Progress in Probability and Statistics Vol. 8
Edited by Peter Huber Murray Rosenblatt
Birkhauser
Boston Basel Stuttgart
Philippe Bougerol Jean Lacroix
Products of Random Matrices with Applications to Operators
Editors Philippe Bougerol Universite Paris 7 UER de Mathematiques 2 Place Jussieu F-75005 Paris/France
Jean Lacroix Universite de Paris XIII Departement de Mathematiques F-93430 Villetaneuse/France
Library of Congress Cataloging in Publication Data Bougerol, Philippe, 1951Products of random matrices with applications to Schrodinger operators.
(Progress in probability and statistics ; v. 8) Bibliography: p. 1. Stochastic matrices. 2. Schrodinger operator. II. Title. I. Lacroix, Jean, 1944- . III. Series.
QA184.B68 1985 ISBN 0-8176-3324-3
512'.5
85-15695
CIP-Kurztitelaufnahme der Deutschen Bibliothek Bougerol, Philippe: Products of random matrices with applications to Schrodinger operators / Philippe Bougerol ; Jean Lacroix. - Boston ; Basel ; Stuttgart : Birkhauser, 1985.
(Progress in probability and statistics ; Vol. 8) ISBN 3-7643-3324-3 (Stuttgart ... ISBN 0-8176-3324-3 (Boston)
NE: Lacroix, Jean:; GT
© 1985 Birkhauser Boston, Inc. Printed in Germany ISBN 0-8176-3324-3 ISBN 3-7643-3324-3
CONTENTS
"LIMIT THEOREMS FOR PRODUCTS OF RANDOM MATRICES"
PART A
INTRODUCTION
CHAPTER I
1
THE UPPER LYAPUNOV EXPONENT
-
.
5 5
1. Notation
2. The upper Lyapunov exponent
6
3. Cocycles
8
4. The theorem of Furstenberg and Kesten
11
5. Exercises
13
CHAPTER II
-
17
MATRICES OF ORDER TWO
1. The set-up
17
.
2. Two basic lemmas
19
3. Contraction properties
24
4. Furstenberg's theorem
30
5. Some simple examples
33
6. Exercises
.
36
7. Complements
.
38
CHAPTER III - CONTRACTION PROPERTIES
.
43
1. Contracting sets
44
2. Strong irreducibility
48
3. A key property
4. Contracting action on direction
50
P(]Rd) and convergence in 55
v
5. Lyapunov exponents
60
6. Comparison of the top Lyapunov exponents and Furstenberg's theorem
64
7. Complements. The irreducible case
68
.
CHAPTER IV
-
.
COMPARISON OF LYAPUNOV EXPONENTS AND BOUNDARIES
77
.
1. A criterion ensuring that Lyapunov exponents
are distinct
77
2. Some examples
81
3. The case of symplectic matrices
87
4. p-boundaries
93
CHAPTER V
CENTRAL LIMIT THEOREMS AND RELATED RESULTS
-
1. Introduction
.
101 101
2. Exponential convergence to the invariant
measure
103
3. A lemma of perturbation theory .
111
4. The Fourier-Laplace transform near 0
116
5. Central limit theorem
121
.
6. Large deviations
129
y
7. Convergence to
134
.
p
8. Convergence in distribution without normalization
135
9. Complements equations
140
CHAPTER VI
-
:
linear stochastic differential
PROPERTIES OF THE INVARIANT MEASURE AND APPLICATIONS
145
1. Convergence in the Iwasawa decomposition
147
2. Limit theorems for the coefficients
155
3. Behaviour of the rows
159
4. Regularity of the invariant measure
161
5. An example : random continued fractions
166
SUGGESTIONS FOR FURTHER READINGS
173
BIBLIOGRAPHY
175
vi
"RANDOM SCHRODINGER OPERATORS"
PART B
INTRODUCTION
CHAPTER I
183
THE DETERMINISTIC SCHRODINGER OPERATOR
-
.
187
1. The difference equation. Hyperbolic structures
187
2. Self adjointness of H. Spectral properties
190
.
3. Slowly increasing generalized eigenfunctions
195
4. Approximations of the spectral measure .
196
5. The pure point spectrum. A criterion
200
6. Singularity of the spectrum
202
CHAPTER II
ERGODIC SCHRODINGER OPERATORS
-
.
"
.
205
1. Definition and examples
205
2. General spectral properties
206
3. The Lyapunov exponent in the general ergodic case
209
4. The Lyapunov exponent in the independent case
211
5. Absence of absolutely continuous spectrum
221
6. Distribution of states. Thouless formula
224
7. The pure point spectrum. Kotani's criterion
232
8. Asymptotic properties of the conductance in the disordered wire
234
THE PURE POINT SPECTRUM
237
1. The pure point spectrum. First proof
238
2. The Laplace transform on
240
CHAPTER III
-
Sl(2,]R)
3. The pure point spectrum. Second proof
247
4. The density of states
250
CHAPTER IV
-
.
SCHRODINGER OPERATORS IN A STRIP
.
253
1. The deterministic Schrodinger operator in
a strip
253
2. Ergodic Schrodinger operators in a strip
259
3. Lyapunov exponents in the independent case. The pure point spectrum (first proof)
2 62
4. The Laplace transform on
267
Sp(i.,IR)
.
5. The pure point spectrum, second proof
vii
272
APPENDIX
275
BIBLIOGRAPHY
277
viii
PREFACE
This book presents two closely related series of lectures.
Part A, due to P. Bougerol, is an introduction to the works of Furstenberg, Guivarc'h, Le Page and Raugi on products of random matrices. Only invertible independent identically distributed random matrices satisfying an irreducibility condition are considered. The purpose is to prove in detail the analogues of the classical limit theorem (e.g. law of large numbers, central limit theorem). This part is based on a course given at the University of Paris 7 in 1983.
Part B, due to J. Lacroix, deals with the spectral theory of random Schrodinger operators, where the products of random matrices play a crucial role. It presents a rigorous and unified treatment of the main known results in the one-dimensional discrete case. Since we are aware that some readers are mainly interested in Schrodinger operators, any notion or result needed from part A is clearly restated (but of course not proved again !).
The book is self-contained and should be accessible to readers with minimal background. Since we feel that these topics deserve a large audience we have "tried" to write it English. It is not sure that we have succeeded and we beg the indulgence of the native speaker.
It is our pleasure to thank Yves Guivarc'h, Francois Ledrappier Emile Le Page, AlbertRaugi, Jose de Sam Lazaro and Bernard Souillard for enlightening conversations concerning the material presented here and for their encouragements.
ix
Finally we thank Joelle Navarron and Catherine Simon for their typing.
Paris, May 1985.
x
PART
A
LIMIT THEOREMS FOR PRODUCTS OF RANDOM MATRICES
INTRODUCTION
This part is devoted to limit theorems for products of i.i.d. invertible random matrices. The subject matter, initiated by Bellman, was fully developed by Furstenberg, Guivarc'h, Kesten, Le Page and Raugi. This text is intented to serve as an introduction to their work.
We have chosen to keep the level as elementary as possible. This had sometimes led us to write lengthy proofs when shorter ones are available and to omit some important topics. On the other hand the text is self-contained and should be accessible to readers familiar with probability theory as usually developed at graduate level. In particular, no prior knowledge of group theory is assumed.
Let us roughly describe our general line of approach. We consider a sequence matrices of order We set
So = Id
d
Y1,Y2,...
of invertible random
which are independent and identically distributed.
S1 = Y1,....Sn = Yn...Y2Y1.
,
Our purpose is to prove that under an irreducibility condition, the sequence Log 1Sn xll
satisfies, for any
x # 0
;
n = 1,2,...
in ]Rd,analogues of the classical limit
theorems for sums of i.i.d. random variables (e.g. law of large numbers, central limit theorem, ...).
The strategy we shall adopt is based on the following observation.
For any class of F
:
x
x # 0
in
]Rd, let
in the projective space
Gl(d,]R) x P(]Rd) -> ]R
x
defined by 1
denote its direction (i.e. the
P(]Rd)). Consider the function
2
_
IlYxil
F(Y,x) = Log
.
IIx 11
Then IIS n
Log
n-1
xil =
II S
1+1
Log
E
IISixII
i=o n-1
IIXII =
E
i=o
x11
S. x Log IIYi+1
1
II
IISixII
n-1 E
i=o Therefore
Log IISnxII
of the Markov chain
(Y
i+11
F(Yi+1' Six)
can be written as a simple functional
Six)
on
Gl(d,IR) X P(1Rd)
.
Taking into
account our irreducibility assumption, the first conclusion we can draw is the law of large numbers. Namely, there exists a real the upper Lyapunov exponent, such that for any
n-
, called
a. s.
,
lim n Log II Sn xII = -Y1
yl
x # 0
The other results require a detailed study of the Markov chain (Y1+11`6.1 x)
.
Its essential properties will be derived from the fact
that, loosely speaking, the random matrices
Sn
have asymptocally a
contracting action on the directions. To be more precise, let the natural angular distance on
6
be
P(JRd)
6(x,y) = Isin{angle(x,y)}I
We shall prove, following Guivarc'h and Raugi [34], that under fairly weak assumptions,
lim sup n Log 6(Snx, Sny) < 0
a. s.
(1)
n-
f or any fixed x and y in
P (]Rd)
This property is of primary importance. For instance it yields easily that the Markov chain
(Yi+1' Six)
has a unique invariant
probability measure. More crucially we shall see, following Le Page [49]
,
that (1) is the cornerstone on which rest the proofs of the
central limit theorem, the estimate for large deviations and many further results. We might say that, in some sense, this inequality replaces Doeblin's condition under which the precise limit theorems for functionalsof Markov chains are usually available.
Let
yl > y2 > with the sequence (Yn)
.
> Yd Since
be the Lyapunov exponents associated
n
lim sup n Log d (Sn X , our first task is to show that
Sn Y)
< Y2 - Y1
Y, > Y2 .
The main purpose of the first three chapters is to derive the inequality
from a careful analysis of the qualitative
yl > Y2
behaviour of the Markov chain
(Snx).
In Chapter I we introduce the basic properties of the upper Lyapunov exponent
yl
.
In Chapter II we restrict ourselves to
2 X 2
matrices. For
pedagogical purpose we first develop in this simple setting the general argument leading to a proof of this inequality. Most of the results which are needed in part B can already be found here.
Chapter III treats the case of matrices of arbitrary order.
In Chapter IV we digress from the main line. We apply the preceding results to the study of all the Lyapunov exponents. We also briefly present the link between these exponents and the boundary theory of Furstenberg.
In Chapter V we derive the main limit theorem on from the inequality
y1 > Y2
.
Log IISnxll
Further results, such as limit theorems
for the coefficients of the matrices
S
n
,
are proved in Chapter VI.
All the main results proved in this text come from Furstenberg
[21]
,
Guivarc'h and Raugi [341
and Le Page [49] . We hope that our
exposition entices the reader to go back to these profound original works.
Despite its importance, we have chosen not to consider Osseledec's theorem. The reason is twofold. Firstly we have tried to keep the prerequisites to a minimum and to give a self-contained account of the subject. Secondly Ledrappier has already given in r46]
a beautiful treatment of the applications of this theorem to products of i.i.d. random matrices. We felt there was nothing to gain by a
4
reproduction of this material. In our opinion, the reader who wants to have a full picture of random products has to read Ledrappier's monograph. It will then be an easy and useful exercise to him to check how some of our proofs can be shortened by making use of his results.
We have not considered positive matrices (except in some exercises). We understand that Joel Cohen is writing a book on this subject.
For the sake of simplicity of notation we have restricted ourselves to matrices with real entries. But all the statements are also true in the complex case (the proofs carry over immediately to this case by replacing everywhere
IR by
M)
.
Some chapters contain "complements" sections. They develop some additional material which is not used elsewhere and the reader can skip them. Some of their proofs are only outlined and may be of a more advanced level.
CHAPTER I
THE UPPER LYAPUNOV EXPONENT
In this chapter we define the upper Lyapunov exponent
y
which
gives the exponential rate of growth of the norm of products of
independent identically distributed (i.i.d.) random matrices. In order to prove the analogue of the law of large numbers we develop some basic results on G-spaces which will often be used in the sequel.
I.1. Notation
We shall write
M(d,]R) for the set of
d x d
is the set of invertible elements of
entries. Gl(d,IR)
M
the set of matrices with determinant one. If in
matrices with real
is in
M(d,]R), Sl(d,]R)
M(d,]R) and
x
]Rd, Mx will be the vector equal to the product of the matrix M
by the column vector
x
.
The transpose of
M
is
M
In general the results we present do not depend on a particular norm chosen on
]Rd or
M(d,IR).
For convenience we shall use, unless
tho contrary is explicitely stated, for
x e ]Rd and
M c M(d,]R)
d jxII
E
_
i=1 IM11
= Sup{ JIM-11
:
x E IRd
jjxjj = 1}
Similar definitions hold for matrices with complex entries.
5
6
It will be understood that the random variables are defined on a
probability space random variable
(0, A, 1P) , 1E (X) will be the expectation of a = J X d1P, and if
X, ]E(X)
F
is a sub
a-algebra of
A, 1EF(X) will be the conditional expectation with respect to F A topological semigroup is a topological set associative product is defined, the map
T
on which an
(M1,M2) -> M M I
T
being continuous. For instance A topological group is a group
mappings
(g,h) -> gh
G
T2
to
with a topology for which the
g -> g-1
and
from 2
is a topological semigroup.
M(d,]R)
are continuous.
I.2. The upper Lyapunov exponent
{Yn, n >_ 0 of i.i.d. random matrices with
Consider a sequence common distribution
p
.
IISnII < Therefore if Log+ IISnII
]E(Log
Sn = Yn...Y1. For the chosen norm,
Let
is integrable. If
n,p ?
]E (Log II S n 11 )
for
Sup(f,0))
IE(Log IIYn+p...Yn+1II) +]E(Log IIYn.Y1II)
Thus the sequence fIE(Log IISn II),
n
f
then
1
IE (Log
1
IIYI II
is finite (we write
IIYIII)
IE(Log IISn+p II)
...
IYn_, II
IIYn II
IIS
n
1
1}
1)
IE (Log IISnII) is subadditive and
converges to inf 1m]E (Log I I S m II ) in
IR U
m?1
(see exercise 5.2).
DEFINITION 2.1.
If 1E(Log+ IIY1II) < 00 , the upper Lyapunov exponent
associated with
pi
is the element
Y=1 n
n
y
of IR U {_co}
defined by
]E(Log IIYn...Y1 I1)
Since all norms on the finite dimensional vector space are equivalent, (when
Y1
y
M(d,]R)
is independent of the chosen norm. We shall prove
is invertible) the stronger result that almost surely (a.s.)
Y =1imnLogllYn...Y1I n-_
7
Although this turns out to be an easy consequence of Kingman's subadditive ergodic theorem (see e.g. Ledrappier [46]) we shall follow the original proof of Furstenberg and Kesten [25]. The reason is twofold. On one hand it is a good illustration of the techniques we shall use. On the other hand it gives us some information that will be needed later. Let us first describe two examples.
Suppose that each matrix Y
Example 2.2.
is a diagonal matrix
n
diag(a1(n),...,ad(n)). This means that the entries
(Yn )1,J It is clear that
_ I
0
if
i # j
ai (n)
if
i=j
y = Sup1E(Loglai(1)l)
satisfy
(Yn)i,J
.
and
i
y = lim
Log IIYn . .Yl II n
by the usual law of large numbers.
Example 2.3.
The preceding example is degenerated. Far more interesting
is the following due to Cohen and Newman [15]. Let G1(d,1R)
Y1,Y2,...
be a sequence of i.i.d. random elements in
such that the distribution of
IIY X ll 1
x # 0
in 1Rd. For each vector
x
does not depend upon
Ilxll
0, the real random variables Y
Log IIY X 11,...,U
U,
1
11- 11
= Log IIY
n
are independent and identically distributed a-algebra generated by
n1 :
.Y x
n-1 n-
for, if
Il,
1
...
lYn-1...Y1xil
F
n
is the
U1,...,Un, for any Borel bounded function
f
on 1R F 1E
{f(Un+1)} = 1E{f(Log llYn+lyll)} = 1E{f(U1)}
Yn...Y1x where
y =
.
Remark that
IIYn. . .Y1x ll
Log IIYn...Yjxll = Un + Un-1 + ... + U1 + Log IIxII Therefore we may apply the classical limit theorems for sums of i.i.d. random variables in order to obtain results on the sequence
3
{Log IY....Y1xII} . If 1E(Log+ IIY1II) law of large numbers
IE(U+) < - and, by the usual
,
<
lim n Log IIYn...Y1xII = IE(U1)
a. s.
nNote that IE(U1)
is independent of x . Thus using
IIYn...Y111 where
{ei,
dIIY....Y1II
1
is the canonical basis of
d}
i
1
< d SupllYn...Y1ei1I o <_
n Log IIYn...Y1
IRd
=1E(U1)=1E (Log
we obtain
I)
Y1
IIxII
n y = IE(U1). In general there will not exist such a simple
This yields
expression of
in terms of
y
Y1.
1.3. Cocycles
The proof of Furstenberg and Kesten's result requires a simple lemma on cocycles. Since it will be often needed in this book we present it in a general setting.
DEFINITION 3.1.
Let
be a topological semigroup. We say that
G
acts on the topoZogicaZ space each
(g,x)
in
of
an element
G X B
B
in such a way that
for
If G
is a group with unit
DEFINITION 3.2.
Let a
g1,g2 e G
G :
,
e
is said to be an additive cocycle if a(g2,x)
x e B.
G
to
S
is a G-space if we set
For
x e B.
be a topoZogicaZ semigroup acting on a space B.
G X B -> IR
When it does not depend upon IR .
,
and if moreover
a(g1g2,x) = for any
g1,g2 c G
G-space.
a
A continuous map
e
G
if one can associate continuously to
B
G = Gl (d , IR) and
x S
,
a
is just an homomorphism from
the unit sphere
{x E 1Rd ;
IIxII = 1}'
9
x e S
M c Gl(d,IR) ,
IINxII and
is an additive cocycle we shall frequently meet.
a(M,x) = Log IjMxll
If
is the set of positive matrices in
G
whose entries satisfy on
S+
same formula
in
acts on
G
M
is a topological semigroup acting
> 0) , G
M.
x
the set of
,
(i.e. matrices
M(d,1R)
with nonnegative coordinates. With the
S
and
S+
defines an additive cocycle on
a
GxS Let
DEFINITION 3.3.
p * v
be a topoZogicaZ semigroup acting on
G
is a probabitity measure on
(resp. v)
the distribution on
which satisfies
B
f(x)d(p * v)(x) =
G
for all bounded BoreZ function
p * A
on
G. Note that
distribution on
and
a
distribution
If v
Let
p
G
,
the ordinary
f(gh)du(g)da(h)
(p * A) * v = p * (A * v)
G
if
v
is a
be a topological semigroup acting on the space G x B
.
Consider a sequence
of independent random elements of
G
with common
B
such that
.
p-invariant distribution on
is a
o(g,x)Idp(g)dv(x)
JJ
JI
an additive cocycle on
{Yn, n ? 1}
.
B.
PROPOSITION 3.4. B
.
is defined by
Jfd(p * A) = f
B
are two probability measures on
A
convolution product
for
on
f
(1)
B
p-invariant if p * v = v
is
v
and
p
If
J
I
B
We say that
B. If p
we denote by
(resp. on B)
G
is finite, then
if
lim n a(Yn(w) ... Y1(M ), x) n-
exists for 1P®v-almost all Proof
(w,x)
and in L1(W®v)
.
Without loss of generality we may suppose that Yn e G} , A
0 = {w = (Yn)n:21
a-algebra generated by the
being the
'
coordinate maps Y
's i
Yn
and
IP
the probability measure for which the
are independent with distribution
p
.
1G
Let E = Q x B
and
Q=IPOV
.
Define
0
: E -> E
by
0(((Yn)n>l,x)) = If
is a Borel subset in
A0
Q{((o,x) E Q x B
and
B
Borel subsets in
G,
(A1 X A2 x ... x An x ...) x Ao}
0((w,x)) e
;
Al,A2,...,An
= (1P ®V) {y
c A1,...,Yn+1 c
= (IP0V) {Y1
F- Al,...,Yn c An...,x e Ao}
E Ao}
= Q{(A1 X A2 x ... x An x ...) x Ao) This proves that
preserves
0
For each integer
p
Q
,
0P(((Yn)n>1.x)) = ((Yn+p)n>1, so that if we set
F(w,x) = G(Y1,x)
for
w = (Y.).,,,1, x e B, then D(Yn-l*** YI'X)
G(Yn...Yi,x) = O(Yn'(Yn-l'** n =
E
.. Y
a(Y , (Y P_1
p
p=1
1
n =
E
F(0p((w,x)))
P=1
Therefore Birkhoff's ergodic theorem (see e.g. Breiman Theorem 6.2.1) implies that
Q(Yn...Y1,x) = n n converges
distributions on power of LEMMA 3.5.
F(0p(w,x)))
L1
is separable and compact there exist
B
If
a.s. and in
1P R V
E
P=1
B
:
(p1
p
= u,
Let
indeed if pn denotes the u2 = p * p, ...) , we have
be a compact separable
B
11-invariant
nth
G-space and
convolution
p
be a
distribution on G For any distribution m on B , each limit point n p1 * of {! E m , n ? 1) is a p-invariant distribution on B .
n i=1
Proof
:
n Let
V
n
=
E
-n i=1
}11 * m
.
Since
B
is separable and compact
there exists a weakly convergent subsequence of
(vn )
and its limit,
11
say
v
,
is a probability measure. For any 1
*V n
}1
n
i+1 *
n
m
n i=1 n
lal*m+n{un+1*m-u*m}
nE i=1 Vn
+
1
{un+1
n so that, letting
n -> m
along a subsequence,
p * V = V
1.4. The Theorem of Furstenberg and Kesten
We can now prove the following result of Let
THEOREM 4.1.
1E(Log+IlY1II) < -
[25I
be i.i.d. matrices in
Y1,Y2,...
G1(d,1R). If
then with probability one
lim
Log
II
=Y
n-
Fix an integer m
Proof
p 2;0
and for every integer
n , write n = pm+q,
0 S q < m. We have
,
II < n Log 1 n Log flYn...YJ
IIYn...Ypm+1 II +
n
n
E
i =pm+1
Log IIYpm...Y1 II p-1
Log I!Yill + n
E
j =o
Log IlY(j+1)m...Yjm+1 II
Therefore by the usual law of large numbers 1711" 1
Log IIYn...Y1li
a. s.
m 1E(Log IIYm...Y1Ii)
n lim
and
Log IIYn...Y1 II < lim m.,m IE(Log 1IYm...Y1 II) = Y
(2)
n This proves the theorem if Suppose now that
y
y = -- .
is finite. Let
B
be the compact space of all
d x d
matrices whose norm is equal to one. If for
M
B
in
Y
in
G1(d,IR) and
we set
YM IIYMII
B
becomes a
G1(d,1R)-space. Let m
be the probability measure on
B
12
equal to the Dirac mass at the identity matrix distribution of on
n
Y1. For any integer
Id
and
p
the
consider the distribution Vn
defined by
B
n V
If
n
=
*m
p
E
n
i=1
is a convergent subsequence of
Vn(i)
(Vn)n>1
its limit, say
v,
p-invariant distribution (see lemma 3.5).
is a
Define
a
:
Gl(d,IR) x B -> 1R by Y e G1(d,IR) ,
a(Y,M) = Log IIYMII ,
For any Y11 Y2
in Gl(d,1R) and M in
MeB
B
G(Y1Y2,M) = Log IIY1Y2M11
= Log IN IIYZM II11 + Log IIYZMII Y 2M
=
is an additive cocycle.
a
We have and; for
a
a(Y,M) <_ Log IIYII IIMII < Log IIYII = Sup(a,0)
,
a+(Y,M) < Log+ IIYII
We have assumed that IE(Log+IIY1 ) < 00 bounded continuous function on B and
,
I
is thus a
a+(Y,M)d}i(Y)
11
lim JJ a+(Y,M)d}1(Y)dvn(1)(M) _ 1J a+(Y,M)dp(Y)dv(M)
On the other hand, for lim J
a
= Sup(-a,0)
and any
k > 0
inf{k,a (Y,M)}dj1(Y)dvn(.)(M) =f inf{k,a (Y,M)}dp(Y)dv(M)
thus
lim
II a (Y,M)dp(Y)dVn(i)(M)
>_
JJa (Y,M)dp(Y)dv(M)
<
IJ G(Y,M)dp(Y)dv(M)
and J
a(Y,M)du(Y)dVn(i)(M)
.
