T satisfy this condition it is necessary and sufficient that (T¢(A) denotes the value of T¢ at A) T')
¢~O ~
4
T¢(X) ~¢(X)
The necessity is trivial as II¢II = ¢(X), ¢ ~ O. That the condition is sufficient follows if we use the well-known fact ([7], Ch~p.VI)
¢
= ¢t
that every
¢Q 0'
is the difference of two finite measures
are oontractions of ¢ and therefore c 0')
_ ¢- (¢t and _¢
such that II¢II = ¢ +(X) + ¢ - (X). In fact T¢ = T¢
t
-
- T¢,
+IIT¢1i ~ IIT¢ II t IIT¢ II
....
4),
and, by virtue of the positivity of T and of T
The most important operators of this kind are the Markov operators, the positive linear operators T with the property that N(x) '" ¢(X) .
They are intimately associated with Markov processes in X : If ¢
~
0 , ¢(X)
=1
, is a probability distribution in X then T¢ is
the probability distribution which results through the process. A Markov operator possesses a representation T¢eA) =
J
P(x,A)¢(dx)
X
where P(x,A) can be interpreted as the probability of transition of x into A ([14], §2). The proof given in [14] works also for 53
- 6 E.Hopf the general T and furnishes the same representation (3) where P is still
~
0 and
a-~dditive
in A but where
P('x, X) ~ 1
instead of being
E
1.
We want to state two equivalent ergodio theorems, the first in terms of set functions and the second in terms of point functions. The proof of the second theorem is given in the following paragraph. Here we discuss .solely their meaning and .we prove their equivalence. In order to state the first we need a weak form of the splitting of a finite a-additive set function ¢ into absolutely continuous and singular part with respeotto a finite or a-finite measure: There exists a (perhaps vacuous) set N such that 7T(N) :; 0 , ¢
-<
7T
wi thin
Within X-N there exists therefore a
X-N .
Radon~Nikodym
derivative of
¢ with respect to .7T. It is, of course, determined only up to a set with 7T :; O. The ambiguity in the choice of N and, also, of the derivative in X-N will have no effect in what follows. Let us now.return to the situation faced above. We have a linear space ¢' of finite a-additive set functions as stipulated and a positive linear operator T of ¢' into ¢'. Let ¢f.¢', 7TE¢', 7T
~
O. The ergodic theorem is concerned with the sequence of pairs
of set functions (the second is > 0 as T is positive) S ¢ n
(4)
n
:;
1, 2, ...
n-1 :;
~ 0
J)
T ¢
S 7T n
n-1 :;
~
J)
T 7T > 0
,
0
They are, of course, all in ¢' • Take, for each n,
any set N such that n
54
-
-
~
E.Hop f
=0
S 7T(N ) n n
, S ¢ n
-<
S 7T n
wi thin
X-N
n
and take any Radon-Nikodym derivative q (x) (within X-N ) of the n
n
first with respect to the seoond set funotion (4 ) , S ¢(A) ..
(6)
J
q
nAn
7T (dx), A, X-N
(x)
S
11 '=
0,1, ·...., n-l .
n
n
(5), first part, means th!lot II
T 7T(N ) -0, n
Th e ergodio theorem is oonoerned with the pointwise limit .behaviour of the sequenoeof Radon-Nikodym derivatives q (x),
q (x),
1
2
The only points x that .matter .in this context are those .in which at
le~st
a tail end of this sequenoeis well defined. They are
preoisely the points of the set
u1 n (X-N i ) CD
CD
CD
=
n
U 1
CD
n1
(X
CD
U n
Ni
or, in other words, o! the set X-NI,
(8)
(~ )
NI n
NI n
CD
= U n
N i
implies that II
T 7T (NI ) n and finally, as (9 )
CD
= n 1
NI
II
=0 ,
NI C NI n
T 7T (N I )
= 0,
,
0 ~
II
< n
n > 1
,
that
II
= 0,
,
1, ...
Result: The Radon-Nikodym derivatives qn(x), n = 1, 2, ... , a r e at least tailwise defined outside some set NI whioh satisf ie s (9) .
55
- 8 E.Hopf The same argument .makes it olear that they are not only defined but oompletely determined in this sense: Any otherohoioe of sets N
and derivatives '1
n
n
affeots values ,in tails only in
.':1
set with
jihe property (9). In partioular, ,the limit behaV:iour of ,the '1 (x), n
n
-<
r:JJ ,
is independent .of' suoh a iohoioe outside of some suoh set.
With these faotsin mind weoannow state the FIRST ERGODIC THEOREM. Let ,0 1 denote a linear spaoe ,offinite a-add. set funotions v'e linear ,opera.tor from
':IS
stipul!ted above and let T ,be a positi-
·0' ,to ,0',. 'Let ¢,.C/J', 7Tc. 0 1"
7T ~
o.
Oonst-
n-1 " ,11 . der' the operators·S = ",-·T and denote, for eaoh n = 1, 2, ... , n o by Nn any set with the property (5) and by qn(X) any Radon~Nikodym derivative in X-N norm IITII~ 1
n
of S ¢ with respeot to S.n . Then, if T has a n n
J
exists and is finite in eaoh point outside some set N whioh oontains the set NI defined by (8) and whioh satisfies T 1I n(N) 11
=0
,
= 0, 1,... . In virtue of what was said before it is olear that in order
to prove the theorem, it suffioes to prove it for some sequenoe, Nn , 'In' We have to remember this later. An important .oase of this theorem is the one in whioh ¢
-<
n
(in X): It was shown i.n [14], § 3, that (both in X) (10 )
For the sake of oompleteness the simple proof of (10) is mentioned here a little farther below. From (10) it follows that
n-1
L: o
11
T ¢-<
n-1
L: a
11
T n
in
56
X .
- 9 E.Hopf In fact,
the vanishing of the right hand side for some set A im-
plies the vanishing of .eaoh of its terms and, by virtue of (10),
~ T'1I¢(A):;: 0
T 1T(A) :;: 0
whioh in turn implies that the left hand
,side.is zero. So, under the additional hypothesis ¢
-<
11
,
all the
=0
derivatives q (x) .may be taken asdsfinsd .in all of X and N' n
maybe assumed. We add here the proof of (10). Use the representations (3) of 'T¢ and T1I and observe that T1T(A) :;: 0 . . 'lTGB} :;: 0, B :;: [x /P(x,A) > oj . Therefore, tr(B') :;: 0, B'C B. From the ;premise in , (10) it follows now that ¢(B') :;: 0, B' C B. This and the representation of T make it obvious that T¢(A) :;: 0 . (10) :is .hereby proved. It was mentioned above that the ·simplest of the spaces 0' oonsidened ,is a spaoe ,O
where
° Co °
j.J.
,is a finite ,measure, simplest in
}J.
the sense ,that
fi. '0'
.::>
j.J.
and that every
°
j.J.
is itself a space
0'. The following ,l'emark ,:Ls :teohnically .useful to us. LEMMA. ,Under ,the ,hy.potheses of ·the first ,theonemthene exists a subspaoe
°
j.J.
of 0' with some finite measune
that the
,j.J. f~'suoh
relations ,mentioned ,in these hypotheses staytnue ,with '¢
in plaj.J.
oe of
0 To see this ws must pnove the existsnce of some
'j.J.
e.¢' suoh
that (11 )
whel'e ¢ and
¢ 11
E
°
T¢
j.J.
j.J.
C
¢
j.J.
ane the set funotions ,given in the hypothesis of the
theorem. For ,this ,punpose we ,use again .the repoesentation ¢
= cj/
of ¢ as a differenoe of two finite measures whioh are both in We olaim .that :themeasure de:£ined by
57
¢'.
- ¢-
- 10 -
~
=
~
~ 2
-II
II
n' = ¢ + ¢ + n
n' ,
T
E.Hcpf
+
1
fits the bill. Obvicusly , ¢
<
jJ-
is finite because (seeT
4»
TJ;!71' (X) < n ' (X).
n', n~ 71 ' and
( 12 ) The first twc relaticns are, by virtue of (2), identical with the first tilO relations (11).,
and the thiI'd of the ,latter relations
follows from the third relation (12) and from the basic fact (10) ,'IS
applied ,to rj;,
jJ-.
In ,order to formulate ,the ,ergodic ,theorem ,in terms ,of ,point funotions we stant from a fixed ,measune
' jJ-
(defined ,in
.f ) ,which
need not be finite but whioh is supposed ,to be ,a_finite. Consider the Banach , space ,L
with the norm
jJ-
II til
=
J If (x) I ~( dx ) X
and denote ,by ,T a positive linear operator from LjJ- tOLjJ- with a norm
II T II ~
1. Strictly speaking, argument and ,value ,of
Tare
both jJ--equivalence ,classes of ,jJ-.,integrable functions ,f(x) ,in X. We may, however, regard ,f and f keep in mind that f
I
I
,in f
= Tf as funotions ,if we
I
is determined only up ,to a set with
=0
and that a change of f in a set withjJ-
,it = 0
has no effect on the
jJ--equivalence class fl. SECOND ERGODIC THEOREM. ,Let
f
and let
a norm tients
II T II
T
be a a-finite measure in X ,
J:L
be a positive linear operator ,from .Lto L
~ 1. Suppose that f
jJ-
e ,L
4 q (x) = q ( x; f, p) = n n n-1
L 0
58
with
, P ~L,p ~ 0 . Then the quo-
n-1 0
jJ-
II
T f(x) TII¢(X)
E.Hopf n
= 1,
2, ... , have a finite limit in each point outside some set Cl) v N which contains the set NI where I T p(x) = 0 and which satio dies J.l(N-NI) = O. Before discussing the .implications of these theorems we show that th.e first th.eorem jor the case ,0 1 = ·0 .wher:e J.l is a given
finite.mBasur:e in
f
J.l
CT-
and . the,second theor:em.aremer:e~y different
forms of the same fact. This implies the equivalence in question beoause, 'ly virtue of the lemma, the fuJl first theorem is already a consequence of ,its ,speCial case in which 0 1 = 0
J.l
a finite measure in; • The proof of equivalence rests upon ·the
with J.l being
one-to~one
linear i-
sometry between a spaoe0J.l with a given a-finite measure J.l and the spaoe LJ.l of all J.l-integrable point funotions in X or, rather, of the J.l-equivalence olasses of these funotions. The isometry is furnished by the relations
(13)
¢(A)
=f
f(x) J.l(dx) A
(f = Radon-Nikodym derivative of ¢ w.r. to J.l) and
A linear operator T from L
to L
J.l
(14)
T¢(A) =
a linear operator T from
f
J.l
A
induces by means of the formula
Tf(x) J.l(dx)
0 to 0 and vioe versa. The use of the J.l
J.l
same letter T in both spaoes is permitted sinoe the distinotion in the notation for set functions and point funotions makes it olear in eaoh oase whioh Tis meant. It is also olear that positivity of T in one spaoe implies the positivity in the other. Sinoe oorresponding funotions ¢ , f have the same norm the two Tis have the
59
- 12 -
E.Hopf s!1.me norm. For either T we use the positive linear ,oper!1.tors
(15)
S
n
:;
n-l v L T
n >0
o
which had !1.lready been introduoed before in one C!1.se. For corresponding functions ¢ , f there ,holds
(16)
S
¢(A) :; f S f(x) J.L(dx) nAn
and, for corresponding ,non-negative funotions
If, for given
1T,
it is
that
o~ious
p,
71,
p, a set Nn: is defined by
J.L
~
S
n
holds within X-N n . Consequently,
1T
(19) If we define
Snf(x)
(20)
S
in X-Nn
th~n
p(x)
we can infer from (16) and (17) that
Sn,¢(A) :; holds for any A € by (20) is a
n
fA q (x) S p(x) nn
t ,
J.L(dx) :;
fA
qn(x) Sn
Radon~Nikodym
derivative of S ¢ with respect to S n
n
n
n
OJ
N'
(dx)
A C X-N n . In other words, qn (x) as defined
within X-N . The sets (18) form a desoending sequenoe n the sets N', N' formed by them ,through (8) are N' :; ,N (21)
1T
v
:; [xl L T p(x) :; 0]
i>
To this we add the ,remaIlk that
60
1T
Therefore, n
and
- 13 E.Hopf Q)
(22)
1/
L T ?T(N) = 0 o
#
where N' is the set (21) and where n
,u.(N-N') = 0
N;:l N', ~ 0
and p
~ 0
are correspon-
ding functions, n(A)
=J
A
p(x) ,u.(dx) ,
The truth of this is evident from (21) and from the identity Q)
1/
L Tn(N) = o
Q)
J Nl
1/
Q)
1/
L T p(x) ,u.(dx) + I N_N, L T p(x) ,u.(dx). 0 0
Now it is obvious that the second theorem and the special first theorem, 0' = ,0
where,u. is the ,u. in the premise of the se,u. oond theorem, say exaotly the same thing in the following sense,
The funotions f, ¢ and p, n as well as the two operators T correspond 'to each other, respectively, under the isometry between L,u. and 0,u.. and the first theorem is understood to be stated for Bome speoial sequenoe of sets N satifsying (5) and Radon-Nikodym derin vatives qn(x) in X-N n , namely for the N defined by (18 ) and the n qn defined by (20). Remember here that the first theorem necessarilyholds for any
qn satisfying (5) if it holds for so-
sequenoeN~,
me suoh sequenoe. The equivalence is hereby established.
lb. IMPLICATIONS OF THE ERGODIC THEOREM. The case
II T"
< 1 is of no interest because then every series eO
5.. f" l.1 T"'f itself cOllverges in norm and, therefore, almost everywhere. This situation can arise even in the ·limit oase liT 11= 1 and even in the case of a Markov operator T. Closer attention will be paid to the question ·of oonvergenoe or divergence farther below. Let us briefly examine how muoh the theorem says for general Markov ,operators T.
61
- 14 -
E.Hopf There are two extreme cases of Markov processes. In the first case the transition from one stage to the next possesses a smooth spl'ead . More preoisely, P(x, A) :;
JA
K(x, y) /1-(dy)
where K is a continuous function of both arguments. In this case, .oonsiderably sharper theorems exist about ,the limit behaviour of the powers Tn. The same is .true ,if,more generally, the operator T is completely .continuous ,in some topology naturally oonnected with the process (the uniform ergodic thecremof Yosh:Lda andKakutani [20]). At the other extreme of Markov pl'ocesses theneis the strictly deterministic case, sharp point to point transition. It is in this case and incases closely bordering on :Lt that the theorem has real significance. Let T denote a single-valued point transformation x'
= Tx
of
the space X preoisely into X ,
TX
=X
Again, the use of the same letter T in various contexts will not oause oonfusion if olose attention ,is paid to ·the argument of T . Tx need not be one-to_one. ,It is olear what ,is ,meant ,by TA, AC X. We define T-1 A by [x I Tx ~ A).. Observe that'T- 1 exaotly preserves the elementary set operations, formation of union and intersection, and also the property of disjointness of sets. If T ,is one-to-one the same .is true for Titself whereas this is in general not .the -1
case if T
is
many~valued.
We wish to assooiate a Markov operator
with each of the point transfol'mations T and T- 1 . -1
Consider first T suppose that T-1 is
~
sets of the O"-field
f'
by writing
r"
(regal'ded as a transformat:Lon of sets) and
_measurable, in other words that it sends again into sets of
62
f .
We may express this
- 15 E.Hopf (23)
T
-1
tC~
If we express the desired Markov operator as an operator on finite O"-~dditive
set functions (i.e. on elements ¢ of 0) then the answer
is simply (24)
¢ E 0,
In faot,
A ~f
linearity and positivity .are obvious, and .the ,right hand
side is an element ·of ,0 ,whenever.¢ ,is. We ,may, therefore, apply the first ergodic theorem with 0'
= ,0, in pa11ticulan, the ,case Iwhe-
11e ¢.( 17 . The result is essentially the ergodic theorem of HurewiCIil
[16]: Let
f
be a a-field of subsets of 1 inoluding 1 itself. Sup-
pose that T is a 1 suoh that T
-1
single~valued
transformation of 1 into precisely
1s .; -msasurable. -Suppose, furthermore, ,that ¢, 17
r
-<
and that ,17 ~ 0 ·, ¢ 17. n-1 _v Then the Radon..,Nikodym derivative qn(x) of I ¢(T A) withrespeot o n-1 ._v to I 17(T A) has a finite limit as n • w in every point x ·ofl-N o w -v where I 17(T N) =0 are finite cr-addi t1ve set funotionsin
o
Of oourse, if T is one_to_one and ;
-measurable then the theo-
rem holds with all the negative powers or T replaced by the oorresponding positive powers. This was ·the oase ·considered by Hurewioz. If, however, T
-1
is not single-valued then it is more interesting
to oonsider T .itself. To find a Markov operator assooiated with'T itself is less easy. T¢(A) =¢(TA)oannot be ,the answer beoause the right hand side .need not be additive ,in A. ·The answer ,which is this time best expressed in terms of .polntfunotions ,is ·obtained in the following way. We assume again (23) :
,T- 1 .sends 63
,sets of.f
again .into sets
- 16 E.Hopf of
f
We assume furthermore,
/l ,in
r
the existenoe of a
such that IleA) is 3.bsolutely continuous ,with ,respect to
the me3.sure /l(T
-1
A) and vice versa. Ifwe ,observe that·T
ag3.in a C"-field and that /l(TA') .is a measure .in T
Ii )
measure
.a~finite
-1
f
-1
fis
(but not in
we can evidently phrase that assumption as follows. /l(TA')
is absolutely continuous ,with respeot to!J-(A') withinT
-1
rand
vice versa. The Radon_Nikodym theorem therefore implies the existence of rex)
,-1
Ii
T
.;
_measurable and almost everywhere ,finite function
0 such that
~
/l(TA') =
~ " \ ( c..J;
J
A'
-1
r(x)/l(dx)
A' E T
r·
The assumption of the absolute continuity of /l(T
-1
A) with respect
to ,IleA) implies that ,r > O. The condition of T- 1 ;
_measurability
determines r(x) uniquely .in the .sense .of /l-equivalence. However, i f .this condition is .we3.kened .to ·the oondition of f,_measurabili-
ty - the latter is weaker .byvirtue of (23) - then .there exist also other funotions r(x)
.~()
whioh .satisfy (25). An example illua,-1
trates this farther below. What .does 'T r? It means that every .set ,r'
~,c
;
..,measurability,mean for
is of the form·T
-1
A. This .in turn
me3.na thatr(x) is of the form s(Tx) where a(y) is single_valued and
f
..,me3.surable in X.
(25' )
IleA) =
(25) may be written
JT-1 Arex)
/l(dx) =
J
X
rex) e (Tx)/l(dx), A€~ A
is the ohar3.oteristio function of A. A In other words, the relation where e
(26)
J
X
holds for every
f(x) /l(dx) =
f'
J
X
rex) f(Tx) /l(dx)
_me3.surablecharacteriatic function. Well-known
arguments show .then that (26)muat hold for every
64
f..,measur3.ble
- 17 E.Hopf and
~-integrable
funotion. f(x) . Simultaneously the integrability
of the right hand integrand is inferred. We oan express these resuIts simply by saying that the positive linear operator Tf(x)
( 27)
= rex)
is a Markov operator from L
}l
f(Tx)
to L
}l
as it was defined in oonnection
with the seoond theorem . To complete the story it must be shown that (27) is aotually an operator between
~-equivalence
classes,
that it transforms a funotion that vanishes except in a set with
}l = 0 again into suoh a funotion. This, howsver., follows at on" from the assumption that }l(T
-1
A) is absolutely continuous with re-
speot to }leA). The seoond ergodic theorem oan now be applied. ,Let. T denote s single-valued mapping of X onto 'X and let be a cr"'field of subsets of X suoh that A' ~ ~ T
-1
AE~
furthermore, }lbe a measure on ,. suoh that }leA) and }l(T
f
. Let, -1
A) are
both cr-tinite and absolutely oontinuous with respeot ·for eaohother. ·Determine an
f
-measurable, finite function rex) ~ 0 such
that (26) is satisfied by every r (x) o Then, i f f € L}l
= 1, , Pe
= reT
r (x) n L~
S f(x) q (x) = n n S p(x) n have a finite limit as n
,
f
P > 0
-measurable f(x) ELand let
}l
n-1
xlr
(x)
n
> 0
the quotients
S g(x) = n ~ 00
n-1
n-1 ~ 0
i r. (x) geT x) 1
in eaoh point of X-N where }l(N)
= O.
The hypothesis that p > 0 oan, of course, be relaxed to Soop(x) > O. If T is one-to-one this is the ergodio theorem of Halmos [6] or, rather, its generlllizatio'n byOxtoby [19}. HIIlmos supposed that SooP
= 00
•
We mention, by the wily, thllt the theorem as
stated above was established by us in 1946 in an unpublished paper.
65
- 18 -
E.Hopf In the important case in which the measure -1
der T (or, rather, under T
is invariant un-
~
),
(28)
we may take r
=1
and
s g(x) = n
n-1 ~
i
geT x)
o
In this case the theorem .is the author's generalization of G.D. Birkhoff I S ergodic theorem, and .in .the ,case .where the ,invariant me~sure
is finite,
beoause then p
~(x)
= 1E
L
~
< ro , it
b~comes
the latter theorem itself
. In .these theorems, however,
onlyone~to
one transformations T were oonsidered. Consider the following example of a
two~to-one
transformation
(Caratheodory [1)). X is the circular line of all numbers x mod 27T,
f
is the field of all Borel sets in X . T is defined by -1
x' = 2x mod 27T . T
f '. the
-1
totality of all sets T
A, A,
-t ,
con-
sists of all sets A whioh are invariant under the oovering transformation
x* = x
t 7T mod 27T, -1
invariant under T
The ordinary Borel measure ~ in X is
(25) says that t rex t 7T)
rex)
=2
holds for almost all x . The solution r = 1 -1
only one which is T
~
-measurable, in other words, which is mea-
surable and invariant under rex
t
is essentially the
the~overing
transformation,
7T) = rex). There are, however, infinitely many) ... measurable 2
solutions so, for example, r = 2 sin (x/2). The ergodic theorem says that 1 n
n-1 II 2: f(2 x) o
66
- 19 E.Hopf has a finite limit almost everywhere if f is of period 2w and integrable. Let us return, for a moment, to the general seoond theorem with T being a Markov operator between point funotions with
J.L
€ Land J.L
being a given measure. What does ,it mean to saY that
J.L
is
an invariant finite measure of the ,Markov prooess? As the assooiated operator T on set funotions
¢(A) =
J
dJ.L
¢
is determined by
T¢(A) =
A
it means that TJ.L = is €.
L~
J.L
JA
TfdJ.L
or,in other words, that the funotion f = 1
(finiteness) and that T1,= 1 (invanianoe). In this oase
the second theorem states the ,oonvergenoe almost everywhere of 1 n-1 v .- I T f (x) n
o
LJ.L' is the largest funotion spaoe in whioh this theorem oan be expeoted to hold. 'So, for .instanoe,if 'Tf(x) = f(Tx) with T one ... toone and stion is
J.L~preserving
J
fdJ.LIJ.LCX)
X
a~d
Tmetnioally transitive the limit in que-
whioh limit would be
00
if f ~ 0 and
The theorem was proved by us for the full spaoe L
J
fdJ.L =
X
00.
[14]. Previously,
J.L
it had been proved for smaller spaoes, by Kakutani [17] for bounded f and by Doob [3] for f ~ LP, p > 1. In our paper [14] we had stated the general second ergodic theorem for Manko v operators as a oonjeoture. It was) however) only very reoently that the seoond theorem was proved by Ohaoon and Onnstein [2}.
67
- 20-
E.Hopf 10. THE WORK OF CHACON ·AND ,ORNSTEIN. PROOF ,OF ,THE SECOND ,ERGODIC
THEOREM. What follows is a simplified presentation [15] of the ingeneous proof ,of these authors. ,Sinoe
:X,'r,
J.L
are given ,onoe ,for all we need not ,refer to
them explioitely. We write
JAf
and simply Jf .if A = X. The proof
makes use of ,the lattice .operations f ... f +, f ... f L = L
II'hiohcanry
into itself,
J.L
+ f (x) = eup(f(x~jO) , f-(x) =-lnf(f(x},O}.
(29)
There holds f = f
(30)
+ f , f (31 )
are both f
= f2
+
~
-f
° but
If I
=f
+
+ f
never both > 0.
- f l' f 1 ->
~ f2
°
-
f
+
~
0 ,
fl - f
> 0
follows from this remark as at least one of the two last inequalities must hold and as ,their left hand sides are equal. Fnom + Tf = Tf - Tf , the positivity of T and from (31) we infer that (Tf)+ < Tf+
, (Tf)- ~ Tf
ITfl ~ T If I
Recall that liTH ~ 1 is equivalent to the statement ,that
The proof of the seoond ergodiotheorem ,is oarried through in several steps. The first (prinoipal) lemma we formulate as follows (statement and proof are somewhat more transparent than .in Chaoon's and Ornstein's paper).