But JJ
a(Y,M)dp(Y)dvn(M) = n
E
i=1
JJ a(Y,M)dp(Y)d(pi * m)(M)
(3)
13
E
= n
1E
i=1 =
n
IE (Q(Yn...Y1,Id))
E
i=1
= n 1E (Log IIYn...Y1 II) . Therefore i lim II U(Y,M)dp(Y)dVn(i)(M) = Y
and, by (3)
,
f f o(Y,M)dp(Y)dV(M) ? y This proves that
for some f n with
:
is actually in
a
L1(p 0 v). By Proposition 3.5
n ->
U X B -> IR , as
(4)
.
,
Log IIYn()...YI(w)MII = n a(Yn(W)...Y1(w),M) - > f(w,M)
IP 0 V probability one. With (3) we obtain
f(w,M) < lim inf n Log 11Yn (W) . . .YwII < Y 1
n-
and, using (4)
,
y = JJ f(w,M)dIP(w)dv(M) . So
f(w,M) = Y
and almost surely
y < lim inf
n-
Log
II
n
This, together with (2), proves the theorem.
1.5. Exercises
Using the notations of I.1, prove that
Exercise 5.1. 2
(a) IIMII
IIM*
=
II2
= the largest eigenvalue of
(b) If Idet MI = then IIMII ? is orthogonal (i.e. MM = Id.). 1
Exercise 5.2.
Let
for any integers
{an, n
m,p ?
a
m+p
11
1
a m + ap
1
, and
IIMII =
M *M. 1
if and only
M
be a sequence of real numbers such that
14
n = mp + q,
Writing
p
>_
prove that for any m >_
0 < q < m
0,
1
a m
a
lira-nn <
m
n--
Deduce that
a a lim nn = inf min
n
n-
{Yn , n >_ 11
Consider a sequence
be two positive integers.
d1, d2
[26],[351. Let
Exercise 5.3.
of i.i.d. random matrices which can
be written as B
Y
where
(resp. C
6.n
d1 X d
random
n
n
0
C
is in
)
n
n
G1(d1,]R)
matrix. We suppose
2
(resp. Gl(d2'IR)) and
]E (Log+IY 11
)
I
Bn
is a
and denote by
< `°
the upper Lyapunov exponents associated with
Y, Y1, Y2 (C
n
=
(Yn)' (n
)
a. Prove that
Y1 < Y
Y2
_<_ Y
A'
A"
B,,
n
Yn...Y1 =
b. Write
n
and
Cn
0
B"
n
n
Y2n...Yn+1 =
Cn
L 0
I
Prove that a.s. for each
T >
1
and
IIAn II
<_ exp nTy1
, IIA" 11
II Cn II
< exp nTY2
II Bn II
Deduce from this that
n
large enough
< exp nTY1
, II Cn II
<_ exp nTy2
< exp nTY
, II Bn II
< exp nTy
IIY2n" .Y111 5 8 exp{T Sup(y+y1
, y+y2)}
Prove that y = Sup (y1 ,y2). Let
Exercise 5.4.
be the unit sphere in
S
Lebesgue measure on
1Rd and
m
the normalized
gives the element of surface area and
(dm
S
m(S) = 1). a. Let
be a matrix with
Y
m
image of
under the map
d(Ym)(x)
T
=
:
Ym
Idet YI = 1. Show that if
S -> S
is the
T(x) = Y ,
defined by
then
IlYxll
IIY-1x-d
dm (Hint
:
the measure
p(A) = To
(rx)r
1
S
p
A
defined for each Borel set d-1
dr dm(x)
A
of
d IR
by
is a multiple of Lebesgue measure.
15
Hence it is invariant under multiplication by b. Suppose that
Y
that the distribution of Example 2.3). For
Y).
is a random matrix with IYxJI
does not depend on
such
Idet YI = 1
x
in
(see
S
y = IE(Log IYx1I) verify that d(Y-
Y = - d
f J f
Log
dm )(x)dIP(Y)dm(x)
Using Jensen inequality show that if
IP (Y
is orthogonal) <
1
then
Y > 0. Exercise 5.5.
Suppose that
(IE (Log+ IIY11 II ) <
]E(Log IIY1)
(y >
the reverse implications are false.
is finite. Show that
(]E (Log IIY1 II) > --)
but that
CHAPTER II
MATRICES OF ORDER TWO
We present here the main arguments we shall use for studying products of random matrices of arbitrary order, but in the case of 2x 2 matrices. Our main interest is not the theorems in themselves and actually shorter proofs are available when dealing with
2 x2
matrices.
We rather intend to explicit in this simple situation the general approach, valid for matrices of any order. This chapter is thus introductive. Although all the results are particular cases of more general statements to be proved later, we give them in full detail for the convenience of the reader who is only interested in matrices of order 2.
II.1. The Set-up
Consider a sequence
{Yn, n ? 1}
of i.i.d. invertible random
Sn = Yn ... Y2 Y1. As explained in the introduction
matrices and set
our first task is to show that under broad conditions, for any non zero x and y , the random vectors S x and n up" in the same direction exponentially fast.
vectors
We say that two non zero vectors direction if for some relation
r
on
d I R
x, y
in ]Rd
Sny
tend to "line
have the same
A E IR, x = ay. This defines an equivalence { 0 }
0}. Formally,
17
18
IRd-{O}/r
P(IRd) defined as the quotient space
For For
If
denotes its direction (i.e. its class in
x
x E IRd-{0},
M
1Rd is the projective space
The set of directions in
DEFINITION 1.1.
P(lRd)).
G1(d,IR) we set Mx
in
and
x
are unit vectors in
y
]Rd
we define
<x,y>2}1/2
6(x,Y)
It is the modulus of the sine of the angle between
x
and
x = (x1) x2
In this case, if
d = 2
on we suppose that
y
and
From now
.
y = (Y1) Y2
1x1y2 - x2yll (1)
6(X,y) =
Ilxll
IIYII
It is easily seen that this defines a distance converges to
to prove that
Note that if we replace each
n
Yi
0
by
on
P(1R2) and we want
exponentially fast as Idet
Yil-1/2
Yi,
is
unchanged, we can thus suppose without loss of generality that Idet Yil =
1
.
In that case, by (1)
-
-
,
IIxiiIIYII
6(Sn x, Sn Y) =
6(x,y)
(2)
IISnx1IIISnYII so we are looking for conditions ensuring that exponentially fast, i.e. essentially that
IISnxll
y > 0 , y being the upper
Lyapunov exponent.
This is the content of the now so-called "Furstenberg's theorem". The original proof of Furstenberg in
[21]
has important consequences
and we strongly recommend the luminous account of it in Ledrappier [46]. But for to
d > 2
it doesn't help to prove that
0. So we rather follow Guivarc'h
route is longer than
[21]
converges
6(S
and Raugi [34]. For
d = 2
this
but it is worth seeing it first in this
simple situation. If we spend some time here we shall see that the general case
(d > 2)
will be just a careful copy of this one.
Let us outline the general idea of the proof in [34]. We need the following.
NOTATION 1.2.
If M is in
P(]Rd) we denote by
Mm
Gl(d,]R) and m is a distribution on
the probability measure on
P(]Rd) defined by
19 /
f d(Mm) =
dm(x)
1
f for any bounded Borel function
y > 0
If
on
f
P(IRd).
P(IR2), then for almost all
2
P(IR ),i.e. w
m({-x)) = 0
for any
d_
(recall that
n
Sn). Conversely, if this holds for a continuous
_
distribution and if the distribution of
Z
is continuous, a simple
deterministic result (see Proposition 3.1) shows that, for any n
is
in
is the
S
Z(a) transpose of
x
the sequence
,
converges weakly to a Dirac measure
IIS
m
geometric considerations (cf. section 7) show that if
a continuous distribution on
xll > - with probability one, which in turn implies
x # 0
y > 0.
So we are led to see if, for some continuous distributions
m ,
t
S m n
converges to a Dirac measure. The trick is that if we choose m as
an invariant distribution under the common law of the
Y
's
,
i
then
S m n
always converges weakly. If it happens that there is such a m which is continuous we will prove using the path indicated above that either y > 0
or all the
the sequence
are isometries. When m
Yn's
is not continuous then
has particularly simple properties and the Lyapunov
(Yn)
exponent, in this case, can be explicitely computed.
11.2. Two basic lemmas
We use the notations of 1.3
.
The following lemma due to Guivarc'h
and Raugi [34] improves a result of Furstenberg [21].
LEMMA 2.1.
Let
be a topological semigroup acting on a 2'
G
locally compact space
B
.
Consider a sequence
independent random elements of defined on
(0, A, IP). If v
then for almost all
w
G
countable
of
{Xn, n >_ 1)
with a common distribution
p
is a p-invariant distribution on
there exists a probability measure
such that the sequence (X1(w)X2(w) ... X(w)g v ,
converges weakly to respect to
A = E" n=o
vw
as
2-n-lun
n ->
n = 1)
for almost all
B
vw on
g e G with
B
20
Moreover for each bounded BoreZ function
f
f dv}
.
J f dv Proof F :
Let
f
=
1E {J
on
B
be a continuous bounded Borel real function on
B and
G -> IR be defined by F(g) = J
dv(x)
g e G
,
.
B
Set
Mn = X1X2...Xn.
If
Fn
is the 0-algebra generated by
X11 ...,Xn
then
F 1E n {F (Mn+1) } = J f (Mng) dp (g) =
du(g) dv(x)
JI
= J
-x) dV (x)
f (M
F(M ) n (we have used the independence of the
u-invariant). This shows that bounded,
F(Mn)
X.'s
and the fact that
(F(Mn), n= 1}
converges a.s. to some
1E {rf } = IF {F (M1) } = We have for any positive integers
Ff ( I
f dv
.
k, r,
{F(Mk":.) 2) + 1E {F(Mk)2}
= 1E
(F(Mkr)2} - IE {F(Mk)2)
E
IE { IF(Mk+r)-F(Mk) I2) < S
p
1E {F(Mk)2} +
E
2r Sup IF(x) I2
This yields E
1E{( IF(Mkg) - F(Mk)I2 da(g)} _
E
k=1
x
k=1
- 21E {F(Mk+r)F(Mk)}
k=1
k=1
is
and
1E { IF(Mk+r)-F(Mk) I2} = IE
Hence, by cancellation, for any
V
is a martingale. Being
.
lE {F( k+r)2}
21
E
E
2-r-1
E
E
IF(Mkg) - F(Mk)I2 dur(g)} _
1E{I
k=1 r=o
J
g-r
r-1 1E{IF(rrk+r)
2
- F(Mk)I2}
k=1 r=o So
r=o
is finite and
IF(Mkg) - F(Mk)I2
E
Sup IF(x)I2 x
r
E
F(Mkg)
converges to
rf
k=1 almost surely.
1P ® A
Choose a dense sequence
continuous functions on
A
measurable subset
of
in the space
{fq, q -> 1}
C0(B)
of
which vanish at infinity. There exists a
B
with
S2 x B
OP ® A)(A) = 1
such that for
(w,g) rA, dv(x) -> rfq(w)
.
J If
vW
is a limit point of
{Mn(w)g v ,
for the weak topology,
n >_ 1}
g
f fq dvw,g = Ffq(w)
This shows that
V
w ,g
does not depend on
We therefore denote it by
Mn(W)g V .
Vq >_
,
.
and is the limit of
g
Vw .
1
Since
1 fq dv = lE{rfq} for any bounded Borel
f
f dV
=
1E {J
f dv
1
and
Vw must be a.s. a probability measure.
For proving that Lyapunov exponents (or difference of Lyapunov exponents) are positive we shall need the following.
LEMMA 2.2.
Let
on
G x B
.
be a
B
nrydom elements of
G
G-space ,
a sequence of independent
(Yn)n =1
with distribution u and
Suppose that
v
is a
a
an additive cocycZe
W-invariant distribution on
B
that
IJ 0+(g,x) dp(g) dv(x) < - .
(i)
(ii)
For
1P ® v -almost all
(w,x)
,
I'm o(Yn(w)...Y1(w),x)
n
Then
a
is in
L1(1P 3 v)
and
(r IJ a(g,x) du(g) dv(x) > 0
such
22
Using the arguments developed in the proof of Proposition 1.3.4 this follows at once from the following lemma. Its proof is borrowed
from Dekking [16]. LEMMA 2.3.
Let
be a probability space and
(E, F, A)
a measurable transformation which preserves so that
n i=1
E
)f dA > 0
and
f E L1(da)
Proof
Jim
n-o
f+ dA < W and
J
f
A
If
.
0
:
E -> E
: E -> IR is
almost everywhere then
01 = oo
,
f
.
By the Ergodic Theorem (see Exercise 2.6) n
limn where J = {A E F
;
f
E
n-)oo
a.s.
01 = IEJ(f)
o
i=1
n
A(A A 0-1A) = 0}
Since
.
fo01
E
i=1 E`J(f)
so
>= 0
f
is actually integrable.
If f f dA = 0 then n n'
o 01 -> 0
f
E
a.s.
(3)
.
i=1 n
For each
c > 0
IE(t) = Et-, t+E]
let
,
Sn(x) =
E
(f
o 01)(x)
and,
i=1
for
m
being the Lebesgue measure on
1R,
n
IE(Si(x))}
R6(x) = m{ U i=1
By (3), for A-almost all such that, for
k > no
Rt(x) < Rn (x) + 2nd
and any
x
ISk(x)I
,
and
n Rn(x) -> 0 a.s.
.
there exists
Hence, for all
RR (x) < 26
lira
naao
O
6 > 0
<= kd
for all
no = no(x,d)
n 6 > 0
.
Therefore
n
and
limn 1E (Rn) = 0 n-_ Note that, since
Sn o 0 = Sn+1
- S1
n+1 Rn+1 - Rn o
0
= m{ U
=
(4)
n I(Si)} E - m{ U I(Si+1-Si)} E
i=1
i=1
n+1
n+1
m{ U i=1
I
E
(S
.)} - m{ U
1
i=2
I (S.)} E
1
It is plain that
Rn+1 - Rn o 0 ? 2E
1
{ISi-S1I>2E, i = 2,...,n+1}
23
so
lE{Rn+1}- ]R{Rn o6} B 2e A{x ; ISi(x)-S(x)I > 2E,
and, since
preserves
6
i = 2,...,n+1}
A
1E{Rn+1} - 1E{Rn} B 2E
IS.(x)I > 2e, i= 1,...,n}
A{x ;
This implies that limn IE {Rn} B 2e A{x
;
> 2E
ISi(x) I
Vi >_
,
1}
nBy (4), for any
E > 0
IS.(x)I > e
A{x ;
The processes
,
{S>_.,
i
and
11
>_
{S
A{x
i
i+P-S
distribution. Therefore for any integer
0
=
Vi >_ 1}
,
1}
>_
F' p > 0
have the same
and
ISi+p(x) - Sp(x)I > E
,
Vi ->-
ISi+p(x) - SP(x)I > e
,
Vi >=
c > 0 ,
1} = 0
'
A{x ;
and
This contradicts the fact that
= m
lim S
1
for some
p} = 0
a.s.
in
Exercise 2.4.
group and
that if m gm = m
T
be a compact separable semigroup contained in a
is a distribution on
for any
contains T
Let
a probability measure on
11
in
g
T
H
If
gm = m
show that
T
.
whose support is
T
such that
T
T . Prove
11 * m = m , then
is the smallest closed group which
for any
g
in
H. Deduce from this that
is actually a group.
Exercise 2.5. 1
x
0
1
Yi
Let
i
{Yn, n >_ 1}
be i.i.d. matrices of the form
Prove that one can have
.
lim
IIYn...Y111
a.s.
limn Log IIYn
but
=0
.Y1 II
m>_
Exercise 2.6. With the notations of Lemma 2.3 we want to prove that n f+ dX < under the sole condition f o 61 = Ej(f) lim E
n
J
i=1 (i) For any K < 0
EK = {x
let
;
1E`T(f)(x)
> K}
.
Using that
n
EK e J prove that
lim n E i=1
(ii) Show that on {x ;
f
o
6' = lEJ(f) on
lE`T(f) (x)
EK
n
limn E f i=1
o
0i
24
(iii) Conclude.
11.3. Contraction properties
Consider a sequence order 2 with
of i.i.d. random matrices of
Y1,Y2....
= 1. We intend to show that under broad
Idet Y.
on zero
conditions, for any
x
IR2, IIYn...Ylxll -> W with
in
n -> - . The following deterministic result makes
probability one as
it clear that we have to look at the action of
Y1...Yn
on continuous
distributions on POR2). PROPOSITION 3.1.
with
Idet A n
Let
= 1
I
{An, n >- 1)
S
z
.
2 x 2
matrices
Suppose that there exists a continuous
.
distribution m on measure
be a sequence of
P(1R2) such that
A m converges weakly to a Dirac n
Then
lim IIAnII = lim IIAnII = and if
x
in
is a unit vector in
z
(5)
n-)
n->-
1R2 with direction
z
, then for any
IR2
IIAn4II lim n-wo
We first prove LEMMA 3.2.
A
Let
of order 2 and
= I<x,z>l
(6)
I I An *11
:
be a non zero (non necessarily invertible) matrix
m
be a continuous distribution on
P(1R2). Then the
equation f d(Am) =
dm(x)
I
J
valid for all bounded BoreZ functions, defines a probability measure Am
on
If Anm
P(IR2). (An)
is a sequence of non zero matrices which converges to
converges weakly to
Proof of the lemma is well defined. Since
Am
.
For any A # 0
x
in
1R2 such that
Ax # 0, Ax
there exists at most one direction
y
A,
25
such that if
is a vector with this direction,
y
is defined for all
except
x
continuous and therefore
Since
Ay = 0. Hence
m({y}) = 0
but
m
since
is
A -> A then for
can be defined. If
Am
outside a countable set
x
all
y
n , An x exists and converges to
D
m(D) = 0 , Anm -> Am.
Proof of the proposition
to some matrix A
Suppose first that
By the lemma (note that
.
converges
II-1 A
IIAn
n
IIAII = 1
A # 0)
so
,
Am = d
z
If
det A # 0
m
,
continuous. Hence
A
is the Dirac measure at
and is not
and
det A = 0
det An
0 = Idet Al = lim
1
I = lim i
n-,°°
II An II 2
11 An 112
proving (5).
A
Moreover the range of direction of this line is
z
canonical basis of IR2, Ae1 For any
x
is just a line and since Thus if
.
is the
Ae2 = ± IIAe2IIz
= ± IIAe1 llz and
,
IA*xII2 =
= <x (
Using
Am = d_ the
{e1,e2}
and
1
IIzII =
IIAII =
1
one obtains *
lim
,
e1>2 +
,
+ <x
Ae1>2
1A
Ae2>2
,
and
IIAe1 II2 + IIAe2II2 = 1 2
<x,z>2
= 1
e2>2
IIAe1 II2 + IIAe2II2) <x,z>2
IIAn*112 II=IA*xII2
n-
,
n
This is true for any convergent subsequence of
IIA
n II-1 An
,
so we have
proved the proposition.
*
So we are led to show that
*
Y1...Ynm
converges to a Dirac measure
at least for one continuous distribution m and let (i.e.
p 11
be the distribution of
* V = J MV d11(M) = v)
converges by Lemma 2.1
.
X1
.
If
on v
P(IR2). Put is
we already know that
Actually we have
X. = Y.
p-invariant X1...Xn V
26
PROPOSITION 3.3.
on
X1,X2,...
Let
be independent
and the same distribution
Idet XiI = 1
matrices with
2 x 2
Suppose that there exists
.
p-invariant distribution
P(1R2) a continuous
support of p
p
v .
Then, if the
is not contained in a compact subgroup of G1(2,]R),
there exists with probability one a direction X1(w)...Xn(w)v
converges weakly to
such that
Z(w)
dZ(w)
Moreover the distribution of Z is v and is the unique p-invariant continuous distribution on
Proof
P(]R2).
An = X1...Xn. From Lemma 2.1 we know that there exist
Let
Vw
probability measures
such that, for almost all
w
,
An(w)V -!L-> Vw w
A(w)MV
and
> vw
for
p-almost all
Fix such an w . For any limit point
M of
A(w)
IIA
n
(w)II-1 A (w) we have,
n
by Lemma 3.2
A(w)V = A(w)MV = vw for
M
p-almost all Notice that
H = {M E Gl(2,]R) ; MV = V
IIMnII -> W
case
and
CV = v
H
is a compact
such that C
.
In that
and
n1I)I Mn
=limJIM
v = Cv
sends all ]R2 onto a line and
C
in
converges to some matrix
IIMnII-1 Mn
IdetC( =lim Idet(JIM so
Idet MI = 1}
,
Mn
subgroup. For otherwise there would exist
1II2=0
n
must be a Dirac measure,
which is absurd.
is invertible then
A(w)
If
p(H) =
1
MV = v
for
which contradicts our assumption. So
matrix. If Vw = 6Z(C)
This proves that
Note that
A(w)
J
the distribution of
v
.
Since by Lemma 2.1
P(]R2)
on
f do = 1E { I f dvw} Z
is
and
A(w)v = vw
X1(w)...Xn(w)v W > 6Z(w)
does not depend on
Z
Borel function f
M
is a rank-one
is the direction of its range, then since
Z(w) .
p-almost all
V
and
v
]E {f (Z) } is unique.
for any
27
We do not say that there exists a unique 11-invariant
Remark 3.4.
P(1R2), but only that there is at most one such
distribution on
distribution which is continuous. Remark 3.5.
is contained in a compact group, the result
Supp(p)
If
does not hold. We have seen in this case that
M E Supp(p)
X1...XnV = V
so
My = V
for each
.
We now prove the following basic result
THEOREM 3.6.
Let
be a sequence of independent random
1}
{Yn , n 2-!
matrices of order 2 with the same distribution Suppose that
Idet Yij = 1
(a)
u
:
a.s.
(b) The support of
is not contained in a compact subgroup of
pi
G1(2,1R) . 11 -invariant distribution
(c) There exists a continuous P(1R2) where
pi*
Then (i) For each
is the distribution of in
x, y
on
Y1
P(1R2), with probability one Yn...Y1.y) = 0
lim
.
n(ii) If
x#0
, there exists
]E(Log+IIY1Ij) <
y > 0
lim a Log IIYn...Y1x1I = lim n Log nn-_ and if
v
is a p-invariant distribution on
(( Y= Proof
Set
distribution on set
such that for each
,
with
00
II
II
a.s.
=y
P(1R2)
dp(M)dv(x)
x
Sn = Yn...Y1
.
If
m
is a continuous
u -invariant
there exists by Proposition 3.3 a measurable
P(1R2) 1P (S10)
IIMxII
Log )
II
=
1
and a random direction
such that, for
wEl20 , *
lim Sw)m = By Proposition 3.1 if
x E 1R 2
,
then for all
Z
dZ(w)
is a unit vector with direction
w c 920
Z
and
28
lira IS.(W)
(7)
n-
'IS()xIl n
lim n>°°
and
For each fixed
= II
(8)
Sn(W) ll is the direction orthogonal to
u
x # 0, if
x
IP((w; = 0)) = 1P((w; Z(W) = u}) =m({u}) = 0 since
m
is continuous.
x # 0 , by (7) and (8)
So for each fixed
lira IIS.(w)xII
n-
ts n (W)xll
lim
and
+
a. s.
(9)
# 0
a.s.
(10)
n- Sn (W) II We thus obtain by (2), for
x,y c IR2-{0}
IIxII IIxII a(x.Y) lim 5(S n-wo
S
n
lim
n
n-5
= 0
a.s.
,
IISn(W)xII IISn(W)yII
Suppose now that IE (Log IIY1 II) < °° . (Note that Log IIY1 II >= 0 since Idet Y1I = 1 , see Exercise 1.5.1). We define an additive cocycle a on Gl(2,1R) x P(IR2) by IlYxll
G(Y,x) = Log
,
Y C Gl(2,1R) ,
x e IIt2-{0}
IIxII If
v
is a µ-invariant distribution on
P(1R2), then by Fubini's
theorem and (9) 1P® v-a.s.
lim a(Sn(w),x)
nSince
f a+(Y,x)dp(Y)dv(x)
Lemma 2.2
0
is in
<-
J Log IIYII dp(Y) < - we can apply
L1(IP 0 v) and if IIYX II
y=
J
Log
II
y
is defined as
d (Y) d\) (x)
x II
then y > 0. Moreover, by Proposition 1.3.4, for some
(11)
limn Q(S.(w),x)
limn Log IIS.(w) x II = IIxII
((
: 0 x IP(1R2)->lR
IP 0 v - a.s. with ff 0(w,x)d1P(w)dv(x) = y
_
(w,x)
29
Fix an
such that (11) holds for almost all w. By (10)
x
1
lim In Log IISn(w)II = lim n Log IISn()XII = (w.x)
nSo
(w,x)
is the upper Lyapunov exponent
directly that
(one can also remark
y
0-algebra of the
depends only on the tail
1(w,x)
sequence of the independent random variables
Y1,Y2,...
and hence is
constant).
y # 0 , by (10)
Finally, for any fixed
lim n Log ISn(w)YII = lim
a.s.
Log IIsn(w) II = y n
proving the theorem.
One must note that the almost sure convergence statement in (i),
resp. (ii), is only valid for
x, y, resp. x, fixed. For instance, as
one can guess from the proof,
lim n Log IIYn(w)...Y1(w)xjj
to
y
x
if
is in the direction orthogonal to
is not equal
Z(w). The content of
Osseledec's theorem which we will prove in Chapter VI is that this limit is actually
-y
Let us show this on a typical, although
.
degenerate, example.
Example 3.7.
Suppose that
is. a sequence of i.i.d. bounded
(b.).
non constant random variables and set for 2
Yi
i ? 1
b1 .
=
0
1
n If
an =
2-1bi
E
and
Sn = Yn...Yt
then
i=1
2n[20
Sn = 2 -n/2 O
For
a. =
E
i=1
2-1b.
we have
1
2o
[bo Since
a
1
0 1
=
J
bi
[I
1
a l
2bo
has the same distribution as 2 bo +
the distribution of the direction of Since
__
a-
this shows that
(a ) is 11*-invariant. 4
is not constant it is easy to see that this latter
distribution is continuous. So we can apply the theorem. If we choose
30 -ate
x =
y = 2 Log 2
we find
.
But for
x =
( 1
Snx = 2 -n/2 and 11S
x112- Zn(1 + { E i=1
n
L2n(an_ao)1
bn+1}2
So
2i
2-n
<- 2-n SupIb.I <
11Snx112
lim 1 Log IISnxl] = - 2 Log 2 = -y
and
11.4. Furstenberg's theorem
We will state this important result in several equivalent forms.
(Furstenberg's theorem, first form)
THEOREM 4.1. Let
be a probability measure on
p
smallest closed subgroup of
G1(2,]R) such that if
G11
is the
Gl(2,IR) which contains the support of u,
the following hold :
(i) For M in (ii) GP
GP ,
Idet MI = 1
is not compact.
(iii) There does not exist a subset
L
of
1R2 which is a finite
M(L) = L
union of one-dimensional subspaces such that
for any
M in
G u
Then the conclusions of Theorem 3.6 are satisfied for matrices with
distribution
p
distribution
v
.
Moreover there exists a unique p-invariant on
P(1R2) and
v
is continuous.
This is an immediate consequence of Proposition 3.3, Theorem 3.6 and the following lemma, where
p
denotes the image of
p
under the
transpose map.
LEMMA 4.2.
p
Let
be a probability distribution on
G1(2,]R) for
which (iii) of Theorem 4.1 holds. Then each p-invariant distribution and each p *-invariant distribution on Proof
a > 0
Let ,
v
P(1R
2 )
are continuous.
be a 11-invariant distribution on
(x c P(1R); 2 v({x}) > a}
is finite. Thus if
P(1R2).For each v
is not
31
continuous there exists
such that
> 0
(i
v({-}) < (i for any x e P(]R2) F = {x E P(1R2); V({xl) _ (3} If
x
0
E F
and since
( = V({xo}) =
V=
p
rr
1{x
IJ
V({M
I
1
0
JJJ
and M
b
so
v({M
This shows that
GP
is contained in the closed subgroup
V({M
But
is finite and non empty.
(S
E F
for
11
almost all M.
H = {M E Gl(2,1R); If
L = {0} U {x E 1R2-{0}
dimensional subspaces of
;
for any
E F
x E F} , L
1R2 and
x E F}
is a finite union of one-
for any M
M(L) = L
in
in
,
G11
contradiction with the hypothesis. Therefore
is continuous.
v
To prove that each p -invariant distribution on continuous we verify that
P(]R2)
is
satisfies (iii) too. Suppose there exist
G * u
one-dimensional subspaces V11V2,...,Vr
such that for each i E{1,...,r}
and M in G * u M(Vi) = V. for some
j
If
E
Wi
(12)
is the orthogonal of
Vi
,
(12) gives
M (W.) = W. *
Therefore for each M E G1, _ {M (iii) doesn't hold for
J
1
u G
i=1
One can in fact weaken the hypothesis (iii) using
PROPOSITION 4.3.
If u
r
r
; M e G *} , M(U Wi) = U
and
W.
i=1
:
satisfies (i) and (ii) of Theorem 4.1, (iii)
is equivalent to (iii)' For any
x in
M E Gu}
P(]R2),
has more than two
elements.
Proof
(iii) _> (iii)'
suffices to verify that if
is obvious. To prove the converse it x1,...,xr
are distinct points of
P(1R2)
with {x1 ,. ..,xr}
,
for each M e Gu
32
M
r 5 2. Each
then
{x1,...,xr}
H = {M E GV
i.e.
subgroup of
of
induces a permutation (M)
GP
is a group homomorphism. The kernel
and
x.
;
and
GP
Gu/H
,
Vi _=
x1, x2, X3
H
is finite. By (ii)
consider three non zero vectors
r > 3
of
c
,
is a closed normal
...,r}
1
of
H
is not finite. If
x1, x2, x3
with directions
We can write
.
with a# 0
x3 = ax1 + x2 M
and for each
in
H
for some
,
(i # 0
,
,
0
A.
i = 1, 2, 3
Mx. = A.x.
.
This yields
A3ax1 + A313x2 = a3x3 = Mx3 = aMx1 + (MMx2 = ax1x1 + W, and
aX
= aa1, a3 =
3
Idet MI =
and each M
1
finite, so
r
<_
2
of
H must be
2X2
and M = A1Id. By (i),
So Al = a2 = A3
2
1
Id
-Id
or
.
But
H
is not
.
In some applications it is more convenient to use the following equivalent form
THEOREM 4.4.
(Furstenberg's theorem, second form)
be a sequence of independent invertible matrices with
Y11Y2'...
Let
:
the same distribution P(]R2) such that
support of p
.
p .
Suppose there is no distribution m on
mm = m for each M in the group generated by the Then if
Idet Y1I = 1
Theorem 4.1 are satisfied. In particular p-invariant distribution
unique
Proof
:
the hypotheses of
a.s.
y > 0
and there exists a
v .
To check (ii) of Theorem 4.1 one can either use Remark 4.5 or,
more simply, notice that if
then for all
x
GP
in P (]R2) , and
is compact and M
in
MU *6X) =(6M*a)
A
is its Haar measure,
G1
6x=A*6X
This contradicts the hypothesis.
Suppose that (iii) doesn't hold. We then could find in
P(]R2)
such that, for any M
in
,xr} _ {x1>
Gu ,xr}
{xi.,x2,...,xr}
33 1
M(-
But this would imply that
r
r E
i=1
d_
x.
r
1
)
=
i
r
d
E
i=1
x.
, which is absurd.
i
11.5. Some simple examples
Given a sequence
{Yn, n >>-
1)
Gl(2,]R) with the same distribution
of independent random matrices in p
satisfying Furstenberg's
theorem one would like to compute the upper Lyapunov exponent or the p-invariant distribution on classical law of large numbers with respect to
y
y
and/
P(1R2). Unlike in the case of the cannot be written as an integral
only. Therefore this computation is most often
p
unfeasible.
y
we have described one of the only known models where
1.2
In
has a simple expression. We give here two related examples due to Furstenberg, valid in any dimension and a third one which appears in the study of Schrodinger operators.
We shall see more intricate examples in Chapter VI.
5.1. Rotation invariant distributions
Let SO(d) be the group of orthogonal
determinant one and mK mK(SO(d)) = 1.
measure on V
on
For any
d x d
the Haar measure on x
in
which satisfies
P(1Rd), and = mK * dx
P(7Rd) which does not depend on z
P(1Rd)
matrices with
SO(d)
.
is a probability
For any distribution
,
mK * V = J MK * dX d) (x) = J and dv(x) = md and
m
d
is the unique distribution on
P(1Rd) with satisfies for any
Mmd = and
It is clear that
(13)
M e SO(d)
.
is actually the image of the probability measure
and
on the unit sphere which gives the element of surface area under the canonical projection onto
P(1Rd).
As in Exercise 1.5.4 one proves that if
d(Ym dm
Let
d (x) =
IIYx11-d
x
is a unit vector
for each Y s.t. Idet YI =1.
Y1,Y2'... be independent matrices in
Gl(d,1R) with
34
distribution
Then
p
Suppose that the distribution of
.
(in other words
k c SO(d)
for all
m
d
k e SO(d)
ak * P * d
is a p-invariant distribution on
k-1
P(]Rd)
is the same
kYIk 1 = p
, Vk e SO(d)).
(Proof
:
for any
,
d)=(ak*p*a
U *and =P * (ak so, by (13)
k-1*ak)*md-(ak*u)*and
,
u *and = J (dk *p) *mddmK(k)=(mK*p)*md =mK * (11 *md) = md) In this case the distribution of
IIY1xll
-
does not depend on
x
on
the unit sphere. So this example is a particular case of the one introduced in 1.2.3
.
An other example where the invariant distribution can be calculated is the following. Suppose that there are two probability
measures a1
and
other words each
Gl (d, IR)
such that
A2
on
Y
can be written as
.
1
11 = X 1 * mK * A2 ,
A K B
1 1 1
independent matrices and the distribution of on
dSO(d)). Then
Al
where
.
K.
A1.,K1.,B
(in .
1
is the Haar measure
is the unique p-invariant distribution d on
* and
P(IR ),because for any probability measure
V
on
P(IR ), by (13)
(A1 *mK*A2) *v =]`1 * (MK * (a2 *v)) =A1 So if
V
is p-invariant
v = Al
* and
and this distribution is
invariant. We thus have Y _
IIYX II
( - IJ
Log
I
dp(Y) d(A1
IIx
md)(x)
II
_
AKB A'x
(
Log
dA1(A)da2(A') dmK(K)da2(B) dmd(x)
IIA'xll
Writing
Log
IIAKB A'xll = Log IIAKB A'xll
one finds
- Log IIA'xll + Log IIB A'xll
IIB A'xll
IIA'xll
are
IIB Axll ( d), 2(B) dA1(A) dmd(x) y = J Log
llxll One can also check this directly by noticing that exponent associated with the distribution of
K2 B2 A 1
is the
y K1
.
,
35
5.2. The Cauchy distribution
Define a bijective map
Tr
:
as follows
P(1R2) -> IR U {co} x
Xi
x1
x = (
if
x2
is non zero then
)
7r(x) =
L°° Gl(2,IR) on
One can transport the action of
for t E IR u {-}
1 (t)}
.
if
x2 # 0
if
x2 =0
2
P(IR2) by setting
If g = (a
b)
at+b ct+d
(14)
This is just a concrete realization of
DEFINITION 5.2.
The space
P(1R2).
endowed with. the action of
IR U {W}
G1(2,IR) defined by (14) will be caZZed the projective Zine.
The image under
of the probability measure m2
Tr
is the image of the Lebesgue measure on the circle 1
defined in 5.1 , under
?Tr 1[0 21 d0
Thus it is the Cauchy distribution, i.e. the probability
.
tan B measure on
IR defined by the density
1
Tr(1+t2) For
in
z = a + i(S
distribution of
C
with
where
(SU + a
U
(i
for the
Cz
> 0 , write
is a Cauchy-distributed random
variable. Using the fact that each matrix
g
Gl(2,IR) can be
in
written
`a
b
0
a
one sees that
In z > 0
,
g C
i
with
k
g =
=
and more generally, for any
Consider now the random matrix Y = (X variable with distribution
x E IR , z' E C x (1
So if
U
in
z
T
with
if g' i = z g Cz = gg' Ci = C gg ',i = C g.z
any
k E 0(2)
1
-1
Cz. Denote by
pz
.
O1)
where
X
is a random
its distribution. For
,
o)CZ, = (0
0 1)
(1
-1
o)CZ, = (0
is a random variable with distribution
1)G-1/Z' Cz,
,
independent of
36
X
,
the distribution of
Cz * C_
is
that is
1/z,
Cz_
1/z'*
We
can thus write *
uz and if
z = z' +
Cz, = C, 1/z' is a
Cz,
It is easily seen that by Theorem 4.1, if
pz invariant distribution on
]R U
satisfies Furstenberg's theorem. Therefore
}1z
is its upper Lyapunov exponent,
y(z)
IIM() II dpz(M) d Cz'(t)
Y(z) if Log I(1)f
2
Log (tx-1) +t
fJ l
if
2
1
2
dCz (x) dCz,(t)
+ t
If Lag{(tx-1)2+1}dc (x) dcz,(t) + + 2 f Log t2 dCz,(t) - 2 J Log(1+t2)dCz,(t)
But since
pz * Cz, = Cz,
,
for any bounded Borel
if f(ttt)dCz(x) dCz,(t) therefore
f
on
JR
f(t)dCz,(t)
Y(z) = f LogltldCz,(t).
By computation or using the fact that upper half plane and that
Cz,
Loglzl
is harmonic in the
is the Poisson kernel one obtains
y(z) = Loglz'I
11.6. Exercises
Remarks : After the exercises (6.4), (6.5) and (6.6)
it should be easy
for the reader to study the Lyapunov exponent of arbitrary distributions on p
*
Gl(2,1R). (6.5) shows that in Theorem 3.6 one cannot replace
11
by
in the hypothesis (c). Exercise (6.6) implies that it does not
suffice to suppose that
GP
does not leave any one-dimensional subspace
invariant to obtain Fustenberg's theorem 4.1
(6.1)
Let
{An, n >_ 11
.
One really needs (iii).
be a sequence of matrices in
that for some continuous distribution m
on
P(]R2), A
Gl(2,]R) such
nm
converges
37
weakly to
d_ . z
Show that this is actually true for all continuous
distributions.
Verify that working with
(6.2)
instead of
P(r12)
P(1R2), all the
results of this chapter are valid for matrices with complex entries (with the same proofs).
(6.3)
Prove that Theorems 4.1 and 4.4 are equivalent.
(6.4)
Let
be a probability measure on
pi
det M = 1}
Denote as usual by
.
Sl(2,IR) = {M E G1(2,IR);
the smallest closed subgroup of
GP
Gl(2,IR) which contains the support of
Prove that one of the
p
following holds is finite.
Gp
(i)
For any
(ii)
(iii)
QG
(iv)
Q GP
some
Q
has more than two points.
P(IR2)
,
a_1)
,
a#0
a-1)
,
a
a
e{(
c
Q-1
{(0
b e IR},for some QtGl(2,IR)
,
# 0} U t(0b
1
b
b # 0}
,
for
.
Let
(6.5)
in
x
Q-1
be independent matrices with distribution
{Yn, n _> 1}
p
of the form b.(w)
a.(M )
Y.(w)
=
1
1
0
a. Prove that if for some Q
1
x e IR, a x + b. = x 1
invertible matrix
a.(w) # 0
,
1
such that each
a.s., there exists an
1
Q Yi
Q-1
is diagonal (the
converse is clear).
b. Suppose that for any
1E(Log 1IY1II) < m (i) Assume
Q
,
Q Y. Q-1
is not a.s. diagonal and that
.
]E(LogIa1l)< 0
Prove that
.
u =
ai...an-ibn
E
n=1
exists with probability 1 and that the distribution is p-invariant. For
of
e1 =
prove that
((1))
v
of the direction
{a de + (1-a)v i
0
_<- a <_
1}
is the set of p-invariant distributions on
(ii) Assume
distribution on
Prove that the only 1i-invariant
IE(Log Ia1I) ? 0
.
P(1R2)
(Hint
is
de1
P(1R2).
:
if not, there would be a
continuous p-invariant distribution and Theorem 3.6 would lead to a
38
contradiction).
(iii) Prove that, on all cases, limn Log (Hint
:
= Sup(O,IE(Loglall))
IIYn...Y1 II
use Exercise 1.5.3). G =
(6.6) Let
01)
{(a 0
a. Prove that {Yn
b. Let
,
b #
,
0}
0
Gl(2,IR).
be i.i.d. matrices in
n 2 1}
1
)
-b
a
is a subgroup of
G
b
0
a # 0} U {(
G with distribution
p
Suppose that
))
±(0
or
IP(Yi = ±I
1
and
is diagonal) # 1
IP(Y1
2
Prove that there is no continuous u-invariant distribution on (use Proposition 3.3). Deduce that
)
is the only
V = 2 (de + de ) 2
1
11-invariant distribution. Show that
P(IIt
lim n Log IIYn...Y1II = 0
(this
latter question is more involved).
11.7. Complements
G = S1(2,IR), the matrices of
Let
H-- {z = x+iy, x e IR, y > 0} g = (a
For
d)
in
G
and
and z
in
Gl(2,IR) with determinant 1
8H = {z = x+iy, x E: IR, y = O} U {oc}. H = H U 8H
we set
az + b cz -+d
With this action H
and
8H
(15)
are G-spaces. Notice that
aH
is the
projective line introduced Definition 5.2.
PROPOSITION 7.1. lim inf
that
Let
(ii) If v
z
in
x
8H
converges to x as
H , A1A2...Anz
in
is a continuous distribution on
converges weakly to
:
be a bounded sequence in
G
such
such that
n
n-
(i) For any
Proof
{A n, n 2 11
Log IIA1...An1I > 0. Then there is
d
x
n + ' .
2H , A1...Anv
.
It is more convenient to work with the unit disk
D = {z e C
,
be defined by
Izl < 11
3D = {z e G
and
T(z) = y+ii
,
Izi = 1}
This map extends to .
3H
.
: H -> D
Let
T
and
T(8H) = 3D
39 -i)
i
and G'=(ab
Consider the map
G -> G'
Let
Q = (1 1
b)
a
a, b e C
:
lal2 - lbl2 = 1}.
,
defined by
geG
$(g) = Q gQ-1 g = (b
If for
a)
in
and
G'
D = D UaD we set
in
z
az + b
g'z = _
bz + a it is clear that
Therefore it suffices to show that if (i)' for some (ii)' if
x e DD
bn
lim inf n log 1IA1...An11 > 0
u
Ibn12
-
, A'l...A'v > 6x
DD
a 1
n
.
lim inf n Log lanl > 0
z e D
Since
and for some
large enough
n
Ia For any
Ian12
with
bn) an
(an 1
and
for any
x
,
is a continuous distribution on
v
Write
a > 0
A' = ip(Ai)
in
n
I
> en a
D
a u+b
b
n _n I
bnu+an
and if u e D
Iul
- -°l =
b
an
(16)
IanI2 Ian u+1I n
, 21
A'...A'.OI <
(17)
Ianl {1-IuI}
is bounded, there exists
Since
(An)
So if
zn = Al
I z
and (15)
n+1
-z
n
=
I A '
1..*
for any v
z
in
D
p <
=
x = lim zn
be a continuous distribution on n
suppose that for some subsequence
ni
1}
.
is in
DD
and by
, proving (i)' and (i).
x
n >_
x
,
-2na 1-p
la I2(1-p) n
lim
point of the sequence
p
< e
1
A'A' 0 - A' 1... n n n+1
,
such that
1
yields
is a Cauchy sequence. If
(zn)
Let
I
(17)
and m a weak limit
aD
b re an
Since
and some
is bounded we can
y
m i
and
aniibn1,
-> y
as
i ->
40
By (16), if
u E DD
and
yu +
# 0 , A..
1
x .
and m = d
\)({-y 1}) = 0, A1'...A' V -> d ni x proves (ii) and (ii)'
Therefore, since
which
x
.
COROLLARY 7.1.
If
are i.i.d. bounded matrices in
Y1,Y2,...
lim inf n
and if
there exists a random
0
Sl(2,7R)
x in
DH such
that, a. s. (i)
(ii)
Y1...Ynz -> x
for any
H
in
x
Y1...YnV -> 8x for any continuous distribution
We bring to the reader's notice that since of proof of Fustenberg's theorem, due to
on
v
aH
aH = P(1R2) , our method
[34], consits simply in trying
to find a converse of this corollary.
Consider independent matrices same distribution such that k t SO(2). Then for any g
g
Y1,Y2....
in
Sl(2,IR) with the
has a distribution independent of
k Y1k 1
Sl(2,IR) the sequence
in
Z0 =
is a Markov chain on the upper
g
half plane. To verify this first note that if we set, for any Borel set A
H
in
and
h
in
Sl(2,1R)
A)
=
(18)
J
then if J
because
h-1h' e SO(2)
.
depends
Therefore
really only on Let
Fn
be the 0-algebra generated by
a-algebra generated by
F IE
Zo,...,Zn
n{1A(Zn+l)}=
.
Y11 ...,Yn
One has for
and
Gn
the
h = g YI...Yn
=P(ZnA)
and
G G F IE n(lA(Zn+l))= IE n{1E n(1A(Zn+1))} = P(Zn'A)
41
So
Z
is a Markov chain with transition probability
n
P
By
.
Proposition 7.1 this chain converges a.s. to some point of the boundary aH
From 5.1 and 5.2 the distribution of this point is
.
CZo
.
An example of such a Markov chain is obtained by choosing, for a
on
mK
g E Sl(2,IR), p = mK * 8g * mK where
fixed
SO(2). Starting from
x E H
{z' E H ; d(z,z') =
circle
is the Haar measure
it goes to a point of the hyperbolic (see Exercise 7.3), this point
Log Ilgll}
being chosen with uniform distribution. Remark that if
g,h
is defined by (18), then for
P
in
Sl(2,IR),
P(gh.i,g.A) = f
So for any
in
z
H
,
P(z,A)
Conversely consider a transition probability (19). Define a distribution in
Sl(2,IR) and
B =
.
k, k' E SO(2)
p(kBk') = P(i, so
is a Borel set
B
g E B}
p(B) = P(i,B) If
H which satisfies
on
P
S1(2,IR) by, if
on
p
(19)
.
Ek * 11 * Ek, = u
P(i,B) = p(B)
and it is easily seen that if
i.i.d. matrices with distribution
P
chain with transition probability
P
,
.
Since
hypothesis required in example 1.2.3, if
lim
are
is a Markov
satisfies the
p
and
Zo = i
llxll = 1
Log IlYxlldp(Y) = IE {d(i,Z1)}
d(z,Zn) = Y =
n
{Yn, n >_ 11
Zn =
J
11
n where
d
is the hyperbolic metric introduced in Exercise 7.3.