68
- 21 -
E.Hopf LEMMA 1. If f , Land .if .sup S f(x) > 0 n>O n n-1 Sn = 2:o T i, holds .in eaoh point x of a set A( e t ) then, , to every € > 0, there exist funotions h € L, g E" L suoh that a)
h
~ f
y)
Jh
~
(3)
=
h
+ Tg - g
f
J
g
~ 0 ,
Jf
PROOF. The role ,of (3) will emerge farther below: The additional term Tg-g has no ,influenoeon the limit behaviour of
h= Snt/SnP'
S) means that h is praotioally ~O in A. a) implies .that ,h ~ ,f
as .long as f' <:,0 whereas y) implies a limitation ,on h ,in .the opposite sense. To prove the lemma we define a sequenoe of funotions hi €
L
reoursively, by applying T to the positive part only,
= Th,1+
h
o
= f
and we show that each .h i has ,the properties a)- y) and that ,it satisfies S) for sufficiently.large 1. From (34), the positivity of T and from (31) it follows that .h i ~ 0,
i +1
~
h,
1
whioh implies a) as h h
i +1
= h
i
0
+ Th
= f. (34) may be written + i
h
+ i
Hence, by summation, i
+
o
II
2: h
69
- 22 E.Hopf and this proves ~). ~l .implies y) by virtue of (33). If (36) is combined with
=
h
+ i+1
~
h
i+1
there follows
i (n
~
O. As T is order-preserving this implies, by induction, that
> 0) i
o
+ + and, since go = ho = f
~
n
T f + T g
o
f , that
Obviously, this holds for n
=0
too. This inequality and the hypo-
thesis of the lemma imply now,; In each point of A at least one of
+
the h 4 , i ~
is
= O.
~ 0,'
is > 0, or, in other words, at least one of the h.
1
Hence and from h i ~> 0 and from (35) it follows.that h-n ~O
holds in A and, consequently, that
f
h
A
n
... O. Lemma l.is hereby
completely proved. It is possible to eliminate the
au~iliary
function h of Lem-
ma 1 by means of the properties a) - 8). The result is .the LEMMA 2. If f eLand .if sup S f(x) .? 0 n>O n
x
E A ,
then f and A satisfy the inequality
f Af
+
Lr A
+
~ 0
A=X-A.
PROOF. Suppose first that the sup in the hypothesis .is > 0 in A. For any given E > 0, lemma 1 may then be applied. On using 8), a)
70
- 23 E.Hopf and y) we infer that -
~nd
E
-
I Af -
<
- IA-h - = - Ih - ~ Ih ~ If =
I Ah -
I Af
+
Lf
A
hence that (with E > 0 arbitrary)
The lemma is hereby proved under the stronger hypothesis, sup
> 0 in A. Under the original hypothesis, .sup
~
0 in A, we need
merely use a funotionp EL, p > 0 and apply ·the foregoing result to f + 7JP, 7J > 0, in .plaoe of f. The existenoe, by the way, of such a function p follows from the sic measure
~.
The proof of
lem~a
aS8umeda~finiteness
of the ba-
2 beoomes oomplete if we let
Ti ... O.
REMARK . It should be noted that lemma 2 is oompletely equivalent to the "maximal ergodio theorem»,
If> 0
B
B -
=
[xl sup S f(x) ~ol, n>O n
established by us previously and used in our proof of the ergodio theorem for a Markov operator with a finite invariant measure [14J. The equivalenoe beoomes olear .if three simple faots are noted. Firstly, the hypothesis of lemma 2 says that A
C B . Seoondl;'l, the
left hand side .of the inequality of this lemma oan be written se that B
= B~A+A,
A
=B
I Bf
+
(u~
+ B - A)
I B-A f-
> I f - B
Thirdly, in B there holds sup S f I': 0 and, in particular, f = n + S f < 0 or, in other words, f = o. <37 ) obviously holds also if 1 B is the set in which sup S f > O. Lemma 1 .is thus seen to furnish n a simple proof of (37). However, we .have mentioned lemma 2 because
71
24 -
~
E.Hopf it is a handier form of (37). The following lemma is a known consequence of (37) but it is more easily derived from lemma 2. It deals with the quotients
(38)
q (x) n
= qn (Xi
f, p)
S f(x) n S p(x) n
=
n > 0
Note that, for any constant UJ,
LEMMA 3. If f
e
L, PEL, P
~
0 then
sup jqn(Xi f, p)j <00 n>O holds almost every.where in the set where p > O. PROOF. As jqn(Xi f, p)j < qn(Xj
jfj, p) holds it suffices to pro-
ve the lemma for the case that f
~
0, and without the absolute va-
lue sign. Let A
=
[xjp(x) > 0, sup qn(Xj f, p) n>O
= 00]
•
By virtue of (39) sup qn(Xi g - UJP, p) > 0 n>O holds everywhere in A, for any constant UJ > O. In other words supS fl (x) > 0 n>O n
f
I
-
f - UJp
must hold in A. Hence, by virtue cf lemma 2,
I A(f From f
-
- UJp) +LU A
-
UJp)
+
~ 0
UJp < f, f
~ 0
it follows that (f
UJ
IAp
~
Therefore
IAf
+
I_f A 72
=
If .
-
wp)
+
< f
- 25 E.Hopf As w > 0 is arbitrary and as p > 0 in A it follows that A has measure zero, and lemma 3 is proved. For the following lemma of Chaoon and Ornstein we give an abbreviated proof. LEMMA 4. If f , L, P € L , p lim n~
~
0, then
Tnf(x)
=
S p(x}
0
n
holds almost everywhere in the set where p > O. PROOF. It suffioies to prove this for the oase that f a number
€
~
O. Piok
> 0 arbitrarily and oonsider the functions g
o
= f
whioh satisfy the relation (40 )
= Sntl I, I = identity. We need merely show that, in n almost every point of the set [p » ol, gn < 0 holds for all suffi-
since TS
oiently large n or, in other words, that!e n converges where en . is the oharaoteristio function of the set [gn ~ lation e nt1 gntl
t
= gntl
we infer from (40) and (33) that
On adding up the resulting inequalities from n
fg
t
n
t
€
0]. Using the re-
nt e i ~ go
JP !1
J
and hence
73
=0
on ,
- 26 E.Hopf -1
€
Jgo+
.
Lemma 4 is herewith proved. The next lemma oontains the prinoipal part of Chaoon's and Ornstein's proof of the ergodio theorem. LEMMA 5. Let f
~
L, PEL, P
O. If in ee.oh point of a set A
~
there holds p
> 0 ,
and
W
qn (x; f, p) < a < b < lim qn (x; f, p)
where a, bare oonstants then A has measure zero. PROOF. Write qn (f) for the quotients qn and note that, in oonsequenoe of part of the hypothesis, sup q (f - bp) n>O n
lim q (f - bp) > 0 n
~
holds in the set A. Choose
E
> 0 arbitrarily and apply lemma 1
to f-bp in plaoe of f. Write the h of this lemma in the form h-bp.We infer from a) that (41)
< (f -
b )
P
and from S) that
JA (h
(42)
- b) P
<
€
holds. (41) and a < b imply that
S (f • ap)
(h - ap)
(43)
holds a fortiori. In faot, that h
~
(41) is equivalent to the statement
f i f h < bp. Therefore h
that (43) holds. Now use
/3)
~
f i f h < ap, and this means
of lemma 1, h = f + Tg - g,
74
- 27 E.Hopf
g)/s p . n
On applying lemma 4 to g, p and on using the hypotbesis that Sol
= 00
on A we may infer that qn (h) and qn (f) have the same up-
per and the same ,lower limit (as n .,. (0) in eaoh point of A exoept in a nul set whioh may safely be negleoted in what follows. In partioular, there
ho~ds
or
lim qn (h) < a
-
lim qn (ap-h) > 0
within A. Consequently, lemma 1 may be applied to ap-h in plaoe of the f there. write the new auxiliary funotion h of this lemma in the form ap-f'. Then it follows from y) that (44)
or
J(ap-f') < J(ap-h}
J(f'-ap) ~ J(h-ap)
and hom (I.) that
(45)
or
(f'-ap)
+
~ (h-apr
+
and, finally, from 0) that (46)
We now have a set of inequalities involving beside the given f two auxiliary h, f'. h can be eliminated from them in the following way. Split the integrands in the seoond inequality (44) into positive and negative parts, (47)
< J(h-ap)
-
+ J{(f'-ap)
+
+
- (h-ap) }
By virtue of (45) the secQnd integrand on the right is
~
O.
(47), therefore, stays valid i f the inte.gral is taken over A.
75
- 28 -
E.Hopf
(h-ap)
+
~
h-ap = h-bp + (b alp > - (h~bp)
+ (b-a)p .
Hence the right hand side in (47) is < J(h-ap)
+ J (fl-ap) A,
+
+ J (h-bp)- - (b~a) J p A
On applying (43) to the first term,
A
(~6)
to the seoond and (42)
to the third term we obtain that (48) By the same reasoning ae above (theuee of lowe that qn(fl) and qn(h) have the same
fi)
of lemma 1) it fol-
upp~r
and the same lower
limit in almost each point of A. Thereby the following result is obtained. If f satisfies thE! hypothesis of lemma 5 then, to any given
E
> 0, there exists a function fl whioh satisfies the same
hypothesis and, in addition, the inequality (48). This result may in turn be applied to fl and we get a function fll whioh satisfies the hypothesis of the lemma and (48), with f, fl replaced by fl, fll, and so forth. On writing these inequalities underneath eaoh ther and on adding up the first n 'of them
we
find, on neglecting
the non-negative integral remaining on the lett, that n ,{(b-a) J
A
p ..
2d < J(f..,ap)
must hold for any integer n. Hence
Lemma 5 follows from this since E was arbitrary and since a < b, and p > 0 in A.
76
0-
- 29 E.Hopf COMP~E~ION
OF ·THE :PROOF ·OF ;THE ·SEOOND ERGODIC 'THEOREM. 'We pro.
ve first that the qnhave a finite limit almost .every"here in the set[p >0]. We may suppose that f ~ ·0. Lemma 3 ,implies that the finite or infinite limit if it
e~ists
must be finite almost everY-
whene. So "e need only pnoveexistenoe.Within .the ,part ·of the set
.h
> 0) in "hioh
i ~ T p
00
a
,plies that the senies
this is obvious beoause ,lemma 3 .then im-
<00
~T
i
p
"ith non-negative terms has a .finite
sum alma st every"her e . in t hi s part. In the remaining ,pant i
00
[ ~ T P a
= (0)
of [p > 0), ho"ever, the positivity of the ,measure of
the set in "hiGh .lim
-
rational numbers a < b suoh that the set [Um < a
= ffi
must hold almost every"here .in that .remaining .pant.
Now we ·oan easily .provethefull theorem: ·The qn have .a .finii
te .limit .almost everywhere in the set [ A~ p >
01. Observe that this
'.set ,is ,the .union ·of .the sets ' [p> 0] and
sk+l
p > 0, Sk P
k
In the latter set, T p =' S
k+1
=.:: 0)
, k
P -- Sk P ;> 'Oh On using two ·new funotions
"etind ·that in that set (49) and for n > S
n-k
f'
sn-k p'
=
>0 .
= -
k
..
From this formula and from the previous result as in ·(49) it tollo"s no" that the q
n
ap~lied
to fl, p'
have a finite li.italmost eve-
ry"here .in the set (49). The seoond ergodio theorem is hereby .oom·plately proved.
77
- 30 E.Hopf 1d. CONSERVATIVE AND DISSIPATIVE PART OF A MARKOV PROCESS. Consider again a positive linear operator T from
L~
to
L~
with a. norm IJTII = 1. We now turn to the simpler question of convergence or divergence of the infinite series Scof =
co i L T f o
f
€
L~
DECOMPOSITION. X is the union of two disjoint sets, the convergence set Xc and the divergenoe set Xd , with the following properties. If P f v ~ rywhere
L~,
P ~ 0, P > 0 in Xd , then Sco P
in Xd in the sense of the measure
= co
holds almost e-
~
the series Scof oonverges absolutely almost everywhere in Xo' PROOF. Let q > 0, q ELand denote by Xd and X;, respeotively, the sets where Sco q
= co,
< co . Consider a funotion p with .the properties
stated .above and apply lemma 3 to the quotients Snq/SnP' As p> 0 and Sco q = co holds in Xd it follows from this lemma that ScoP = co must hold a.e. in Xd ' Now let f ( L~ and 90nsider the funotion p' = q + I f I € L~, p' > O. Apply the same lemma to the quotients S p' /S q. As Scoq < co holds in X it follows that also ·S~p I < co n n o w holds a.e. in I . The sam, holds therefore for scolfl. The rest of o
the lemma follows from Ifnfl
~ Tnlfl.
This result shows that within X the ergodio theorem is trio vial and uninteresting. Both denominator and numerator oonverge separately a. e. inX o ' It is possible to separate Xd from Xo in suoh a way that the operator T somehow splits .into two operators one of whioh aots on and produoes funotions that vanish .in Xo whereas the other does the same relative to Xd? In other words, is the implioation (50)
f = 0
a. e. in
Xo
78
'9
Tf = 0
a. e. in Xc
- 31 E.Hopf valid? And is the analogous implication valid relative to Xd? (50) means that the values of Tg, for general g, wi thin Xc do not depend on the values of g within X . From the representation ford
mula
J Tf A
d~ =
J
X
P(x,A)fd~
we easily infer another equivalent of (50): P(x, X) = 0 holds a.e. c
in X . d
(50) is valid [14] and its proof is given below. However, the other implication regarding Xd does not generally hold good. It, is certainly valid if T is generated by a
on~-to-one
point transforma-
tion but in the other extreme case where T is a completely continuous operator the resufts of Yoshida and Kakutani [20] show that it is invalid or, in other words, that Xc has an influence on Xd . PROOF OF (50). We may assume that (51)
f ~
0 .
Consider a fixed q > 0, q , L . By the deoomposition theorem, J.L
a.e. inX d
(52)
q and that Note that S q = S n ntl TSnq = S q - q < ntl
S~q ~
.
On letting h (x)
n
= inf
we can infer from (51),
(55)
(f(x), S q(x» n
(52) and the premise of (50) that
o,Shff n
79
- 32 E.Hopf holds in Xc as well as in Xd . We show that a.e. Th h
In fact
f
n.
< h
n+l
f
~
n
~..., "?'
t
Tf •
Thn < Th
(Tf - Thn)d~ = f T(f-hn)d~ < X X
n+l
f
~
Tf . and, by virtue of (33 ) ,
(f-hn)d~,ltO .
X
almost everywhere. This holds for any f whioh satisfies the hypothesis and, therefore, also for
€
-1
f,
€
> 0 ,
almost everywhere. This holds for any f which satisftes the hypothesis aM, therefore, also for £-I f , e: > 0,
As Sooq <
00
holds a. e. in X .i t now follows that Tf whioh is c
~
0
must be = 0 a.e. in X. (50) is herewith proved. c
In one speoial case the sets Xd and Xc oan be characterized in another way . Suppose that ·T is induced by a one .. to .. one mapping x'
= Tx
-1
of X onto itself such that T as well as T
and map sets with ~
= 0 (~given
are measurable
and o: .. finiteon!;) again onto su c h
sets. It was above that such a mapping may be regarded as a Markov operator T from
(57)
L~
to
L~,
Tf(x) = r(x).f(Tx)
where the ~ ~measurable rex) > 0 is essentially uniquely determined. The setsX d , Xc are thereby well defined. We reoall the familiar
80
- 33 E.Hopf concept of a "wan.dering set" A with. respect to the mapping: AGt, n
the images T A, n
~
< 0,
are all disjoint. The following known theo-
rem is worth mentioning. X
c
00
= -00 LJ
i
T A where A is a wandering set. X oontains no wanded
ving set of positive . measure
~.
The equali tyis, of course, understood to h01d up to a set with
~
= O.
It follows, by the way, that Xc and X
d
= X-X 0
invariant under the .mapping. The theorem oharaoterizes X essentially largest among all set.s .of the form
00
U
-00
.
o
are both as the
T1B , B wande-
ring. A set of this form is usually oalled "dissipative" with respeot to the mapping, or the mapping is oalled dissipative within this set. A set, however., whioh is invariant under the mapping and .whioh oontains no wandering set of positive measure is sometimes oalled ·conservative·with respeot to the mapping. Or the mapping is said to be oonservative within this set. Adopting this nomenolature wa oan say : The mapping is dissipative within X
o
and con-
.servative within Xd . A classioal faot : If
~
is finite and invariant under the point
transformation then X = ,Xd ' If, X = X
o
~
how~ver,
is ·infinite and invariant
~
is possible. Example: X = infinite line, T = translation,
= Borel
t
measure. If a ooptinuous group T n
dered irlsteadof the disorete group T
of mappings is oonsi-
then there is the example
of the geodetio flow on a oomplete surfaoe of oonstant negative curvature of the seoond class whioh is a dissipative flow. An open problem : Is the phase motion in the olassioal problem of three bodies dissipative? This would imply that the maximal distanoe between the three bodies tends, in general, to infinity as t -
00.
Let us return, for a few moments, to the theorem stated above and to its proof. Only the barest outline of this proof is gi-
81
- 34 E.Hopf ven here. We start with dissipative sets in X and bring in the sets Xc' Xd , later. The proof splits into the following steps. a) The union of two dissipative sets is again a dissipative set. b) If is not finite use an equivalent (same nUlsets) finite measure oonsider the supremum M of the measures
~'(D)
~r,
of all dissipative
sets D C X and show that there exists a dissipative set DH
with
= M. 0) X-D* is oonservative. d) Show that, for any P, f L~,
~' (D#f
p > 0
~
,
SooP <
00
holds a. e. in D~ . e) Show conversely that, i f
p 6 L 0 and i f SooP < 00 holds in a set A, ~(A) > 0, A must j.t' P > oontain a wandering set of positive measure. It is olear that these five steps make up the oomplete proof of the theorem. We merely prove d) and e). Reoall that the assooiated opera-
f
tor on set funotions is T¢(A) = ¢(TA), A' 'Tf'(TA)
=
JA
Tp d~
if
?T(B) =
J
and that pd~
B
where T is given by (57). The left hand identity holds for all pon
wers T , n (58)
0, too. Henoe
~
00
i
L ?T(T A) = J o
A
*
,
To prove d) take for A a wandering set such that D is the union i of all its images T A, i < O. As these images are pairwise disjoint it follows that the left hand side of (58) is .::. ?T(D*) .::. ?T(X) <
00.
Consequently, SooP is a.e. finite in A. Since, however, every set TnA, n ~ 0, is wandering too the same must hold within this set < and, therefore, in their union D*. To prove e) observe that the set A mentioned there necessarili contains a set B, 0 < j.t(B) < ?T(B) > 0, such that SooP is bounded in B.On applying (58) to B we infer that
(59) 82
00,
- 35 E.Hopf co
Now let On;:
U TiB
and note that B ;:
n
virtue of (58), co
.
7T(On) < 2: 7T(T 1 B) n
oan be made arbitrarily small as n gets large, oertainly < 7T(00) for some n ;: k.
Oonsequently~
00 - Ok has positive measure
same must be true for at least one set A' ;: 0.
~
. The
0i+1' i ;: 0, ... , k-l.
1
Now observe that m
T A' ;:
m
>
°,
and that these sets, m > 0, are disjoint with A'. As is wellkhown this (and the assumption that T is one-to-one) implies that A' is a wandering set. The proof of the theorem is thereby finished. For the validity of the theorem the hypothesis that the mapping is one-to-one is absolutely essential. Recall again that the sets Xc' Xd are determined by a Markov operator in the first place. Now, if the mapping is one-to-one there is only one Markov operator which can be reasonably connected with the mapping, and so it is not surprising that Xc' Xd can be characterized by the mapping alone. The situation is entirely different in the case of a many-to-one mapping, We have seen above that, in general, the representation of such a mapping by a Markov operator is no longer
uni~ue.
That this
ambiguity has an influence on the sets Xc' Xd is clearly illustrated by the following example. Let X be the set of all natural numbers, x ~
::z
1, 2, . . . . Let
be the a-finite measure which is one for each such x. Oonsider
the point mapping x >1 , x ;: 1 ,
83
- 36 E.Hopf ~
of X onto X. 1 consists of all subsets of X, T
-1
f
only of those
subsets which contain either none of the two numbers 1, 2 or both. A Markov operator associated with T is of the form (57) where r
~
0
is such that (26), the conservation property of Markov operators, is satisfied for each f 00
~ -00
L ,
~
f1-
00
f(x) ::
This requirement is .
2: r(x) f(Tx) .
-00
read~ly
r(1) + r(2) :: 1 ,
seen to be equivalent to the relations r(3) :: r(4) ::
=1
.
So we have infinitely many Markov operators (57). There holds (60 )
n
n
T f(x) :: r (x) f(T x) , n
r
:: r(x) r(Tx) ... r(T
n
n-l
x).
First case. r(1) :: 1, r(2) :: O. We distinguish between the two parts (61)
[x Ix >
1J
:: 1, Soof ::
00.
[xix:: 1),
of X. In the first set, r
II
f(l). However, in the se-
oond set, r (x) contains, for all large n, the factor r(2) :: 0 . n
Therefore Soof is finite. It follows that the first set is X
d
and
the second X . c
Second case. r( 1) < 1. No matter what x we start from the factors of rare:: r(l) from a certain place on. Hence, n
rn(x) < C(x)r(l)
n
and, oonsequently, Xo:: X, Xd:: O. Note, by the -1
way, that the unique r(x) which is T
f
-measurable, r(l) :: r(2),
comes under this seoond case. It is instructive in this connection to consider another example in whioh the spaoe X is not oountable. This example was briefly mentioned already
x is an angle variable mod. 27T and the
84
- 37 E.Hopf mapping is X'
= Tx = 2x mod 2n .
We found that each measurable function rex) 2 0 with satisfies the relation r(x) + r(x + n)
=
2
furnishes a Markov operator T , Tf(x) = r(x).f(Tx) , for the mapping T. In order to study the behaviour of the iterates Tn of the operator we begin with the simplest case r already mentioned that, for g € L
Ji.
= L 1,
lim
1 n-1 v L g(2 x)
n~CD
n
= 1.
It was
'
the limit
0
exists almost everywhere. We need the fact that the limit function g(x) is constant a.e., and that the constant equals g(y) dy . The simple reason for this is : For almost all x there holds g(2x)
= g(x),
g(2 k x)
= g(x)
where k .is an arbitrary positive inte-
ger. The Fourier expansion of g must therefore reduce to the constant term. Now we return to our question regarding the associated operator rf(x)
= r(x).f(2x)
where rex) is any admissible function> O. X is the set on the circle in which every series CD
i
L T p(x) o
85
- 38 E.Hopf 1
P > 0, PEL, oonverges a. e .. Now, n
= r n (x)p(2
T p(x) and ro(x)
= 1.
n
x) ,
= r(x) ... r(2
r n (x)
n-1
From the immediately preoeding result, g
it follows that log
x)
= log
r,
a.~.
1
n
V r (x) n
21T'
21T
f l o g r(y)dy . 0
As the log is oonvex the right hand oonstant is ~ log
f-
1
21T
f 21T 0
1
r(y) dy
and equal to it if and only if r is oonstant a.e .. The last named expression has the value zero as follows readily from the admissibility relation for r(x)-. Henoe we infer tha.t, for almost every x,
r'--r-
n
has a limit if r
=1
n-=-(x-:)
< 1 unless rex) is oonstant a.e. (whioh oan happen only
a.e.). If we oombine this with the faot that the set xo
is essentially independent of the ohoioe of the funotion p, p > 0, 1 n p , L , (we may take p = 1, T P = rn) we obtain the following ourious result. Any admiss-ible- funotion r, r > 0, rex) + rex + 1T)
= 2,
furni-
shes a Markov operator "assooiated" with the mapping. In the oac e where r
=1
a.e . we have X
o
=0
funotion "one" then we have X
o
but if r is ~
n~t
equivalent to the
Xl
We had notioed this faot already in 1946. There is, of oourse, the question wether there is an "opt"imal" Markoy operator whioh desoribes the future behaviour of the iterates of a many-to-one mapp~ng
better than the other assooiate operators. On oomparing the
86
- 39 E.Hopf results of the two examples mentioned here we can see that a generally valid answer cannot be trivial. However, the beautiful applications of information theory to ergodic theory which Kolmogorov has initiated and to which we were introduced in this Seminar by Paul Halmos may throw light on this question. Let us finally return to the general second ergodic theorem itself. We wish to conclude this section with the question (not answered as yet) about the nature of the limit function lim q (x). This n
question is, of course, interesting only in Xd . If we operate in Xd only, more precisely, if we confine ourselves to functions f vanishing in X then (50) has the following c
implication. Within Xd , T represents again an operator of the same kind! positive, linear and of
L~
- norm
~
1 within Xd' Furthermore,
if T is a Markov operator in X so is T in Xd' We may therefore assume, without loss of generality, that in
X if
P > 0 .