For instance let
Pt
be the transition semigroup with 2
2
on H i.e. the LaplaceA = 2 y2 {a a2 + a } x ay Beltrami operator associated with the hyperbolic metric. Since each g
infinitesimal generator
of
Sl(2,IR) acts on
Pt
satisfies (19). It follows from the preceding results that if
H
as an isometry
A
commutes with
is the associated Markov process starting from
i
,
g
lim X t
t
and each Xt
Xexists ao
t-)m
with probability one and is in is obvious
:
the operator
A
DH = IR. But in that case this result is a scalar multiple of the ordinary
42
Laplacian
2 {a2Z +
a22}
1
ax
with
Bo = i
is the complex Brownian motion
Bt
for some clock
Xt = BT
,
So if
.
ay
T = inf {t > 0
Bt E a H}
z =
and
So
X. = BT where
.
a. Prove that
Exercise 7.3.
b. If
.
Tt
t
z'
i} = SO(2).
G ;
with
=
d(z',z) = Log Jig Prove that this defines a distance on
g,g' E Sl(2,IR)
1g, H
set
IrI
(the so-called hyperbolic
distance). c. Let
d(zn,zn+1) in
aH
with
Exercise 7.4.
{z n, n >_
C
lim zn = x
n-
Let
IE (Log IIY1Ib< w . IIYn(w)II < nr
1}
for
be a sequence in H
lim inf n d(i,zn) # 0
and
(Hint : write
Y1,Y2,...
such that for some
with
zn =
r > 0
C
Prove that there exists
be i.i.d. matrices in
Prove that for any n
.
G
An E G).
with
and almost all
w
large enough. Deduce the analogue of Corollary
7.2 under these hypotheses.
x
CHAPTER III
CONTRACTION PROPERTIES
Y1,Y2,...
We now consider i.i.d. invertible random matrices arbitrary order, say
d
.
of
This chapter is devoted to the study of the
basic almost sure properties of the products
Sn = Yn...Y1
. We shall
derive their salient feature, which is the contracting action of S n
on the set of directions. In particular we shall give (see 3.4, 4.3 and 6.1), following Guivarc'h and Raugi [34], a condition ensuring that
(a) For any x # 0 in
Iltd ,
lim
n
a. s.
Log II Snx II = Y
(b) There exists a unique invariant distribution on
(c) For any x, y in P(]Rd) lim
P(lRd)
a.s.
,
Log
0
m
where
d
n is the natural angular metric on
P(lRd).
(d) The two upper Lyapunov exponents associated with
(Sn)
are
distinct.
This result is basic for further developments.
A close look at the proof of these statements in Chapter II d = 2) leads us to introduce the following conditions. Let
11
(for
(resp. u
*
be the distribution of semigroup in
Y1
(resp. Y1)
and
Tu
be the smallest closed
Gl(d,]R) which contains the support of
require that
43
l1
.
One has to
44
(i) There exists a that
m(V) = 0
p -invariant distribution m on
for any proper projective subspace
(ii) There is a sequence
(Mn)
such that
Tu
in
P(IRd) such
V . nm
converges
to a Dirac measure.
In the first two sections we develop natural and tractable conditions on
T11
itself ensuring that (i) and (ii) hold. In Section 3
we generalize the method introduced in Chapter II. Namely we study the action of
S*
n
on measures on
P(IRd). We derive from this analysis a
key property from which we shall deduce the main results and we prove statements (a) and (b).
In Section 4 we give, under an irreducibility assumption, a necessary and sufficient condition ensuring that a.s. as
0
n -> - . For later use we shall be needing the stronger
statements (c) and (d). They are proved in Section 6.
(The
d
Lyapunov
exponents are introduced in Section 5).
We end this chapter with a complement section, which can be skipped at first reading, where we slightly improve Furstenberg's theorem. For the reader's convenience we only consider matrices with real entries. But all the given results are valid with essentially the same proofs for matrices in
Gl(d,C). Finally we must say that the method we
use is influenced by the paper [32] of Guivarc'h.
III.1. Contracting sets
We first define what we will call a polar decomposition of a matrix. Let
diag(al,...,ad)
denote
d x d
the
matrix A whose entries
A.
satisfy
ra.i 1,J We write
0(d)
DEFINITION 1.1.
LO
if if
i=j i#j
.
for the set of orthogonal matrices, 0(d) = {M E Gl(d,IR);
M*M = Id}
Let M be a matrix in
Gl(d,IR). We say that
.
M = K A U
is a "polar decomposition" of M if K and
U
are in
0(d)
and if
45
A = diag(a1,..,ad)
LEMMA 1.2.
with
2 a2 a ...
a1
>> ad > 0.
Any invertible matrix has a polar decomposition. Moreover are necessarily the square roots of the
a1 a a2 >_ ... a ad > 0
eigenvalues of M*M. Proof
Let
:
{f1., i
<_
1
i = 1,...,d
{e., 1
< i _< d}
be the canonical basis of
IRd and
*
be an orthonormal basis such that
5 d} .
1
Define the orthonormal matrix
U
by
2
M Mf. = a f 1
Uf1 .
= e1 .
.
1 1
,
15 i<- d.
,
We have
a21
UtA2Uf. = U*A2e. = 1
1
so that
1
1A 1
M = KAU , M
= A
1A2A
1
= a?f. = M*Mf. 11 1 1A 1
K = MU
U*A2U = M M. If we set
K K = A 1UM MU
for if
1e.
then
K
= Id , whence
is orthogonal. Since
has a polar decomposition. The last statement is obvious
M = KAU
then
U- 1
A2
U.
Recall the rank of a matrix is the dimension of its range. DEFINITION 1.3.
of
Given a subset
as the least integer
T
in
{Mn, n >_ 0}
LEMMA 1.4.
Let
a1 (M) a a2(M)
24
for which
T
We say that
T
T
of MM . Then, if
converges to a rank p matrix.
IIMnII-1 Mn
is contracting when its index is one.
T
...
of Gl(d,]R), we define the index
such that there exists a sequence
p
be a subset of G1(d,IR). For any 2: ad(M) > 0 p
M in
T ,
let
be the square roots of the eigenvalues
is the index of T ,
sup{ai(M)-1 a1(M) ; M e T} = o0
if and only if Proof
:
i > p
Fix an integer
{Mn, n a 0}
in
T
i
,
2 5 i
IS d
,
and consider a sequence
such that
lim ai(Mn)_1 a1(Mn) =
sup{ai(M)-1
n-,
a1(M)
, M e T}
M = K A U is a polar decomposition of M we may always assume, If n n n n n passing to a subsequence, that Kn (resp. U ) converges to some n orthogonal matrix K (resp. U) and that 11A II-1A converges to some n n diag(a1,a2,...,ad) with a1 a a2 a ... > ad a 0 But orthogonal .
matrices are isometries, whence
IIMnII = IIAn1I
.
This implies that
46 lim
IIMnII-1 Mn = K diag(a1,...,ad)U
n-*By assumption the index of
is
T
so that
p
K diag(a1,...,ad)U
(since the rank of
ap # 0
ap+1 = 0
and
Max{i;, ai # 0}). The
is
conclusion comes thus from the relation
lim ai(Mn)-1 a (Mn) n-
=
c1
m on
A probability measure
1.5.
proper if for any hyperpZane
in
H
P(lRd) is said to be
IRd
x e H - {0}) = 0
m(x c P(1Rd)
The following result shows that the word "contracting" refers to measures on
the action on proper
Let
PROPOSITION 1.6.
be a contracting subset of Gl(d,IR). For any
T
proper probability measure (Mn
in
, n ? O}
as n-> m Proof
:
m on
P(1Rd) there exists a sequence
Mnm converges weakly to a Dirac measure
such that
T
P(lRd).
.
Let
(Mn)
be a sequence in
to a rank one matrix, say
For any
M .
lim Mn°x = But since m n-*_ is a hyperplane, we see that
such that
T
x e ]Rd such that
is proper and since
z
; Mx = 01
is the direction of the range of
M
for m-almost all
x
nMnm
Mx # 0 d
x
lim Mn - x = z so that
{x a lR
for m-almost all
lim M - x = n n-
It is now clear that if
converges
IIMnII-1 Mn
dz
converges weakly to
It is often difficult to compute the index of a given set. But we shall use the following.
Let T
PROPOSITION 1.7. T
is equal to
d
then
compact subgroup of Proof : Let
be a semigroup of Gl(d,IR). If the index of MI-11dM
{Idet
; M C T}
is contained in a
Gl(d,]R).
T' = {Idet
MI-1/dM
semigroup with index equal to
; M E T} d
.
.
It is clear that
Furthermore its closure
T' C
is a
is itself
47
a semigroup and it is easily seen that its index is also show that
is compact. If
C
is a sequence in
(M 1)
subsequence, denoted also by M
d
.
we can find a
C
for convenience, such that
n
Let us
IIM
n
II-1 M
n
converges to some matrix M . We have
Idet MI = lim Idet (JIM Mn1I)I = lim
IIMnII-d
n Since the index of shows that
IIM
n
the limit is in
is nonsingular and this relation
d , M
is
C
converges. Therefore M C
since
C
is in fact a compact group.
It is useful to notice that under the hypothesis of
Remark 1.8.
[
Proposition 1.7 there exists a scalar product ,
IRd such that
on
is orthogonal for this scalar product. This
Idet MI-11dM
is due the fact that if A
C
is a compact semigroup
C
G1(d,IR). By Theorem 9.16 of Hewitt and Ross [36] or by
Exercise 11.2.4 this implies that
for M E T
G1(d,IR) and
is closed. So every sequence in
C
contains a converging subsequence and contained in
converges in
n
II
C
is a compact subgroup of
G1(d,IR) and if
is its normalized Haar measure, then each element of
C
is
orthogonal for the scalar product defined by
[x,y] = J Exercise 1.9.
<Mx,My> da(M)
Suppose that a semigroup
.
T
Gl(d,IR) contains a
in
matrix M with a unique eigenvalue of maximal modulus, this eigenvalue being simple. Show that
T
is contracting. (Solution
Show that any real matrix of order
Exercise 1.10.
decomposition. Prove that any M
in
Gl(d,V
M* M , and where
Corollary IV.2.2).
has a polar
d
can be written as M=KAU
where A = diag(a1,...,ad), a1 ? a2 > ... ? ad > 0 roots of the eigenvalues of
:
K
being the square
and
U
are in
U(d) = (M e Gl(d,C) ; M*M = Id} Exercise 1.11.
Consider two polar decompositions
of a matrix M
in
G1(d,IR)
.
*
(i) Prove that
M = KAU , M = K'AU'
*
*
U el,...,U ed
are eigenvectors of
(ii) Show that there exists an orthogonal matrix
AB = BA ai
and
K = K'B
a the entries
,
Bi
U' = BU . If and
B.
B
A = diag(a1,...,ad) of
B
M M .
such that
prove that if
are zero. If a1> a2>...>ad
48
with
B = diag(E1,...,Ed)
verify that
ci = ±1
be a sequence in
Let (M ) n following are equivalent
Exercise 1.12.
Gl(d,IR). Show that the
(i) For some proper distribution m on to
.
P(7R
d )
Mnm
converges
d_ . z
(ii) For any proper distribution m to
on
d P(]R ), Mlm
converges
6- . z
(iii) The limit points of
range
1
IlMnll
Mn
are rank-one matrices with
lRz .
111.2. Strong irreducibility
Given a subset
DEFINITION 2.1. (i)
of
is irreducible, if there is no proper linear subspace
S
]Rd such that (ii)
of Gl(d,IR) we say that
S
M(V) = V
for any M in
S
V
.
is strongly irreducible if there does not exist a finite
S
family of proper linear subspaces of
1R1, V1,V2,...,Vk
such that
M(V1 U V2 ... U Vk) = V1 U V2 ... U Vk for any
M in
S.
For instance
S =
0
1)}
is irreducible but not strongly
irreducible.
NOTATION 2.2.
Given a probability measure
on
Gl(d,1R) we write
Tu
for the smallest closed semigroup (resp. subgroup) of
(resp. Gu)
G1(d,]R) which contains the support of p Remark that if matrices
p
M
subgroup of Supp p
V1,...,Vk
such that
.
are subspaces of
]Rd, then the set of
M(V1 U ... U Vk) = V1 U ... U Vk
is a closed
Gl(d,IR). Therefore
strongly irreducible <_> Tu
strongly irreducible
<_> Gu
strongly irreducible
The importance of strong irreducibility comes from the following
49
result. Recall that a probability measure
V
on
is said to be
P(]Rd)
p-invariant if V(A) = JJ A
for any Borel set
on
Gl(d,R)
be a probability measure on
p
such
is strongly irreducible. Then any }p-invariant distribution
GP
that
P(]Rd).
in
Let
PROPOSITION 2.3.
dV(x)
v
P(fld) is proper.
Proof
Let
:
be the set of
P(R)
k- dimensional subspaces of
]Rd.
Consider
R0 = inf it < d
for some
V(V) # 0
;
V
in
F(R)}
{x e P(]Rd); Y, E V-{0}})
(where
r = sup{V(V)
;
A = {V F P(20 )
and If
are in
V2
and
V1
and the definition of V1,V2'...,Vk
A
V c P(Ro)} ;
v(V) = r} V1 = V2
either
are distinct elements of
dim(V1 n V2) < to
or
v(V1 n V2) = 0
implies that
to
Therefore if
.
A
k
v(V1 U V2 U ... U Vk) = E v(Vi) = kr i=1
A must be finite.
so
On the other hand, for each
V
A
in
r = v(V) _ if 1V
dp(M)
V(M J
V(M
But
r
M -1-V
is in
V
A
in
.
A
.
,
Write
L
,
{M E Gl(d,]R); M(L) = L} p
.
strongly irreducible. Therefore
R0 = d
V
and
Prove that a closed subgroup
G
G/H
T11
.
If
is not
T11
is proper.
of
Gl(d,IR)
irreducible if and only if each closed normal subgroup that
is a closed
It thus contains
is a finite union of proper subspaces and
Exercise 2.4.
and
r
for the set of vectors which belong to some
From the foregoing
subgroup which contains the support of
to < d, L
M V(M
so that for p-almost all
H
is strongly of
G
such
is finite is irreducible.
Exercise 2.5.
Show that
Gu
11-invariant distribution on
is strongly irreducible if and only if each
P(]R) d is proper.
50
111.3. A key property
The main properties of product of random matrices will be consequences of the following.
THEOREM 3.1.
be independent random matrices in
X1,X2,...
Let
p , defined on (0, A, IP) .
GI(d,IR) with a common distribution suppose that
is strongly irreducible and write
T1
Then, if Mn = X1X2...Xn , for almost all p-dimensional subspace
Z(w)
Proof
:
V(w)} = 0
is orthogonal to
v
Mn(w)v
P(IRd) and
on
w
dZ(w)
for almost all
.
Vw
,
Henceforth we fix such an { JIMn(w)II-1 Mn(w), n > 1)
P(IRd). By Lemma
on
w
2-n-1 Un
A = E' n=o Consider a limit point
.
A(w)x # 0
If
n -> - ,
as
,
Gl(d,IR) w.r.t.
in
g
v
a probability measure
,
M(w)g V -> Vw
A(w)
of
when
,
goes to infinity along a subsequence. By Proposition 2.3
A(w)g x = 0}) = 0
N) ({-x E P(IRd) ;
A(w)g V = lim Mn(w)g V = Vw
course define
support of
A
being
X- almost all
for
by Proposition 2.3
A(w)g V
11.3.2). It is clear that
Consider now a sequence converges to a matrix is
p). For any
{gn ,
g
in
for all
A(w)g gn V = Vw
,
p
is closed. The
in
Tu
.
(1)
T such that IIgn II gn u (which exists, since the index of
ggn a Tu .
g
in
n a 11
T1
(We can of
.
see the analogous Lemma
this yields
,
of rank
h g
,
{g e G1(d,IR); A(w)g V = Vw}
T1i U {Id}
A(w)g V = Vw
Tu
.
converges weakly to
Consider a p-invariant distribution
such that, in the weak topology
and
For
.
is any non zero vector of V(w)
11.2.1 there exists, for almost all
n
V(w)
is contracting there exists a unique p-invariant
Tp
distribution where
matrix with range
p
x ,
IP {x
When
w there exists a
Rd such that any limit point of
is a rank
{ IIMn(w)II-1 Mn(w), n >_ 1}
any non zero vector
of
V(w)
We
for its index.
p
so that
51
A(w)gh x # 0, A(w)g
If
converges to
Therefore if
A(w)g
v({x E P(IRd) ; A(w)gh x = 0}) = 0 then
A(w)g gn v
2.3
v
converges weakly to
Since by Proposition
.
is proper, (2) yields that
for any
A(w)gh x = 0
If (3) were true for all by
A(w)g h v
(2)
g
in
in
x
the subspace
Tp ,
(3)
IRd
of ]Rd spanned
V
g E T11 , x c Im h} would be contained in the null space of
{g x ;
A(w) , and hence would be proper. But for any
in
g
Tu
and
g V = V
this would be in contradiction with the strong irreducibility of We thus can find
vw = lim A(w)gngv = A(w)ghv . Let By (4)
,
be the linear span of
V(w)
for the rank of
(Supp
h
is
p
(4)
{x E IRd - {0}
is contained in the range of
V(w)
x e Supp vw}.
;
A(w)gh and
dimV(w) <-p
.
M
On the other hand the support of the distribution of p)n , hence
IP(Mn E Tp) =
and
1
definition of the index, the rank of vw = A(w)v
,
V(w) = Im A(w)
and
We have thus proved that
x
in
IP(Mn E Tp
p
We know (see Lemma 11.2.1) that
.
1
.
By
Since
.
is a p-dimensional subspace which IIMn(w)II-1 Mn(w). d
IR
is
n
, Vn >_ 1) =
is at least
A(w)
dim V(w) = p
V(w)
is the range of each limit pointd of
For any non zero
Tu
for which (2) holds and then
g c Tp
set( H(x) = {y E P(IR )
,
<x,y> = 0).
v = J Vw d]P(w). Therefore
IP(x is orthogonal to
V(w)) = IP(Supp vw C H(x))
= IP(vw(H(x)) = 1 )
= v(H(x)) and this is equal to zero for Finally if V(w)
1
dZ(w) dIP(w) = v
is proper.
is contracting then
Tu
is reduced to a point
does not depend on
V
V , v
.
Z(w)
in
p = 1
by definition and
P(IRd). By construction
Since the distribution of
Z(w)
Z(w)
is
must be the unique p-invariant distribution.
The following consequence is of prime importance. We shall make an intensive use of it.
PROPOSITION 3.2.
Let
Y1,Y2,...
be i.i.d. random elements of G1(d,]R)
52
with common distribution
Sn = Yn...Y1
and
p
.
Consider a polar
decomposition S
Kn and
with a1(n)
>_
If
in
Un
.
is a strongly irreducible semigroup with index
T1
with
An = diag(a1(n),...,ad(n))
and
0(d)
>_ ad(n) > 0
...
= Kn An Un
n
{Un(w)e1,...,U*(w)ep}
(a) The subspace spanned by
to a p-dimensional subspace
V(w)
p , then
converges a.s.
.
(b) With probability one lim
ap+1(n)
n-
ap(n) = 0
and
> 0
Inf
n
1Sn11
(c) For any sequence
{xn, n >_ 1}
in
IISnII 1Rd which converges to a
non zero vector , 11 Sn (w) 11
a. s.
<
Sup
IISn(w)xn11
n->_1
We first prove
If
LEMMA 3.3.
is the distribution of
p
equal to the index of Proof
:
and
Tp
TP
V.) = U
so that the index of
Looking only at the
V
is.
TP
of
IR d
such that
for each M E T
V.
i=1
i
i=1
TP
(recall that rank M = rank M ). Suppose
Tu
that there exist proper subspaces Vi,V2,...,VR M( U
is
TP
is strongly irreducible if
TP = {M*, M E TW}
It is plain that
is equal to the index of
the index of
Y1
i
P
with maximal dimension we may suppose that
's i
they all have the same dimension. Clearly {MV1 ,
for any
M
in
implies that
Therefore
TP
.
... , MVR.} _ {V1 , ... , Vk}
If
is the subspace orthogonal to
W.
{M*W1,...,M'`WR} _ {W1,...,WR}
M( U i=1
W.) = U
W. i
i
for
M
in
Tu
M
for any and
in
V.
this
TP
is not strongly T11
i=1
irreducible.
By the lemma we can apply Theorem 3.1 to the
Proof of the proposition random matrices V(w)
X. = Yi
.
There exists, a.s., a p-dimensional subspace
which is the range of each limit point
M(w)
of {IISn(w)II
Sn(w),
53
n >-
11
.
Since
IISnII = a1 (n) S*
*
n
a (n)
a (n)
diag(1,
=U n IISn II
a1(n))Kn
...
a 12 (n)
1
Each component of this product lies in a compact set. If a2(w), ..
are limit points of
ad (w)
a (n)
Uu(w), K-(w)
Un(w), Kn(w),
a (n)
a2
(n) (w) ,
..
then
ad (n) (w) ,
,
M(w) = UU(w) diag(1, a 2(w),...,ad(w))K:(w)
is a limit point of rank
M(w) = p
S*(w). Therefore with probability one
IISn(w)II
and
a2(w) 2 ... >_ ap() > 0
i.e.
,
p+1 (w) _ ... = ad(w) = 0
{U*(w)e1,...,U*(w)ep}
The linear span of
.
is thus equal to the range of
M(w)
follows readily from the relations
, proving (a). Assertion (b)
ap(w) J 0
ap+1(w) = 0
and
.
In
order to verify (c) write
IIAn Un xnII2
I(sn xnII2 11S
11A
n II2
1
a (n)
II sn xn II 2 so that I
If I
Sn
xn -> x
II
p
2
> {aP(n)}
2
and
2 E1{a1(n)} 2 < xn, Un e1>
2
n
a.(n)
d
1
E1
*
<xn
, Un e i >
2
i
is the orthogonal projection of
y(w)
x
onto V(w),
then
lim inf n-
a (n)
I(s (w)xnll >
inf{p(w)} Ily(° n a1 (n)
II
n
IISn(W)II
By Theorem 3.1 IP (II y (w) II = 0) = IP (x
is orthogonal to V (w)) = 0
and (a) follows using (b) and Lemma 1.4
.
We now deduce from this proposition an easy, but important, corollary due to Furstenberg. Actually we shall see in Section 7 an analogous result when
COROLLARY 3.4.
matrices in
G11
is merely irreducible.
Consider a sequence
{Yn, n ? 1}
Gl(d,IR) with common distribution
(a)
1E(Log+IIY1II)
(b)
Gu
< oo
,
is strongly irreducible
of independent random p
.
We suppose that
54
Then (i) For any sequence
which converges to a non zero vector,
(xn)
a.s.
limn Log IIYn...Y1 xn11 = Y v
(ii),If
is a p-invariant distribution on
y=
11 Mx II
Log
P(lRd), then
_
dp(M) dv(x)
IIx 11
(iii) If moreover lE (Log+ IIY1111)<-, then lE{n Log IIYn...Y1xII} converges to y uni f ormZy on {x e IItd ; II x II = 1 } Proof
and
Sn = Yn...Y1
Let
:
roots of the eigenvalues of
a1 (n)
S S
.
n n to some non zero vector there exists
... > ad (n) > 0
>_
be the square
By Proposition 3.2 if (xn) converges such that, a.s.
C(w)
Is n (w)xn II
0 < C (w) <
II Sn (w) II
Therefore
lim 1n Log
S
n
(w)x
n
I
= lim n
Log II S n
II
a. s.
=Y
which proves (i).
be a p-invariant distribution on
V
Let
((
JJ Log
is finite, one can deduce from Proposition
dp(Y)dv(x) IIXII
1.3.4
IIYXII
a(Y,x) = Log
using the cocycle
converges
_
IN
Y,XII
IIrII
foregoing this limit is
y
this proves (ii).
,
{x a ]Rd; IIXII = 0 is compact it
Finally we prove (iii). Since suffices to show that if
(x
n
)
is a converging sequence on the unit
sphere then
lim n,. But we have, by (i)
lim
n
lB(Log uISnxnII) = Y
,
1
Log
ISnxnII=Y
and
ILog IIYXII so that
Log
n
II
(see also exercise 11.2.6). Since by the
dp(Y)dv(x)
Log
X
lP ® v a.s. to a random variable whose expectation is
IIYXII
f
1
, that II
1
P(lRd). Since
+ IIYXII
I
< Log+ IIYII + Log+
IIY-1
II
55
n <
in Log IlSnxnll
I
{
Remark 3.5.