We need the notion of the dual operator T associated with the given operator T, (D)
explicitely,
'"Tg(x) = fp(x, dy..)g(y) . '"T is actually an operator from
~-equivalence
'"
class of bounded func~
tions to again such a class [14). T is positive, and T1 = 1 holds if T is a Markoff operator. We assume here that T is a Markoff operator (this property is used in the proof of one of the subsequently mentioned facts).
87
- 40 -
E.Hopf A bounded and
~-measurable
funotion hex) is said to be inva-
riant under the Markoff prooess if it is invariant under T, (62)
Th = h
(up to a set with
~
= 0).
The theory of these invariant funotions
was developed in [14], however, under restrotion to bounded funo,.. tions. The main results are the following (see [14], §9). The invariant funotions form an algebraio field. Also, Ihl is invariant if his.
If h is invariant T(ll w)
( 63)
there holds (almost everywhere)
t~en
=h
T(w)
for every bounded and measurable function w(x). Conversely, of oourse, the latter property of h implies invariance of h since T1
= 1.
The proof of (63) is given in [14], §9 . The property (63) is oompletely equivalent to the following property of the original operator T :
=h
(64)
T(h.w')
T(w')
holds for every
~_integrable
funotion w'(x). The equivalenoe of
the two properties follows easily by means of the duality relaHon (D).
Let us now return to the question of the limit funotion f* = lim
S
n
f
n-+oo
of the general ergodio theorem. It has not yet been proved that this limit funotion is invariant and that it is independently charaoterized by the oondition that pf* is ~~integrable and that
88
41 -
~.
E.Hopf holds for every bounded and invariant funotion h. We repeat that this conjecture is stated under the hypothesis that T is a Markov
0-
perator. In order to prove it is first of all necessary to extend the notion of invariant function from bounded functions to functions h' for which
JIh 'I
pd,u
is finite. This should not be difficult. We also would like to call attention to another way of proving the general ergodic theorem, namely, by generalizing a simple trick which F. Riesz onoe introduoed to prove the classical mean ergodic theorem. Consider first functions of the form Tg - g. The funotions h which are orthogonal to all of them must satisfy the relation
o = J(Tg
- g) hd,u
'"
= fg(Th
- h)d,u
for all g or, in other words, h must. be invariant, Th = h. Show that every f can be approximated by a sum of a function
~g
- g
and an invariant function h. Then prove the ergodic theorem in the two simple cases 1) f
= Tg
- g, 2) f = h. In the first case use
lemma 4 (with g written in place of f) to show that
S (Tg - g)/s p~o. n n
In the second case use t.he relatibns i T (h.p)
=h
i
T P
which follow by repeated application of (64). In this case there would hold
for all n. Finally, it must be shown that, for a function f' of
89
- 42 -
E.Hopf small norm, the funotion lim sup n~oo
is also small exoept in a set of small measure. This oan be inferred from our maximal ergodio lemma (lemma 2). This proof of the ergodio theorem, if oarried through oompletely, would not only prove the existenoe of the limit funotion but also its invariance.
90
- 43 E.Hopf 2a.
TWODIMENSIONAL~¥PERBOLIC
GEOMETRY.
The well-known Poincar' model of the hyperbolic plane is the interior of the unit circle endowed with the metric ds
( 1)
2
=
4
dX~
+
dX~
which has curvature minus one. The isometries in this geometry are those Moebius transformations that map
2
2
< 1 onto itself. 1 2 An isometry is oompletely determined by the requirement that it x
+ x
oarry a given line-element (such elements are
understoo~'to
be di-
reoted) into another such element. Of course, isometries leave all quantities unohanged which are determined by ds only., angles (= euclidean angles), element of area dA
(2 )
Geodesics are carried into geodesios. They are the
al"OS
2 2 in xl +)(2 < 1
of the cirQles orthogonal to the unit oirole. Hyperbolic distanoe between two points x = (x' ,x ), x' = (x',x') is denoted by s(x,x'). 12 1 2 Consider now thethreedimensional space of line~elements e in the hyperbolic plane. Each isometry of this plane induces, of course, a mapping of that spaoe onto itself. We introduoe as metric in the
line~element
dowhere
d~
2
2
= ds'
spaoe the expression + dX
2
is determined in the following way. Consider two nearby
elements e, e' with bearer points x, x'. Move e from x to x' by parallel displacement, in other words, move the element to x' along the geodesic from x to x' in such a way that its direction makes always the same angle with the geodesic.
d~
is then the an-
gle between the element e in its final position and e'. Obviously,
91
- 44 -
E.Hopf
Id X. I
is independent of the order of the two elements. dO" is there-
fore a Riemannian metrio in e-space. dO" is even invariant under the mappings induced by the isometries of the plane since
the operation
of parallel displacement .is .invariant underisometries. In other words, those mappings are themselves .isometries in e_space relative to dO" . We denote .by O"(e,e') the invariant distance defined by dO" in e-space. The invariant element of
volum~-meaeure
induoed by
dO" in e-space is found to be (4)
dm = d¢dA
where d¢ is the angle-differential in a point of the plane. If we consider in
e~space
the motion along the geodesios with
speed ds/dt = 1 we obtain a one-parameter group of mappings Tte, t +s t s t = T T , of e-spaoe onto itself: T e is the position attained T by e after it has moved t units along the geodesic determined by it. This is the geodetic flow in the e-space of the hyperbolic plane. It is a well-known faot of differential geometry that the geodetio flow on a surfaoe leaves the element of measure (4) in
e~spa-
oe invariant. The same is, of course, true about the Lebesgue measure m determined by it in e:space. m is invariant not only under isometries but also under the geodetic flow. We need the following simple ooordinate-representation of the geodesics and the flow along
t~em.
We assume throughout that
I
"dro-
desic" is a direoted geodesio. A geodesic is oharaoterized by Its points of infinity
e , e+ on
an angle (mod 2n). And a
the unit circle, ~
line~element
e
~s
~
e+,
whero
8
is
characterized by the geo-
desic determined by it and by the position of its bearer point on this geodesic
(orthog~nalcirdular
characterized by the distance
>
arc) .. This p08ition is, in turn,
.s =·0 ,of the point from the eucli.de"n <
92
- 45 E.Hopf midpoint of this
dir~oted
arc, t
e = (8 , 8 , s).
The geodetic flow is then simply
t
(6)
T e
. . t = (8 , 8 , stt).
The invariant measure-differential dm in e-space has the simple form
(7 ) That p is independent of s follows from the invariance of dm under t
T . Positivity and continuity is the only property of this function P which is needed in what follows.
We also need the following fundamental fact about hyperbolic geometry: To every geodesic and to every point x there exists precisely one geodesic through x that has the same point plus infinity on the unit circle. Two geodesics with the same plus-infinite point we call positiveLy asymptotic (to each other). The asymptotic character is expressed by the following. LEMMA 1. If two line-elements e, e ' determine positively asymptotic geodesics then there exists a number a (depending on e, e l ) such that t
er( T
ta
t
e, Tel) ... 0
as 2
2
This is most easily proved if we map the interior of xl tX2 = 1 by a Moebius-transformation onto the upper half plane of a y-p1ane, y = (y l' y 2)· We do this in such a way that the common point t of the two geodesics goes into y straight lines 1, I
I
= <Xl.
<Xl
The two geodesics become two
orthogonal to the line y 2 = O. Consider the
93
- 46 E.Hopf two tangent elements e on I, e' on 1'. Evidently the desired value of a is the one for whioh Tae and e' have the same ooordinate y~.
,
It is geometrioally obvious that the non-euclidean distance Jds t til. t between the bearer points of the elements T e, T e' tends to zero as t -+ co . At the same time the geodesio aro joining them becomes straighter and straighter in the euolidean sense. Now, the o-(T
t til.
dist~noe
t
e, T e')
is not greater than the length of any path in e·spao, joining the two elements. Choose for this path the geodesio aro betwoen the
hOi
rer points plus parallel displaoement of the first element along it (do-
= ds),
and then the remaining rotation of the element into
the seoond element (do-
= Id t I).
This makes the lemma obvious.
2b. COMPLETE SURFACES Y OF CONSTANT NEGATIVE CURVATURE. It is a classical fact that every abstract twodimensional Riemannian manifold of ourvature minus one whioh is complete (P.Koebe, H.Hopf and W.Rhinow, every geodetio aro oan be oontinued on the manifold to all values of the length-parameter s) oan be realized as follows. Let G denote a group of isometries S . We suppose that it is disorete OD, in other words, that it does not oontain isometries arbitrarily olose to the identity. Well-known oonsequenoes of t disoreteness are: 1) G iEl oountable, 2) the Elet of distinot Sx, S
~
G
oongruent to a point x within the unit oirole
doe~
have a oluster point within this cirole, 3) the fixed poin isometries S
€
3
S,i)'
.ets
not of the
G different from the identity form a oountable set
of single paints and orthogonal aros. G possesses a fundamental domain D within the unit cirole. The olassioal oonstruotion of a simple fundamental domain is this. Choose a point
94
XO
within the unit
- 47 E.Hopf oirole whioh is no fixed point of any S f G different from the identity. Then the set of all points x whioh satisfy the inequalitit;! s
o
0
s(x, x ) < s(x, Sx ) for all S
e
G different from the identity is a fundamental domain
D = D(x o ) for G • D is geodetioally oonvex. Its boundary is formed by oountably many geodetic aros and, perhaps, parts of the unit oiro
ole itself. This is so beoause the equation s(x,x )
= s(x,x 1 )
de-
fines a geodesio in hyperbolio geometry. Fixed points can obviously ocour only on the boundary of suoh aD. We obtain a Riemannian surfaoe y of ourvature minus one if we identify all points oongruent to a given point x and if we regard the set p :;; [Sx
IS
E Gl
as a single point p . Distanoe between suoh a point and another suoh point p'
= [Sx'
Is £
Gl
is defined by S(p,pl) =
inf
s(Sx, S'x') = inf
S ,G
S 6 G, S' r; G
The last two equations hold
s(Sx,x') = inf s(x,S'x'), S"
G
by virtue of the invarianoe of s . Si-
milarry, the direoted line.elements P on yare defined by identifioatiop of oongruent line-elements on y , and distanoe of two suoh elements P is defined by u(P,
P') =
inf u(Se, S'e') S€G,S'£G
where S, S' mean the isometries induoed in
e~spaoe
and G means
the oorresponding group. The threedimensional spaoe of line-ele-
95
- 48 -
E.Hopf ments P on 'Y is denoted by
n.
All the other quantities whioh are
defined within the unit circle and invariant under isometries define similar quantities on 'Y • Plane measurability of a set of points p on 'Y means measurability of the set of 2
2
1
2
a~~
representants
x in x + x < 1 of all those p . A set in 'Y is said to have measure zero i f that set of all representant points has this property., Of course, measure of a general measurable set on 'Y has to be defined somewhat more carefully. It is defined as the measure fdA of the ,intersection of that set of all represeritants x with a fundamental domain D
of
G. It is an easy consequence of the invariance of
fdA under isometries that the measure defined in this way is independent of the particular D . Measure zero, according to this definition, agrees with measure zero as defined a moment ago because the set of all represent ants x is the union of all oopies under G of those intersections and beoause these copies are countable in number. In a perfectly analogous way m-measurability and measure m are defined in the space
n of
line-elements P on 'Y •
The geodetic flow Ttp is unambiguously defined on 'Y or, ther, in
ra~
n because t
ST e
= Tt Se
holds for all induced isometries S . The measure m in
n is
inva-
riant under Tt. A geodesio on 'Y is said to be positively asymptotic to another geodesic on 'Y if some representant of the first is pas. asymptotic to some representant of the second in the unit t t circl,e. Two streamlines T P, T pI are said to be pas. asymptotio if the geodesics represented by them on 'Y have this property. t t LEMMA 2 . If the streamlines TP, T pI through two points P, pI of
D are
pas . asymptotio then there exists a number a suoh that
96
- 49 E.Hopf cr(:r
t+a
t
P, T PI)..., 0
as
t ...
<Xl
•
This is an immediate oonsequenoe .of lemma 1 and of the faot that cr(Q, QI)
cr( e, e I) holds for any representative e, e I of two
~
n.
points Q, QI of
We add here a little remark whioh has to be used later, LEMMA 3. Let
~
be a set of direotions in a point p of a surfaoe y .
Consider a seoond point p' on y and a geodesio through p' whiohis pos. asymptotio to some geodesio passing through p in a direction of
~
. Denote by
~I
the set of direotions in p' of all these geode-
sios through pl. Then, if
~
has angular measure zero so does
~I.
To prove this we oonsider first two points x, x' .in the cove2 2 ring surfaoe xl + x 2 = 1 . Any geodesio through x determines a unique geodesio through x' which is pos. asymptotio to the first. In this way a
one~to_one
oorrespondenoe is established between the di-
reotions in x and those in x I. It is bi-analytio and, therefore, it oarries a set of direotions in x of angular measure zero into a set of the same sort in x'. Now oonsider two points p, p' on the surfaoe y . There are, in general, many geodesios through p' pos. asymptotio on y to a given geodesio through p. Pos. asymptotioity on y means that, in the oovering surface, some representant of the seoond geodesic is pos. asymptotio to some representant of the first. This implies that, on y, the direotions in p' oorresponding to any given direotion in p are furnished by oountably many one-to_one oorrespondences of the kind mentioned above. The lemma is thereby proved. 20. THE TWO CLASSES OF SURFACES y. We oall it positiveLy
Consider a geodesio gon a surfaoe y
divergent on y if, for t ...
<Xl ,
s(Pt' po) ...
97
<Xl
where Pt is the point
.:. 50 E.Hopf
t running along g, more exaotly, the p-coordinate of T P and where pO is an arbitrary fixed point on y . Of oourse, if the statement holds true for one po then it does so for any other fixed pO. Quite analogously, a streamline Ttp in the P-space 0 over y is oalled pos . divergent if, for t
~
00 ,
~(Ttp,
po)
~
00
where pO is an arbitrary
fixed point in 0 . It is easily seen that both ooncepts mean the same thing for a geodesio in y and the corresponding streamline in
o.
In faot, this follows from the general inequalities S(p,p')
(8)
~
u(P,P')
~
S(p,p') +
7T
where p, p I are the bearer points of P, pI, respeoti vely. The seoond follows by parallel displaoement of P along the geodesio arc from p to p' and by a rotation of the line-element in pl. An important oonsequenoe of lemma 2 : If a geodesio on y (streamline in
OJ is posi ti vely divergent .in y CO) so is every
0-
ther geodesic (streamline) whioh is positively asymptotic to the first. This statement remains, of oourse, valid if the word "positively" is replaoed by the word "negativelyll throughout (t
~
... 00).
Lemma 3 in oombination with this last statement permits the use of the abundance of divergent geodesics for a subdivision of the olass of surfaces y into two subclasses. DEFINITION. A surface y is of the first olass if the divergent geodesios issuing, frolll a point, pof y form a set oJ directions in p of angular measure zero. If this is true for one point of y it is true for every other point of y . Y is of the second class if it is not of first c],.ass. The possible surfaoes y of first class represent, topologically, a vast variety of surfaces. There are many types of closed surfaces. For instance, every closed orient able surfaoe of genus
98
- 51 F..Hop!
> 1 occurs among them. A closed surface y is of first cl 'lsS ;',"'<" us e , obviously, no divergent geodesic exists on suoh a su!" fa o e. 1A 0re generally, the surface is of the first olass if the group G of covering transforma t ions possesses a fundamental domain D the 0 102 2 s ure of which is entirely in x + x < 1 . Many surfaoes y wi t h boun1 2 d a r y f311 into this oategory. In this oase the problem of the geodesic s is a non-euolidean bi'l'liud problem with refleotion at the boundary. Take, for
eXB ~ ple.
an equil a teral non-euolidean triangle wit h
the three interior angles equal to 2rr!2n
= rr/n
where n is an
ger > 3 (the sum of the three angles must be < rr) and take
ir :~
t~9 t t r e!
hype r bolio refleotions at the three sides as generators of a ir nup G . The oondition on the angle insures that the images under G of the triangular domain oover the hyperbolio plane simply or, i n
0-
ther words, that this triangle is actually a fundamental domain for this group G . The surfaoe y generated by G is then this tr i angular area, and the geodesios problem is the hyperbolio billiard problem. Quite generally, we oan say that a surfaoe y is of first olass if its area A is finite. The reason is this. If y has finite area its P-spaoe fl has finite volume m(fl)
= 2rrA.
By virtue of Poinoa r/
:3
reourrenoe theorem, almost all streamlines of the geodesio flow in
Q (with .invariant measure m) are reourrent. However, if y were of second olass then, by virtue of the theorem of the next paragraph, almost every streamline would be divergent whioh is a oontradiotion. There are many types of surfaoes y of finite area but of infinit e non-euolidean diameter. A well-known example is that of the
mod ul ~c
group G whioh has a geodetio triangle for fundamental domain w~t h 2 2 one oorner or, rather, cusp at infinity (on Xl + x 2 = 1). Tb.", cusp
99
- 52 E.Hopf does not prevent the domain from having a finite area. A surfaoe y of seoond olass is realized if the fundamental domain D of the group G has on its boundary an open aro of the unit oirole. Any geodetio ray that ends up on this aro must be divergen t on y . In faot, suoh a ray stays ultimately in D and the point ru n ning along it gets farther and farther removed, in the sense of the distanoe s, from the finite part of the boundary of D. .
0
its minimal distanoe from all the pOlnts Sx , S to a fixed point x
o
~
Conse~uently )
identity, congru ent
inside D (they are outside of D) tends a fortio-
ri to infinity. Obviously, the initial directions of the geodetic rays issuing from a fixed point and ending on that aro fill a whole angle. To the arc cn the unit oircle there corresponds an infinite funn.l cf the surface y generated (at least this is true if y has no finite boundary points). Hcwever, surfaoes of second olass oan be much more oomplicated in that they can reaoh to infinity in muoh more intrioate ways. We mention that the olassioal two "kinds" of surfaoes are not the same as our two "olasses". "kind" is a purely topological and, as such, extremely natural motion. From the standpoint of ergodio theory, however, the natural division is the one into the two "c15sses". A more detailed discussion of the relation between "kinds· and "classes" can be found in our memoir [11].
2d. ERGODIC THEORY AND THE TWO CLASSES OF SURFACES Y We are now ready to begin with the proof of the two principal theorems on the geodetio flow on surfaces y . FIRST THEOREM. For a surfaoe y of first class the geode t ic flow is ergodic. In other words, i f f(P) and g(P) > 0 are m-integrable in
n then 100
- 53 E.Hopf
fOf dm
lim "t'-oo
=
holds for almost every P E 0 in the sense of the measure m. The same holds for the limit as
T
~
-00
•
SECOND THEOREM. For a surfaoe y of second class the geodetio flow is dissipative or} in other words} for almost every P E 0 in the t
sense of the measure m the streamline T P is divergent in 0 } posi' tively as well as negatively. Both these theorems have a common root} namely} the existenoe of asymptotic geodesics and the relations between them peculiar to hyperbolic geometry. Their effect is expressed in the following. PRINCIPAL LEMMA. Let B+ and B
be two m-measurable sets in the line-
element space 0 of a surface y . Suppose that they satisfy the following conditions. a) Each set is invariant under the geodetio flow . b) With every streamline in B+ every .streamline pos. asymptotic to it is in B+i the same holds for B
with respect to neg. asymptoti-
oity. c) The set of all points P in one set but not in the other has measure m
= O.
Then} under these conditions} either B+} B
have both measure m re m
=0
or their complements in 0 have both measu-
= O. First we prove this lemma and then the two main theorems. It
is interesting to observe that the lemma is true if in the hypothesis c) and in the oonolusion the statement "set of measure m
= 0"
is replaoed by the statement "empty set". In this altered form the lemma is not only true but almost trivial. In faot} 0) means then that B+ B+
=B
=B =n
} and the oonclusion says that either B+
= B_ = 0
or
This modified lemma is true simply because to two ar-
bitrary geodesics on y there always exists at least one geodesio on y whioh is neg. asymptotic to the first and pos. asymptotio to
101
- 54 E.Hopf the seoond (on the oqvering surface there is preoisely one suoh
=B
geodesio). If Bt
is non-void
this argument oan be applied
the~
t
to an entirely arbitrary streaml.ine and to a streamline in B . However, we need the lemma in the form stated previously beoause, in hoth our applioations, the exceptional nUlsets in the hypothesis c) are, in general, non-void. The proof of the principal lemma rests upon the same simple gument but then the exceptional sets of measure zero
re~uire
~r-
care-
ful handling. A set of complete streamlines in the element-space
n of
a surface y can be represented in two ways, as a flow- inv ari atL
point set in
n
e+)
and as set of pairs of points (8 ,
on the un::t
circle which is invariant under the simultaneous transformations
E G . Remember that t ffde de is a measure
S
n has
n.
= 0
t
measure m = 0
only i f the corresponding set of pairs of points
ffde-Jet
_
in the space of those pairs
ve first: A flow-invariant point set in
sure
n and (e , e ). We
m is the flow-invariant measure in
(e , e ) +
that pro-
i f and
has mea-
To see this -denote by 0 the invariant set in
Let 0' be the set of all elements e in the covering surface
2
2
Xl + x 2 < 1 which represent the elements contained in O. We know that m(O) = 0 implies m(O') = 0 and vice versa. 0' is invariant under the geodetic flow in the covering surface. On using the coordinates
(e-- , e+,
s) in (6) for the elements of 0' we see that 0' is
a cylindrical set, with its base in
-
t
(e , e ) -space
e~ual
to th ft
set of pairs of points. By virtue of ( 7 ) , m(0' ) = o means that that - + set has melisure de = 0 As P is > 0 and continuous this - + The statement is thede = 0 means that that set has -measure
ffpde
ffde
reby proved. The prinoipal lemma is now proved as follows. We assume that
102
- 55 E.Hopf m(B ) > 0 and show that B
=
n-
and B+ =
B
n
-
B+ have measure m = 0
Re-
taining the letters B , B+ for the sets oorresponding to these sets + in (e , e )-spaoe we infer from the hypothesis b) of the lemma that
-
both those sets are produot sets in that spaoe ~
(10)
B
=b
X
Q,
B+
=Q
X b+
where Q is the e-line and where b_, b+ are oertain measurable subsets of it. Striotly, we would have to exempt the diagonal-line
e
=
e+
GX G
from
sinoe no geodesios oorrespond to the points
on' this' line. Sinoe, however, it is of measure
IIde +de +
0 it may
safely be disregarded in this proof. Hypothesis 0) says that B • B+ = b X b+
(11 )
have measure zero. This may be understood in the sense of the produot measure
IIde-de+.
By virtue of (9), b_ has non-zero measure
on the e-line. Consequently, sinoe the first set has produot measure zero, b
+
has e-measure zero. Sinoe the seoond set has produot
measure zero it now follows that b
has e-measure zero. In
other
words, the oomplements of botli sets (10) have measure zero. These
-
oomplements in produotrspaoe oorrespond exaotly to the sets B_, B+ in the spaoe m
= O.
n.
Consequently, these latter sets have measure
The lemma is herewith oompletely proved.
We prove the seoond theorem first. Denote by B_(B+) the set of all elements P 6
n
on neg. (pos.) divergent streamlines . As y
is of seoond olass the angular measure
Ide
of the "pos. divergent"
direotions in a point p € Y is positive. Henoe
JJ fit
d¢dA =
I
y
{fd¢}dA > 0 .
103
- 56 E.Hopf That the two sets Band B+ satisfy hypothesis c) of the principal lemma follows from a general theorem ([g] or [10]) If Ttp, TOp = P, Tt +s = TtT B, is a continuous flow in a comple-
te metric space
n with
invariant measure m
(m u-finite) then the
set of all streamlines that are divergent in one direction but not the other has measure m
= O.
That hypothesis b) is satisfied was remarked before. Therefore, the conclusion of the principal lemma holds, and the second thearem is thereby proved. By virtue of lemma 3 we have obtained a sharper result: If Y is of second class then, in any point p
~
y, the geodesic
rays issuing from p are divergent for almost all initial directions . We now prove the first theorem. Just as in the beginning of the preceding proof we infer that, for a surface y of first class, the set of all pos. or neg. divergent streamlines in sure m
= O.
n is
a set of mea-
We have to mention now that the general theorem
refer-
red to in the preceding proof is part of the following general theorem ([g] or [10]) : Under the same hypothesis as in that theorem
n splits
into
two invariant parts, the conservative and the dissipative part. The first contains almost no pos. or neg. divergent streamlines. The second consists almost excluBively of pos. as well as neg. divergent streamlines. In the conservative part 00
(12 )
J
o
t
geT P)dt
=
00
holds for any g(P) > 0 almost everywhere. In the first chapter of these lectures this decomposition was mentioned already for the case of a single mapping T (however, without the part about divergent streamlines).
104
-
~7
E.Hopf
In our present case, geodetic flow on a surface y of first class, the flow is purely conservative. For a conservative flow with invariant measure m the ergodic theorem states this [10) : If f(P), g(P) > 0 belong to L (m-integrable) in 11 then m 7 f o f(Ttp)dt lim q (P) = f 'It' (P) , (13) q (p) = 7
7: ....00
7
f7 g(Ttp)dt o
exists almost everywhere (m) in 11
gf If €
Lm ' f ll'
t
is T -invariant
and satisfies the relation
f
(14)
11
=f
gf*hdm
- 11
fhdm t
for every bounded and measurable h(P) that is invariant under T . The average in the past, lim
(15)
q (P)
7 .... -00
= f /tit
(p)
7
"'t -t exists almost everywhere too (apply theorem to the flow T = T
Tt), and · which has the same invariant f unc t lons as the same .relation (14) as
f
11
g (f
f~~ sa t"lS f"les
f~. Consequently,
** - f'f) dm
= 0
must hold for every bounded invariant h, so for instance, for h
= sign
(f** - f*). Hence, f
(16)
*"*" (P) = f~
a. e.
(p)
must hold. Our aim is to show that f h
= 1,
*" =
fW-*
is constant a. e .. From (14)
it would then follow that this oonstant has the value ff/fg ·
Tc prove constancy of f *for any f
'L m
it suffices to prove this
for every f in a set of f's that is dense in Lm' Reason: The linear operator fit" = T*'f (g' is kept fixed) satisfies
105
- 58 .. E.Hopf
J
n
* I dm .s f nI f Idm
g If
(apply (14) with h = sign (f~ )}. We use this fact as fJllows in
n).
our present oase (y,
We choose the fixed function g > 0 such
that g(P') - g(P) (17)
holds
uniform~y
a(P,P') - 0
as
g (P' )
with respect to P, P'. In the important speoial ca ..
se where y has finite area we may simply choose g
= 1.
For funct io ns
f we take only those f's which have the similar property that (18)
holds
fep')
~
f(P)
o
g (P' ) uniform~y
in
n
a(P,P') ... 0
as
In the case where y has finite area and whe-
re g = 1 this simply means that f is uniformly continuous in
n
It can be shown by means of claSsioal arguments that the set of those f's is dense in L . m After these preparations we now turn to be main point of our proof. Consider g, f, as indicated above. We show first that ( (19 )
f(Ttp')dt
ro g(Ttp')dt
have the same limits as
if the two streamlines occurring he-
T ~ 00
ft' g , f', g' t t t integrands, respectively. Then by virtue of lemma 2 and in consequenre are pos. asymptotio. Write briefly
ce of (17) and (18), the quotients at =
tend to zero as t
~~.
l' '
t
-
-
f 't
bt
g' t
=
g' t
-
gt
g' t
The differenoe of the two quotients (19)
may be written as
106
- 59 E.Hopf T
I
T
0
T
Io
atgtdt,
t
Iog'dt t T
Iobtg'dt t
f dt t
t
g dt t
0
g'dT 0
T
This equality and (12) show that the difference tends to zero as 1)
Obviously, the same is true as
if the two stream-
T ~ -00
lines are neg. asymptotic. Consequently, the two invari,lllt sets
=
Bt
[p If*' (P) .? c]
satisfy hypothesis b) of the principal lemma no matter what the value of the constant cis.
(16) implies the validity of hypothes i s
c) . From that lemma we can, therefore, infer that either Bt or its complement has measure m in other words, f~
=0
. This holds whatever value c
h~s
o~,
is constant a.e., the first theorem is thereby
completely proved. Lemma 3 permits again to state the theorem with a
r
clusion but a stronger hypothesis: If
sh~rper
con-
is of first class, if f(P),
g(P) > 0 are in Lm and if these functions have the continuity properties (17) and (18) then lim qT(P) point p f
r
= IfdmlIgdm
holds in every
for almost ; all directions (P) in p.
2e. ADDITIONAL RESULTS AND PROBLEMS. The previous results have
str~ightforw~rd
complete n-dimensional manifolds
l7t of
generalizations to
scalar curvature minus on e .
The universal oovering space is the interior of the unit sphere 2
2 x
1)
1
tX
2
2 t".tX
n
<1
This conclusion is right provided that the first factor of the second term remains bounded as
107
T
~
ro . This . I.
- 60 E.Hopf endowed with the metric da
2
=
dV
=
Ita isometries are the n-dimensional Moebius transformations that map that oovering space onto itself. In the (2n-1)-dimensional
sp~-
oe of the directed line-elements a measure m is defined by d1l\
where
d~
= dwdV
is the element of volume on the sphere of direotions fram
dV. It is invariant .both under the induced
isomet~ies
and under
th~
geodetic flow. In our memoir [11] the reader finds the preceding theory completely developed in n dimensions. He also finds there the following theorem and its proof : If
bt
nt
is an n-dimensional manifold as stipulated above and if
has finite volume v(nt) then the geodetic flow on ~ is atrongly
mixing: For two arbitrary sets A, B. of finite m in the line-element spaoe
n of
nt
there holds lim mCB'T\) t...oo
=
m(A)m(B) 1tl(O)
It is natural to conjecture that, in the more general case where
71L
is of first class but not necessarily of finite volume the geo-
detio flow is still strongll mixing in a properly generalized senee. If m(h)
= 00
the
~reced1ng
teresting. Is this
relation is presumably still valid but unin-
r~la'ion
still valid if the expressions on both
sides. are replaoed by quotients of them using four sets A, H; AI, Blf --------~----~~--
is certainly tr.ue for almost all P. However, the interference of one more nulset, can be avoicted if we argue as follows. We can restrict the tunctions l' still further by requiring that fig be bound~d. The hctor in,question.is th~n bounded by the same bound for any P. Moreover, this additional restriction on l' does not invalidate the statement that the fls. ape dense in Lm'
108
- 61 E. Hopf Let us return to n
=2
dimensions. Our surfaoes 1 furnish a
large variety of interesting flows but they are restrioted by the oonstanoy of their ourvature. In our paper [il] it is shown th'>t ex aotly the same ergodio theory remains vali.d if 1 is a oomplete two-dimensional Riemannian .manifold of variable negative curvature (1 is sumed to be of differentiability olass
C'~.
88-
There is the same basic
diohotomy into two olasses, and both first and seoond theorem remain literally valid provided that the ourvature remains .between negative bounds. The mixture property has, however, not yet been proved for variable curvature. We would like to return onoe more to our surfaoes 1 of
' oon~
stant negative ourvature. We mentioned above that the purely topologioal subdivision into two kinds is different from our subdivision into
two~lasses .
The first kind is aotually a larger totali-
ty than the first olass. Take a surfaoe 1 of first kind but of seoond olass. Paul Koebe had already proved, in his famous memoirs on
non~euolidean spaoe~forms,
that on any '1 of first kind there e-
xists a quasi-ergodio geodesio (streamline dense in D).l) For a general flow that .isergodio (f* = oonst. for every f) the same thing is true; even "most" streamlines are quasi_ergodio. What 1)-------~--·
-
Koebe's topologioal analysis of the geodetioal flow on surfaoes 1 has been oarriedo'n, and several of his results sharpened, by Gottsohalk and Hedlund [51. There is the following result on"topologioal" mixture on surf~oes y of the first "kind" (this is sher~ar thari Koebe's rrsu1t mentioned above): If A is a non-void open subset of n then. TA 'oeoomesdenserand denser inD as t -> ro. To Hedlund [8] the ergodio theory of the geodesios on surfaoes 1 owes the first (and an ingeneous proof it is) proof of the JIleasure-theoretioal,mixture of the flow on a surfaoe1 'of finite area. The author~s proof of the same faot for n_dimensional manifolds 17t(quoted above) would not have been possibLe without Hedlund's beautiful idea. The author's .first proof ,of ergodio mixture for surfaoes 1 of finite area [13] had yielded mixture only in a less stringent sense.
109
- 62 E.Hopf ~bout
the oonverse question: Does the existenoe of a
qU~Biergod10
streamline imply ergodioity of the flow? The answer is "no". On a surfaoe y of first kind but seoond olass the geodetio flow is' d1s· sipative .
110
E.Hopf REF ERE NeE S
1.
C.CARATHEODORY: Bemerkungen zum Ergodensatz von G.Birkhoff. Sitzungsber. B"Iyr. Akad. Wiss", Math. -Naturwiss. Abt. 1944(1947) p.189-208.
2.
R.V.CHACON and D.S.ORNSTEIN: A .gener"ll ergodic theorem. Illinois J. of M:lth. 4 (1960), p.153-160.
3.
J.L.DOOB: Asymptotic properties of Markoff transition probabi1 i tie s. T r:l n s . Am e r. Mat h . Soc. 63 (1948), p. 393 - 4 21
4.
Y. N. DOWKER: A new probf of the. general ergodic theorem. Act'l. Sci. Math: Szeged 12 (1950) p.162-166.
,5.
W.H.GOTTSCHALK 'l.nd G.A.HEDLUND:Topological dynamics . Amer. M.' l.th. Soc. Colloquium Publicati.ons 36 (1955).
6.
P. R.HALMOS: An ergodic theorem. Proc. N'>t. Ac,>d. Sci. p . 156-161.
7.
P.R.HALMOS: Measure theory. New York 1958.
8.
G,A.HEDLUND: The dyn!l.mics of geodesic flows. Soo. 45 (19~9), p.241-260.
Bull.
32 (1946)
Amer. M:lth.
" Zwei Satze uber den VerI auf der Bewegungen dynamischer Systeme. Math . Annalen 103 (1930), p . 710-719.
E HOPF:
10.
EL HOPF:
11.
" E. HOPF: St:ltisti* der geod'ltischen Linien in ,Mannigfal tigkeiten negativer Krummung. , Beriohte S~ohs. Ak'l.d. Wiss." Math.N9.turwias. Klasse 91 (1939), p.261-304.
12.
Ergodentheorie. Berlin 1937.
" II E.HOPF: St9.tistik der Losungen geod'l.tischer Prob.leme vom unst9.bilen Typus II. Math. Ann'l.len 117 (1940), p.590-608.
E. HOPF: Beweis der Mischungscharakters der geod~tischen Str~ mung auf vollst~ndigen FlBehen der Krilmmung minus Eins und ~u dlioher ·OberflMche. Sitzungsber. Preuss. Akad. Wiss." Phys.math. Klasse 30 (1938).
14.
E.HOPF: The general temporally discrete Markoff process. J. R:ct Meeh. An'l.l. 3 (1954)., p. 13-45.
15·
E.HOPF: On the ergodic theorem for positive linear operators . J , reine und angew. Math. 205 (1960), p.101-106.
16.
W. HUREWICZ: Ergodic .. theorem without invuiant me'l.sure. 45 (1944), p. 192-206.
17.
S.KAKUTANI: Ergodic theorems and the M'l.rkoff process with a stable distribution. Proe. Imper . Acad. Tokyo 16 (1940); p.49-54.
111
Ann. Math
- 64 E.Hopf 18.
S.KAKUTANI: Ergodic theory. Proc. Int. Congress Math. Cubridge, Mass., Vol. 2 (1952), p.128-142.
19.
J.C.OXTOBY: On the ergodic theorem of Hurewicz. Ann. Math. 49 (1948), p.872-884.
20.
K.YOSHIDA and S.KAKUTANI: Operator-theoretioal treatment of Markoff's prooess and mean ergodic theorem. Ann. Math. 42 (1941 ) , p.188-228.
112
CENTRO INTERNA'ZIONALE MATEMATICO ESTIVO (C.I.M.E.)
J 0 S E'
MAS S ERA
LES EQUATIONS DIFFERENTIELLES LINEAIRES DANS LES ESPACES DE BANACH
ROMA - Istituto Matematico dell'Universitl - 1960 113
- 1 -
J.Massera INTRODUCTION Nous nous proposons de faire un expose sommaire du oontenu de p1usieurs Memoires eorits en oollaboration aveoJ~J~Sch~ffer et qui ont ete pub1ies sous 1e titre general Linear differentia~ equations
and functionaL anaLysis, Parties I, II, eto. [2, 3, 4, 5, 6, 7, 10, 12]; 1es theoremes, formu1es, etc. de ces Memoires seront oites
P!H
1a suite de 1a fa90n suivante: Theorhe II.5.4, formu1e 1.(2.1) , etc. II nous sera aussi necessaire de resumer les principaux resu11\
tats de J.J.Sohaffer dans son Memoire Function ipaces with
transLa~
tions [11], que nous oiterons F.3.2,F.(4.1), eto. L'obje .t d I etude . sont les equations
ou
t €
x
+ A(t lx = 0
(1)
x
+ A(t lx = f(t)
(2 )
x
+ A(t lx = h(x,t)
0)
= [0,(0); x, f, h
J
A(t) ) pour chaque
t
E X , un espaoe de Banaoh queloonque ;
fixe, est un endomorphisme (continu) de X ;
A(t), f(t) sont desfonotions integrables (Boohner, au sens cbnven a.ble) dans ohaque sous-intervalle borne JI C J . Dans les annees 1930-35,
o. Perron [aJ,
K.P. Persidskii [9] et
r . G.Malkin [1] etablirent· l'equivalenoe des p~oprietes suivantes (dans Ie cas dim X < (p 1) Pour ohaque
00 ,
A(t) oontinue):
f
oontinue ' bornee dans
J (11 espace for me
par ces functions, avec la norme . duo supre/llum, sera designe dans ce qui suit par ~), toutes lea soluti~ns de (2) lerons des
€
t
(nous les appe-
C -solutions).
(p ) Pour ohaque 2
h
oontinue, Ilhll ~ (3,' IIh(x',t) -h(x",t ) ll..::
.:s A Ilx'
- x" II, aveo (3,A auffisamment petites, toutes 1es s o l ut i ons
de (3)
<;
'G 115
-
2 -
J.Mas s era (P 3 ) II existe une fonotion de Lyapunov V(x,t) ddfinie posit i ve, ayant uneborne supdrieure
infinimen~
petite (c .•. d~, i1 existe des
= 0,
fonotions oontinues positives a(r), b(r), nulles pour r que
te l le s
II) ), telle que la Urivde V'('x, t )
a(1I x II) ~ V(x,t) ~ b( Ilx
de V Ie long des solutions de (1) est ddfinie ndgative. (p 4) Il existe des oonstantes posi ti ves N ,
toute solution x(t) de (1) et tout oouple t II x(t) II ~ Ne
-1I(t-t )
~
t
o
11
~
telles que, pou r 0 , on a
Ilx(t ) II .
0
o
(p ) La solution x
.5
= 0 de (1) est uniformdment asymptotiquemen t
staLle, o . • ;d., il existent une oonstante A st une fonotion positive T(~) telles que, pour toute solution x(t) de (1) on a II x (t) II < A l'lx(t pour
o
)11 pour t ~ t 0 ~
to ~ 0 , t -> t En
r~alit',
0
+ T(€)
0 Jet
II x(t) II
.
Perron a ddmontrd l'dquivalence des propridtds
suivantes, plus gdnerale que l'equiva1enoe de (p 1 ) et (P 2 ) : (Ql) Pour ohaque f € (0 ~ k ~ dim X) de
(Q2) Sous les metres de
t
il y a une famille • k parametres
~-solutions de (2). hypoth~ses
e-solutions
Dans nos travaux
de (P 2 ) i1 Y a une famille • k para-
de (3).
I~III
nous nous avons posd la question de
formuler adequatement des proprHtes analogues de hOon a rdtablir l'd,qu:uvoahn!oe 6lv,ed
I
a (p 3 ), (p 4 ), (P5) (Q2) . )la'uxl'tl.-
mement, nous avons gdneralisd oes propridtds au Cas
o~
dim X =
00
et l'hypothsse de oontinuitd est remplaode par des hypothdses du type de Caratheodory. Troisiamement, nous avons fait quelque s progras dans Ie sens de remplaoer l'hypothase "pour ohaque f E
t
dans (p ) et (Q ) par "pour ohaque fappartenant • oertains esp ac e s 1
1
fonotionnels (tels que
t,P, eto.) ". Finalement, nous avon s envis').
gd plusieurs applioations des thdoremes obtenus aux ca s pdr iodique
116
- 3 J.Massera eto.
presque-periodique~
Plus tard, dans les travaux F et IV, nous avons oonsidere une oategorie generale d'espaoes fonotionnels qUi oonstituent le oontexte naturel des problemes qui nous oooupent; pour ohaque oouple de tels espaoes
1b
,1)
,
nous avons enonoe la propriete (Ql) de la
faQon generale : "pour ohaque f (2) appartenant at)
'73
il Y a au moins une solution de
(une 1) -solution)", et etudi6 l'equivalenoe
de oette propriete avec les autres enonoees plus haut. Nous avons aussi elimine plueieurs restriotions superflues et envisage de nouveaux types de oomportement asymptotique analogues a (p 4 ) (stabilite en moyenne et par tranohes). 11 y a enoore un vaste plan d'etudes oom'p16mentaires dont quelques resul tats ont ete deja publies [6, 7, 10, 12].
No us voulons souligner que la generalisation des resultats obtenus
~u
oas dim X
= 00
et aux A(t) disoontinues n'est pas du tout
l'aspaot le plus important de nos travaux : presque tous les theoremes gardent leur interet dans le oas olassique dim X <
00
•
Le pas-
sage a la dimension infinie est banal dans plusieurs oas; quelques fois il implique, au oontraire, des diffioultes teohniques plus ou moins grandes. Pour un petit nombre de resultats l'hypothese de dimension finie est essentielle. Nous appelons
U(t) la solution~operateur (endomorphisme) de
l'equation
ir +
A(t) u =
a
qui satisfait U(O) = I (identite); il est faoile de voir que U(t) est inversible pour ohaque t ~ J , La solution de (1) par (x ,t ) o
est alors x(t)
0
La solution de (2) est donnee par
la methode de la variation des oonstantes :
117
- 4 -
J.Massera
+ It to.
-1 U (r)f(r)d-Tl.
(4 )
Des oalouls elementaires (Lemme de Gronwall, eto.) conduisent aux inegalites ( IV. (2. 1»
II x ( t ) II
(IV. (2. 2) )
Ilx(t) II ~ (1
~ (II x ( r) II + -
-1
I JI
I JI
II f ( r) II d r ) • ex p C!J I II A( r) II dT)
Ilx(r) Ildr + expC!
I JI II
JI
( 5)
f(r)11 dr).
II A(r) Ildr) ,
(6)
ob x(t) est une solution de (2), JI est un sous-intervalle de J de 10ngueur.1 et t, r € J I. Dans ce qUi suit nous utiliserons la notion de distance angu-
Lairede deux vecteurs non-nuls de l'espace X , definie par r(x,y)
= Ilxllx
11- 1 _ yllyll-1 11 .
ESPACES FONCTIONNELS Nous designerons par
t
l'espace des fonctions (plutot : clas-
a
ses dlequivale~oe) de~inies dans J integrables dans
ohaq~e
valeurs reelles, mesurables et
sous7intervalle fini JI C J , avec la topo-
logie de la convergenoe en moyenne dans chaque JI
C'est un espa-
ce vectoriel localement oonvexe de Frechet (oomplet et metrisable). Soit V et U Iorsque
un espaoe vectoriel localement convexe de Hausdorff
un espaoe norme. Nous dirons que U
)j
U est algebraiquement oontenu dan~ V et Ia convergence
dans Ia topologie de de V
est pLus fin que
U
implique la oonvergenoe dans Ia topologie
Alors, pour chaque semi..,norme
stante positive a 71 telle que
71 (y)
~
de ))
.71
i1 existe une con-
a 71 II y 11v.' y ~:U
mame est norme, i1 exis·te une constante
118
a
Si
telle que IIYIIV
'J
lui-
~
allyll~
!
- 5 J.Massera yF.U
dans oe oas nous eorirons
L( JJ.) de
U
2L
aU
~
(la boule unitaire
est oontenue dans a fois Lev) ).
Soit f(t) une fonotion definie dans J IT> 0 . Les trans~a-
tees (a droi te, a gauohe) de f
+
I
T" f T
=g
f
-f = h
sont definies
T
par get)
=0
get)
= f(t
0 ~ t < T - T)
t
~
T
h(t) = f(t + T)
t
~
0
Nous definissons maintenant les classes d'espaoes fonctionnels que nous util1$erons par la suite: les espaces fonctionneLs admet-
tant des trans£ations(cf.
Fl.
tions (plut8t : olasses d'equivalenoe) definies dans J reelles appartient ala olasse la norme
a)
valeurs
s'il est norme (notation pour
b) f~'J
" f(:'f
"1
I
il existe un nombre posit i f a(J') tel que
JJ,lf(T)ldT~a.(J')ldt
0)
rr
a
de fonc-
I I~ ) et ei les oonditlons suivantes sont rempliee pour chaque sous-intervalo. a. d 7 est plus fin que
16 fini J' C. J
hl~ ~
1
Nous dirons qu'un espaoe
J
Ig(t)1 ~ If(t)1 presque putout implique g~
J
7
ttl, 1 f.
{o}
Si lIon a auesi
> O. implique no us dirone que
~
T- f T
~
1
appartient a la claese
Noue pouvons etendre la definition de ces olasses d'espaces au cas des fonctions
a
valeure dans un eepace de Banaoh quelconque.
Pour une telle fonction f noue definissons la fonotion norme II f II par la relation Ilfll (t) = II f(t) II; c'est une fonction a valeurs
119
- 6
.~
J.M3sser3 reelles. Alors, designons par
'1
etant un espace de classes
1 (X)
11 eapaoe des fonctions f
fortement mesurables, telles que prietes plus importantea de
1 est
ai
complet,
r
II file 1s I etendent
T"J,
T
a
nous
valeurs d'lns X ,
En general, les pro-
a
'1 (X)
(par exemple,
1(x) l'est auasi).
Pour les applications aux equations differentielles (les solutions etant toujours continues) il sembleraitnaturel de considerer dea espaoes de fonctiona continues ayant des proprietes analogues.
'"r
On peut dElfinir une telle claaae
c
(of. F ) mais la condition d)
implique 11'1 reatriction peu naturelle f(O) = 0 . D'autre part, on peut montrer qu'il exiate une relation etroite entre ces espaces et des espaces
1r
oorrespondants (Theoreme F.6.3), de telle fa90n
e
qulon peut remplacer les premiers par les seconds (par exemple, par Lea eapaoes olassiquea demment 11 11'1 01a88e
m : eapaoe
T'.
p p ,
~
1
~
P
~ 00 ,
appartiennent evi-
DI autres exemples im,portanta aont :
des fonotiona tellea que
sup
.Utt +1 I f(T) IdT
: t f J} < 00,
avec 11'1 norme du aupremum;
t
00
o
:
soua-espaoe de
t 00
forme par les fonctions dont 11'1 limite es-
sentielle 11 llinfini eat nulle; : aoua..,eapace de ~ forme par lea fonctions telles que t +1 If(T)ldT = 0 . Toua oeseapaces aont complete. lim J t ....oo t La relation d I ordre "plus fin" dEl fin i t dans rrf 1 une
r [
structure de treillia dont nous designerons les operations par
V.
La
A
A,
d I une famille d I eapaces est form6e pa,r les fonctions
appartenant 11 l'interaeotion.pour lesquelles le supremum des normes eat fini; on prend oe supremum comme norme dans
A.
La
V
d ' une
famille d'eapaces est 11'1 aOmme de 11'1 famille, 11'1 norme d'une fonction etant l'infimum de 11'1 somme des normes pour toutes les decompo-
120
- 7J.Massera sitions possibles de la fonotion donnee dans les espaoes de la famille. On peut demontrer (Theoreme F. 4.2) que la
'f [ 'If]
d' espaoes de olasse aussi de olasse
'T [ TI]
existe toujours et ou bien elle est
ou bien
= .i0}
.§tant exolue pour les famil1es finies; ces de olasse
r-t'
appartient aussi"
'r
espaoes sont de olasse
V
la
aT,
appartient
Ad' une fuille
a
, la derniere possibilite
V
la
T",
,d upe famille d' espa-
si elle existe; si les
et la famille est totalement ordonnee,
si elle existe . La
oes oomplets est aussi oomplete (pour 1\
0'
1\ [Y) de deux espaest aussi vrai meme si
la famille est infinie).