Idet Yli =
If
implies that
IE (Log+ IIY11
d-1
that
index
d-1
Ild-1
= IIY
be a proper probability measure on is a closed subgroup of
{M e GI(d,]R) ; Mm = m) d
IIY1II) <
i=1 m
Let
ai(d) < a1 (d)
II
=
IE (Log
since
)<m
ad(1)-l
Exercise 3.6.
by the law of large numbers,
L1
a.s. the condition
1
IY,111 =
.
1
is uniformly integrable and (iii) follows.
1}
n >_
Log ISnxI
IYII + Log+ IIYi
=1
The right hand side converges in hence
(Log
iE
n
P(1Rd). Show
GI(d,IR) with
.
Exercise 3.7.
Show that the conclusion (a) of Proposition 3.2 is not
necessarily true if
111
is merely irreducible (Hint
:
use the example
introduced in Exercise 11.6.6). Exercise 3.8.
Show that if
1E (Log' IIY11 II) <
and
IE (Log
IIYIII) <
then the conditions
lE (Log Idet Yl I) > -- are equivalent.
P(IRd) and convergence in direction
111.4. Contracting action on
Recall the definition of the natural angular distance on DEFINITION 4.1.
if
x, y
are unit vectors in
P(IRd).
IRd with direction
x,
y We set <x,y>2}i/2
6(x,y)
For any orthogonal matrix
K
,
6(K-X,
6(x,y)
.
To check that
is a distance it thus suffices to show that
6(y,z) <_ 6(y,x) + 6(x,z) for
x = ell y = yle1 + y2e2, z = zlel + z2e2 + z3e3
being three unit
vectors. In that case
6(y,z)
{(y1z2-y2zl)2+(y1z3)2+(y2z3)2}1/2 {(ylz2- y2z1) 2
+
Z3} i/2
6
56
< {(y2+z2)2 + Z2
< ly21+ (z2+
Z2
6(x,y) + 6(x,z) For any
LEMMA 4.2.
M
in
GI(d,IR), if
are the square
a1(M), a2(M)
roots of the two top eigenvaZues of M M then 6
11X II
< a (M)a (M) 6(x>y)
iIMxllIIMyll
2
1
lly it
This follows from an elementary computation (See also Lemma 5.3 and (9)).
Now we prove the main result on the behaviour of the random product
S
n
available without moment hypotheses. It states that
ultimately contracts the distance direction of
S
n
S
n
with probability one while the
d
converges in probability to a random limit
x
x (we shall see in VI that under stronger assumptions
independent of
this latter convergence holds also almost surely). For instance all the column vectors of the matrix
tend to align in the same direction
Sn
and the directions of the rows converge. This important result is due to Guivarc'h and Raugi [34]. It is remarkable that it holds under a condition which depends only on
THEOREM 4.3.
matrices in
Let
Sn = Yn...Y1
Tu
.
where
Gl(d,IR) with distribution
{Yn, n > 1)
are independent
p . Suppose that
TP
is
strongly irreducible and contracting. Then x, y
(i) For any
in
P(IRd), with probability one
lim 6
Sn -Y) = 0
n-
(ii) There exists a random direction converges in probabiZity to
Z
Z
such that x c P(IRd)
, uniformly in
(iii) There is a unique p-invariant distribution
and 'or any continuous function f
on
P(IRd)
Sup 11E {f(S x)} - 1 f dvl -> 0 n
v
S *-x
n
.
on
P(IRd)
,
as
n -> c
x
Proof
: Let
a1(n), a2(n)
be the square roots of the two top
(x ) and (y ) are two sequences of unit n n n n vectors which converge in direction, Lemma 4.2 gives
eigenvalues of
S S
.
If
57
d(Snxn , Snyn)
a1(n) a2(n) II Snxn II
I Snyn II
a2(n) IISnII
I
IISnII
IISnII
I Snxn II
I I Snyn II
Therefore using Proposition 3.2 we have, a.s. d
n
lim n-
(5)
= 0
n 'yn)
which, in particular, proves (i). *
Let
be the common distribution of the random matrices X.= Y.
p
1
and
Rn = Xn...X1. I claim that as
n -> -
Sup IE X,y For if (6) does not hold we could find (yni)
(Xn
(6)
0 e > 0, a subsequence
n.
and
for which
ni'Yni)} > e
Without loss of generality we may assume that converge. By Lemma 3.3, (5) holds for
R
.
and (yn1.)
(xni )
and
n
lim d(Rn xn. , Rn yn ) = 0 i-)oo
1
1
1
1
which leads to a contradiction. We know (Theorem 3.1) that if distribution on
measure
P(lRd) , X1...Xnm
*
m
is the unique U -invariant
converges weakly to some Dirac
6Z . We have
IE {d(Sny,Z)} *
whence, since
_
Sup xEP(IRd)
]E
S
n n
has the same law as
R n
Sup 1E {6(R x,R y)} + 11E n
n
1
n
X,Y
The first term of the right hand side tends to
0
by (6), the second
one is equal to
lE {J 6(u,Z)d(X1...Xnm)(u)}
IE {J which converges to If
f
IE {6(Z,Z)} = 0. We immediately deduce
is a continuous function on
P(IRd), f
continuous. Therefore, using the foregoing
,
(ii).
is uniformly
1
58
Sup IE{If(S n x) - f(Z)I} -> 0
n ->
as
x
Thus
f dmI
Sup x
f(Z)I}
<_ Sup
x
converges to zero.
Similarly, working with
Y.
instead of
, one obtains (iii).
X.
The assumptions of this theorem are rather weak. For instance let us show, using its notations
If
PROPOSITION 4.4.
if and only if LEMMA 4.5.
is only supposed to be irreducible,
T1
(i) holds
is strongly irreducible and contracting.
T1
If
is not strongly irreducible we can find proper T11
V1,...,VZ with the same dimension such that
subspaces
(a) If i # j
i e {1,...,R}
(b) For any
j
e {1,...,R.}
Proof
:
Let
and M in
Tu , MV. = V.
for some
be the least positive integer such that there exists a
r
of subspaces with dimension not greater than
L
for each M
M(L) = L
with dimension
r
.
Note that, if
in
Tu
.
Let
V1
r
such
,
be one of these subspaces
V2'...IVQ
It is clear that there exist
{MV1, M E Tu}
such that
.
.
finite union that
V. fl V. = {0}
,
{V1,V2,...,Vf}. We suppose the
L
in
distinct.
V.
M E T11 '
M( U
i0J Since
dim V.
fl V.
i
< r
(Vi n vi)) = U 1 J 10J
(Vi n vi)
.
J
1
the minimality assumption on
r
implies (a).
(b) is obvious.
Proof of the Proposition
:
Let
m be any proper distribution on
(see Definition 1.5). If (i) of Theorem 4.3 holds one can find and
ni
to some
such that z
and, as
i -> 0
and
This implies that and
Sni(w) E Tu
Sni(w)m
for
,
m-almost all
y.
converges weakly to the Dirac measure at
Tu is contracting.
x
converges
,
IISni(w)II-1 Sni(w) converges to a matrix with range
This proves that
P(IRd),
w
z
z
59
Suppose that
is not strongly irreducible. Let
Tu
V1,...IVR
x E Vi} , where
V. n V. = o
lemma. Since
are the proper subspaces introduced in the ,
a = inf{d (Vi,V. ) If (i) holds we can find
Vi= { E P(]Rd
x c V1
,
i<j
1
and
V 2
,
S R} > 0 Sn(w) E Tp
such that
0. But (b) of the lemma implies that for
lim some
E Vi
i # j
and
E V.
for each positive integer
a > 0
.
Thus
n
It is usually difficult to know whether a given semigroup
Remark 4.6.
is contracting or not. The study of this problem remains to be done. Remark 4.7.
For arbitrary positive matrices, i.e. matrices whose
entries are
> 0
]Rd
a stronger statement holds. Let
,
= {x = (x1I...,xd) E ]Rd; x. > 0
,
called the
with the topology,
following property
:
if
and
<_ d}
5 i
1
C = {r. E P(]RO,x E ]Ra}. There exists on
a distance
C
0
compatible
Hilbert projective distance, with the
M > 0
there is
0(M) <
1
such that
0(M) 6(x,y) for any
x, y
(7)
C. Individually each matrix contracts
in
0
The following exercises provide the proof of this result due to
G. Birkhoff [6] (one can find another one in [64]). If
Sn = Yn...Y1
is a product of positive i.i.d. matrices it is almost
immediate to obtain the analogue of Theorem 4.3 where (C,0) take the place of
(P(]Rd),d), see Exercise 6.5
Exercise 4.8. and
(case d = 2). Let
L = [0,co]. For
u, v
in
L
L
.
denote the open half line
(0,-)
T(u,v) = ILog u - Log vi, this
set
defines a distance. Consider a projective map p (this means that ax+b such that p(L) C L. We want to prove that if A is the p(x) = cx+d) diameter of
p(L)
for the distance
, then
T
T(P(u), p(v)) S {tanh a. Using that
with
cx+d
b. Prove that
(v))
a > 0, b > 0, c > 0, d > 0
Sup dx Log p(ex) = ]R
(8)
show that one can suppose
T(p(u),p(v)) = T(p(1u),
p(x) = ax+b
A =Logy .
T(u,v)
v+2 c+1
if
and
v =
ad-bc # 0 ad
and that
60
c. Prove the inequality (8). d.
If
0 (x
,
x1
Y) = Log
define
Y1
- Log
.
I
y2
x2
matrices.
2 x 2
Show that (7) holds for positive
IR2
are in
x = (x1,x2), y = (yl,y2)
d
Exercise 4.9.
(case d
>_
H = {x = (xl,...,xd) E lRd, E x i= 1}
3). Let
i=1
and
B
x, y in H fl 1R+ we consider
the boundary of H fl IR+ . For
the two points
{ax + (1-a)y, a e IR} which lie on B,
on the line
a, b
and define
8(x where
[a,b
C ;
,
Y) = Log [a,b
x,y]
;
is the cross ratio (if
,
x,y] _ p'). For any
x = as + lib, y = A'a + }1'b
[a,b] = {aa+ (1-(x)b, 0 < a <
on
z
1}
we
set 9'X - (z) = Log [a,b
alb
;
,
z
Y
xi
0(x,y) = Log{ Max
a. Show that
xi } { Min 1
}
lid yi b. Prove that isometry from c. Let
0
is a distance on onto
([a,b] fl c, 0)
M
T)
be a positive matrix. Show that the diameter
is finite. For
x, y
in
prove that
C
P(u) _ TM,X,M. (M is a projective map and that 0(M) = tanh
,
Y'_ _ is an x,Y (see Exercise 4.8).
and that
C
(L
p
L -> L
A
of
defined by
1Y(u))
9'
p(L) C L
.
Prove that (7) holds with
(Use Exercise 4.8).
111.5. Lyapunov exponents
We shall now define the
d
Lyapunov exponents associated with
(Yn). They will in particular help improve Theorem 4.3. This is easily done if we work with the exterior powers
APlRd. For the reader's
convenience we recall the definition and simple properties of these spaces.
Let
E
be a
d-dimensional vector space with dual
E
.
For any
61
positive integer
E
u1,u2,...,uP
For
.
denote the vector space of alternating
APE
let
p
p-linear forms on
in
E,
in
f1, ... If
E
we
P
set
(u1Au2A...Au(f1,...,f) =det[{fi(uj)}i j] It is clear that
is an element of
u A ...A u
AP]Rd , we shall call
P
1
it a decomposable
.
p-vector.
The following facts are readily checked (see e.g. 7.7 of Loomis,
Sternberg [52]). LEMMA 5.1.
{e1,...,ed}
(i) If
{ei A ei A ... A ei
is a basis of E , then
is a basis of APE.
f i1 < i2 < ... < ip f d}
1
p
2
1
,
(ii) For u1,u2,...,uP in E, u1Au2A...Au is non zero if and P only if u1,u2,...,uP are linearly independent. (iii) Two independent
{ui
p-upZes
,
1
f i f p}
{vi,
and
1
have the same linear span if an only if u1 A ... A uP = A (v1 A ... A vp ) for some
X
.
One deduces from this lemma that the image of the set of decomposable p-vectors in the projective space
NOTATION 5.2.
For
,
APIRd
APIRd
< i2 < ... < iP
will be the one associated
{ei A ...A ei 1
,
p
is an orthonormal basis of
<_ d}
Jul A u2 A ... A u
]Rd we
v1 A ... A vp> = det [{}i, j]
Throughout the book the norm on
f i1
.
by the formula
with this scalar product. Note that 1
E
being the usual scalar product on
define a scalar product on
that
can be
P(APE)
identified with the set of p-dimensional subspaces of
is the
11
APIRd
,
and
p- dimensional volume of the
P
parallelogram generated by
in IRd
u1,... u p . We have, for any non zero
x, y
,
6(X>y)=
Ix Ayll
(9)
Ilxll Ilyll
If
M
is in
GI(d,]R) we define an automorphism
APM
(AP M) (u1 A ... A up) = Mu1 A Mu 2 A ... A Mup
of .
APIRd by
62
Ap M will denote either the linear mapping or its matrix in the basis (ei1 A ei2 A ... A eip)
We set
.
= Sup { II(ApM)wUI, w e ApIRd, IIwII
IIApMII
Note that Ap(MN) _ (ARM) (ApN) so that If K is an orthogonal matrix
hence
A ... A Ku p
IIApMNII <= IIARMII IIAp NII
A ... A Kv > = det{} = det{}, p 3
Kv
,
11
=
1
is orthogonal.
ARK
The importance of exterior powers in the study of random matrices comes from (9) and the following.
LEMMA 5.3.
M be a matrix in
Let
a1(M)>_ a2(M)
>_
the square roots of the eigenvalues of
>_ ad(M) > 0
...
M*M. Then for any
p,
< p < d
1
IIApMII
G1(d,IR), and
,
= a1 (M) a2(M) ... ap(M)
.
Proof : Write a polar decomposition M = K A U with
A =
( M ) ,...
APIRd
,
ad(M))
,
Since
.
AP K and
1
and
But
IIAp MII = II (AP K) (ARA) (ARU) II = IIARAII .
(ApA)(ei
K, U e 0(d)
are isometries of
AP U
A ... A ei ) = ai (M)...a. (M)e. 1 p p
A ... A ei
p
1
whence
IIApAII = Sup{ai (M)...ai (M) LEMMA 5.4.
For
Log IIApMII
1
I
1 < i1 < i2< ... < ip
in Gl(d,IR) set R(M) = Max(Log+ IIMII, Log+ IIM 1II).
M
Then for any p,
;
p
1
< p < d , and any unit vector w of < p Q(M)
and
ApIRd
ILog IIApMwIII pi(M)
Proof : With the notations of the foregoing lemma we have
... ap(M) < a1 (M)p <
IIARMII = a1 (M)
so that Since
Log IIARMII < p Log IIMII < p Log+ IIMII
II ARM 1 ARMII
= IIAp Idll = IIARMII-1
so that Therefore
<
IIApM 111 5 IIM
I
< p k(M)
we also have
1
- Log IIARMII < p Log IIM Log IIARMII
IIMIIR
1
1Ilp
11 < p Log+ IIM
< p R(M)
.
1
11
< p R(M)
63
, for llwll = we also have
Since
1
, we have
ILog IIAPM w MI
Remark that if
I
Idet Ml =
We can now define the
and
IIApMwJI < IIAPMII
1
< p Z (M)
.
i(M) < d LoglIMII
,
IIAPMwII-1 < 119M-1 II
,
(see Remark 3.5).
Lyapunov exponents associated to a
d
sequence of random matrices. DEFINITION 5.5.
Y1,Y2,...
Let
with 1E (Log+IIY1I)< - . to
be i.i.d. random matrices in
The Lyapunov exponents
are defined inductively by
(Yn)
P Y
E
i=1
1
Y1,Y2,...,Yd
and for
y = Y1
G1(d,]R) associated
p ? 2
= lim nn IE(Log IMAp nY.. Y J 1
P
In other words
E
is the upper Lyapunov exponent associated
Yi
i1 with the matrices
(APYn). As such it exists and is equal to the almost
sure limit of n Log IIAPYn...Y1II
ambiguous if for some p
(see Chapter I). The definition is
PE 1 y1 .
_ -
,
in this case we put
i=1 P
p+1 = ... =
=
d
=
Note that if IE(Log+ IIYI1) that for any
is also finite, Lemma 5.4 implies
p
IIE
yiI '- Sup 1E(Log+IIY1
P 1=1
I
,
Log+ IIY11 I) <
so that all the exponents are finite.
PROPOSITION 5.6.
If
a1(n) ? a2(n)
roots of the eigenvaZues of
>_
... ? ad(n) > 0
are the square
(Yn " .Y1) (Yn " .Y1), then with
probability one,
Yp = limn IE (Log ap(n)) = lim n Log ap(n) Proof
:
If
y1 + ... + Yp-1
ap(n)p-i < a1(n) ... ap-1(n)
implies
(p-1) limn Log ap(n) <_ limn Log a1 (n) ... ap-1 (n) which yields, using Lemma 5.3
(p-1) lim 1 Log ap(n) < limn log IIAP Yn...Y1 II = Y1 + ... +YP-1
=°
64
y
so
= - CO = lim 1 Log a (n). n p
p
If
Yi
+ ... + Yp-1 # - °°' at(n)...ap (n)
lim n Log a (n) = lim 1
1
n
Log
(n)...a
a
p
1
lim n {Lag
(n)
P-1
IIApYn...Yt1I - Log
IIAp-1Yn...Y1II}
= Yp
It is thus clear that fact that we have called We have
Y1 > Y2 > ... ? Yd , which justifies the y = y1
the upper Lyapunov exponent.
Idet(Yn...Y1)I = a1(n)a2(n)...ad(n)
,
therefore
Y1 + Y2 + ... + yd =lE(Log Idet Y1 I)
and all the exponents are finite if and only if
Logldet Y1I
is
integrable.
111.6. Comparison of the top Lyapunov exponents and Furstenberg's theorem
We now prove the main results of this chapter. Under the assumptions
IE (Log+ IIY1II) < -
and
Tu
strongly irreducible we
determine following Guivarc'h and Raugi the least
p >
1
such that
Y1
> yp. In particular we recover Furstenberg's theorem. As before
Y
? Y2 ?
1
following is
[34]
are the Lyapunov exponents associated with
? Yd
11
.
The
maybe the principal result of this book. It appeared in
.
Let
THEOREM 6.1. Log
be a probability measure on
is finite. We suppose
IIMII dp(M)
J
p
Tu
Gl(d,IR) such that
irreducible. Then
y I
if and only if Proof
is strongly irreducible and contracting.
Tp
Suppose that
:
>Y 2
is strongly irreducible and contracting (i.e.
T
U
the index of x
in
IRd
with law
,
p
Iu if ,
is one). From Proposition 3.2. we know that for any Sn = Yn...Y1, where
then
{Yn, n _>-
1}
are i.i.d. matrices
65
a2 (n)
IISx1I n Inf
n
and
# 0
IISnII
11A2SII n
= lim
Jim
n-
= 0
.
n- IISnII2
IISnII
This implies that for each non zero matrix M , a.s.
IISnMII2
lim
= +
(10)
2
n-
11A Sn II
We shall use Lemma 11.2.2 to deduce from (10) that
Y1 > Y2 Consider each matrix as an endomorphism. From the proof of the theorem
of Furstenberg and Kesten (Theorem 1.4.1) we know that there exists a }i-invariant distribution
of
IRd ,
B1
the set of norm-one endomorphisms
,
such that lim
n-_ n IP ® v1 -almost all
for
on
v1
there exists a
Log IISnMII
= Y1
(w,M). In the same way, working with
11- invariant distribution
v2
on
B2
(A2Yi)
the set of
,
norm-one endomorphisms of A2IRd, such that
lim n Log 11A 2
n
for 1P ® v2 -almost all Consider
B = B1
S. N II
= Y1 + Y2
(w,N)
X B2
.
It is a
G1(d,IR)-space if we put
2
(A Y)N
)
Y c G1(d,IR),
IIY MII
u1
{n
E i=1
on
B1
Let
a
(v® ® v2)
;
n ? 11
,
N E B
2.
1
By Lemma 1.3.5 it is clear that each limit point n
M E B
II (A 2Y)NII v
of
is a }1-invariant probability measure
X B2 :
Gl(d,IR) X B -> IR be the cocycle defined by a(Y
,
(M,N)) = Log
Since the projections of V
on
IIYMII2 IIA 2
and
B1
YNII B2
are
V1
and
v2
IP ®v a.s. lim n a(Sn, (M,N)) = lim n Log IISnMII - "m n Log IIA2SnNII n,_ n
n
= 2Y1 - (Y1+Y2)
= Y1 - Y2 On the other hand, by (10)
,
.
we have,
66
,IS"M112 lim a(S y1
> y2
i->
= -
lim Log
n-
n
nand
(M,N))
II A 2S
n
11
by Lemma 11.2.2.
Since a direct proof of the converse would be tedious, we make use of the fact that when
is irreducible, for any non zero
G11
x
in
1Rd
a.s.
lim n Log 1IYn...Y1x1I = Y1
nWe will show this in Proposition 7.2 that if
Y1 > y2
Taking it for granted we see
for each unit vectors
,
"m 6(S
n-'°°
.
This in turn implies that
IIA2Sn(w)11
S "m
n
n
x, y < 0
a.s.
n-'°° II So (w)x 11 1 S. (w)y 11
is strongly irreducible and contracting
Tu
(see Proposition 4.4).
More generally we have
PROPOSITION 6.2. that
Let
p
is finite. If
f Log+ IMII dp(M)
if its index is
p
be a probability measure on
is strongly irreducible and
Tu
then
Y1 = Y2 = ... = Yp > Yp+1
Proof
:
Gl(d,1R) such
It is obvious from Lemma 1.4 that
.
Y1 = Y2 =
Yp
By
Proposition 3.2 for any non zero matrix M , a.s. IIp+1
lim
II SnM 11Ap+1S.11
=
n-'°°
n-'m
From this relation one deduces that
_
II Sn 11
l im
a p+1 (n) using Lemma 11.2.2
yp > Yp+1
a proof similar to the proof of Theorem 6.1
.
In particular we obtain the following result of
(Furstenberg's theorem)
THEOREM 6.3.
on
If U
[211
I
(b) GP
Log
IIMII dp(M) <
is strongly irreducible and non compact
Then the upper Lyapunov exponent is positive
.
is a probability measure
{M c Gl(d,7R), Idet MI = 0 such that
(a)
by
(y > 0)
67
Proof
By Proposition 1.7, the index
p < d
of
Tp
satisfies
Therefore by Proposition 6.2 (recall that if
GP
is strongly
:
irreducible then
p
is also strongly irreducible)
T11
yp > Yp+1
But
.
the definition of the Lyapunov exponents gives Y1 + Y2 +
'"
+ Yd = lim n Log
n-
IIAdSnl
lim n Log Idet S,J 0
y
If
Yi
were not positive then p + 1
and for
< i < d
.
since
,
y = Y1
, yi < Y p
that
yi < 0
E
0
p
d
,
for
<-
1
i
<_ p
This would imply
.
which is absurd.
i=1
Our analysis of the central limit theorem rests upon the following result of Le Page [49]. Recall that
i(M) = Sup(Log+ IIMIJ PROPOSITION 6.4.
Suppose that
ii
.
1
II
)
are i.i.d. random matrices
{Yn, n ? 1}
in Gl(d,IR) with common distribution if
Log+ JIM
,
If
IE(i(Y1)} is finite and
is strongly irreducible and contracting then, for Sn = Yn...Y1
Tu
(i) For any x, y in
P (IRd)
1
lim n Log n-
< 0
a.s.