1-
Une notion importante est oelle, de jerflleture ~ocIlLe. Soit un espaoe de olasse semble de 111.1
.c , est
T [
7'<#];, la boule L(
oonvexe, balanoee (si f
t , A ~ O}
1)
l)
(dans
L(
1 ),
Af € r(
1 . II
meture looale de
to = t , OO
00
'1) ,
1)
avec
est fermee).. La fermeture de
ayant les memes proprtetes, on peut la oons i derer
oomme la boule d' un nouvel espaoe
10
€
oonsideree oomme en-
= 1), bornee, radialement fermee (l'interseotion de r(
une demi-droite {Af : f " r(
~),
1011/,=
eas des espaoes
tP
F.4.4 et Oorollaire eomplet; que si
,'IJ.
et
1 E- 'T'
rj,
sera di ,t looalement ferme; tel est Ie
I?t
F.4.~)
t-J.
1~, 10; , 10 10 iT= log . Un e-
On a
1 = 101
spaoe pour lequel
que nous appelerons la fer-
est faoile de voir, ' par exemple, que
"Il1o
0
101,
On peut montrer (Theoremes F.2.2 et que tout sspaee looalement ferme est
[ 1" 1 ,
10
d-f: T'
['i'4t] ; que les
A
et
V
de famillesfinies d'espaoes'looalement fermes Ie sont aussi. On peut demontrer que parmi les espaees de olasse il
Y en ' a un, a savoir
17l,
qui est le moins fin de tous. Quant
aux espaees les plus fins,
la situation est plus oompliquee : il I""i'""t (1) n 'yap as d' e s,p a 0 e 1 e pI u s fin dan s I] ma is 1 a sou S -
IT [
(i)-------------
Ceoi a ete demontre apres la reunion de Varenna.
121
- 8 -
J.Massera olasee dee eepaoes looalement fermes dans
rl
a un eepaoe qui est
Ie plus fin de tous (Theoreme F.4.19). Oet espaoe, que no us desi-
C,
gnerons par 00
est oelui dee fonotions f telles que
.s
.s
C
L BUP ess {If(t)l: n t n + 1} < 00 ; 1" norme de est equin.=o valente Ii la somme de la serie (1a somme elle-meme ne satisfait p a s 1a oondition d) ).
Soit
1 E ,f que loon que
ohsse
l'
l' ensemble des fonations f
fg dt I : g € L(Y)} < 00 ; on peut o aveo oe supremum oomme norme, ~st un espaoe de
mesurablee telles que sup { If verifier que
l'
et 00
rrtlooalement !'erme (Theoremes F.4.16 et F.4.17).
1-
e ,' appelle I' espace assode Ii re F. 4. 7)
1"
'1'
(ou dual de K~the). On a (Oorollai-
= 10'if ; I' assooiation est un anti-automorphisme in-
volutoire du treillis des espaoes looalement fermes de olasse T~
"fYi'
= ~,
Z;
,
'="J't(Lemmes F.4.9 et F.4.10 et Theoremes F.4.17 et pi
F.4.18); on a aussi (Lemme F.4.l1) (",)' =
t
00
00
,(t)' =
0 1
I..;
Finalement, on peut voir que les espaces d'OrLicz sont loaalement fermes de olasse
T*'(Theoreme F. 5. 1), que I' assooH d' un
espaoe d'Orlioz est equivalent, en norme Ii l'espaoe d'Orlioz defini par la fonotion de Young oonjuguee (Lemme espa-oes d' Orlioz plus fin
f 1 JJ
£ f\
,00
IV
et
1
L
V
t
00
F.5.3) et que parmi les
sont, respeoti vement, Ie
et Ie moins fin.
ADMISSIBILITE' Nous dirons que 1e oouple
(13,b)
(on devrait plutot eorire
(P.1(X), ')j (X)) ) d'espaoes fonotionnels est admissibLe pour l'e qua tion (2) si pourohaque l' ~
'15
(X) 11 Y a au moins une solution de
(2), x E .1)(X) (une V-solution).
1.> 122
et ~
seront toujours par la
- 9 J.Massera suite des espaoes oomplets de olasse
T ,
quoique quelques rElsul-
tats sont aussi valables dans des oonditions plus g'n'raies. Nous dirons que Ie oouple que Ie oouple
(13 2, ~2)
si
(1J 1 ,
'p~U$ fort
[hibIe)
~1 est moins [plus) fin que
~2
)J 1 est plus [moins) fin que oi 'quivaut
(10 1, ;01) est
a. 1->1 ~ ~2 ' ~ 1
~
at
2
Les espaoes 'tant oomplets, oe-
C ;0 2
L'admissibilit' du oouple
'J) ) est alors une hypothese plus forte sur I' 'quat ion que 1
l'admissibiliU de (~2' Z>2) Soit XoD
l'ensemble des valeurs initiales x(O) des
~-solutions
x(t) de l"quation homogene (1); olest une vari't' lin'aire dans X et si, pour une f donn'e, 1 ~ 'quat ion (2) admet une
1) -solution
x(t) , l'ensemble des valeurs initiales de toutes ses
)J-solutions
est x (0) + Xot) ferme); 1.4.1 est un exemple de oette oiroonstanoe pour Dans la suite nous ferons toujours l'hypothese que X
oJ)
et quill admet un
sous~espaoe
oompl'mentaire
1)
=
e.
est ferme
Xl~
peut se passer de oette derniere hYFothese, mais au prix de complexites que nous voulons 'viter dans oet expose. Dans oes oonditions" si (1;, lJ ) est admissible, pour ohaque il Y a une et une seule x(O) {X
et la oorrespondan~e
m
.o-solution x(t) de (2) aveo ~~V
ainsi d'finie est 'videm-
ment lin'aire. Un resultat fondamental est qu l elle est aussi bor-
THEOREME 1 (Corollaire IV.2.2). Si (7.>, ~ ) est a,d1llissib~e, d te une consta,nteK >0 te~ Le que pour cha,que f
f'f.>
exis .-
i ~ y a, une e t u-
ne seuLe so~ution x(t) de (2) sa,tisfa,iSllnt x E'iJ , x(O) E X 1X), et on a,
I xI~
~
d d~
La demonstration de oe r'sultat repose sur Ie
theor~me
de gra-
phe ferm' suivant qui peut se · deduire des inegalites (5) et f6) :
123
- 10 -
J. Ma ssera
t
THEOREME 2 (Lemme IV.2.1). Soit{f}C n
(X) et x
n
une soLution de
L
i + A(t)x = f n . Si .{f n } et .{x n } convergent dans respectivement ·versf } x , on a x + A(t)x = f et x -- x uniformernent ·dans chaque n
sous-intervaLLe borne.
DICHOTOMIES La generalisation naturelle de la propriete (P 4 ) est Ie type de oomportement des solutions de (1) que nous avons appele di c hotomie 8 (simples ou exponentielles) et que, comme nous verrons en s u i te , est etroitement lie
a
l'admissibilite de oertains couples pour l'e-
quation (2). En vue de (P 5 ) on peut aussi designer oes types de com portement par stabiLite conditionneLLe uniforme (simple ou asympt otique) de la solution x
=0
de (1); Ie oas asymptotique est essentiel-
lement equivalent, pour les equations lineaires, a oe que N. N.Kr asovs kii appelle comportement non-critique uniforme (of.
Theor ~ me
I .3. 5,
Corollaire I.3.1 et Exemples I.3.3 et I.3.4). ( 1)
Nous dirons que les sous-espaoes oomplementaires Y J Y de X o 1 induisent une dichotomie expon~ntieLLe des solutions de (1) s'il existe des oonstantes positives N , N' , oonditions suivantes sont
V
,
v' , y
o
telles que le s
rempli~s
(Eil Pour ohaque solution x(t) de (1) _aveo x(0) € Y Ilx(t) II
~
Ne ~V(t-to7 IIx(t ) II, t o .
~
t
0
~
on a
0
o;
(Eii) Pour ohaque solution x( t) de (1) avec x (0) v' (t-t ) IIx(t) II ~ N'e 0 IIx(t ) II , t ~ to ~ 0 ;
~
Y
1
on a
0
(Eiii) Pour ohaque oouple de solutions x. (t) de (1) avec ~
~ y , t ~ 0 . 010 Si dans (Ei), (Eii) on supprime les faoteurs exponer:tiel s, on
x (0) £ Y. , i = 0, 1, on a
y. (x (t), x (t))
obtient les proprietes (DU,
(DiU, lesquelles, avec (Diii ) = (Eiii),
i
(1)
1
On peut formuler la definition de fa90n a ne pas fai r e i nte rveni, Y1
124
-
11 -
J.Massera ddfinissent Ie oomportement appeld dichotomie (simp'e) des solutions de (1) indui te par les sous-espaoes Y ,Y o
1
.
On peut demontrer (Lemme . 1.5.4) que si A f
M alors (Eil et (Eii)
impliquent (Eiii). Dans Ie oas Ie plus simple, on a : LEMME 1. 81 dim X · (00
,
A = const. et si 'es pa.rties reeHes des
ra.cines ca.ra.cteristiques de -A ·sont
non~lIuLLes,
on a. une dichotomie
exponentieLLe; et reciproquement.Si a.ux ra.cines ca.ra.cteristiques purement ima.gina.ires correspondent ·des diviseurs eLementa.ires Linea.ires, a.Lors on a. une (etPeut~6trePLusieurs) dichotomie simpLe; et reciproquement. On peut aussi envisager desoomportements "en moyenne" ou "par tranohes" du type suivant : On a la sta.biLite a.symptotique uniforme en moyenne s'il existent des sous-espaoes oomplementaires Yo ' Y1 ' des oonstantes positives v , v' et des fonotions positives M(tl), M'(tl) ne dependent que de tl > 0 , tels que : (Mi) Pour ohaque solution x(t) de (1) avec x(O) (Yon a o
t ttl
J
t
Ilx(u) II du
(
)
t ttl
~ M(tl) e -v t-t o J
0
to
Ilx(u) II du
t
~
t
> 0
o
(Mii) Pour ohaque solution x(t) de (1) avec x(O) f Y. on a 1
t+tl
J
t
Ilx(u) Iidu ~ M' (tl)e
On a la vement
sta.bi~ite
v'(t-t ) to+tl 0
a.symptotique
J
to
Ilx(u}lIdu,
~niforme
t
~ t
o
~
0 .
pa.r tra.nches (relati-
a un espaoe fonotionnel donne lJ · oontenant les fonotions
oaraoteristiques des sous-intervalles bornes) s'il existent Y
o
Y1 ' v , v' , y(tl) , M' (tl) comme plus haut tels que :
qui peut meme ne pas exister comme complement de Y Yo est ferme est, par oontre, essentiel. 0
125
Ie fait que
- 12 J.Massera (Til Pour chaque solution x(t) de (1) avec x(O)
I x[ t,t+AJ
x' ~
-v(t-t ) 0
M(6)e
)j
I t rto,to+AJ xl, X)
€ Y
on a
o
t~t
~O,
o
~
(ou
designe la fonction oare.oteristique de 1 I ensemble E C J); E (Tii) Pour ohe.que solution x(t) de (1) aveo x(O) E Y l on a
t
[t,t+AJ
x,
~
~M'(A)e
v'(t-t)
01%
[t ,t +A] o 0
xl,
t~t
X)
o
~O
.
8i l'on supprime les fe.oteurs exponentiels, on obtient les oomportements simp~es respeotifs.
II n'y a pas apparemment de oorre-
sponde.nt naturel de la propriete (Eiii) dans les oas envisages maintenant.
THEOREME8 FONDAMENTAUX On peut demontrer les theorbmes suivants THEOREME
3 (Theoreme!) IV.6.1 et Iv.6.2). Si (~,n) est un
cQuPbe admissibLe d'espaces decLasse
r
r
(un
-coupLe), iL y a
stabiLitt uniformeen mo,enne.etpar tranches des soLutions de (1) Yo
= Xo~ , Yl = Xl,.:}' Si (tJ, 1)
, Go ) ·00
n' est pas pLus faib Le que
La stabiLitt est mIme asymptotique.
THEOREME4(Theoreme
admissibLe PLus fort que (
IV.S.l). Si
t1
e
,
OO
,X
)
(1D,V) 02J
,X
est un
lt)
T-coupLe
induisent une dicho-
tomie des soLutions de (1). THEOREME
admissib Le et
5 (Theoreme A
e: 7ft ,
rv.S.2).Si
(!b,iJ) est un
T-coupLe
Xod)' Xl~ induisent une dichotomie des so Lu-
tionsde (1). THEOREME
tions de (1),
p~es
6
(Theoreme IV.S.3).
Le coupLe
(f} ,
eoo )
PLus faibLe$)j si dim Y <
admissibLe.
o
00
S'iL y a une dichotomie des soLu est admissibLe
(et tous Les cou-
mIme Le coupLe (
126
J,\
,too) est 0
- 13 J.Massera THEOREME 7 (Theoremes IV.9.1 et 1V.9.2). Si (~,.D) est un
T -coup~e
admissibLe qui n'est pasp~us faibLe que (
t 1,
too) et o
si ou bien H y a une dichotomie des soLutions de (1) ou bien A~
rr;;
,aLars Xo.o' X1X)
so~utionsde
induisent une dichotomie exponentie LLe des
(1).
THEOREME 8 (Theoremes V~5.2 et 1V.9.3). J'iL y a une dichotomie
exponentieLLe ,des so~utions de (1), pO'll-r que Le soit adlliissibLe
impLique F
12;0
tres forts
(7lb,
T -coupLe
('1;),;)))
m
n
suffit (et, si Af1Yb , iL Le faut) que f€ au F(t) : Jttl If(u) Idu . BTl 'particuLier, Les coupLes tool 1;/) , ('M, l ), ,IC) sont admissibLes. "'Clo
(e
0
11 est impossible de donner ioi meme un apper9u synthetique des demonstrations de oes theoremes, qui sont assez longues et compli:quees. Nous voulons seulement indiquer l'idee maitresse de la demonstration. Pour les theoremes direots (4, 5 et 7), si x(t) est une solution de l'equation (1) aveo, disona, x(O) E Xor; (o.a.d., une
D-solution)
at si ¢(t) est une fonoti~n soalaire
dont la de-
rivee est nulle en dehors d'un sous-intervalle borne et telle que ¢(O): 0 , la fonotion yet): ¢(t)x(t) est une solution de (2) oorrespondant a f(t) = ~(t)x(t) ; aveo un ¢ oonvenable on obtient f E 7?J
et, puisque y(O) = 0 € X1a:;) , on paut appliquer le Theoreme
1, on obtient I¢x~
~
d¢x1t3
at les propri6tes de type diohotomi-
que s'ensuivent. Pour les Theoremes reoiproques (6 at 8), la base de la demonstration est la formule de la variation des oonstantes (4) eorite oomme il suit :
ob Qo ' Q1 sont les projeotions oomplementaires sur Yo' I1 U(t)Q oU- 1 (T)f(T) et U(t)Q1U-1(T)f(T) etant, pour ohaque T fixe, des solutions de l'equation homogene qui partent, respeotivement
127
- 14 -
J.Massera de Y et Y • on peut alors leur appliquer les
o
indgalit~s
1
oar~ct~ri-
stiques des dichotomies.
AUTRESRESULTATS Les theoremes suivants peuvent donner une idde des applioations de .la methode et de quelques compements possibles. a) Theoremes sur l'existence de solutions presque-periodiques [2, 6) :
THEOREME 9 (Theoreme I.6.2). S'i~ y a une dichotomie exponentie~~
Le et si A(t) que
estpresque~p.riodique,pour
LJ~quation (2)
admet une et une
quasi~lindaires
(15,.0)
un
I-coupLe admissibLe et h(x,t)
II x II
une jonction 'Mjinie pour t c: J , x EX,
< a (0 < a ~
vaLeurs dans X , te He que pour chaquB jonction
<
a
on ait h(x(t), t) €
7b
.Soit
qu'iL existe une constantepositive x"
XII E f)
n ~ 00,
Hors, si (3,
pour chaque ;
~
o
'x" 00'
Ix
II
/3=
~teLLe
e: tJ
nt
h(o, t )1 16
EX, 0;0
Ii; 0 II
pas mains jaibLe que
;0 + ;1 ' ;1 €
(pj,;[)) un 00
.
(t, t ). Soit
a(O < a ~ 00);
J
a
00
et admettons
b
>
teL que
0
< b , iL y a une et une seuLe soLution
m et 1
)
que, pour chaque coupLe
sont sujjisa,m'fltent petites, iL existe un
COROLLAIRE. Boit A'
x, II x II <
x
00
((Xl < a ,on a i t
x(t, ;0) f)J de (3) tene que x(O, ;0)
x ~
(of. propriete
["2, 7)) :
THEOREME 10. Soit
Ixtoo
presque~p~riodi-
seubesobutionpresque~periodique.
b) Thdoremes sur les equations (Q2) et
chaque f
o
X1D .
T-coupLe admissibLe :11.i n eS
h(x,t) definie pour
t
e J)
a,dmettons que pourchaque x fixe e LLe
128
- 15J . Massera
est une fonction ,de
t
continue et born.e et qu'il existe une constan-
t e . Ate LL e q1.1 eli h ( x ' , t) - h ( X II, t )
II·x ' Ii , II XII II
< a,
t
II
~ A II x' -
XII"
ex,
pour x' , x"
~ 0 : Ators, la condusion du Theofleme 10 reste
valabLe si L'on refllptace tJpar ~. THEOREME 11. Joitx r.fLexifetsupposons qu'il y a un. dichotomieexponentieL~e
.des :soLutions IdeO). Bait h(x,t)unejonction
jaib Lement continue :de :X X 'J Idansx (c. a.!d.!,continue borsque :Xest ;muni :deta topobogie !aibLe),· ad1llettons , que"hb:,t)~' ~¢(IIxll), ¢(r) .• tant unejonction ' continue d.finie pour r
r ....
(Xl
•
'Hors, pour chaque
t
~y
'0
" 0
~
= oCr)
0 , ¢(r)
pour
"equation (3)admet ,au moins
une :soLution born.e x(t, go) tdLe que x(O, go)::
to
+g{, giG y.1- )
Desresultats analogues sont valables pourl'existenoe de ,lutions
presque~periodiques
d'equations
so-
quasi-lineaire~p~esque-pe-
riodiques. 0) Grossierete [2]
THEOREME 12. Boit (~,
t. ) admissible (Xl
pace des endo1ll0~phismes JdeX) • .oor~, si tit e, ( ~,
t
(Xl )
est a dId s sib Le ,pour
~
~
etE f
IE Ie
73(x)
(X : L'es-
est suffisamment pe-
+ (A (t) + B( t ))x= f (t ) .
On peut aussi voir (TheoremeI.8 . 2) quel'ensemble ,0 €-1'f2,
des
A(t) .pour lesquels il y adiohotomie exponeniielle est un ouvert et que les elements oaraQteristiques de la dioho,omie (y , Y , v, v', '0
1
eto. ) possedent des proprietes de oontinuite dans leur dependence par rapport
.A
e,O
d) Methode de Lyapunov Cof. propri6te (p. ) et [4]) : . . ,3 THEOREME 13 (Theoremes IIl.3.1et III.3.2). 'SoH A f 7fU. Gon-
dition-n.c.ssaire et suffisantepour
qu~n
y ait une dichotomie ex-
ponentieLLe est .qu'iL existe une jonction de Lyapunov (g.n.raLis.e) ;V ayant une borne sup.rieure injini1llent petite et telLe que V' soit d~finie n.gative.
129
- 16 J.M'Issera THEOREME 14 (Thdor.mes111.4.3 at 111.4.4).Soit dim X <
Condition n~c~.saire et sUffisante 'pour .qu'iL y ait une dichotomie est. qu' i L exists .deux jonctionsds .Lyapunovnon ... negatives v0' Vi'
positivement homogenes en x du memedegre, tdLes que'V soit definis positive et I~atite
r
poss~de
o
+
V
1
une borne superieure infiniment
et V'0 ~ 0 V'1 2 0 . '
e) Cas ou A(t) est p.riodique (rdsultats essentiellement nouveaux seulement lorsque dimX =
OJ)
[3, 7] ;
THEOREME 15 (Thdoreme 11.2.1). 'Soit A(t) periodique de periode log 4 ; aLors iL existe une repr~stntation de noquet Bt des so~utions de (1) ; x(t) = P(t)e x avec P periodique de perio -
1 et
IAI1YI/
o
de 1 , B = const. La reprdsentat:Londe Floquet n'est pas toujours possible si dim X
= OJ
et IAt~ > ~ (Example 11.2.1).
THEOREME 16 (Corollaire 11.3.3).Soit A(t) periodique de periode 1 at supposons
que La farmeture ,de 'Xo'e
so it refLexive. Etant donne
une fperiodique de peri ode 1 I, si L'.equation (2)admet une soLutionbornee, e,LLe admetunesoLution periodiqus Ide ·periode 1 THEOREME 17 (Ths9remeII.3.5). 8'iL exists un T-coupLe admis-
sibLe.qui n'est pas Hus faiHe que
('1.\
e:)
et si A(t), f(t) sont
.pedodiques de p.edode 1 , L'equation (2) admet une et une seuLe Lution
p~riodique
50 -
de p4riode 1.
f) Casou A est oonstant (of. Lemme 1 et
[10]) ;
THEOREME 18. Si A = canst., pour qu'iLy ait dichotomie exponen-
.tieL'e des soLutions de (1) iL faut et iL suffit que Le spectre de A
n. coupe ·pas L'axe imaginaire.
130
- 17 J.Massera
BIB L I 1. LG.MALKIN,
a
GR A PHI E
On stabiLity in the first .a;pno:dmation, Sbornik
Nauonyh Trudov Kazanskogo Aviaoionnogo Instituta, 3 (1935),
7-17
(en russel.
,.
2. J.L.MASSERA.and J.J.SCHAFFER, Linear differentiaL equations and
junctionaL ,anaLysis, I, Annals of Math., 67 (1958L 517-573.
3.
J.L.MASSERA andJ.J.SCHXFFER, Li~ear differentiaL equations .and
junctionaL ,anaLysis,II. Bquations ,with ·periodic coefficients, ibid, 69 (1959>, 88-104. 4. J.L.MASSERA and J.J.SCH~FFER, Linear differentiaL equations and
functionaL anaLysis, III. Lyapunov's second .method in the case of condicionaL stabiLity, ibid, 69 (1959), 535-574.
5.
J.L.MASSERA and J.J.SCHXFFER, Linear ,differentiaL equations and
junctionaL anaLysis, IV, Math.Annalen, 139 (1960),287-342. 6. J.L.MASSERA, Un criterio de e%istenciade soLuciones casi-perio-
dicas de ciertos sistemas de ecuaciones .diferenciaLes casj-periodicas, Publ.lnst.,Mat.Estad. (Montevideo}, III (1958), 99-103.
7
J.L.MASSERA,
des Mat.
1ur
L~e%istence
de.soLutions bornies et piriodiques
syst~mes . quasi~Ljn.aires , d~iquations
diff.rentieL.es, Annali di
(IV) 51 (1960), 95-106.
8. O.PERRON, Die StabHit8tsfrage bei DifferentiaLgLeichungen, 'Math. Zeitschrift, 32 (1930}, 703-728.
9. K.P.PERSIDSKII, On the stabiLity of motion in the first a,pproxi-
ma,tion, Mat.Sbornik, (N.S.) 40 (1933}, 284-293 (en russel. 10. J.J.SCH~FFER, Ecuaciones diferenciaLes LineaLes con coeficientes constantes en espados de Banach, Pub1. Inst. Mat. Estad. (Montevideo), III (1958), 105-110.
131
- 18 -
J.Ma sser a II
11. J.J.SCHAFFER, Function
sp~ces
with transLations, Math. Ann.le n,
137 (1959}, 209-262; .4ddndulr J ibid, 138 (l959}., 141-144. 12. J.J.SCH~FFER, Ljne~r Dffjerenti~L equatjons ~nd
naLysis, V, Math.Annalen, 140 (1960),3 0 8-321.
132
functionaL a-
CENTRO INTERNAZIONALE NATEMATICO ESTIVO 'C.J.·M"E. )
LUIGI AMERIO
FUNZIONI QUASI .. IIERIODICHE AS'llRA'l'TE E PROBLEM I DI PROPAGAZ]ONE
ROMA - Istituto Matematico dell' Universita :- 1960 133
FUNZIONI QUASI-PERIO VI CHE
ASTR' ? T ~
E PROBLEMI
DIPROPAG AZIONE di L.AMERIO
1.- Esporremo alcuni recenti risultati sulle soluzioni quasi-periodiche (q. p.) dell' equazione delle onde, e, piu in generals, tie ll,} equazioni
differenz i~ l i
Rioordiamo, (1. 1 )
y
inn~zi
astratte.
CJ . p.
.tutto, che una junzione continua
= f(t) ,
t E J
= (-
a valori in uno spazio B , di Banach,
gni
10
e
OJ
+
OJ)
,
quasi~periodica
se ad
0-
>·0 pub jarsi corrispondere un insiellle T ·, rdativalllente . 10
denso, di numeri r (~uasi-,pHiodi) per ciascuno ,dei . quaLi r;isuLti (1. 2)
Sup
lint + r ) - f(t)
II
~
10
•
Q1,lesta definizione si riduce, se B
e
t EJ
euolideo, .a quella,
classical di Bohr. Vale .inoltre, . anche nel caso astratto, il fon. '. " . e su ff'1dam ent.le crlterlo dl Boohner (1) : con d Iztone necessar;za
dente perche f(t), continua inJ, sia ".p.
e
cheda ogni suc-
cessione reaLe .{h n } possa estrarsi una sottosuccessione{h'n } taLe che La successione{f(t -I-h'}} converga unijormemente In J n
Per quanta si dir~ nei §§2 e
3, i nteressa supporre B hiL2
1
bertiano: preoisa.mente .interessano .gli spazi L , HO' E di cui
0-
ra rioorderemo le definizioni. 2 I) L e 10 spazio delle funzioni reali y = y(x) a quadra t o integrabile in un insieme
n (aperto,
spazio euolideo Xm (x = xl"'"
x m),
di prodotto seal are :
135
Limitato e connesso ) dello con la consueta definizione
- 2 -
L,Amerio (y (x), z (x))
(1. 3)
=
L2
f
0
y (x) z (x) dO ,
II) Hlo e 10 spazio delle funzioni reali y = y(x) a quadrato integrabile in 0 insieme aIle derivate prime (nel sense della teoria deLLe distribuzioni) e nulle (in senso generaLizzato) sul190 frontiera
U
di 0 ,
1 Assumeremo oome prodotto soalare, in H , 190 quantita 0
(1. 4)
(y(x), z(x) 1 = H 0
f
l--m oy oz { L + a(x)y(x)z(x)}dO a (x) Ox 'Ox k jk OJ, k j
---
Nella (1.4) Ie ajk(x) = akj~x), a(x) Bono funzioni misurabiLi e 1--m Limitate, e a(x) .? 0 e' La forma quadratica L aJ'k(x) ~J' k
e
j, k
soddisfa aLLa Limitazione a
(x) ~
jkj
gk
m
>
~
/J.
r
2
L1J' 'J'
1
Rioordiamo ohe H· si ottiene compLetando, rispetto alIa norma in-
o
dotta dalla (1,4), 10 spazio delle funzioni oontinue in 0 insieme aIle loro derivate prime, e nulle in un intorno della frontiera U
,
E'
' 1 tre Hl0 C L2 ,
lItO
Ill) E e 10 spazio prodotto oartesiano di H1 per L2
o
E = Hl X L 2
o
Ogni elemento Y(x) E E
e
peroia oostituito da una coppia 1
2
{Yo(x), Yl(x)} di funzioni : Yo(x) E HO' Y1 (x) E L , 8i assumera oome prodotto soalare, in E, 190 quantita, (1. 5)
(Y(x), Z(x))
E
= (y (x), zO(x) 1 + (y1(x), z (x)) 2 '
0
H
o
oui oorrisponde 190 norma (1. 6)
136
1
L
- 3 L.Amerio Chiameremo E 10 spazio deLL'energia e la metrica (1.6) la
metrica deLL'energia. Inf~tti
Ie quantita
1
2
-2 Ily 0 (x) II H1
o
1
1
2
2
rappresentano rispettivamente l'energia potenziaLe,
L'energia ci-
netica e L' energ'ia totaLe di una membrana i1 , col borde fisso, in cui YO(x) designi 10 spostamento del punto x, Y1 (x) la veLocita del punta medesimo.
2.- II
de
(0
(2.1)
misto, secondo Hadamard, per l'equazione deLLe on-
problem~
equazione deLLa membrana vibrante) -/y at 2
=
l--m a oy I ) (a (x) Ox jk j,k ox . k J (t ,
e
-
~(x)y
+ f(x, t)
J, x \7 11) .
stato oggetto di numerose ricerche, anche recent 1,
nostr~
eaposizione
interess~
(2)
.
Per la
considerare Ie .oLuzioni deboLi del-
la (2.1) : ci riferiremo inoltre al problema definito dalle con-
dizioni iniziaLi (2.2)
y(x,O) :.: YO(x)
con YO(x), Y1 (x) funzioni assegnate) e (2.3)
y(x , t)1
dal1~
condizione ai Limiti
=0, (J'
corrispondente alIa vibrazione di una membrana coL bordo fisso. Le soluzioni deboli si associano alIa teoria variazionaLe, anziche differenziaLe, della
membran~
vista, Ie soluzioni deboli ci
vibrante: sotto questa punto di
~ppaiono
137
come Ie vere soluzioni,
- 4 -
L.Amerio risultando, in piu, soluzioni della (2.1) (soluzioni da dirsi for-
til quando soddisfino, unitamente alIa frontiera
IJ
,
a oonvenien-
ti oondizioni di regolarita. Soluzioni deboLi della (2.1) saranno dette, preoisamente, le soluzioni y(x,t) dell"equazione variazionaLe deLLe onde (oon 1"1. quale si traduoe it principio di Hamilton): (2.4)
/3
J{Yt (x,t), ¢ (x,t)) 2 - (y(x,t), ¢(x,t)) 1 + ~ t L HO
+ (-f (x, t) J ¢ (x, t)) 2} d t = 0 . L
Questa deve essere verifioata (comunque si prenda l"intervallo ~ /--1 (3) in oorrispondenza di tutte Le variazioni ¢(x, t),
nuL Le per t =
~
e per t = (3 . In modo preoiso, oeroheremo Ie so-
luzioni del problema misto nella olasse
r
delle funzioni u(x,t)
soddisfaoenti aIle seguenti oondizioni: 1
A)
u(x,t) sia continua, oome funzione di t a valori in HO;
B)
u(x, t) sia derivabi Le, oome funzione di t a valori in L , e
2
la derivata ut(x,t) risulti continua in J . El" ohiaro ohe, se vaLgono Le A) e B),
La funzione U(x,t) =
- {u(x,t), ut(x,t)}, a va Lori in El, ha ivi, come traiettoria,
una Linea continua. Sulla variazione ¢(x,t) si fanno anoora, nell"intervallo
~H (3, Ie ipotesi A), B); si supporra, in piu, ohe sia 11¢(x,~)
II
1 =
11¢(x,/3) II
H
1
=
o .
HO
o
Quanto al termine noto f(x,t), ammetteremo ohe sia a 2
in L , per quasi tutti i t , e ohe La norma
II f (x, t )11 2 L
val~ri
sill, in-
tegrabiLe in ogni intervaLLo Limitato. Infine, per quanto riguard aid at i in i z i al i, sis up po r r a
0
he, nell e (2 . 2 ), s i a
138
- 5 L.Amerio Y 1 (x)
e
L
2
ei08 Y (x) 0) ~ E •
8i pub allora dimostrare (efr. L.AMERIO) Loe. cit. in (2» ehe ~a so~usione Y(x)t) = {y(x)tL y (x)t)} dena (2.4») con La t
condisione
ini8ia~e
esiste in tutto J )
e unica
e dipende con continuita dai dati, in
virtu della formula di maggiorazione (2.5)
La soluzione medesima pub caleolarsi) seguendo 10 schema classieo delle soluzioni elementari) mediante la formula risolutiva un(x)
00
2: . 'Y (t) 1n n
(2. 6)
An
'Y1(t)
00
2: n .:..::..- u (x)} . \
1
I\n
n
Nella (2.6) le un(x) e le eostaoti An rappresentano rispettivamente Ie autosoLusioni e gli
dell'equazione
= A2 (u(x») hex») 2
(u(x») hex»~ 1
(2.7)
autova~ori
H
L
o
ehe deve intendersi soddisfatta per tutte le hex) €
Hi
o
La (2.7) ammette una suceessione {A } di autovalori) per i
n
quali risulta
o
< Ai < A <... < A < ... ) lim A = too 2 - n n~oo n
e Ie corrispondenti autosoluzioni sana legate dalle condizioni di ortogonali til. u (x)
(u (x»)
r
u (x»
s
L
2
= Srs
=(2A
r
139
- 6 L.Amerto Inoltre la suooessione {u (x)} n
e compLeta
eia in L
2
ohe in
Risulta infine
r n (t) =
(2.8)
a.
oos A. t n n
+ b
t
sin An t
n
+
J
o
f n (7))sin "n Ct -7))rl1),
essendo (2.9)
a
=
n
u (x) (Yo(x), _n_) A H1 ' n f (t) n
(2. 10)
bn
°
= (f(x,t),
= (Y1 (x),
u (x)) n L2
u (x)) 2 n L
Dalle (2.9) segue: (2.11)
OCJ
(2
1
n
2: n a
+ b
2) n
2
2
= Ily (x) II 1 + IIY1(x) IIL2 = lIy(x,O) H
0
II
°
2
E
< + co .
Rioordia.mo infine ohe, se Y(x,t) e Z(x,t) sono due soLuzioni
corrispondenti ai termini noti
g (x, t), va,Le L' egua,gLian-
f (x, t),
za : (2.12)
d
dt
(Y(x,t), Z(x,t))
La. (2.12), per f(x,t) (2.13)
d
dt
E
= (f(x,t), Zt(x,t)) 2 + (g{x,t),y (x, t)) 2 L
= g(x,t) = 0
, Y(x,t)
L
t
= Z(x,t)
diventa
2
ilY(x,t)1I
=0
E
ohe esprime il principio di conservazione deLL'energia.
3.- La quasi-periodicita degli integrali dell'equazione omog enRG delle onde (f(x,t)
= 0) e
stata dimostrata, in ipotesi sempre pili (3) (4) generali, da vari Autori: Muokenhoupt (per m = 1), Bochner Boohner e von Neumann
(5)
,Sobolev
(6)
.
,LadyzenskaJa
(7)
.
Consideriamo ora l'equazione non omogenea, supponendo iL ter -
mine noto f(x,t) funzione q.p. di t , a vaLori in L 140
2
. Ze.idman
(8,
- 7 L.Amerio ha dato, in questa oaso, per il primo, una oondizione sufficiente perche Ie soluzioni Y(x,t) siano q.p., come funzioni a valori in E : valendosi, precisamente, di un teorema di Phillips sui semigruppi di Hille, Zaidman ha dimostrato la
quasi-periodicit~
del-
Ie soluzioni Y(x,t) a traiettoria relativamente compatta, supponendo f(x,t) dotata di derivata continua f (x,t). Nel lavoro cit at
0
(2)
.
~n
t
,
ho eliminato questa ipotesi, ma non quella di com-
pattezza. Suocessivamente ho potuto rimuovere anche quest'ultima ipotesi (molto restrittiva e di non chiara interpretazione fisi. f( ) q.p., a va L0ca ) pervenendo al seguente teorema (9) : SJa x,t
2
ri in L , e sia Y(x,t) una soLuzione deLL'equazione variazionaLe
deLLe onde, Limitata in J . ALLora Y(x , t) vaLori in
e
q.p.,come funzione a
E
La dimostrazione di questa teorema e stata ottenuta in due modi distinti, di cui ora indicheremo Ie linee fondamentali . a) Nella prima dimostrazione (L.AMERIO,
100.
cit. in (9»
ci si vale, in modo essenziale, di una proposizione sulle successioni monotone di funzioni q.p., proposizione che present a analogia con quella, notissima, del .Dini. Sia, dapprima,{f (t)} una successione di funzioni q.p., rean
1i
0
Ii a nJ
complesse. Diciamo A ·la famiglia di tutte Ie successioni rea-
= {a k }
regoLari rispetto a {fn(t)} : tali cioe che, per ogni
sia
(3.1)
lim f
k .... oo
n
(t
+a ) = k
f
n,a
(t)
uniformemente in J . La nuova successione {r
n., a
(t)} sar~ forma-
ta, come la precedente, da funzioni q. p. Inoltre, una arbitraria successione reaLe ~
una sottosuccessione a
E
= {~k}
contiene
A (la tesi segue immediatamente dal cri-
141
- 8 -
L.Amerio terio di Bochner, applieando il proeedimento diagonale di Hilber t) . Supponiamo, ora, ehe la successione {f (t)} sia reaLe) monon
tona, e Limitata indiPendentemente da in
t : risulti, ad esemp i o,
J
f(t»f n
-
(t»m>-oo.
nt1
-
Si osservi ehe, presa comunque a E A , la successione {f
n, a
(t :· ~
gode delle medesime propriet. : se infatti vale la (3.2), risul t 1 per 1a (3.1),
0.3)
f
n,a Dalla (3.3), posto f
(f)
~
f
(t)
=
f
(t)
=F
nt1,a
>
(t)
m •
(t),
n,o si deduce che, per ogni a € A , esiste il limite 1 im f n .... oo
0 , 4)
n
n,a
a
(t)
eio premesso, si dimostra che La successione {fn(t)} converge
uniformemente in J se Le funzioni F (t)· risuLtano q. p.) per ogni a
a EA. La dimostrazione del teorema dianzi enunciato si fonda
lora I)
sull~
al~
seguenti eonsiderazioni.
Se e, per t E J
IIY(x,t)11 si deduce dalle (2.6),
E
~M,
(2.8) che gli integrali t
t
J
J
o
o
sono limi tati. PoieH f (t) e q. p. (per la (2.10)) gli integrali n
medesimi risultano g.p.
: tali sono allora Ie funzioni Yn(t),
Y' (t) . n
Per provare il teorema basta allora dimostrare che la serie
(2.6) converge uniformemente in J , cioe che La serie di funzioni 142
- 9 L.Amerio
q.p. non negative 2 Ln (y (t) t
00
(3.6)
1
=
n
IIY(x,t)11
2 E
converge uniformemente in J II) 8i dimostra che l'energia tot ale
1
2
Ily
(x,t)11 2 E
e
q.p.
Per questa basta osservare che, per la (2.12), risulta d dt
IIY(x,t) II
2 E
=
2 (f(x,t), y (x,t)) 2 00
=
2 Ln f (t)
n
1
t Y' (t ) n A. n
L
=
e si riconosce che la serie a secondo membro oonverge uniformemente in J . IIY(x,t)"
2
, limitata e dotata di derivata q.p.,
E
e
allora q. p. III) 8ia a =
{a } una arbitraria successione reale. 8i pub, senk
z'altro, supporre a regolare rispetto alla successione 2
e tale che sia, uniformemente in J ,
{y (t) n
lim f(x,t t a k )
k ...oo
lim y (t t a ) k ...oo n k
=
y
n
= g(x,t)
(t),
,
1 im Y' (t t k ...oo n
\ a ) k
= Y'
n
(t.)
risultando
=
anoos
t
A. t t b sin A t t J g ('T])sin A. (t-'T])d'T] n n non n
g (t) = lim f (tta.) = (g(x,t),u (x)) 2 n k"'oo n ~ n L La funzione
143
- 10 -
L.Amerio
z
ro (x, t ) = {~n Y (t) n 1
un(x)
OJ
Y' (t)
"1
....ll-A. n
~n
An
u (x) } n
e allora soluzione dell'eguazione variazionale delle onde, con termine noto g(x,t), e si ha " Z (x,t)
II ~ E
M.
Ne segue, come in II), che la fun2iione ro ( 2
= ~n
.:y
-'2 y
n
1
(t»)
+~
(t)
A2 n
e
q.p .. Di qui, per la proposi2iione, gi3. enunci9.ta, sulle successio-
ni monotone di fun2iioni q.p., si deduce la convergenz1 uniforme, in J , della serie
(3.6). (10)
b) La seconda dimostrazione nozione di
¥u~si-periodicito
utiliz2ia, innanzi tutto, la
deboLe. Dato uno spazio B , di Banach,
qualsiasi, diciamo B* 10 spazio duale, formato dai funzionali lineari continui, definiti in B . La funzione y
= f(t),
a valori in B , si
te '1.p. (d.q.p.) se per ogni b* E.
e q.p. per ogni
dir~
allora deboLmen-
B~ L~ fun2ione nU111eric~
< f(t),g'>
8e Be hilbertiano, questa equivale ad ammettere che,
b (; B ,
iL prodotto
sc~Lare
(f(t), b)
In ogni caso, si dimostra che condi2ione
ciente perche f(t),
dd.P., sia ".p.
tori'l, in B , risuLti
rdativ~mente
e
B
risuLti ".p.
necessari~
e suffi-
che La corrispondente traiet-
compatta.
Cib premesso, la dimostrazione si svolge nel modo seguente. I') 8i rioonosce, innan2ii tutto, come in I), che se vale la (3.5), la fun2iione Y(x,t) e d.g.p. II') 8i dimostra che, se vale la (3.5), presa ad arbi trio una successione reale a = {a k }, si pub estrarre da questa una sottosuccessione
a'
= {a'} k
tale che la successione {y (x,t +
144
a'» converga dek
- 11 -
L. Amedo bolmente e uniformemente in J a una funzione Z(x,t), d , q.p. e soluzione dell'equazione variazionale delle onde, oon termine noto
= lim
g(x,t) EI
f(x,t + a k' ) .
k-oo
allora Sup
t <; J
IIZ(x,t)11 E
~
Ily(x, t)
Sup J
~
t
II
E
e inoltre (siooome la suooessione {Z(x,t
al)} oonverge debolmenk
te a Y(x,t» Sup
t (; J
Sup
Sup
t co J
II Z(x, t) II
t , J
E
Vale pertanto
0.7)
~
II Y(x, t) II
E
l'equaglianz~
II Z(x, t) II
E
=
Sup
t"
J
III') Si dimostra ohe Y(x,t) ha traiettoria relativamente oompatta, provando ohe, se oOSl non fosse, la Z(x,t) definita in II') soddisferebbe a una limitazione del tipo Sup
IIz(x,t) II <
t f J
oib ohe
e assurdo, per
la
Sup
t E J
IIY(x,t) II
0.7).
Nella dimostrazione sono utilizzate, in modo essenziale : la struttura hilbertiana dello spazio E (attraverso il teorema del parallelogramma), la oostanza della norma delle soluzioni dell'equazione omogenea (prinoipio di oonservazione dell'energia), il teorema di dipendenza oontinua dal termine noto.
4 .. - La dimostrazione data in b) si pub estendere all'equazione a-
stratta (4.1)
(11) d u
dt
= A(t) u(t)
t
145
f(t) .
- 12 L,Amerio Nella (4.1) A(t) e una famiglia di operatori illimitati quas iperiodici (in un certo senso) ed f(t) una funzione q.p.)
IJ.
valori
in uno spazio Y ; la soluzione u(t) si suppone a valori in uno
8p~ -
zio X S Y uniformemente oonvesso e tale che l'immersione di X in Y sia continua. In questo caso) assai generale) non si puo piu ammettere La quasi-periodicita delle soluzioni dell'equazione omogenea) ne la costanza) in J , della lore norma. E' possibile pero pervenire ad una estensione del secondo teorema di Favard, facendo su tali solu zioni e su quelle delle equazioni associate w' (t) = B(t) u(t) (ove B(t) appartiene alla chiusura della famiglia {A(t + h)}) delle ipotesi di comportamento asintotico analoghe a quelle poste dal Favard : in sostanza la traiettoria di wet) non deve avere 10 zero come punta di accumulazione. Si dimostra allora che se La (4.1) ha
una soLuzione Limitata ne ha una 4uasi-periodiea (si tratta di soluzioni deboLi e all'ipotesi di limitatezzasi aggiunge quel1a, assai poco restrittiva, di uniforme continuita debole in J). Preoisamente risuLta q.p.
La soLuzione (ehe esiste ed
e uniea)
earatteriz -
zata daLLa eondi.zione ehe sia minima La quantita Sup
t <:; J
II u(t) II
(teorema di minimax). Questo teorema, gia dimostrato per
l'equ~zio
ne delle onde (L.AMERIO, loo.oit. in (2)), ha, in tal oaso, un significato fisioo notevole, poiche afferma L'esistenza di una e una
soLa soLuzione .per cui L'estremo superiore, in J , deLL'energia, abbia iL piu PiccoLo vaLore possibiLe. 146
- 13 L.Amerio Osserviamo che, in un assai recente Lavoro
(12)
• Lo Zaidman ha
applicazione dei precedenti risultati, al problema di Cauchy per quazione delle onde in tutto Lo spazio, con termine noto q . p. tratta proprio di un problema .inouil'equazione omogenea ha del30luzioni limitate ,che non sana quasi_perlodiche. La stesso Auto~a. 10
infine, ampiamente studiato .il problema delle soluzioni, che
distribuzioni quasi..,periodiche, ·delle equazioni .iperboliche,
termine nato distribuzione quasi..,periodica.
S.BOCHNER, Abstrakte fastperiodische Funktionen, Acta Math.,61 (1933) . O.A.LADYZENSKAIA,
IL probLema misto per L' equazione iperboLica (in _russo), Mosca-Leningrado, 1953; B. Sz.
NAGY~
'Vibrations d'une cordenon homogene,
BulL Soc. Math. de France, 1947 ; L. AMERIO,
ProbLema misto e quasi periodicita per L'equa zione de LLe ,onde non omogenea, Ann. di ¥at.}
29,
(1960).
C.F.MUCKENHOUPT, ALmost periodic functions and vibrating _sy- . stems, Journ . of Math. and Phys . , MIT, 8 (1929 S. BOCHNER, Fastperiodische L8sungen der Ie LLen- GLeichung, Ac t? Math., 62 (1934). S.BOCHNER, J. von NEUMANN, On compact soLutions of oper at ionaL differentiaL equations, Ann , of Math 36 (1935) · S.SOBOLEV, Sur La presque-periodicite des soLutions de L' equation des ondes, I , II , III, Comp o rend . Ac . Sc . URSS. 1945·
147
- 14 L. AmeriL
(7 ) O.A.LADYZENSKAIA,
(8)
L.AMERIO,
oit.
in
(2 )
Sur La presque~p~riodicit'des soLutions de L)~qua tion desondes.non homog~ne,Journ. of Math. and Meoh., 8 (1959).
S.ZAIDMAN,
(9 )
100.
degLi integraLi ad energia Limitata deLL'equazione deLLe onde, con termine noto quasiperiodico,I, II, III, Rend . .Aco.' Naz. dei Linoei, 28 (1960}; ofr. S.BOOHNER , ALmost ·periodic soLutions 'of the in homogeneous 'ave equation, 'Pr.oo. ·of Ithe Nat.
Quasi~periodic~ta
Ao.
(10) L.AMERIO,
(11) L.AMERIO,
of So .
(1960).
.SuLL'equazione debLe onde con termine dodico, Rend. diMat . , 9 (1960).
S.ZAIDMAN,
S.ZAIDMAN,
J
10 (1961).
SoLutions presque-periodiques dans Le probLeme de Cauchy, pour L'equation non homogene des
(13)
quasi-pe-
SuLLe equazioni dijjerenziaLi .quasi-periodiche astratte,Rioerohe eli Mat.
(12)
noto
Na z.
de i Lin 0 e i
(i nco I' S 0 d.i
S
tam p a ) •
SoLutions presque~periodiques des equations hyperboLiques(ancora da pubblicare).
148
CENTRO INTERNAZIONALE MATEMATICO ESTlVO (C.I.M.E. )
LAW R ENe E MAR K U S
SISTEMI DINAMICI CON STABILITA' STRUTTURALE
ROMA - Istituto Matematico dell'Universit.
149
~
1960
SISTEMI DINAMICI CON STABILITA' STRUTTURALE di LAWRENCE MARKUS
ESEMPI E DEFINIZIONI. Consideriamo .oome esempio l'equazione differenziale •• 2 xtbxtkx=O,
b
~
0 , k
2
> 0
oostanti reali
ovvgro il sistema del 1 0 ordine x
2
Y = - k x - b Y
= y
Si tratta di un sistema differenziale nel piano reale R2 od anohe sulla sfera S2
= R2
+
00
(mediante l'ordinaria proiezione ste-
reografioa). Se b = 0 si ha l'osoillatore armonioo. Ogni soluzione, qualunque siano le oondizioni iniziali, e periodioa. Si ha oosl una famiglia di ourve ohiuse intorno all'origine del piano. Se b > 0 si ha l'osoillatore oon attrito. Nel oaso b
=0
non vi e stabilita strutturale poiohe un oambia-
mento della b , per quanto piooolo, oambia radioalmente la oonfigurazione della famiglia delle ourve integrali. Inveoe il oaso b > 0 presenta la stabilita strutturale poiohe un oambiamento suffioientemente piooolo di b in b(x,y) > 0 e analogamente per il coeffioiente k
2
da anoora un osoillatore oon attrito. Consideriamo, in generale, un S i
x
1
iii
si~tema
n
= f ( x , .•. ,x)
n
1
con Ie f (x , ... , x ) ~ C
in
8 ,
i
differenziale reale
= 1,2, ... , n
insieme aperto e limi tato dello
Rn euolideo (reale).