,
6 (x,y)
lim sup n IE (Log
(ii)
n-
0
X,y
6(X,Y)
Proof : We have already noticed that under these hypotheses
are finite. By (9), for any x, y in IRd- {0} IIsnxll IISnYII Ilxll IIMII
6(X,y) Since
IIsnxll IISnYII IIxAYII converges a.s. to
n Log I1A2Sn
`
S
snll
yi + y2
and Y2
IIXII IIMII
2
IIA
yi
,
IIsnxll IISny1I
and n Log
IIsnxll
II x II
converges to
y1
(see Corollary 3.4), we have
lim n Log n
6(Sn X,sn y) 6(X,y)
< y1
+Y2 - 2y1 = Y2 - Y1
By Theorem 6.1, Y2 - y1 < 0, proving (i). We also have by Corollary 3.4,
68
lim
Sup
n-
x,y
1
]E {Log
n
6(s X,s y) n n } d(x,y)
lim n ]E (Log IIA2Sn II) + 2 lim {Sup n Log
n,
n-'°°
x
114 }
11 Snx li
.
This completes the proof.
of order
d
.
be a sequence of i.i.d. positive matrices
Y1,Y2,...
Let
Exercise 6.5.
With the notations of Remark 4.7 prove that
(i) For any
x, y
in
C
,
lim
0
.
nZ (ii) For any continuous function
f
with compact support on
his n
0
C
.
(iii) Deduce from (ii) that there exists a unique 1l-invariant
distribution on
C
.
(iv) Show that
Sup -nlE (Log
lim n-
1
)
x,yEC
<0
6(x,y)
111.7. Complements. The irreducible case
So far we have most often assumed that
is strongly
G11
irreducible. In this section we consider the case where
GP
is merely
irreducible. We first prove a technical lemma which is useful to deal with this situation in various contexts. We then apply it to give a necessary and sufficient condition ensuring that there are at least two
distinct Lyapunov exponents. (Recall that when irreducible then amenability of
by Theorem 6.1). This is related to the
Y1 = Y2
G11
.
is not strongly
G11
Therefore we obtain as a by-product a complete
description of all closed amenable subgroups of Gl(d,]R).
LEMMA 7.1.
elements of
Let
Sn = Yn...Y1
where
are i.i.d. random
{Yn, n >_ 11
G1(d,]R) with common distribution
p
.
Suppose that
GV
is irreducible. Then there exists a direct sum decomposition IR
= V1 ® V2 ® ... ® Vk and a closed normal subgroup
the following properties
:
H
of
G11
with
69
(i) All the
have the same dimension, say
V. 's
Gu/H
(ii) The group (iii) For any
is finite.
M in
and
< i I k
1
r
(iv) The distribution of
H , M(Vi) = V.
T = inf{n
exponentially decaying tail. For any
1
1
,
has an
Sn E H}
1 i _I k
S1) denote an
let
matrix which represents the endomorphism of V.
r x r
restriction of
to
ST
in a fixed basis of
Vi
is the distribution of S1)
obtained by
Then, if
.
A(i)
is a strongly irreducible
Gx(i)
,
V.
subgroup of G1(r,IR). 1E(Log+
(v) If moreover
the upper Lyapunov exponent
11 Y1
y(i)
)
is finite then, for
1
M
(i)
associated to
i <_ k,
<_
equal to
ST
1E(T)y Proof
:
Consider (see Lemma 4.5) subspaces
same dimension 1
l i < i
such that for
r
i ¢ j
V1,...,Vk
We shall choose
is strongly irreducible and the lemma is obvious
H = Gu). Therefore we suppose
have proved the assertion ill
Since
i
..
(11)
.
minimal among the integers for which such subspaces
r
r = d, Gu
exist. If
for any
and for
,
{MVi; M E Gu} _ {V1,...IV R}
(take
1Rd with the
of
Vi fl V. = {0}
,
,
r < d
.
P
V. C ® V.
j, either
In the same way that we
in Lemma 4.5 one sees that
V. fl V. = {0} P
V.fl ® V.
or
j
j R s=1 s=1 is is irreducible the linear span of .U V. p
G
can renumber the
Vi's
is
IRd and we
1
i=1
11
= {0}
is
in such a way that for some
k
V1 ® V2®... ®Vk=]Rd Consider now the map group over
I
i
(12)
GP -> PR , where
:
PQ
is the permutation
letters, defined by
if
{'Y(M)}(i) = j It is clear that
V
M(Vi) = V.
is a group homomorphism..Its kernel
H = {M E G11
;
M(Vi) = Vi
is thus a closed normal subgroup of
If we write each matrix Y
1 .
Gp
,
and
i = 1 ,... ,R} Gu/H
is finite.
in a different fixed basis all the
properties we are looking for are preserved. We may therefore suppose for convenience that if
{e1....,ed}
is the canonical basis of
1R, d
70
spans
{e1,...,er,}
V1 ,
write any matrix M
of
M
V2
spans
{er+l,...,e2r} H as
and so on. We can
(1)
M =
(13)
M
0
where each
r x r
is a
M(1)
We define inductively It is easy to see that
matrix .
T(1) = T, T(n+1) = inf{m > T(n);
H}.
are i.i.d.
T(1), T(2) - T(1),...,T(n+1) - T(n)
random variables. Notice that since finite states Markov chain
Y'(S
n
is the first time when the
T
returns to the identity, its
)
distribution has an exponentially decaying tail. It is also clear that are independent random matrices in
ST(1)' ST (2) S-1 )'...'ST(n)ST(n-1)
H with a common distribution property of
1E(Log
T
. Making use of the stopping time
A
one sees that if
,
1E (Log
IlYlll)
is finite then
is also finite.
11ST11)
We shall now prove that for each irreducible subgroup of
1
< m <_ k
is a strongly
, Ga(m)
Gl(r,1R). For this we first remark that
(since Gu/H is finite, H is open in G11 and M is in if and only if for any neighborhood 0 of M in H , 1P (3 n > 0, S. E 0) j 0 , i.e. IP ( 3 n > 0 , ST (n) E 0) # 0 . This
TX = Tu fl H
Tufl H
M E TA). On the other hand there
latter condition is equivalent to t1,t2'...,tp
exist
in
such that
T,,
Gu = t11H U t21H U ... U tp1H for, since
F(T-1)
is a finite semigroup in the permutation group
is a subgroup and
Y'(T11
contains
Tu
tt,...,tp
Tu
in
Suppose that
m E {1,...,k}
.
Y'(TU1)
with
and it suffices to choose
Y'({tai, ..,tp1}) =
Ga(m) is not strongly irreducible for some
M
G.
in S
s
W.) = U 1
For any matrix A
in
This implies that
W1,...,Ws
of
Vm
,
M( U i=1
.
= T(GU)
Then we can find proper subspaces
such that, for any
t.A E H
must be a closed subgroup which
Y 1(T(TU1))
Therefore
.
P
W.
i=1
1
there exists some index
Tu
A E TX
t j
(since
j
TX = TW fl H)
such that and that
71
(t.A)(W1) C U
W.
A WiC U
Therefore
.
M1,...,Mq
and we can find
t.1 W.
i,J
i=1
such that for each A
Tu
in
A{M1 W1, ..., Mq W1}
{M1 W1, ..., Mq W1}
in contradiction with the minimal assumption on
r
.
Finally we prove M. Using (iv) and Corollary 3.4 we see that for
any
x . FV .
,
x. # 0
,
II = lim 1n Log lim !n Log IS(i) -c (n)
IIS(i)
y(i)
x II = i
T(n)
(14)
This leads to
lim
n and to, if
xi
Log II ST (n)
= Sup limn Log i
I
IIST(n) II = Sup
i
is the orthogonal projection of
lim n Log
x
y(1)
Vi
on
ST(n)xll =Siup lim n Log IIST(n)x.
y(i)
_Sup {i;x ¢O} i
If
, making use of the usual law of large numbers we have
]E (T) = a
so that by (14)
lim T(n n) = a
lim T1 Log
for any
II ST(n)x
n
n
For any positive integer
,
T(0n) < n <
T(0n
in
]Rd
y(i)
Sup
=
define
x # 0
{i,xi#0} On
by the formula
+ 1)
It is readily seen that lim
n-
T (en) n
a.s.
= 1
from which we deduce that for any
lim n Log IISnxII n-
in
x
]Rd, x J 0
= lim T(, j Log IIST(6 )x11 = n
n-
n
Sup
y(i)
{i;xi#0}
a
(use the relations
Log II ST
(en+')x
II
-
T(On+1) Log+ IIYi II
E
< Log IISnxII
i=n+1
n
and
Log IISnxII < Log IIST(0 Now suppose that for some
1
n )x
II
+ E i=r(0n)+1
< i < k
,
Log+ IIYi II)
y(1) 0 ay
.
Then
(15)
72
a.s.}
W = {x E IRd; lim n Log IISnxtI < y
a.s.
Gu
for each M of
, MW = W
implies that
{0}
IRd different from
is a linear subspace of
Gu
.
Since Y(W)W = W
.
The irreducibility assumption on
and this leads to a contradiction for if
in W
vi,...,vd
W = IRd. We thus can find a basis
IIMII = Sup{ IIMviII
;
1 <j
limn Log IISnII < Y (recall that this limit does not
we find that
depend on the chosen norm). The proof is complete.
The end of the foregoing proof shows that for each lim
Log
I Snx I
x # 0
,
(see (15)). One readily deduces from this the
= y
following result of Furstenberg (see the proof of Corollary 3.4, (ii)).
Actually this proposition can be proved directly in a short way using some arguments of Markov chain theory (see in particular Furstenberg
and Kifer [26]). PROPOSITION 7.2.
Let
where
Sn = Yn...Y1
random elements of
1E(Log+IIY1I)
Gu
is irreducible and that
are i.i.d.
{Yn, n >- 1}
G1(d,IR) with common distribution
p
.
Assume that
is finite. Then
x # 0
(i) For any
a. s.
n-(ii) For any pi-invariant distribution
(( y = JJ Log
on
v
.
P(IRd),
11 MX II
II
x
dp(M) dv(x) II
Let us state without proof an important consequence of this proposition (see Furstenberg and Kifer [26] or Hennion [35]).
COROLLARY 7.3.
Let
(pn)
be a sequence of probability measures on
Gl(d,]R) with upper Lyapunov exponents
fII g 11 -:-N
Log' IIg II dun(g) + J
converges to zero as
lim y(u
n
)
Log' IIg-1 II dun (g) IIg-1 II >_N
N -> - uniformly in
weakly to the probability measure
n-
such that
y(un)
p
and if
n .
GP
Then if Pn
exists and is equal to the upper Lyapunov exponent
associated with
p
converges
is irreducible then
73
Let us notice that when each is easy
Gu
n
is irreducible then the proof d
11n-invariant probability measure on
is a
Vn
if
:
then each limit point
of
v
P(1R )
is p-invariant and
(Vn)
HIM- 11
his Y(un) = him f Log - dun(M) dVn(x) n
n-
II
IIMx II I I x
Log - du (M) dv(x) = Y(u)
_
IIx II
But the conclusion of corollary can be false when irreducible (consider the case where 1)} = n)
11n{(,
}1n{(02
is not
Gu
0_
and
= 1 - n
.
We now return to the main line of this section and state its main result, namely
THEOREM 7.4. GP
Let
p
be a probability measure on
is irreducible and such that
is finite. Then
if and only if there exists a closed normal abelian subgroup
Y1 = Yd
A of
f Log+ IIMII du(M)
G1(d,IR) such that
Gu/A is compact.
such that
G11
Y1 = Yd
We shall see in the proof below that with the notation (13)
if and only if,
,
M(k)I-1/r M(k))
{(Idet M(1)I-1/r M(1),...,Idet
is contained in a compact subgroup of
; M e H}
{Gl(r,IR)}k. This condition is
easily tractable on concrete examples. We thus have a complete description of the probability measures for which
Proof
:
H
Consider the subgroup
distribution of
S
and the stopping time
. We first prove that
given in Lemma 7.1
Yi = Yd
which are
T
GA = H , where
is the
A
_
T
We have seen in the proof of this lemma that Gu = Tu1H Tuf1H = TX
Tuf1H
so that
is contained in
that
Tu
find
N
Thus
A = NM
in
Tu
and
is in
A
is contained in Tut GX
in
Tuf1H
H
A
Tut GA
M
and that
This first implies is in
we can
Tu
M = N -t A , using
is in
identity matrix is in the group G , Tut Therefore
.
(because if
such that
so that
G
G = T u iH
GA). Secondly, since the
is contained in
is a semigroup which contains
Tu
and
Tut GA Tut
, which
74
yields that
A = N 1M
write
Tpf1H
in
H = GX
with
N
is in
N
,
T-1
is contained in
G11
and M
Tu
in
A
and
G),
GA . Pick now an A
GX. Since
in
itself is in
G
.
in
H
and is
N = A '
This proves that
.
We now prove the theorem itself.
may assume that in that case
,
is carried by
p
Without loss of generality we
(M E Gl(d,]R)
is equivalent to
yi = yd
Suppose that for some
i
1
,
; M E H}
is not confined in a compact subgroup of H = GX
(13)). Then since
,
and,
the set
<_ i < k ,
M(i)I-1/rM(i)
{Idet
Idet MI = 1}
y = 0
Gl(d,]R)
(we use the notation
we have by (iv) of Lemma 7.1 and by
Furstenberg's theorem 6.3 (I)
Y
> 1 IE(Log Idet S(1)I)
I
T
Therefore, since all the exponents
]E (T)y
< j < k , are equal to
y(])
,
k
rk IE (T)y >
E
lE (Log Idet STI) = ]E (Log Idet STI) = 0
i=1 and
y#0
.
Otherwise consider the homomorphism
H -> (Gl(r,]R)}k
$
defined by M(1)I-1/rM(1),...,Idet M(k)I-1/rM(k))
(M) = (Idet
If its image is contained in a compact subgroup, it is obvious that for
i, 1S i < k ,
any
IE (Log Idet STi) I) so that
rk 1E (T)y = ]E (Log Idet STI) = 0 and
y = 0
.
Let
A
denote the kernel of
an abelian closed subgroup of is easy to check that
A
proof of the fact that if subgroup of ff
:
Gu
G11 -> G)A /A
hence
such that
Gu
such that
is normal in
A
.
GP
.
Gp/A
A
is
is compact and it
Finally we sketch the
is an arbitrary closed abelian normal Gu/A
is compact then
be the canonical map. H/Hn A
it is compact. Let
It is clear that
yl = yd . Let
is isomorphic to
ir(H)
75
then
H1/A1
Al = (M(1) C Gl(d,7R)
;
M e Hf A}
Hl = fM(1) e Gl(r,1R)
;
M c H}
is compact and
is abelian. Using the Markov Kakutani
Al
theorem and the Haar measure on H1/A1 , we can construct a probability measure
m
on
such that, for each M
P(lRr)
M(1) m
H
in
m
=
But this implies (Use Exercises 2.4 and 3.6) that {Idet M(1)I-1/r M(1)
; M e H)
is compact. One then deduces as above that
y1 - Yd
As a consequence we obtain the following result of Furstenberg COROLLARY 7.5.
With the hypotheses of Theorem 7.4 , Yl = Yd
only if there exists a probability measure Mm = m
M
for each
Sketch of proof
:
in
GP
in the proof above. For any
,
x
Mx = x If
A
let in
be the kernel of
constructed
V1
M
for any
is the Haar measure on
p
if and
P(1Rd) such that
.
y1 = Yd
If
m on
.
in
A .
we can define m by
Gu/A
m = J g S_ dp(nr(g))
(where
Tr
:
Gu > Gp/A
is the canonical projection).
Conversely if there is some measure
each M in
Gu
then
Mm = m
for each
we can consider the image ml of m M(1)ml projection. We have = ml for end of the proof above) that the restriction
m'
for each M
H
in
of
Y1 = Yd
m M
on
such that
in
If
.
If
m(V2 e ... ® Vk) # 0 satisfies also
and we argue in the same way with V2
,
m(V2 ® ... ®Vk)=0
and this implies (see the
m on V2 ® ... ® Vk
if possible its projection on
for
under the orthogonal
V1
M c H .
H
Mm = m
m'
Mm' = M.
considering
and so forth.
From the foregoing it is natural to introduce the following class of groups (the reader is refered to Greenleaf [29] and Glasner [27] for details. It is shown in the latter that the definition we give is equivalent with the more common one).
76
DEFINITION 7.6.
A topological group
exists on each compact G-space
gm = m for each
in
g
G
is said to be amenable if there
G
a probability measure m such that
B
.
If there exists a closed normal abelian subgroup of G/A
is compact then
such that
G
is amenable. Therefore it is clear that we can
G
express the foregoing results in the following way.
Let
COROLLARY 7.7.
be an irreducible closed subgroup of
G
Gl(d,IR).
The following are equivalent : (i) G is amenable .
A of
(ii) There exists a closed normal abelian subgroup that
G/A
G
such
is compact.
(iii) For each probability measure
Lyapunov exponents associated with
p
p
G with
on
GH = G , all the
are equal.
(iv) There exists a probability measure
p
on
such that all the Lyapunov exponents associated with
G with p
G11 = G
are equal.
One can find in Guivarc'h [31] a direct proof of the equivalence
(i) <_ > (iv). It is easy to deduce from the equivalence (i) <_> (ii) that an arbitrary closed subgroup
Gl(d,IR)
of
G
(or
is amenable
Gl(d,G))
if an only if there exists a nested sequence {e} = Ho C H1 C ... C Hn = G of closed subgroups Hi such that
any
Hi+l/Hi
of
such that
G
H.
is normal in
Hi+l
and
has a cocompact abelian normal closed subgroup for
0 !i
exists an open normal subgroup such that
H
H
is irreducible and amenable there
G of
can be imbedded in some
G
such that
ARQ X K
G/H
where K
is finite and is compact.
This implies easily that there exists a recurrent random walk on with distribution
p1
satisfying
Gu = G
if and only if
G
G
is amenable
and the rank of the subgroup
A
3 (we say that
is recurrent if for almost all w, (Sn((O))
Sri = Yn...Y1
given in the corollary is smaller than
returns infinitely often in each neighborhood of the identity matrix).
CHAPTER IV
COMPARISON OF LYAPUNOV EXPONENTS AND BOUNDARIES
In the preceding chapter we have given a criterion ensuring that the two upper Lyapunov exponents are distinct. It will give us all we need for the study of limit theorems. But a sharp study of the behaviour at infinity of the random products
Sn
requires a precise knowledge of
the relations between all the exponents. For instance they provide all the limit values of runs through
1Rd
n Log IISn(w)xll, when w
is kept fixed and
x
(Osseledec's theorem) and determine the possible
boundaries towards which
S
converges.
n Therefore we digress from our study of limit theorems and prove
the theorem of Guivarc'h and Raugi which gives us a condition ensuring that two given Lyapunov exponents are distinct. We then study in all detail some important examples. In particular we consider the case of symplectic matrices which is often encountered in concrete applications (see e.g.. Chapter IV of Part B). Finally we proceed with a short
introduction to Furstenberg's theory of boundaries and establish its connection with the previous results.
This chapter will not be needed in the sequel of this part of the book (but has important applications in part B).
IV.1. A criterion ensuring that Lyapunov exponents are distinct
We first generalize the definitions given in Chapter III.
77
78
DEFINITION 1.1. in
in
T
in
T
is contained in
(ARM)(L)
L
.
p-contracting if there exists a sequence
T is
(ii)
such that
T
{Mn, n >_ 1}
converges to a rank-one matrix.
IIAPMn II-1 APMn
In other words
is p-strongly irreducible (resp. p-contracting)
is a strongly irreducible (resp. contracting) subset
{APM ; M E T}
if
be an integer
of proper linear subspaces of APIRd such that, for all M
L ,
p
We say that
is p-strongly irreducible if there does not exist a finite
T
(i)
union
.
be a subset of G1(d,IR) and
T
Let
{1,...,d-1}
APIRd. We shall give in the next
of the set of linear automorphisms of
section some criteria with which one can check these properties on examples.
The following important result is taken from Guivarc'h and Raugi
[34]
.
THEOREM 1.2.
matrices of
Let
where
Sn = Yn...Y1
Y11Y2,...
Gl(d,IR) with common distribution
1E(Log+IlY111)
is finite and that, for some
p
are i.i.d. random
p . Suppose that
in
{1,...,p-1} ,
p-strongly irreducible and p-contracting. Then, if are the Lyapunov exponents associated with
yp # - °°
(i) When
is
T11
>
Y1 > Y2 >
Yd
p , the following hold
YP > YP +1
,
(ii) There exists a unique p-invariant distribution
vp
on
P(APIR )
and AP JJMx Yl
i=1
dp(M) dvp(x)
- II Log
Ilx M
(iii) For any non zero
in
x
lim n Log IIApSnxIl n Proof
:
Let
a matrix in entail that
q
11
APIRd,
p =
Yi
E
be the dimension of
G1(q,IR). If Tp(q)
p(q)
,
a.s.
i=1 APIRd. We may view each
APY.
i
as
is its distribution, the hypotheses
is a strongly irreducible and contracting subset of
G1(q,1R). Therefore, if
exponents associated with
a1
>_ a2
denote the two upper Lyapunov
{APY , n 24 1) n
we obtain by applying
79
Corollary 111.3.4, Theorem 111.4.3 and 111.6.1
to this sequence
that
(a) Al > A2
.
p(q)-invariant distribution
(b) There exists a unique
on
v P
P(APIRd) and ((
11
Mx
11
dp(q)(M) dvp(x)
= JI Log
A 1
IIx II x
(c) For any non zero lim
APIRd
in
a.s.
Log IApSn xli
n It is obvious that the theorem will follow from these assertions once we have proved that
p
)1 =
yi
E
a2 =
and
i=1
P-1 E
-
yi+ y +i P
i=1
A short way to verify this is to write a polar decomposition of S If
Sn = Kn An Un with a2 (n)
a1 (n)
Let
basis
{ei,
1
>_
... ? ad (n) > 0
5 i < d}
.
n 0(d) and An = diag(a1(n),...,ad(n)),
in
, we have
APSn = (ARKn)(APAn)(APUn)
denote as usual the canonical basis of
A ... A ei
{ei
i1 < ... < ip S d}
of
APIRd
,
.
IRd.In the
the
P
1
matrices
Kn, Un
APKn
APUn
and
are orthogonal and
APAn = diag{ai (n)ai (n)...ai (n) 1
1
;
:5 i1 < i2 < ... < ip < d}
2
Therefore the two top eigenvalues of
{a1(n)a2(n)...ap-1(n)ap()}
2
and
(APSn) APSn
are
{a1n)a2(n)...ap-1(n)ap+1(n)}
2
This yields that (see Proposition 111.5.6) P
a1 = limn Log a1()a2(n)...a
n-
P-1
(n)a (n) P
i=1
Yi
P-1
and
A2 = lim n Log a1(n)a2(n)...a
P-1 n)a P +1
n
(n) = i=1 E y. + y P+1
The proof is complete.
Exercise 1.6 provides an example where and 2-contracting and where
Tu
is strongly irreducible
y2 = Y3
As a consequence of this theorem let us prove
COROLLARY 1.3.
Suppose that
Tu
is p-strongly irreducible and
80
p-contracting for any
p ,
S
Then one can find a polar
.
= Kn An Un
n
and An = diag(a1(n),...,ad(n)) , such that
Kn, Un E 0(d)
with
-< d-1
<- p
1
So = Yn...Y1 , namely
decomposition of
1E(Log+IIY11i)
{S**Sn}1/2n
< m , then
U. diag(exp(y1),...,exp(yd))UU
Proof
converges a.s. to
Y1 > Y2 > ... > yd
and
Consider any polar decomposition
:
write for any
APS
p
n
n
n
Unei A ... A Un
x
in
clear that one can find %*
E1
n
1
UnEn e1
n
converges for any
p
decomposition and to
111.5.6)
such that
.
E {1,-1}
E2 n
It is clear that U.
Y1 > Y2 > ... > Yd
converges to
An/n
{1,-1}
x
It is
.
such that 2n,* A Un(e1A e2)
such that
AP Un En...En(e1 A ... A e
converges to some
then
) < -
_
Un = diag(c1,...,Ed)Un
Let
.
Kn = Kn diag(En,...,En)
1E(Log+IIY1
we see that the Choose a unit vector
converges. With this procedure we find
E {1,-1}d
(En, En,..., Edn )
n
In the same way, since
x .
and
S
converges to
Unei
P(A2IRd) one can find
A e2)
A2Un s n E n (e
.
with values in
= E1(W)
of
n
If we apply (a) of
.
converges a.s.
e
actually converges to
converges a.s. in
n
n
)
{APYn, n > 1}
p such that the direction of
IR
= K An U
S
,L n
= (APK )(APA )(APU
Proposition 111.2.2 to the sequence directiond of
U n
Um . Moreover if
converges almost surely to some orthogonal matrix
Sn = Kn An Un .
p
and
is a polar
By Theorem 1.2, when
and (see Proposition
= diag(exp(y1),...,exp(yd)). By
A
definition )1/2n (S*
nn
= (U* A* K* K
n
n
n
U )1/2n = U*(A1/n)U
A
n
n
n
n
n
n
*
UU Ao U-
thus this sequence converges a.s. to
Exercise 1.4.
[56]. Let
1E(Log+1IY1I) < -
Y1,Y2,...
be i.i.d. matrices in
Gl(d,1R) with
.
K , K Y1 K
(i) Suppose that for any orthogonal matrix
and
Y1
have the same distribution. Show that Y, + ... + yp = 1E(LogljY1 e1 A Y1 e2 A ... A Y1 epll) (see I.2).
(ii) We suppose that the
d2
entries of
are independent
Y1
Gaussian variables with mean zero and variance one. Show that the hypothesis of (i) is satisfied. Fix an integer
p
,
1
<- p < d
.
Let
P
31
be the (random) projection on the orthogonal of the subspace spanned by
Y1e1,...,Y1eP1
and
be the orthogonal projection onto the span of
Q
Show that
e1'" .,ed P+1
.
Yp = IE (Log 11P Y1 epll) _ 1E (Log 11Q Y1 epI) d-p+1
Deduce that
yp =
]E (Log
2
p-contracting for any Exercise 1.6. e
4
=
(
0
Td-pp
Z {Log 2 + P
I)
.
Show that
}
-
Tu
is
.
With the basis
{e1 0), e2 = (10 0), e3 = (O (O one can identify the vector space M(2,]R) of ,2 X 2
0)}
0
p
Y1 > Y2 > ... > Yd p
_1/2))
,
(1 )i ,
i=1
Suppose
Exercise 1.5.
1
0),
1
matrices with
Is!
.
requiring that, for
Define A, B
'Y
S1(2,IR) x Sl(2,]R) -> GI(4,]R) by
:
Sl(2,1R), Y(A,B)
in
represents the linear endomorphism (i) Show that (ii) Let
closed in
'Y
Y(A,B)(M) = A M B-1
M(2,]R).
of
is a group homomorphism.
be the group
G
is the matrix which
'Y(Sl(2,]R)x S1(2,]R)). Show that
Gl(4,IR) and p-contracting for
p = 1, 2, 3
.
G
is
(One can use
Corollary 2.2 and the image of a pair of diagonal matrices). Prove that G
is strongly irreducible. (iii) Let
}i1,}12
be two probability measures on
A2 are the top Lyapunov exponents associated with
Sl(2,1R). If and
u1
that the Lyapunov exponents associated with the image under
Y`
are
p
u2 of
a1,
prove
p1 ® u2
X1X2' A11X2' x1x21, x11x21
(iv) Show that one can have
Tu = G
and
11X2
= A1A2
IV.2. Some examples
We first give some criteria which can be used to check that a set is p-contracting. We then study two important examples where all the Lyapunov exponents are distinct.
PROPOSITION 2.1. 1
< p < d (i)
,
T
Given a subset
T of Gl(d,IR) and an integer
the following statements are equivalent : is p-contracting .
p
82
(ii) There exists a sequence a2(n) >_ ...
a1 (n)
in
(Mn)
such that, if
T
are the square roots of the eigenvaZues
>_ ad(n) > 0
lim {ap(n)} ap+1(n) = 0. v(iii) There is a compact subset C of Gl(d,]R) such that
of MnMn , then
P E C , M e T Proof
:
Q e C}
,
{P M Q
is p-contracting.
If (i) holds one can find
such that IIAPMn1I-1APMn
T
in
(M 1)
converges to a rank-one matrix. Write a polar decomposition M =K A U
n
.
n n n
We have
APAn = diag(a1(n)a2(n)...ap(n), a1(n)a2(n)...ap-1(n)ap+1(n),...)
so that
119M nII
(n)
-1
ApAn = d iag(1
ap+1 a (n) P
,
indicate positive numbers smaller than point of
APK
and
APU
...), where the dots
ap(n) lap+1(n). Since each limit
is an orthogonal matrix,
n n converges to a rank-one matrix, i.e.
IIAPM II-1 A PA
n
n
lim ap(n)-la p+1(n) = 0
.
This proves that (i) implies (ii), and (i) follows from (ii) by the same reasoning. Remark that
ap(n)
IIAPMn112
',p+1(n) Thus, if
Qn
and
Pn
IIAp-1Mn II 119+'Mn II
are in a compact set, there exists
a > 0
such
that IIAPPnMnQn1I2
a (n)
p
1
a ap+1 (n) for each
n
a (n)
p
aP+1(n)
IIAP1PnMnQnII IIAp+1PnMnQnII
This implies that (ii) and (iii) are equivalent.
.
The following corollary is particularly useful (Guivarc'h has announced that the converse holds under an irreducibility assumption on
T). Recall that an eigenvalue
if
Ker(M-A Id)
say that
A
of a matrix M
is one-dimensional and equal to
is dominating if
COROLLARY 2.2.
A
A semigroup
IXI > IA'I
T
in
is called simple
Ker(M - X Id)2 . We shall
for any other eigenvalue V.
Gl(d,]R) which contains a matrix
with a simple dominating eigenvalue is contracting. If there exists in a matrix with
T
d
eigenvaZues of distinct moduli, then
p-contracting for each Proof : Let A
.
p
,
1
M be a matrix in
< p < d-1 T
is
with a simple dominating eigenvalue
It follows from the Jordan decomposition that
to a projection matrix with range
T
.
Ker(M - A Id)
.
11M2n11-1 M2n
converges
Since this space is one
83
dimensional, {Mn If the
, n >_
is contracting (see Proposition 2.1).
1}
d eigenvalues of
M have distinct moduli, then
a dominating simple eigenvalue for any is
p
,
1
< p < d-1
APM has
Therefore
.
T
p-contracting.
Tutubalin was the first to show that when p
has a density on
S1(d,]R) with respect to the Haar measure, all the Lyapunov exponents are distinct (see Sazonov and Tutubalin [63], Tutubalin E69]). He
assumed moreover that
is finite, and Raugi (see [59])
IE (Log2 IIY111)
dropped this condition. But, when
has a density, T11
p
is open in
Sl(d,]R). The following thus shows that we recover their results.
If
PROPOSITION 2.3.
contains an open set of
T11
S1(d,]R), then
is p-strongly irreducible and p-contracting for any Proof : Let
V
be an open set in
show that, for
n
S1(d,]R) contained in
large enough,
is contained in
.
p
Let
.
We will <_ n1
have distincts moduli. Since
this will prove, by Corollary 2.2, that
T11
p-contracting for any we can find
Tu
Vn = {M1M2...Mn ; Mi E V
contains a matrix whose eigenvalues
T11
p e (1....,d-1}.
Tu
Vn is
M E V , with the Jordan decomposition
Q E Gl(d,]R) such that Al
Q M Q
A2 1
=
A
0
r I
where, for each A. e ]R, or a
i c {1,...,r} 2x 2
A. = A. R(oi) If E 1
E 1 ,
E2,
..
... Er = 1
,
,
Er
is either a
A.
,
matrix
1x 1
A.
matrix which can be written with
R(6) = (sin
cos 6
o
are real numbers close enough to
1
with
then E1A1 E
M'
2
A 2
E A r r
0
is still in the open set
Q V Q-1
and we can choose
such a way that all the eigenvalues of can find
P E Gl(d,IR) such that
M'
P V P-1
ell
..
, Er
in
are distinct. Therefore one contains a matrix
84 0
B1
B2
(1)
0
with
or
B. = (3i
for instance
B
1
1
such that if
R(a )
(i
r
. We first suppress the rotation terms. If
(ii R(ai) =
B
is open we can find
P V P-1
since
,
E:> 0
1
18- all < e
then
,
b1 R(e) I
B
2
0
is still in
P V P-1
we can find
el,...,en
is a positive integer such that
n
If
.
such that
R(ei)R(82) ... R(en) = Id
.
lei-ail < E
This implies that
is in
If we carry on this procedure we find an integer m contains a diagonal matrix. We then perturb contains which has
P Vm P-1
P Vn P-1
.
Br
0
L
obtain an element of
2n E > 27r
and
such that
P VP-
slightly this matrix to eigenvalues of distinct
d
modulus.
Finally we prove that irreducible. Since which contains
V
Sl(d,]R) is
is easy to check that
Tu
,
or equivalently
G11
,
is p-strongly
is connected, the only subgroup of Sl(d,1R)
Sl(d,1R)
itself. Therefore
GP = Sl(d,]R) and it
Sl(d,]R) is p-strongly irreducible (use e.g.
Exercise 2.9).
M F- Tu}
{ldetMl-1/dM
It is of course sufficient to require that
Remark 2.4.
contains an open set of
{M c Gl(d,]R); Idet MI = 1)
to
obtain the conclusions of the proposition.
We have net in 11.5.1 and 11.7 probability measures which satisfy clear that that unless 1.3 hold for
EK * P * e K-1 = u
KTPK 1
= Tu
for any
is supported by
1i
p
for any
K
in
S0(d)
11 on .
S1(d,]R)
It is then
K . Therefore the following shows
S0(d)
the conclusions of Corollary
(see an interpretation of this in Section 4). Actually,
85
using some analysis one can deduce this result from the preceding proposition. But we prefer to prove it directly since the method can be used in other contexts.
PROPOSITION 2.5.
Let
be a semigroup in
T
Gl(d,]R). If there is a
matrix M such that (i)
IIMII-1 M
is not orthogonal
(ii)
For any
K e SO(d)
then 1
is in
1
T
is p-contracting and p-strongly irreducible for each
T
< p < d
Proof
, K M K
If
:
p
.
is the smallest semigroup which contains {K M K 1
T'
K E SO(d)}, KT'K 1
for any
= T'
K e S0(d)
and
Without loss of generality we will suppose that
,
is contained in
T'
T'-= T
T.
Using the
.
polar decomposition it is clear that one can write M = U
with
B U2 1
U11 U2
in
B = diag(bt,...lbd)
and
SO(d)
Assumption (i) implies that there exists for any
i # j
exists
.
We claim that for any
K e SO(d)
j
where
bi
are real numbers.
such that
K1,...,Kn
in
Ibjl > Ibil
such that
...
K(K1 B K1 1) (K2 B K21 )
(Kn B Kn1 ) c T
(2)
.
For proving this by induction it suffices to show that if K E SO(d)
there is some
U1K 1 P KU11 ET and
K' a SO(d)
such that
T
P C T e T
K'P K B K 1
and .
But
whence
U1 B U2 E T
K U2(U1 K 1 P KU11) (U1 B U2)U21 K 1= KU2 U1
is in
there
SO(d)
K-1
PK BK 1
.
Since
bjl >
Ibil
for
matrices, one can find
it is clear that for
i # j
enough and for a convenient
)
large
K1,...,KR-l among the permutation
of
A = (K1B K1
m
...
(Km B
)
such that
m A = diag(at,...,ad) By (2)
K A is in
,
and
for some
T
positive integer n Kn-1An-1 E T since
,
KnAn
{KnAn
K
is in
a1 > a2 > ... > ad > 0 SO(d)
in
.
.
Let us show that, for any
This is true for n = and if AK = K 1(KA)K e T then Kn 1An-1AK is in T
KnAn = K(Kn-1 An K)K 1
clear that
with
, n >_ 1)
T
.
1
,
is in
T
.
, hence
T
,
By Proposition 2.1 it is now is p-contracting for any
p
86
We now prove that say that the group K2A2
generated by
G
are in
KA
and
is p-strongly irreducible. This amounts to
T
T
U
is obvious that for each A 3K 3K3A c G
A2
so
is p-strongly irreducible. Since
T
G
.
Suppose there exist proper subspaces V1,...,Vr i E {1,...,r)
for any
and
(it
SO(d), U G U 1 = G). This implies that
in
is in
K3A = K(K2A K 1) E G
and
K2A K 1 E G
,
G
in
Y
,
AP1Rd such that
of
for some
APY(V.) = V.
j
J
iff
{1,...,r} (defined by
induces a bijection of
A2
Since
we can find an integer m
= V .)
(APA2)(Vi )
A2(i) = j
Am keeps
such that
J
fixed each of these points, that is, writing D
APD(V. )= V. Since
,
1
Am
for .
is connected this implies in particular that
SO(d)
AP(U Dn U-1 XV
)
= V1
1
and positive integer
U
for any rotation
lim IIAPD-II-1 APDn
n
Notice that
.
is the orthogonal projection
e1 A e2A ...Ae
on
P
n-*-
P
and therefore (APU)P(APU-1)V1 = V1
V1 = APIRd; for notational convenience we just prove this
This yields
for p = 2
.
If w = E i<j
.
e. A e. is in
X.
i
1,3
consider a rotation matrix
and
V
J
A.
1,3
1
U which satisfies
#0
U(e1) = ei, U(e2) =± e j,
Then
(APU)P(APU 1)w = A.. ei A e.
spanned by a subfamily of hold for each
(APU)V1
(if for instance such that see that
U(e
i
)
{ei A e.
where
1
Therefore
.
< i < j
<= d}
and
e1 A e3
V1
2,3
is in
U(e
,
must be
V1
The same must
.
V1= APIRd
then choosing
,
e2+e3
ei
e1 A (e2+e3)
,
V
U E SO(d), which implies that
e1 A e2 E V1 =
is in
, U(e ) _
2
U
e2-e3 , we
3
e1 A e2
(A2U)(V1), so that
and e
Ae 3 1
are in
(APU)(V1)
(APU)(V1)
,
a
Exercise 2.6.
but
e1A (e2-e3) = f(A2U)(e1A e3)
is not in
contradiction). The proof is complete.
Define the map
0
:
APIRd -> Ad-PIRd
by the relation
u A v = q(u),v> for any
is a linear homomorphism.
u c APIRd , v E Ad-PIRd. Show that
Prove that, for any M
in
Gl(d,IR) and any
u
in
(det M) 0 (AP M u) _ (Ad-P M*)-1 0(u)
APIRd .
87
Show that if then
T
is a p-strongly irreducible subgroup of
T
Exercise 2.7.
If
p
k_1 distribution of
be a probability measure on k
A e2 A ... A ep
11-invariant. Show that when
if
Let
only if APIRd
P(ApIRd) where
on
SO(d)
p
< p < d)
is a
ma
ma
is
SO(d)
,
Gl(d,IR). Show that,
is p-strongly irreducible if and
T
,
K
Prove that
P(APIRd).
be a connected subset of (1
.
is p-irreducible (i.e. there is no proper subspace
T
such that
p = 1
exists some
for any M
ApM(V) = V
T). Hint
in
:
V
for any
of p
is connected thus it suffices to consider the case p= 1.
{ApM , M E T}
When
T
is an integer
p
denote the
is not confined on
the only p-invariant distribution on Exercise 2.9.
Sl(d,1R) such that
ma
SO(d) . Let
in
random matrix whose law is the Haar measure on is
Gl(d,]R), prove
(d-p)-contracting.
is
for any
= p
Ek * p1 * E
E T}
1
Let
Exercise 2.8.
is a p-contracting subset of
T
T-1 = {M ; M
that
Gl(d,IR),
(d-p)-strongly irreducible.
is
use the fact that if such that for an
q
T
w
is not strongly irreducible there in
(1d)
is finite.
IV.3. The case of symplectic matrices
Let
Sp(d,IR) denote the group of order
i.e. matrices
M
in
G1(2d,]R)
symplectic matrices,
d
d E
2d
which satisfy
Mei A Mei+d
.
ei A ei+d
i=1 It is clear that the action of
Sp(d,]R) on
A2IR2d
is not irreducible
d
since the space spanned by lEl ei A ei+d
is invariant. Thus Proposition
1.2 is not suited for the study of Lyapunov exponents associated with random symplectic matrices. Since these matrices occur in many applications (see e.g. Chapter IV of Part B, and the references in Tutubalin [68]) it is worth proving a criterion adapted to this situation. Before stating it let us recall some basic facts. It is immediate to see that if
88
I
0
O
where
is the identity matrix of
I
Sp(d,1R)= {M E: G1(2d,3R)
transpose of
*
of M so is
(iii) There exist two orthogonal matrices
>= ad
Proof : We have thus
1
M- J(M )
eigenvalue of
K and U
in
such that
1
M = K A U
M*JM = J
so that
= J
MJM
and
If
M M
= -J
(iii) note that the above
such that, if a
{f1,...,fd, if1I... ifdI
are the eigenvalues of
1
is an
A
relation implies that we can find an orthonormal basis of form
J
(Jv)
To verify
.
.
But
.
v we have
M* J My =
M
1
M 1J 1(M*)-1 = J
= J , proving (i)
M with eigenvector
is an eigenvalue of
Sp(d,IR)
with
A = diag(ai,...,ad, a1'. ..,ad
M*(Jv) _ and
denotes the
Sp(d,IR) .
and a diagonal matrix a1 > a2 > ...
M*
(recall that
Sp(d,IR), then
is an eigenvalue
A
,
.
is in
(ii) If
Gl(d,]R),then
M* J M = J}
Given a matrix M in
LEMMA 3.1.
(i) M
M)
;
IR2d of the
a2 >- ...
greater than one, then
>_ as >
1
M*M f. = ai f. 1
and M M J fi = ai-2J fi
.
Let
denote the orthogonal matrix defined
U
by the relations U fi = ei
U J fi = -(ei+d)
,
.
We have * (U
* J U)fi =U* J e i = -Uei+d =J fi
(U* J U)(J f i )
and
-U* J ei+d
=
=
-U*ei
= -fi = J(J f i )
.
*
Therefore
a11,...,ad M = K A U .
U
J U = U and
Since
and
U
is symplectic. If
K = M U-1 A-'
K
,
A
and thus
A = diag(a1,...,ad
K
are symplectic and
is easily seen to be orthogonal, the proof is
complete.
From this lemma one readily deduces PROPOSITION 3.2.
If Y1 >_ Y2 > ... > Y2d
are the Lyapunov exponents
89
associated with symplectic random matrices, then Yd+i-1
p e {1,...,d}
For any
for
-Yi
be the subspace of
Lp
let
i = 1,...,d
.
A1IR2d
{Me1 A ... A Mep , M e Sp(d,IR)}, which is sometimes called
spanned by
the p-Lagrangian submanifold (it is known that u1 A u2 A ... A u is * p <_ p iff <_ i, j in L u. J U. = 0 for any , but we won't use it). 1
L1 =
Notice that
DEFINITION 3.3.
IR2d.
T of
We say that a subset
Sp(d,IR) is
Lp-strongly
irreducible if there does not exist a finite union W of proper linear
subspaces of Lp
such that
M in
(APM)(W) = W for any
T
.
The main result about random products of symplectic matrices is
PROPOSITION 3.4.
Let
Suppose that that
be an integer in
p
Lp-strongly irreducible and
.
For any non zero 1
lim
in
x
L p
p
Log IJAp Yn...YJ xii = E Yi
There exists a unique
vp
on
(
Log
J
Proof : Let Lp
li-invariant probability distribution
k
II
_
AP MxII
dp(M) dv (x) =
be the dimension of
in
.
1
{f1,...,fk}
and
Lp
M
Y.
E
i=1
p
IixII
to be chosen later on. For any
matrix in
and
{x e P(APIR2d); x e L
P(Lp)
.
i=1
n(iii)
{i.....d},
is finite. Then the following hold
Yp > Yp+1
(ii)
and
2d
is p-contracting and
Tp
IE(Log 11Y111)
(i)
be a sequence of i.i.d. random
{Yn, n >_ 1}
symplectic matrices of order
Sp(d,IR)
let
a basis of
M
denote the
Gl(k,IR) whose entries are
M
1,j
=
f1 .
,
f .>
1
]
9-
i, j 5 k
.
The proof of the proposition is broken in three steps. First step : We shall show that for any M
then
II
II = IIA2 MII
in .
Sp(d,IR), if (M)= APM,
90
Let us first prove this when
fl = el A e2 A ... A ep
p < d. Let
A e
f2 = el A e2 A ... A e
and
P-1
Since all the matrices P+1.
(3)
Y B*-11 11C
with
B, C
in
find such a L P
are in
and
Gl(d,IR)
for which
Y
If we write
.
f2 = APY f1
p < d
since
.
with
M = K A U
A = diag(al,...,ad, a1...... ad1)
Sp(d,IR)
symmetric are in
C
This shows that
K, U
f1
, we can and
f2
and
Sp(d,IR) fl 0(d)
in
a1 > a2 > ... >_ ad > 1, we have,
,
,
II (M) II = a. . .ap = II (APA) (fl) II = IIA fl II < IIAII IIAPAII= II
= IAII ,
but IIAII = IIKAUII = IIMII and
II
IIc(M) II=IIMII
On the other hand IIA2
(M)II = (a1...ap)(al...ap-1ap+1)
(4) = IIA(f1 A f2) II < IIA2 All <
so that IIA2 (M) when
p < d
=
IIA2 All = IIA2 KAUII = IIA2 M11 ,
p = d
If
.
and
such that
C
.
If
Ci
proving the claim but
f1 = el A ... A ed
we choose
f2 = e1 A ... A ed-1 A e2d B = Id
IIA2 (M) II
Y = 0
is the matrix of the form (3) with except
we see that
Cd,d = 1
J
APY fl = fl + f2
.
Therefore
f2
is in
L
.
The same proof as above
P
works with this new choice because (4) is replaced by
a d
Second step
:
Let
p
denote the distribution of
Y1
Let us show that
.
Tu
is a strongly irreducible and contracting subset of
Tp
is
Lp-strongly irreducible, it is clear that
Gl(k,IR) .
Since
T- is strongly
irreducible. On the other hand we know (see Proposition 2.1) that, since T
is
p-contracting, there exists a sequence
(Mn)
n-w-11 n
II (Mn) II 2 IIA2
(Mn) II -1 = `°
lim IIMn1I2 IIA2 MnII-1= co Third step
:
This implies that
in
such that
T
u
so that T- is contracting.
We now prove the proposition. It follows readily from the
statement proved in the first step and from the definition of the Lyapunov exponents (see especially Proposition 111.5.6) that the two upper Lyapunov exponents associated with
(AP Yn)
and with
(Yn)
are
the same. Therefore (see e.g. the proof of Theorem 1.2) they are equal
91
p-1
p E
to
and
Y,
i=1
E
i=1
1
y. + y 1
The proposition is thus an immediate
.
p+1
consequence of Theorem 1.2 applied to the sequence
(with
(Yn)
p = 1
in this theorem).
The fact that this proposition is suitable for studying symplectic random matrices is clear from the proof we give of the following result of Guivarc'h and Raugi [34].
PROPOSITION 3.5. that
Tu
If
p
is a probability measure on
contains an open subset of
Sp(d,IR) such
Sp(d,IR) then all the Lyapunov
exponents are distinct. Proof
:
Taking into account Proposition 3.2 it suffices to show that
for i = 1,2,...,p. For this we verify that the assumptions yi > yi+l of Proposition 3.4 are valid for any p e {1,...,d}
a. We first prove that equivalent to the fact that
T11
is
Lp-strongly irreducible. This is
G
is
LP-strongly irreducible. But
u Sp(d,IR)
Gu = Sp(d,]R) (for
being connected,
Gu
contains an open
set by hypothesis). Applying Exercise 2.9 we see that we have to show V
that there is no proper subspace for any
M
A = diag(a1,...,ad, a1',...,ad1)
,
(Ap An)(V) C V
implies either that
V
for any
If
.
al > a2 > ... > ad > 1
the relation
,
n > 0
V , but then
is in
... A Mep , M e Sp(d,]R)} C V
is not proper, or that
V
(APM)(V) C V
such that
Lp
,
el A ... A ep
{MeI A and
of
Sp(d,IR). Suppose there exists such a
in
is in the orthogonal W
el A ... A e p
of
V
.