151
- 2 -
L.Markus Per i teoremi di esistenza e di unicite per ogni punto di E) passa una curva integrale. Consideriamo la famiglia di tali curve.
e
Allora S e un oampo di vettori in
ed una ourva integrale e una
ourva tangente a tale oampo. Sia ~
10 spazio di Banach ohe si ottiene prendendo tutti i si~
stemi differenziali come S . Preoisamente oampi vettori~li di classe 0 1 in
G
oonsista di tutti i
e sulla frontiera
'&8 (che
supporremo regolare, ad esempio una varieta differenziabile di dimensione n-1). La distanza tra due campi vettoriali S ed S' iii n i :c g (x , ... , x ) sia
II S
- SI
II
;:
max x E <:) ,i
Def. Un sistema differenziale S 1. S non
e
max x~~ ,i,j
€?J
ha la stabili ta strutturale se
tangente alIa frontiera
00
(8 non e nullo, su of)
2. Per ogni € > 0 esiste un 0 > 0 tale ohe quando per 8 1
fID
allora S ed S' sono €..,omeomorfi,
un'applioazione topologica di 6)
su
e
lis - s' II
< 0
vale a dire esiste
la quale porta la famiglia
delle curve integrali (non parametrizzate) di 8 sulla famiglia di quelle di 8', in dalla sua immagine
modo tale .che la distanza di ogni punto di (3
e
€ .
~
Allora se si oambiano di pooo, Ie f
i
, la configurazione glo-
bale delle curve integrali non muta qualitativamente. II concetto di stabilita strutturale
e
molto interessante dal
punto di vista filosofioo pOiohe un sistema dinamico della fisica non deve cambiare qualitativamente allorohe si produce un piccolo cambiamento nei ooefficienti. Ma l'idea
e
difficile a trattarsi dal
punto di vista matematioo. Se l'insieme (j
e
aperto e limitato in una variete Mn (inveoe
che in Rn) la teoria procede in modo analogo. In quel che segue sa-
152
);
- 3 L.Markus
e =M
, variet~ differenzia-
(38
= ¢ (insieme vuoto) e
n
rl considerato il CaSo pib semplice
bile compatta. In tal CasO la frontiera
la condizione 1. nella def. 6 sempre verificata.
SCHEMA DELLA TEORIA. La teoria dei sistemi strutturalmente stabili si pub. descrivere nei seguenti punti : 1. Esiatenza; 2. Denait?> in
7j ; 3. Proprietl;
4. Applicazioni. Ci soffermeremo in particolare sui due ultimi e accenniamo rapid lmente ai .primi lIIue.
L'esistenza di sistemi strutturalmente stabili e stata provata nel caso di un diaoo da Andronov e Pontriaghin (v. Bibliografia) e 2
la prova ai puo estendere ad ogni M ne di S.Smale .citata
nell~
Quanto alIa den,sita in
3
per M ai veda la comunicazio4
Bibliografia. Coaa puo dirsi per M ?
lb ,
vale a dire al fatto che i sistemi
strutturalmente stabili sono densi nello spazio
~
di tutti i si-
stemi, essa risulta finora provata solo in U2 (vedi De Baggis e Pei3 xoto). Cosa puo dirsi in II ?
PROPRIETA' . Elenchiamo adesso alcune proprieta relative ai sistemi strutn
turalmente stabili in unavarietk differenziabile compatta M . 1. Esiste un numero finito di punti singolari (~unti d'equilibrio) ed ognuno di essi e un punto elementare. (Cio significa che se P e un punto singolare,oioe un punta in cui il vettore del campo
~
nullo, esiste un'applicazione topologioa di un intorno di P n
su di un intorno dell'origine di R
che porta Ie soluzioni di S su
quelle di una equazione lineare a coefficienti costan·ti
ici =
a
i
xj
j
153
- 4 L.Mark~s
tale ohe gli autovalori della matrioe (a i ) hanno tutti parti reali j
diverse da zero e sono a
d~e
a due distinti).
n
Per provarlo, in R , approssimiamo S mediante un sistema 'i
x
8'
i
1
n
",p(x, ... , x ) ,
i
= 1, 2, .•. ,
i
n
dove i P (x) sono polinomi reali. 8e i ooeffioienti dei P
i
sono gene-
rioi (nel senso della ~eometria algebrioa) i p~nti singolari sono isolati. Nell'intorno di P soriviamo 8'
Faooiamo un' al tra approssimazione, oon Qi ;;; 0. nell' int~rn~ di P . Per l'ipotesi di atabilita .i
so ohe x
= ajx i
j
str~tturale
. La dimostrazione
e
8
e
qualitativamente 10 stes-
ooal oompleta.
2. Le soluzioni periodiohe sono isolate ed elementari. (Cio
vuol dire ohe per ogni soluzione periodioa esiate un intorno tubolare N nel quale non vi sono soluzioni periodiohe oltre quella oonsiderata. Inoltre gli esponenti oaratteristioi della soluzione sono a due a due distinti e, salvo quella banale, hanno tutti modulo
f. 1). Una questione .importante per ora aperta
e
esiste un numero
finito soltanto di soluzioni periodiohe?
3. 8e s 6 una soluzione posltivamente stabile seoondo Poisson (avente oi06 intersezione non vuota oon l'insieme dei propri punti w~limite)
dioa ad s
0
allora s
e
un punta singolare oppure una soluzione perio-
infine esiate un'infinita di soluzioni periodiohe asintotiohe (Congettura di Andronov).
4. 8e 8 ammette una opportuna mis~ra invariante su Mn allota esiste un'infinita di soluzioni periodiahe. 8egue ahe un sistema di Hamilton oon un numero finito di solu-
154
- 5 L.Markus zioni periodiche non pub eaaere strutturalmente stabile.
5. Ogni traiettoria che non sia nomade
e
una traiettoria cen-
trale. Per chiarire questo punta riccrdiamo ohe seoondo la teoria di Birkhoff una traiettoria s di S si dice nomade (wandering) se e81ste un intorno tubolare di s ohe a part ire da un oerto istante non interseca mai piu la propria immagine. L'1nsieme
dei punti ap-
(0 1
n
partenenti a traiettorie nomadi e aperto in M , quindi il oomplementare M1
n
=M
n
e 1 ,oostituito
dai punti non nomadi
e
ohiuso e quin-
di (essendo M oompatta) oompatto. RifetendoLa oostruzione a part ire dal sistema dlnamioo oonsiderato in M1 ' con La topologia relativa, si ottiene una successione decresoente di insiemi oompatti yn:)
M1 :>
M2 => M3 .)
la cui intersezione M , non vuota,oostituisce quello ohe Birkhoff r
ohiama oentro del sistema dinamico. In esso si trovano i punti singolari, Ie soluzioni periodiohe e in generale Ie soluzioni stabili secondo Poisson. La costruzione di Yr
e assai oomplicata in generale.
L'affermazione fatta all'inizio di questa nO e la seguente : Se il sistema S
e
struttura}mente stabile oocorre un solo passo per otte-
nere il oentro poiohe Ml
= Mr
APPLICAZIONI. 1. Consideriamo il flusso lungo Ie geodetiohe d'una superficie ohiusa a curvatura negativa costante. Questione : Un tale sistema dinamico ha stabilita strutturale? 2. Problema di H.Seifert. Consideriamo un oampo di vettori sulla sfera
s3; senza punti
singolari. Esiste sempre una soluzione periodioa? Nel caso della stabilita strutturale Ia risposta e affermativa. Infatti se s1 segue
155
~
6 L.Markus
una traiettoria i1 oa oppure
3.
e
SUg
insieme w-1imite b una traiettoria periodi-
i1 limite d'una suooessione di traiettorie periodiohe.
Genera1i~zaz1one
del teorema di Poinoare-Bendixon.
Dato in un toro solido dt R3 un oampo di vettori senza punti singo1ari tali ohe que1li assooiati a11a superfioie penetrino tutti ne11 1 interno eaiate una traiettoria periodioa? La riapoata b affermativa ne1 oaao one vi sia stabi1its struttura1e.
156
- 7L.Markus
BIB L lOG R A F I A 1. A. Andronov - L. Pontrjagin,
SystS'II68 grossiers, Dokl. Akad. Nauk
SSSR, 14, 247-251 (1937).
2. G. D. BirkhoU,
3.
Dyn"mica~
Systems, New York (1927).
H.De BaggiB, Dynamica~ systems with stab~e structures, Contrib.
to the Theory of nonl. oBoil1., 2, 37-59, Prinoeton (1952). 4. L.MarkuB, GLobaJ structure ,ofordin"ry differenti"~ equ"tions ~n
the
'p~"ne,
Trans. Am. Math. Soo. 76, 127-148 (1957).
5. L. Markus,StructuraL Ly st"b the Am. Mat h . So 0
."
(
~e
differentiaL systems, Notioes of
1959 ) .
6. L.MarkuB,Periodic soLutions and invaria.nt sets of structuraHy
:stabLe differentiaL 'equations,Proe. of the Oongress on Non1. diff. eq:., Mexioo (1959).
7. L.Markus, .Inv"ri"nt :sets :of structuraHystabLe differentiaL sy'st61ls, ·Proe. 'Nat. Aoad.SoLUSA (1959). 8. 'L.Markus,StructuraHy st"He ·differenti"Lsystems, Ann. of Math.
(1961). 9. :1i.·M.Peixoto, On structuraL stability, Ann. of Math., 69, 199-222 (1959}. 10. M.M.'Peixoto,·S'ofa.e ' exaflpL'es
; pn n~di'll8'fl,sionaL
structuraL stabdi-
ty, Proe. Nat. Aoad. Soi. USA, 49, 633-6360959). 11. G. Sansone - R. Conti, Equa.sioni differensiaLi non Liuari, Roma
(1956}. 12.
H~Seifert,
CLosed integraL curves in 3-space and isotropic two-
dimensionaL dejormations, Proe. Am. Math. Soo., 1, 287-302 (1950). 13. S. Smale, StructuraL Ly st"b Le differentia.L systems on cLosed
3-manijoLds, Proe. of Congress on non1. diU. eq., Mexioo (1959 ) . 157
CENTRO INTERNAZIONAI,E MATEMATICO ESTIVO (C.I.M.E. )
G. PRO D I
TEOREMI ERGODICI PERLE EQUAZIONI DELLA IDRODINAMICA
ROMA - Istituto Matematico dell 'Universit8 ;: 1960
159
TEOREMI ERGODICI PER LE EQUAZIONI DELLA IDRODINAMICA di
In questi ultimi anni mente,
[a)
(1)
G. PRODI
e stato dimostrato ([4} e, suocessiva-
ohe 11 problema ,misto (nel senso di Hadamard) per le
equazioni di Navier-Stokes in due variabili spaziali
e univocamente
risolublle "in grande". Questo risultato permette di impost are problemi di comport amento asintotioo per t - too : stabilita, esistenza di soluzioni peri 0diohe, teoremi ergodioi. Si tratta dei problemi che hanno il maggior interesse anohe dal punta di vista fisico. In quello che segue svolgeremo qualohe oonsiderazione su questi problemi; in molti punti non potremo che segnalare emettere in evidenza le diffioolta ohe 8i incontrano. Esaminati i (poohi) dati che l'analisi attualmente ci fornisoe intorno alle equazioni di Navier-Stokes riohlameremo aloune nozioni di carattere generale relative all'esistenza dl misure invarianti. Ceroheremo poi di adattare 901 nostro oaso 190 teoria di Kriloff e Bogoliuboff [3] ed enunoeremo alcuni risultatl ohe si possono oOSl ottenere.
§ 1.- Indiohiamo oon
n un
insieme aperto limitato del piano, oon
u un vettare reale dl oomponentl
u. (j J
------------..--(1)
=
1,2), funzioni del punta
Le rioerohe qUi esposte sono state finan~iate dall'Air Research and Development Command, United States Air Force, can oontratto AF 61 (052) - 414.
161
- 2 -
G.Prodi Indicheremo oon LP(O) 10 spazio dei vettori oon ccmponenti a p-esima potenza sommabile, oon la norma lui dove luCx) I indioa 11 modulo di u(x). Se zio ha struttura hilbertiana (reale),
10
dotto soalare (f, g);:
f fig i
~
t.: (n.)
;: {f lu(x) I P dx}i/p,
0
p ;: 2 , questo spamediante ilpro-
ind'~iduabile
dx . Soriveremo If I in luogo di
If IL~(R;)' Siano poi, f t g
vettori definiti in 0 aventi derivate prime
(in senso generalizzato) a quadrato sommabile .in og _i ox.
;: I r ~
«(f,g)) 8i'l
Soriveremo
2
, II f II ;:
«f,r)'). ox. J J .)f(O) la varieta dei vettori· indefinitamente differenzia- · fl ij
dx
rr.
bili, a divergenza nulla e nulli fuori di un compatto contenuto in
n.
Ni (0) 1a
Sia N(O) la ohiusura di Jf(O) secondo la norma I "
phiusura di .)f(0) seoondo 1a norma
II II.
1
Gli elementi di N (0) so-
no, in un oerto senao, vettori nulli sulla frontiera di 0 . Indichiamo .oon Co 11 .ooefficiente di immersione lui
!lull-i.
Rioordiamo la importante disuguagUanzs ..dd. .Ladyzenskaia (valida per un oampo O. qualsiaei ) .: .1.uI 2,~ .L 1
Se u, v, 11' (; N (0) poniamo
(lli
II 'ull
< .~.,1/2 Iul
. b,()1.~.");:
I o ijr
u
(vedl' [4J).
OVj
i
-0;[
Wj dx .
i
Si ha facilmentelb(u,v,w)1 ~ .Iul 4 ' IIvll 111'1 ~ . • L (n') L ~.n,} p.
Dato uno spazio di Banaoh B , indlaheremoinfine oon L 10 spazio delle tunzlbhi definite nell'intervallo (O,T),
in
B ,
(O,~;B)
oon valori
a p-esima potenza sommabile.
Le equazionl ohe oonsldereremo .con Ie relative oondizioni al oontorno,
~i
esprlmono )
' ln i t~rmini
cla*sioi, in questo modo :
162
- 3 G.Prodi 'Ou ,
-.l
'Ot
( 2)
div u
(3)
u(X)
'Ou ui - j 'Ox i
~
+
i
-
jJ, 142 u j
=-
'Op
-
f
t
'OXj
j
(j = 1,2)
=0 =0
per
x E
TodD) .
Qui la inoognita p ha i1 signifioato di pressione, jJ, indioa il ooeffioiente di visooeit •. leL seguito noisupporremosempre f = (fl' f 2 ) junsione de LLasoLa, x . Questo per metteroi in una 6ituazione di analogia, per quanto • possibile, oon la teoria dei sistemi autonomi . ordinari. 8upporremo
2
f ' L (0); diremo ohe \lna funziorie. u{t},. iooalmente
.limitata oome funzJoneoon valori in N(O), looalmente a quadrato sommabile oome funzione oon valori in N1 (Ol, • soluzione (debole) delle equazioni di Navier-8tokes se vale la relazione (l)
(4)
J
o
{-(u(t),v'(t)) tjJ,«u(t),v(t)))tb(u(t),u(t);v{t))} dt = (l)
=J
o
(f(t},v(t))dt
per ogni funzione vet) .ohe : 1
(a) sia oontinua oome funzione di t .oon· valori in N (0); (b} si annuiLii fuori d1 un .intervallo
7" ....
7'", oon 0 < 7"
< 7'''< too;
2
(0) abbia derivata rispetto a t v'(t) & L (0, till; N(O)).
8i dimostra ohe : 1) (Vedi [9]) le soluzion1 della (4), eyentualmente oorrette su un
insieme di valori di t d1 misura nulla" sono funzioni oontinue di t nelld spazio NCO). 2)
(Vedi . [9]) la funzione lu(t)
I2 e
assolutamente oontinua · e · soddi- ,
sfa all'equazione differenziale
(5)
1
d
2
dt
Iu ( t ) I
2
+ jJ,
2
II u ( t ) 11= 163
(f, u ( t ) )
- 4 -
G.Prodi 3) (Vedi [7], [8]) Assegnato oomunque un valore una ed una sola soluzione ohe 10 assume per t in tutto l'intervallo 0"'"
= Tt
Posto u(t)
u
00
Uo
=0
E N(O), esiate ; easa
e
definita
•
,T
definisoe una applioa~ione oont~nua: t (Ot-oo) X N(fl) ... N(O). (Nonmi e riueoito di vedere se 11.1 oontinuit •. in questione possa
o
sus~isters
in modo uniforms).
Dalla (5) si rioava faoilmsnts
( 6)
~
1 u (t) 1
JJ. -::Tt 1u
o
1
e
On
t
8i ha dunque, per ogni soluzione, 11.1 relazione
i1iii
( 7)
lu(t)
t-rtoo
1
2 JJ. -1 1f1. Cn
~
Anoora dalla (5) si rioava 11.1 dieeguaglianza JJ.J
r
lIu(t)1I
o
Applioando 11.1 (6) si ha 7
J
(8)
o
II
Ilu(t)
2
2
1
dt <- 2
2
1uo 1
7
+ 1f 1
J
o
lu(t)1 dt .
(1)
2 2 -2 1f 12 7 . dt ~ C r11-2 1f 1 1U ·1 + 2-1 JJ. -11 u 12 + CnJJ.
n
0
0
I'
Da questa siottiene lim
1 _
7 ... +00
7
J II u (t) II 7
2
dt ~
2 -2 C JJ.
0
n
1f 12
Aggiungiamo aloune osservazioni relative alle soluzioni ,coat anti della (4). Un teorema di 11.1
Ladyzenskaia
esist~nza
e stato dato reoentemente dal-
[61. (In ipotesi ptb restrittive e in altra impo-
stazlone un tsorsma di esistenza era stato dato fin dal 1933 da Leray).
(1)------------Diseguaglianze di questo tipo sono state introdotte da E.Hopf in [1].
164
- 5 G.Prodi L'unicita della scluzione si ha certamente per valori di If I piccoli (come constateremo indirettamente tra poco). Per valori arbitrar1 di If I non s1 oonosce nulla 1n propos1to. 8i
e portati a
ri tenere che, in questa oaso, il teoremadi ,unici ta non sussista; tuttavia non m1 risulta ahe sia stato trovato aloun esempio a riguardo. Per le soluzioni cost anti s1 ottiene immediatamente dal1a (9),:
lIull~
Co If I . Fissata f , Ie soluzioni desorivono dun que un 1
insieme 8 limitato in N (0) e peroib (per 11 teorema di Rellioh) oompatto in N(O).
~1~
Consideriamo brevemente 11 problema della stabi11ta delle soluzioni ooetanti. Detta u una, soluzione oostante, u~(t)
=u t
wet)
un'altra qualsiasi soluzione, dalla (4), con considerazioni analoghe a quelle ,ohe si fanno per ottenere la (5), si rioava 1
(10)
2
8i ha 2
8upponiamo ,ora che sia
1/2
Ilullllwll Iwl ~ 2
II
nolds ,generalizzato dal1a ,Ladyzenskaia
€
II w (t )
.,f,
cOllullllwll
2 1/2,CniL,-1 ',11 u II < 1 (11 numeno 2 1/2 CO,u -11111 u
diffenisce per una costante daqueH,D ,ohe
I I =,u Bonendo 2 1/2 cnlul
1/2
,otteniamo
e
1/
datto numeno di Rey-
[5]).
I ,u1,w(t), 112 ,tb(w,u,w) >
112 .
8i ha dunque dalla (10) (1)
8i pub dimostnare che l'insieme 8 delle soluzioni costanti ha ,questa proprieta : per ,ogni ,suo punta u esiste una ,variete. line8,re ad un numeno finito"di dimensioni ohg EI, in un ,oerto sensa, "tan,gente" ad 8 Gin Nl(O) ). in un intorno di u , 8 si ,proietta ortogona1mente (sempre in Nt(O) ) su , questavar~eta senza sovrapposi,zioni. Non e detto perc ,ohe questa proiezione ,sia "su".
165
- 6 G.Prodi
1
d
2
dt
Da questa
e
fao11ededurre ohe, per t ... t a').1 w( t) ... 0
2
too
,
e ohe
Ilw(t)11 dt < ta'). Dunque, se e 21/20 ,u- 1 1Iull < 1 (11 ohe 900n 0 -1 / 2 2 ~2 oade se e I f I < 2 la soluzione u e stabile; segue anohe ,u 0
J
n)
ehe essa
e
unioa.
SUl problema della stabilita ,delle soluz10ni oostantinon si oonosoe ,nullapib di queste se,plioi osservazioni ohe riguardano soltanto il oaso di u "piooolo" (e ;ohe si trovano in blema della stabilita
e
legato strettamente a qUel10
15l).
Il pro-
del1Iunieit~;
vedremo suooessivamente oome questo problema 6 ,legato 901 ,problema ohe oi interessaprinoipalmente.
12.-
Riohiameremo qui alouni risultati ,relativi al1a teonia de11a
,misura sugli spazi ,metrioi e 901190 teoria ergodioa 'olassioa, 90110 soopo prinoipllle di adattarli 901 nostro ,pr.oblema. SiaE uno spllzio ,metrioo sepllrabilecompleto,sia ~(E) ,la famiglia dei boreliani ,di E . Si dioe ,misur.a di pr.obabilitauna fun2lione non negativa definita ,per ,un ,CT-oampo q ... additiva,
tale ohe
E' 1 e ,m(E) = 1
g ,di
ins1em1 di ,E ,
. Noi in1ienderrelllo ,selll'/lr;e"
,ina Hr.e ,che ;m ,sia" i £ 'GollpLetalllento ,a,LLa, ;Lebesgue Idi luna, .misur.a, ,definita, in G!>'(E). Dalla :miBura
111
si ,ottiene ,un integrals, aon ,190 consueta de-
,finizione. Bannemo ,M(r)
=.J
r(u) dm(u) . E
Indiahiall10 aon
e(j\) ,10 ' spuio delle ,run2lioni ,continuee
,1imitate definite in E " .oon , lanorma Irl ~(E) = sup Ir(u) I . v ,u EE ,LI integrale definisae dunque ,in t,(E) un funzionale ,M(r) ,linear e , ,continuo non negativo e ,ahe, inoltre, gode di questa ,propnieta Cdi Daniell) : per ogni suooessi one{r h } di funzioni continue non
166
- 7G.Prodi negative oonvergenti a zero in modo monotono si ha: Vioeversa supponiamo di assegnare in
lim MCf ) = 0 n n_ClJ ~(E) un funzionale ohe gode
di questa proprieta (diremo: un integrale astratto); esso ai puo (1) prolungare oon un prooedimento ,ben nota . ; ai potranno .allora dire miaurebili gli insiemi ,Ie oui ,funzi.oni ,oaratt.enistioherisu1tano sommabili. Se aggiungiamo ,1a ,oondizione di nonmali3zazione : M(1)
=1
Goon 1 indiohiamo. qui ,la funzione ,ohe 'e uguale ad 1 au E), abbiamo una oompleta equiyalenza tramisura (~i probabilit1»
ed integrale.
Partendo da una misura m e definendo un integrale M, attraverso ques.to riotteniamo lamedesima ,misura di parte;nza.
Coa~,
partendo da
un integrale M ,e paasando agli insiemi ,m:Lsura.bili, possiamo riottenere
l'~ntegrale.
La oonsiderazione di un integrale al posta di una misura, e spesso pib comoda. Questo vale ad
ese~pio,per
istituire una topo-
logia nell'insieme della ,misura. Noi dotere,mo l',insieme dellemisune della topologia oheviene subordinata in eeso dalla topologia debole , dello'spazlo
"'.
u . (Identifioando ,lemisune ,oon i relativi' integrali,
ohesono elementi di 211.1
Clf ).
E' ,impo!1tante avene ori.teri di oompattez-
,per ,Ie ,misure. ,La sfera unita!1ia ,dello spazio
~lf(E) ,e, in ,ogni
oaso, oompatta (sempne ,ne:j.la ,topologia debole). -Ora, se ,10 spazio E e oompatto, tutti i t'unf'ionali ,nonnegativi .sono ,misure ~infatti, in virtb del teorema di Dini, sussiste ,la proprieta di Daniell).. Po :Lohe la afar a uni tar ia dell 0 apaz:Lo
~*( E ~ J ,ne 11 a to po log i a de b 0-
le, possiede una baae numenab:He di intorni, si deduoe ,ohe do, ogni
successione
{m }
n
,di!llisur:e di probabi ~ita si ,puo estrarre una
aessione partiaLe aonver,ente v6rso;una,misura di
,suc-
~robabiLita.
Ti)---------Vedere ad esempio, Loomis, Abstraot harmonio analysis, cap.nI.
167
(1953)
- 8 G.Prodi Su questa risultato si fonda 111. dimostrazione dell'esistenza di una misura invariante rispetto ad un gruppo {T } di trasformat zion1 (-00< t < + (0) data d!\ Krylotf e Bogoljuboff [,]. Noi abbiamo bisogno di estendere, in qualche ,modo, 11 risultato enunciato 11.1 CaSO in cui .E non sill. oompatto. Da un risultato di Ulam (richiamato in [11]) s1 ha che, assegnata .in uno spazio E .metrico, completo, separabile, unam1sura m " s1 puo trovare un compatto D tale che ;m(E-D) ~
per ,ogni E •
E
> 0
Noi considere-
remoin E una famiglia di misure di probabilitb che soddisfino a questa condizione
unijo~memente.
Precisamente : sia assegnata u-
na successione {Dn} (n= 1,2,3, .•. ) di insiemi compatt:!:, can D C D . S1&
n
n+1
m la
famiglia di tutte le misure di probabilita
m che soddisfano 11.1111. oondizione : m(E-D ) ~ n
1 n
S1 ha Il.'llora :
La
famig~ia
JY(,
e
compatta, neHa
topo~ogia
Baster1:r. dimoatra:re che lafamiglia It[, so di
e
introdotta.
un aott01nsiemechiu-
'c*(E). Indioando aempre con 'Mil funzionale corrisponden-
te alla .misura ,m"
msi
:la .condizione che1ndividua la famigUa
traduce nella seguente : presoun interon > '01 per ,ogni funzione ¢ , continua, nulla in D
J
n
S1& ora F un elemento di di )tl.
.con 0
~ ¢ ~
1
si ha M(¢) < -
2.. n
'e~(E) appar.tenentealla chiusura
Evidentemente F sara non negativo, tale che F(,l') = 1
Inoltre soddisfere. alla condizione : F(¢) -<2:. , per ogn i funzione 11 ¢ (can 0 ~ ¢ ~ 1) nulla in D • Ba-stere. verificare che F gode de1-
n
111. .propriete. di Daniell. Sill. .{¢ } una successione ·monotona non n
crescente di funzioni limitate. Passiamosupporre ¢n ~ 1 . Fissa.:: to un E > 0 e preso un dominio D
m
~ntero m tale che ~ < ~ , consideriamo il
. Ciascuna delle funzioni ¢
168
n
m 2 puo essere deoomposta in
- 9 G.Prodi que s t
0
modo : ¢
l'1nsieme D m
n.
¢t > 0 ¢" > 0 CP';: ¢ n n ,n 'h
;: ¢' + ¢" ., dove n n
Imp~rremo
.inoltrela oondizione
S U 1-
;:
Poiche, per il teorema di Dini, ¢
converge unifor-
n
memente a zero Sl.\ Dm esiste un intero no tale ,che, ;per ogni n > no s1 ha
.II¢~ II C(E)
;: II¢n II e(D m) ( :
Per
n > no si /lvrs. allora
Si ha inoltre
NdLa topoLogia indotta in Jrl,da
tlt(E)
ogni punto rpossiede
,una ,base numer;ab i Le di intorni. Inhttfi, s h i t
nk
} (n, k ;: 1,2, ... ) un sistema di funzioni oon-
tinue soddisfacenti a queste condizioni : a) Ilf ·gni ' n ·fissato 'Ie restrizioni di f a D nk '
n
I '(M'
).1
- Ml(.)(fnk .
G(Dn)
II f nk II 1I~( Dn ).
0)
m. ,
II~ 1; b) per
gli insiemi formati dalle misurem 1 ('-r
0 -
(k ;: 1,2, ... ) costi tuiseo-
no un insieme denso nella sfera unitaria di
Se m*"f
nk
;
Em,
tali che
(n,k ,r;: 1 , 2,... ) cos t1' tulscono ' .un SlS . t ema
fondamentale di intorni. Possiamo ooncludere
neLLe ipotesi fatte, da ogni successions {m·n-} dimisure appartenenti aLLa famigLia zia~e
nu
si pu6 estrarre unasuccessione par-
convergente. Supponiamo ora ohe sia assegnata una famiglia {T t } di trasfor-
mazioni dello spazio Ein se. Per pat ere camprendere ilcasoche ci
inter~ssa,
supporremo 'ohe ·es.sa sta ·dei':i:nita 'per '0
~ ·t
(+co
e
che costituisoa un semigruppo. (Tutto ,fa 'pensare ·che ole trasformazioni Tt definite nel § 1 non abbiano inversa continua). Supp o rremo (ofr. § 1, propriet1l. ,3) ) e he Tt definisca una applioa z ione continua (0 r- +(0) X E ~ E . Supponiamo assegnata in E una ,misura
169
- 10 G.Prodi di probabilit. m . Essa vi ene , mutata dalla Tt (per ogni valore t .? 0 fissato) in unamisura ,m, seoondo la relazione
t
If(u)dm t valida per ogni funzione Se una funzione f finita : f' (u) = f(Tu)
m
e
e
= If(Ttu)dm f G
e
e
oosl de-
integrabile secondo la m . Diremo ohe
invariante, se, per ogni Se R
la f'
va~ore
di t si ha mt = m
un insiememisurabile e indichiamo con fR 180 sua fun-
zione ,caratteristioa, si hanell' ipotesi che m sia invariante
= If R(Ttu)dm
.If (u)dm
R
. La funzione f (T u)
R
t
e
integrabile ed
e
la funzione caratteristica dell'insieme T- 1 (R) . Dunque : sem T
t
di ogni ins ,ieme
e
t
invariante L'immagine :reciproca 'secondo La
'm1isurabi~e
e misurabiLeed'ha ' La ;ste,ssa :misu-
ra di questa. Se :introduciamo nello spazio (0 t-,+ool X ,E :la mlsura pl'odotto della ;ordinaria ,misura per ,una misura invariante ,m "
abbiamo che
le immagini reciprochedi insiemi .misurabili seoondola .trasformazione :
.:1'
t
u : (0 ~ too) XE ... E, so no ,misurabili.
I l olassioo teorema di Birkhoff, inizialmente dato ,per il
caso di un gruppoditrasformazioni, ,si estende (cfr. F.Riesz [10]) al oaso da noi considerato, di un semigruppo di trasformazioni; ,sia rm unamisurainuariante 'e 'sia f una juna,i;one ,integra-
bHe, aL/;oraH Limite i
'1Ja~ari
lim
1' ...00
1
l'
-:f l' , 0
f(T t u)dt
esiste :per ;quasi ,tutti
:d,i , u'. :La. funsioneLimite einuariante :ede integrabibe.
§3.i- 'Dramo ;ora alouni risultati rJelativi all .'esistellza di .misure di ,probabilit. definite in
NUn, invarianti rispetto al semigrup-
,po di trasformazioni di NCO) in s~, introdotto nel §1.
170
- 11 -
G.Prodi I ~I~·
Indioheremo can I 1110 sfera:
2
tero > 0) l'insieme oompatto.:: Ilull
I Co2 # -1 1 f., oon D
(n in-
n
~ n •
Gli integraliohesoriveremo si intenderanno estesi a N(O), senon
~
altrimentipreoisato.
I. - .Per una ,guaLsiasi l'IIItisura ,diproba·bH·itil invariante ·msi
ham(I)
=1
.
Infatti, detta f una qualunque funzione oont:Lnua, limitata .non neg at i va. 0 hell. s sum a valor e 1s u I , s i h a per
08 nit
> 0
,j.f(u}dm = If(Ttu)dm, essendo m invariante. Peroia per ogni If(u)dm
1
= -:;: I
l'
{ff(Ttu)dm}dt
o
1
= J{ -; I
l'
It
> 0
f(Ttu)dt}dm
0
Teniamoora .presente ohe, in virtb della (?) si ha 1
lim
Sideduoe
.1' ...00
jf(u)dm = Jhim 1
I
1' ...00'1'
l'
o
= 11
f(Ttu)dt}dm
dm
=1
ora
= inf
m(I)
If(u)dm
=1
n.-Per una quaLsiasLmisura di p.ro/ioabiLitlLinvariante:m :S
i ha .mEDn)
~
-1
1,_ n C
Indiohiamo .oon fD
2 -2 12 n # 1f
.
Ill. funzione oaratteristioa dell'insieme n
Dn
Procedendo dapprima oome 'peril teorema 'preoedente, :poi ap-
plioando :Ll teorema di Birkhof#, 6i ha l'
m(D ) = If n
=
On
J
(u)dm
Jhim 1' ... +00
1
o
{If
Dn
(T- u)dm}dt
t
=
l'
I
o
Indiohiamo ora oon ¢ (u, n
sieme dell'intervallo 0
~
t
1')
~ l'
1110 misura (ordinaria) del sottoin2
per oui si hallTull> n (oi08 t
171
- 12 -
G.Prodi
TtU~Dn)' 8i ha
T
-
T
¢n(U,
Io
T);:
f
D
(Ttu) dt; peroio esiste il limite
1 n H~ooTrpn(U, T) per quasi tutti i V!llori di U . D'altra parte,
'1P-
plioando la (9) si ha l i m-
1 ¢ n (u, T) < -l i m -.IeT II Tt U 112 d t < Co2 Ji- -2 1f
n
T~+oo T
12
.
T~+oo T
Peroio ;: j{lim T"'+oo
1 T
T
Io
f
Dn
(Ttu)dt}dm >
J.{1-
;: 1 - n
n
-1
-1
2 -2
CoJi-
2
If I }dm ;:
2 -21 f 12
COJi-
Possiamo ora ohiederoi se esistono misure di
.
probabilit~
in-
e ovvia: ogni misura ohe ha il suppor-
varianti. Una prima risposta
to oontenuto nell'insieme 8 delle soluzioni oostanti
e
senz'altro
invariante. Chiameremo misure banali le misure di questo tipo. Interessera sapere se esiatono misure invarianti non banali. 8i intuisoe subito ohe, nel oaso in oui tutte Ie soluzioni siano asintotiche aIle soluzioni oostanti, non potranno esistere che misure banali. Preoisamente: fissiamo un punto Uo £ N(O). Diremo (aeguendo K. e B.
[3]) che La traiettoria Ttu o Ii asintotica ad un insieme
8 se, detta fA 1'1 funzione caratteristioa di un qualunque insiemc .
aperto contenente 8 , si ha
Se tutte
~e
T
lim.2:.. I fA (T t u )dt ;: 1. Ora ai ha: T... +oo T O O
traiettorie sono asintoticne aLL'insieme 8 deLLe
soLusioni eostanti, ogni misura invariante
e necessariamente
banaLe.
La dimostrazione si ottiene oon 10 stesso ragionamento fatto per il Teorema 1. Vioeversa :
111.- Se esiste una traiettoria ene non sia asintotica aLL'insieme8 esiste aLmeno una
misur~
di probabiLita 172
inv~riante
non ba-
- 13 G.Prodi lIa~e.
Supponiamo .dunque ohe esistano un punta u
ed un insieme a-
c S e ohe
penta A tali ohe A 7
1
lim
o
f
7:::;'tOO
f
o
A
(T u )dt
t
=
0
1 -'1
Detto A I l'ineieme oomplementare di A"
si ha
7
1
(11 )
(oony > '0) .
f
f
o
1>1
(T u ) dt = '1 0
t
Indiohiamo oon m7 la misura definita dall i integraie
=
M (r) 7
1
7
f
f(T u )dt •
t
o
0
Dimostriamo ohe questa famiglia soddisfa alle oondizioni del .oritenio di oompattezza introdotto. nel §. 1, almeno per7 ~ 1 . ·Oonsideriamoinfatti una qualunque ·funzione .oontinua .tf; <, o~tf;,~ 1) nulla in D e teniamopresentela funzione n
Goon
4) n
(u, 7)., ,0
definita come nella dimostrazione del teoremaprecedente.Si 'ha, ,indioando .oon Dlil oomplementare di D n
n
=
M Ctf;) 7
<
T
1
f
tf;(T u )dt < o
1
1
n
7
t
1
0
Applioando la (8) siottiene dunque
= sup M (¢) ¢7 .
<
(T u ) iit ~ t
0
m(N(O) .- D ) = n
1 n
Questo d.imostra ohele condizioni del oriterio di oompattezza sono .soddisfatte. Tenendo presente ,la (11) oonsideriamo una successione {7 } k
divergente a too e tale ohe
173
- 14 -
G.Prodi l'
J
1
lim
k
= 'Y
o
Estraiamo dalla suooessione di misure {m1' } una successione {m1'
}
k
oonvergente.
oon m la misura limite, oon M il
Indi~hiamo
k' relativo integrale. Dimostriamo anzitutto ,ohe 'mnon ebana1e.Sia infatti g una funzione oontinua, !limitata non ,negativa, assumente ,valore 1 nell'insieme A'. Si avra tenendo presente :la (12), M(g)
1
= lim
1'k, .. t oo
k'
= inf g
Peroio m(A') La misura m
l'
e
lim 1'k'""+oo
1
l'
k'
J1'k' o
f
A'
0
M(g) ~ 'Y •
,
poi invariante. Ripetendo infatti il ragionaaento
di K. e B. [3], si ha (indioando oon T f la funzione T f(u) ~
M(T f) '7
=
lim l'
=y
(Ttu )dt
k'
o
1 i
=f(T
~
l'
J
1
.. too
= lim ik, .. +OO
~
k'
k'
f(T
7]+t
u ) dt
o
=
1'k,+7]
J
7]
f (Ttuo)dt
= M(f)
Resta dunque 11 problema di dare criteri ehe
stabi~iscono
quando te sdusioni ,sonocaintotiche ,aLL'insie1!le S deLLe ,soLu.zioni costanti e quando ,si ,vedficlJ, H caso eontrado. Da una misura di ,pCQbabj,liU. invariante non banale ai puc ovviamente ,ottenere una misura di probabilita invariante per aui :m(S) :: 0 .
Si pone allora il problema (ohe si ,potrebbe anoora dire del,la er,odicita della trasformazione) : una,taLe misura • unica? Sarebbe anohe interessante indagare :su ,una :oongettura i!onmulata da E.Hopf ,termini : ,il
[z1, oongettura ohe posaiamo tradunre in guesti
~upporto
di una misunainvaniante non banale
174
e
un~
u)
- 15 G.Prodi variet a di dimensione finita. ;La dimensione oresce aon il dear e saere del ooefficiente di viscoaita. Anohe presoindendo dalla r.isoluzione di questi 'problemi, s eQuendo K. e B.
[3], e apportando lievi modifiohe ai ragionamenti
da lora svolti, possiamo ottenere interessanti riaultati. Ci1imitiamo ad aooennarne qua1ouno 8tito10 di esempio. Seoondo K. e B. ai definiaoe insieme di 'probabilita nulla un inaieme di misura nulla riapetto ad ,una ' qualsiasi misura invariantel ai dioe insieme di probabilita mas sima un insieme i1 oui oomp1ementare ha probabi1ita nulla. Allora l
oome
immediata oonseguenza del teorema di Birkhoff,
ai ha : se f Ii una funaione continua,
cui
i~
Limite
1
lim
T
T-+too
-nuL La.
I
T
f(Ttu)dt
L'insieme dei
non esiste
e di
~unti u
per
?robabiLita
0
Ad esempio, se prescindiamo da un insieme di valori iniziali u
o
di probabilita nulla, 1e soluzioni u(t) Bono tali ohe esiste
illimite
lim
1
T
I
T
o
Se dioiamo quasi
/u(t) /Pdt- per ogni p > 0 rego~ari
i
punti u per oui il limite
~ IT
f(T u)dt esiste qualunque sia la funzione oontinua Tot f , si ha : L'insieme dei punti quasi regoLad e invariante sd hI], l im
T-+too
probabiLita massima.
175
.. 16 ... G.Prodi BIB L lOG R A F I A [1]
E. HOPF - "Ein allgemeiner Endlichkeitssatz der Hydroaynamik". Math.Annalen 11~, 764~775 (1941).
[2]
E. HOPF - "A .matheme.tical .example disple.J':ing fee.tures of tUIlbulenoe ft • ,Camm. ,an ,pure and e.pp-l. ,me.th. 1, 303-322 (1948) .
[3]
N.KRYLOFF e N.BOGOLJUBOFF - '!La .theorie generale de la mesu.re daks ·son _application ,a ,l'etude des syst~mes dy_namiques .de .la,mecaniquenan :lineaire'i. Annals of Math. 38, 65-113 (1937).
[4]
O.A. LADYZENSKAIA- "Saluzianiin .grandedsl _prablemaal _oantornonon stal!lianaIlio, 'per il sistema di Nav.ierStakes in due .vaniabtli sp~l!Ii9li". Dokl.Akad. Na uk. 123 (3) 1128-1131 (1958)" (in Russo)
[5]
O. A. LADYZENSKAIA - "Soluziani .in grande del :problema al oontarno nan ste.zianario, ,per .il sistema .di _..• Camm. an Pure and Appl. :Math .. 12, 429-433 (1959).
[6]
O. A. LADYZENSKAIA - 'iLo ,studio .delle equal!lioni _di Nav.ier..,Stakes .nel ,oaso .del :mavimento stazianario .di un .fluido inoampressibile". (in Russo) Uspehi :Math.Nauk 14, (1959).
[7.]
J. L. LIONS:- "Quelques resuHats.d' existence ,dans _des equations .aux ,dEirivEies :partielles .nan :linEiaires". Bull. Soo .:IlIath.Franoe, 87., 245-273 (1959).
(8)
J. L. LIONS e G. PRODI <- "Un ·theol'~me d I existenoeet unioite dans les equations de Nav.ier~Stokes en dimension 2". Oamptes 'renduedes seanoes ·de,:I;' Aoad. des Soienoes, t. 248, p~3519~3521;
[9]
G. PRODI - "Qualohe risulte.toriguardo alle equaziani di N,,viel' .. Stokes nel ,oaso -b:i:dimensianale". Rend .-Sem.-M>t . Paclava, 30, 1-15 (1960J.
[10] F. RIESZ ,- I'Sur 1a theol'ieergodique'!. Camm.Math. Helvetioi, 17, 221-.239 (1944). [11] S.M. ULAM e I.O. QXTOBY - 'IOn the existence of.a measure inve.riant under a transformation". Annals of Math. 40, 560-566 (1939).
177
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C~I,M,E,)
A. N. FELDZAMEN
THE ALEXANDRA IONESCU TULCEA PROOF OF HCMILLAN'S THEOREM
ROMA - Istituto Matematico dell'Universitl - 1960
179
THE ALEXANDRA IONESCU TULCEA PROOF OF MCMILLAN'S THEOREM by
A. N. FELDZAMEN
The objeot of this leoture is to present a new, reoent proof of MoMillan's Theorem, the first suoh proof for the case of convergenoe almost everywhere, and one whioh also inoludes oonvergence in 1
L
It is due to Dr. Alexandra Ionesou Tuloea , of Yale University. The main idea of this proof is simple, elegant, and illuminating,
but I am almost oertain it will not be published in this form. Dr.Ionesou Tuloea later generalized the result and the methods, and this generalization - rather than the present, somewhat simpler and clearer, versibn - will be published shortly. Professor Halmos
disoussed the history of McMillan's theo-
h~s
rem, and the work of Shannon, McMillan and Breiman, and we turn to the details. A few preliminary definitions, and two theorems, are required in this presentation.
We take fixed throughout a probability spaoe (X, pCX)
=1
If
a is
a finite u-field oontained in
7TW..l for the family of minimal set s, or atoms, of
1,
a,
1,
p) , with
we write and
«a)
for the oardinality of7T(~). THEOREM 1. If ~
c.)o
a.
is a finite u-fieLd contained in
= 0,
c'J., n
Lon -vn+l-J
'1 ,
and
1, ... , . is an, increasing sequence of arbitra-
ry .u-fie Lds, then P ({x I sup
I
(all.
o~n<(l)
)(x) > ic}) < C(Q,) e
-ic
n
PROOF: It is clearly sufficient to prove that for each A we have p(h I sup o~n<(l)
I
(0.1
1:. ) (x)
> ic}
n
181
(l
A)
E: 7T(al
- 2 -
A.N.Feldzamen To prove this, let A be fixed and p~t
{xl
E(A) = F (A) = o
sup
{xl
n
r(Q/t lex) > A}
o~n
t
p(AI
n
0
A
lex) < e- A}
and p(A/e, ,lex) ~ e
-A
p(A/I:. lex) <e k
J
-A
},
for every A > 0 . Then F (A)
n
n Fm(A)
F (A) , n
~
if
= if;
nt min, m = 0, 1, ...
n = 0, 1,... .
n
This is an immediate
oonsequenoe of the definition of oonditional probability. (iii)
f
We reoall that, by definition,
c
fF
n
P (AI ~) dP = P (A (l C)
(A)
C€ ~
if
PCAI ~ ) dP = P(F (A) () A) < e
n
n
Thus
-A P(F
n
(A))
for eaoh n = 0, 1, ... and f.. > 0 Q) (iv) I assert E(A) .= U Fn (A) f) A . For, if x fA, we have n:=o
r(Q,! en) (x)= - log
p(AI
t: n ) (x)
(- log P (AI
sup o~n
~
- log (inf
(sinoe the funotion • log is striotly inf o~n
t: n ) (x ))
pCAI
o~n
~
, n= 0, 1,...
t
n
Thus
> A
)(x)) > A
deoreasing in (0, 1))
p(AI ~ ) (x) < e
-A
n
As this last inequality is strict, there exists k such that P(A/'Ck)(x) < e
-A
. Then x
£
Fk(A) for the smallest k with this
182
- 3 A.N.Feldzamen 00
n
U F (A) A , and reverse inolusion follows n=o n in the same fashion from this ohain of equivalenoes.
.property. Thus E(\.) C
(v)
Finally 00
p(E(A))
= n=o L
= e
P(F (A) n
n
00
A) ~
L
U
F ( A) )
~ e
n
n=o
-A
=
P(F (A) ) n
n=o
00
-A p(
e
-A
and tile theorem is proved.
If T : X ~ X is a measurabLe (that is T- 1 E
THEOREM 2.
1I!Ncure preserving (that is peT
af g.
mation}
-1
E)
is a finite (J-fidd,
if
= peE)
E
if E
€
1)}
E'1 ) transJor-
tn+1 ~ 'J '
tn ~
€J
n
= 0,1, ...
is an increasing sequence of arbitrary (J-fieLds, and
-* h
= sup r(a/~), Q.5n< 00 n
(1) hiE' € LP (X,
(2)
3
1,
then
p) for every p ,
h lIIeasuraHe
'1
1'~ P <
such thath ELP(X,
00
t,
p), 1 < P <
(»,
h
is measure preserving, and n-1
1
~
r(a.fe)T n k=o
n
n-k-1
convdrges to h almost everywhere and in the norm of each L 1
~
P <
Then
F
,
(»
(1)
PROOF
P
j +1
()
let p be fixed,
A
j
l$p
A j
=
{xl
j
F
=
{xl
(h II (x3-l
,
and
j
=¢
Put
< (h*"(x))P < j
A. J
+ 1}
= Fj
F
and
183
j +1
}
j
> j } Thus
= 0,
1, ...
P (A. ) = P(F j l-P(F j +1 J
- 4 -
A.N.Feldzamen Jlh~ (x)
,p
Pdx
~
P(A ) + 2P(A 1 ) + 3P(A 2 ) + ••• 0
=
P(F ) - P(F 1 ) + 2P(F 1 ) - 2P(F 2 ) + 3P(F 2 ) 0
...
00
~
=
j
P (F. ) J
=0
00
~
=
j
p(hl
sup o~n
=0
00
~
j
<
(2)
e
-j
I(a/~ )(x) > n
lip j
})
lip
(using Theorem 1)
:::0
00
To prove the remainder, we must make use of deep results
of ergodio theory. The first of these, an ergodio theorem for a sequenoe of funotions, states the following If
h, n :: 0, 1, ... , is a sequenoe of measurable funotions n
sup Ih I € LP , (1 ~ p < (0), and (b)
lim h n (x) x exists almost everywhere, then there exists a measurable measure p 1 n-l k 1 preserving funotion h E L such that L h (T n - - ) oonverges to ii k=o k h almost everywhere an. in the norm of LP satisfying
(a)
n
n
(See P.Maker, The ergodio theorem for a sequence of funotions, Duke Math. J., 6, 1940; and aleo : L.Breiman, The individual ergodio theorem ot information theory, Ann. Math. Stat., 28, 1957).
= I( 0..1 t ). Hypothesis (a) follows from n n part (1) of this proof, and hypothesis (b) from the faot that p(AI t n) In this theorem let h
is a martingale ( - see Seotions 10 and 17 of Entropy in Ergodio Theory by Paul R. Halmos). Thus the proof is oomplete. APPLICATION. In theorem 2 , let k = 1, 2,...
We reoall that
184
- '5 A.N.Feldzamen I(
k-l
v
j
lS,
T-ja.,)
= IUk)
Tn -
1
=0
n-l
+ L I( k=l
k
(J./
v
j=l
T
-j
n-l-k
0,.;) T
=
by Theorem 2 1 n
k-l
I (V T-jo.,) ... h j =0
n ... co , both almost everywhere and in the norm of the proof of MoMillan's theorem is oomplete.
185
LP , 1 ~ P <