Butthen by (i) of Lemma 3.1, if <Ap M w ,
for any
M
in
w e W
v> = <w, AP M* v> = 0
Sp(d,]R)
and
is in W
Therefore Me A ... A M e
.
= W . This contradicts also the fact that
L
for
p
1
any M
v e V
and
V
is proper.
P
b. We now show that to
verify that
Tu
T11
p-contracting. For this it suffices
is
contains a matrix
M with
eigenvalues whose
2d
moduli are distinct (see Corollary 2.2). We first show that if U an open subset contained in
T11
,
then there is in
U
is
a matrix with
distinct, eventually complex, eigenvalues. We may of course suppose that -1
is not an eigenvalue of the matrices in
U
.
In that case the
92
mapping U
l
U -> M(2d,]R)
:
defined by 1(M) = (Id-M)(Id+M)-'
onto an open subset of
S = {M a M(2d,]R) , M*J + JM = 0}.
M
Since the eigenvectors of
and
to show that each open subset of
eigenvalues. For each M
in
let
S
are the same it suffices
*(M)
contains a matrix with distinct
S
be the resultant of the
P(M)
caracteristic polynomial associated with M
its derivative (see
and
M has a double eigenvalue iff
e.g. Lang [45j).
sends
is a polynomial on the vector space
S
P(M) = 0. But
It can be zero on an open set
.
if and only if it is the zero polynomial. This would imply that each matrix of
has a double eigenvalue, which is false since
S
diag(al,.... ad, -al,...,-ad)
is in
M
So we have found a matrix
-L ''d' a '" ''a 1
Suppose that
Let
of
U
with distinct eigenvalues are the real elements
A2k+1' " ''Xd
d
1
al,...,ad
for any
S
in
{fl,f2,...,fd' hl,...,hd}
be a basis of
2d
such that M(f2j+l + i f2j+2) = A2j+1(f2j+l + i f2j+2)
j = 0,...,k-1
.
M(h2j+l + i h2j+2) = X2j+l(h2j+l + i h2j+2)
j = 0,...,k-1
M fj = A. f.
j = 2k+1,...,d
M hj = A.1 hj
It is immediate to deduce from these relations that for any r,s, r# s ,
f*Jf r f*Jf r r s
and
If we replace by
hr
P(ei) = fi
Therefore if
M'
Sp(d,1R)
.
=0
,
a r
hr
f*Jh =0 r s f*Jh r r = ar # 0. this means that the matrix
P(ei+d)
1
<= i < d
= hi+d , is the matrix which represents
{fl,...Ifd' hl,...,hd} of
,
-1
by
and
=0
, M = P-1M' P
Moreover
M'
,
defined
M
in the basis
is in the open subset
P U P-1
can be written as
rB
0
Bd
M' =
M'
P
is symplectic.
,
B-1 1
Bal
0 where
Bi
is either a real number
Si
,
or a
matrix
2x 2
Si R(Oi)
.
Working as in the proof of Proposition 2.3 (with obvious modifications) we see that for some
n , P Un P-1
,
and therefore
Un
,
contains a
93
Exercise 3.6. then
eigenvalues whose moduli are distinct.
2d
matrix with
Show that if
:
contains an open set of
T11
Sp(d,]R)
is strongly irreducible.
Tp
IV.4. p-boundaries
We now return to the study of general random matrices. We shall be interpreting the results of Section 1 as informations on the behaviour of products
Y1...Yn
towards boundaries of
Consider integers
G1(d,3R).
_<_ pl < p2 < ... < pk < d
1
. Let
F(pl,...,pk)
denote the subset {(x Ax A...Ax 1
of
2
pl
,x 1
A...Ax
pl A...Axp2 ,...,x 1 A...Axpl A...Ax pk ) ... x P(Apk ]Rd)
P(Apl ]Rd) x P(Ap2 ]Rd) x
.
; x
where
(E1,E2,...,Ek)
E.
is a
,
a ]Rd}
It follows from Lemma k-uples
F(pl,...,pk) may be identified with the set of
111.5.1 that
i
pi-dimensional subspace of
3Rd
and
where
E C ... C Ek .
E
For instance
F(1) = P(3Rd)
subspaces of
]Rd.
and
is the set of
F(r)
F = F(1,2,3,...,d-1)
It is clear that each obvious action. Note that
F(pl,...,pk)
ply
is called the flag manifold. is a
e1A...Ae A...Ae
As such it is compact. At last note that
p1
is the group of upper triangular matrices and Gl(d,]R)/S
PROPOSITION 4.1.
is
Let
Y1,Y2,...
p
...,e1A...Ae
).
pk
F
is the homogeneous
p
.
be independent matrices in We suppose that, for any
p-strongly irreducible. Let
the integers
of
.
with the same distribution 111
P2'
0(d)
e(1,2,...,d-1)}
S = {M a G1(d,]R) ;
space
Gl(d,]R)-space for the
is the orbit under
F(pl,..,pk)
e(pl,...,pk) _ (e1 A ... A e
r-dimensional
such that
1 u
is
1
Gl(d,]R)
p e {1,...,d-1},
< p1 < p2 < ... < pk < d
p-contracting. Then
be all
94
(i) There exists a unique F(pl,...,pk)
p-invariant distribution
on
v
.
(ii) With probability one the measures
Y1...Yn V on
F(p1, ...'pk)
converge weakly to a Dirac measure. (iii) If
on
< ql < q2 < ... < qR < d
1
contained in
{p1,. '-1pk}
and
then for any
,
{ql,...,q,,}
is not
p-invariant distribution
F(gl,g2,...,qR) , the probability that
v
converges to a
Y1...Yn V
Dirac measure is zero. pi Proof
:
If we apply Theorem 111.4.3 to the sequence
we know that there exists
in
Z.
converges in probability to
1
Z.
P(Ap1 1Rd)
uniformly in
,
implies that there is a subsequence
(n.)
*
Yn , n ? 1}
{A
such that Apl Y ...Y x 1 n pi d 1R ). This
x e P(A
such that for any
x
p lim A 1 Yl...Yn
n-Therefore pi
x = Zi
is a decomposable pi vector. Repeating this for each
Z.
we find a point
Z
F(pl,...,pk)
in
lim Y1...Yn
nfor any
C
in
such that
= Z
(5)
3
F(pl,...,pk), with probability one. If
p-invariant probability measure on V-almost all
a.s.
3
v
is an
holds for
F(p1,...,pk)' (5)
with probability one, by Fubini's theorem. Therefore
E
lim Yl...Yn V = 6Z
a.s.
A compacity argument yields lim Y1...Yn V
=
6Z
this proves (ii) but also implies (i) (V must be the distribution of Z). To prove
probability projection of
(iii) suppose that # 0 v
,
(resp.
Z)
on
{p1,...'pk}
qi
lim Y1...Yn v = 6Z
If
V
P(Agl 1Rd)
and that, with
(resp. Z) ,
is the
then with non zero
probability
limAglYl...Ynv= 6_Z_. But
v
is proper (see Proposition 111.2.3) and this would imply (see
Exercise 111.1.12) that
Tu
is
qi.-contracting. This is a contradiction.
95
Consider now symplectic matrices of order 1
2d
.
Put, for integers
ep ),...,M(e1A...Aep )),
k
2
1
1
M e Sp(d,]R)}. Notice that
is the
L(p)
p-Lagrangian submanifold.
Using the proof
of Proposition 3.4 one shows as above the following
PROPOSITION 4.2.
Let
{Y , n >_
p e {1,...,d} , TV 1
is
is
< p1 < ... < pk < d
be i.i.d. random matrices in
11
Sp(d,]R) with common distribution
p
.
We suppose that for any
LP-strongly irreducible and denote by all the integers
such that
p E {1,...,d}
Tu
p-contracting. Then
p-invariant distribution
(i) There exists a unique
(ii) Y1...Yn V converges to a Dirac measure, a.s. (iii) If
contained in
< q1 < q2 < ...
1
{r i'" .,Pk}
< qt < d
then for any
v
.
{g1,...,gR}
and
on
is not
p-invariant distribution
v on
L(g1,...,gR), a.s. , Y1...Yn v does not converge weakly to a Dirac measure.
Exercise 4.3.
:
Show that, under the hypotheses of Proposition 4.1,
p-invariant distribution on
(i) There exists a unique
P(Ap1 ]Rd) (ii) If
x
P(AP2 ]Rd) x
... x
P(Apk ]Rd)
is a polar decomposition then
Y1...Yn = Kn An Un
converges a.s. in
F(p1,...,pk).
We shall now define, following Furstenberg, the
11- boundaries. We
begin by a somewhat informal discussion. Let
G
denote the group
random elements of Zn = Y1...Yn
G
and, for
Gl(d,]R).
If
g
in
G
,
are independent
Y1,Y2,...
with a common distribution
P
we put
Zo = g, Zn = g Y1...Yn. It is readily
is a Markov chain on G. Proposition 4.1 tells {Zg, n ? 0) n us something about the behaviour at infinity of this Markov chain. seen that
Namely we have found G-spaces
B
such that if
V
is a
p-invariant
96
distribution on
Zn V ->
and define a map T(x) = d
`Y
denote the set of probability measures on
G U B -> M1 (B)
:
by
F(g) = g v
topology 'r on
which induces on
G U B
and for which
G
and
if
B
g E G
consider the coarsest
x E B. Following Furstenberg [71]
if
x
M1 (B)
almost surely. This yields that
W > 6Z
, Y1...Yn V
B
Let
their own topology
B
is continuous. It is not difficult to see that with
`Y
this topology is open and dense in
(a) G
(b) With probability one, Thus the pair
with p
(B,`7,
GU B
.
converges to
Zn
g.Z
GU B
in
appears as a kind of boundary of
)
associated
G
.
Proposition 4.1 provides us with such boundaries towards which converges a.s.
G-spaces
.
F(p1,...,pk)
n
one can wonder if we have exhausted all possible
"boundaries" associated with
(G U B,')
Zg
If it was, in some way, natural to introduce the
p
In other words, are there other
.
for which (a) and (b) hold which give us more information ?
We want to show that the answer to this latter question is essentially no. This will not be needed later so we shall be rather sketchy. (The
interested reader should consult the works of Furstenberg [20], [22], [23], to have a full account of these problems). We first state this
[24]
question in a more tractable form. Suppose that on
G U B
(a) and (b) hold. It is natural to impose that
,
converges to
have
is a compact G-space such that for some topology
G-space and in this case
itself a
of
B
a.s.
Z
.
so that
{Zn, n >_ 1)
Y1Z(1) = Z
Z E B
.
G , Zn
converges a.s. to some
IEp{f(Z)} = F
Y11 ...,Yp
,
of
Z(p). We
is p-invariant.
Z
f
on
B
,
if p is the
,
dv(x)
J B
The martingale convergence theorem yields f(Zpx) dv(x) = f(Z)
lim
pand (if
B
Zn
goes to infinity (that's the usual case)
For any continuous function
a-algebra generated by
v
is
G U B
(b')
is just a copy
p, {Yp+1...Yn , n > p}
Yp+1...Yn
so the distribution, say
We will suppose that in so that
For any
is equivalent to
(b)
a.s.
B
is separable), Y1...Yn V W > dZ
with probability one.
97
The conclusion of this discussion is that it is equivalent to consider a topology ' ' on
for which (a) and (b) hold or to
G U B
p-invariant probability measure
specify a
on
V
such that
B
{gv, g c G}
is contained in the closure of
(a')
{6x, x c B}
(b')
Y1...Yn v converges weakly to some SZ, with probability one.
Condition (a') takes the place of the density assumption of A pair
for which (a') and (b') hold is called a
(B,v)
Given a probability measure
such that
B
distribution
in G U B.
11-boundary.
it can be difficult
G
p-boundaries. But it is worth describing compact
to describe all G-spaces
on a group
0
G
.
on
v
is a
(B,V)
B
p-boundary for each
11-invariant
In fact this is easy and these spaces are
.
important. We have the following (amenable groups have been defined in 111.7.5).
.
PROPOSITION 4.4. p
Let
be a second countable locally compact group,
G
normal) amenable subgroup of a compact metric
H a closed (not necessarily
and
G
a probability measure on
G-space such that
is compact. If B
G/H
such that
G
is a
(B,v)
p-invariant distribution
v , then there exists a closed subgroup
of G which contains
such that
Proof B
Since
:
H
B = G/H,
is
p-boundary for any H'
.
is amenable there exists a probability measure m
H
with, for any
h
on
H
in
= m .
hm
n. i
Let
(n,)
lim ni- j=1 E u) * m = V
be a subsequence such that
i
By Lemma 1.3.4,
v
is
probability measure
A
exists.
p1-invariant. Let us show that there is a on
such that
G
v = A * m . Let
7r
:
G -> G/H
denote the canonical projection. Since is compact we can suppose ni converges to some probability measure p on G/H. that 1r (n E p3)
If
A
i j=1 is a measure on Since
such that
(B,v)
is a
G
with
gn V converges to some
sequence of neighborhoods of large
rr(A) = p
,
v=A*m
11-boundary there exists a sequence
z
6z in
.
B
Let
with
gn
in
be a decreasing
VP
fl v
p
= {z }
.
p
gn V (vp) = J g. (VP) d (cg. * A) (g)
?_
1-
.
p
For
n
G
98
So we can find
in
g'
such that
G
h
p
, k c K h E H , we may suppose that p p Taking into account the relation
k
,
p
k
G = KH
Write
.
z
.
This yields
.
p
K
P
P some
p
p
for some compact set
lim g'm = 6
g' = k
1
g'm(V ) ? 1 -
P
G . If
in
converges to
p
g'm =kP h m = kPm p
P
we obtain
km = 6z
H' = {g e G H
x1}
,
m
and ,
is supported by
H'
G/H, can be identified with
and
x1 = k
But
is compact
p-invariant probability measure
so it carries a
is equal to
implies that
B
v
.
Condition (a')
The proof is complete.
.
If, as in this proposition, (B,V)
p-invariant distribution
If
is a closed subgroup of G which contains
is a
1i-boundary for every
must be unique. For if
V, V
and V2 are
V1
V +V V3 =
p-invariant,
2 1
Y1...YnVi
6Z(i)
2
is also
11-invariant and if
for, i = 1,2,3
Z(1) = Z(2) = Z(3)
,
and
V1 = V2.
This class of boundaries is thus exactly the one of boundaries which carry a unique
p-invariant distribution. Consider the case where
G = G1(d,]R). We have seen that if
is the subgroup of
S
uppertriangular matrices G/S is the flag manifold.
is amenable and
S, being solvable,
G/S is, by Proposition 4.1, a
G/S is compact. So
p1-boundary. Moreover one can show that
{F(p1,...,pk) ;
1
< p1
< p2 < ... < Pk < d , k = 1,...,d-1}
where
H'
is a closed subgroup of
describes, when
Proposition 4.1 each
p
,
all the boundaries
G T11
(B,V)
exhausts all the
which contains is
S
.
G/H,
Therefore
p-strongly irreducible for
which carry a unique
p-invariant
probability measure. With the notations of this proposition F(p1'...'pk)
is the maximal
p-boundary with this property, the others
being its equivariant quotients.
Now look at the case of matrices in
Sp(d,IR). Let
L =L(1,2,...,p).
It is easy to see that if S = {M a Sp(d,IR); then L
.
S
is solvable (hence amenable) and that
Sp(d,IR)/S
is equal to
Since one can show that all the equivariant quotients of
L(pl,...,pk) all the
when
e1A...Aek , k = 1,...,d}
Tu
,
1
are the
< p1 < P2 < ... < Pk <= d , Proposition 4.2 describes
p-boundaries which carry a unique is
L
p-invariant distribution,
Lp-strongly irreducible for each
p
.
99
The reader who is aware of Lie group theory may guess that if we
G with compact center, the
deal with a general semi-simple Lie group possible
11-boundaries carrying a unique
G/P where
to be found among the
p-invariant distribution will
runs through the set of parabolic
P
This approach is developed in Guivarc'h and Raugi [34].
subgroups of
G
Remark 4.4.
In many cases one can show that under the hypotheses of
.
is actually maximal among all
F(p1,...,pk)
Proposition 4.1,
11-boundaries. In this case it can be identified with the Poisson space associated with
p-harmonic bounded function on
(i.e. each
p
G
has
a unique integral representation on this space). This is proved, under the assumption
when
Tu = GP , by Furstenberg (see [20])
S1(d,IR) and by Ledrappier
absolutely continuous on
p
is
(see [47])
when
is discrete.
G U
Suppose that
Remark 4.5.
such that
is a probability measure on
p
for any
Ek * 11 * E k_1 = p
quotient space
and
Sl(d,IR)/SO(d)
k
Tr
in
Sl(d,IR) => D
:
map. Exactly as in 11.7 one sees that for any Tr(g Y1...Yn) if
11
is a Markov chain on
is not confined on
p1-boundary. For any
normalized Haar measure on
is the
mK
mF
does not
p-invariant measure. Therefore
converges to a Dirac measure on
g Y1...Yn mF
itself is a
F
(where
S0(d)). It is clear that
and is the unique
depend upon
the canonical
By Proposition 2.5 we know that
.
mF = mK * dE
let
F
denote the
D
Sl(d,IR),
in
g
S0(d), the flag manifold
in
E
D
Sl(d,IR)
S0(d). Let
F
.
In this case one may
interpret this as a convergence of the Markov chain in a natural compactification of
D
.
D
Namely let
g E Sl(d,IR)} in the space of probability measures on g mF = g'mF if and only if seen as a dense subset of
.
boundary of
disk
{z E
CC
F ;
D
.
in its compactification
D
to a point of
.
When
Izl <
1}
F
On the other hand
identified with a closed subset of
d = 2
,
and
D
D
F
.
(easy exercise)
Tr(g) = Ti(g')
D
{g mF,
be the closure of
Therefore D
,
and
F
Since can be
D
is clearly
is a part of the
Tr(g Y1...Yn)
converges
can be identified with the unit
with
{z c
thus generalized the results of Section 11.7.
;
Izl
<_
1}
.
We have
CHAPTER V
CENTRAL LIMIT THEOREM AND RELATED RESULTS
In 1970 Kaijser showed in some particular but typical cases that the contrastive action of the random products implies that
Log ISnxll
on
Sn = Yn...Y1
P(]R
d )
suitably normalized converges in distribution
to a gaussian law (see [33] , [39] , [4O] ) This idea was later fully developed by Le Page in
[49] where he proved that, loosely speaking,
behaves like a sum of i.i.d. real random variables and
Log IISnxll
satisfies analogues of the main classical limit theorems. We shall give here a detailed introduction to the important work of Le Page and present, with some improvements, his main results (namely, the central limit theorem with speed of convergence and an estimate of the large deviations of
Log IiSnxll). We also study the tightness of
{Snx, n ? 1}
without moment assumption. This chapter ends with an application to linear stochastic differential equations.
V.1. Introduction
Let us explain the main ideas leading to a proof of the central Log lSnxil , where
limit theorem for
i.i.d. random matrices in
E = Gl(d,]R) x P(7Rd),for any direction x),
X1 = (Y Z , Y 1
that if
a
:
..
,
Sn = Yn...Y1
is the product of
Gl(d,]R). They derive from the fact that if x
,
the sequence
Xo = (Y1,
is a Markov chain on
X_+ = (Yn+1'
E -> ]R is defined by
101
E
.
Notice
102
!tall
G(M,x) = Log
11x 11
then, for 1x11=
1
,
a(Xn-2)+...+6(X0)
Log liSnxll = a(Xn-1) + Therefore chain
an additive functional of the Markov
is just
Log l1Snx11
and we may try to apply the general devices for studying
(Xn )
such objects.
In fact (at least) two methods are useful. The first one, due to Gordin and Lifschitz, relies upon martingale theory. We shall see (cf.
complements 2.8) that it gives a short proof of the central limit theorems. The second method, due to Nagaev ([55]) and first used for random products by Tutubalin ([66]), rests on perturbation theory for operators. It is technically more involved but gives the best estimates.
Since it may also be applied to prove other results (e.g. large deviations) we shall develop it at length. (X ) on E the idea is to introduce n on E acting on some space of functions
For a general Markov chain {P(z), z E IL}
operators
by the formula
z Q(X ) 1E
{e
(X1)} .
1
(1)
The Markov property implies that
{e
1E
z{0(X )+. ..+Q(Xn )} 1
Y
So that, if
9L
(Xn)}
is the function identically equal to one, z{0(X )+...+a(X )}
n
{e
]E
1
Y
Thus these operators
} = [Pn(z) 1L] (y)
play the role of the usual Fourier-Laplace
P(z)
transform.
In our specific model, Tutubalin noticed that since for depends only on 0
on
E
y
it suffices to consider the functions
which can be written q(M,x) =
If we put
(P(z)4)(M,x)
_ (T(z)f)(x) = ]E{e
we obtain, if
z Log IN x 11
f(Y1x)}
and (see lemma 5.2)
zLog11S xll
(Tn(z)f)(x) = 1E {e
n
f(Snx)}
11x11=
1
(2)
103
acts on
T(z)
IL, such that
thorough analysis of the iterates of
IL
T(z)
in such a way that a
is available (at least for
small). Nagaev and Tutubalin supposed that
Izl
P(IRd) which
IL of functions on
The point now is to find a space contains
satisfies the
(Xn)
IL the space of Borel bounded functions.
Doblin condition and used for
But this hypothesis is too stringent for random products. The idea of Le Page, suggested by particular cases studied by Kaijser, is to choose IL
to be a space of Holder continuous functions. T(0)
We shall first see that if T(0) = N + Q
N
where
acts on such a space, then
is a rank-one projection,
spectral radius smaller than one and
NQ = QN = 0
an operator with
Q .
This is really the
fundamental result. From this we shall deduce, using some easy facts of perturbation theory, that for
z
small enough,
is a bounded
T(z)
IL which can be written
operator on
T(z) = A(z)N(z) + Q(z)
where
is in
A(z)
I
,
N(z)
N(z)Q(z) = Q(z)N(z) = 0 than
IE {e
t'`
IX(z)I
i
.
,
is a rank-one projection,
and the spectral radius of
is smaller
Q(z)
This implies for instance that
Log II S nx }
is equivalent to a(-)n as n ->
.
It is then easy to prove the analogues of the classical limit theorems for sums of i.i.d. real random variables which depend only upon the behaviour of the Fourier-Laplace transform near zero, by mimicking the usual proofs.
V.2. Exponential convergence to the invariant measure
Let
be independent random matrices in
Y1,Y2,...
common distribution
11
. We have seen in Theorem 111.4.3
Gl(d,IR) with that if
Tp
is strongly irreducible and contracting then there exists a unique p1-invariant distribution
functions
f
on
V
P(IRd) and
on
X
P(IRd). Moreover, for all continuous in
P(IRd
IE {f (sn x) } -> JIf dv where
Sn = Yn...Y1
'
We shall improve this result under a moment condition on
p
104
Let
DEFINITION 2.1.
Z(M) = Sup{Log+ IIMII , Log+ IIM 1 II} u
Gl(d,IR) . We say that a probability measure
(i) a moment of order
if
p
for M in
Gl(d,]R) has
on
1 Q(M)p dp(M) < -(, T > 0
(ii) an exponential moment if for some
,
J exp Tf(M) du(M)
is finite.
For the reader's convenience we restate Lemma 111.5.4, adding the obvious inequality (i).
(i) For any
LEMMA 2.2.
M1, M2
in
Gl(d,]R)
i(M1 M2) < Q(M1) + R(M2) (ii) For any integer
p
< p < d , M in
1
,
.
G1(d,]R) and w in
Ap ]Rd
ILogIIAYMII and
<
p R(M)
<
p R(M) IIwII
I
ILogIIAp MwII
I
Note that by (i) the convolution powers of many moments as
of
p
have at least as
has.
We first prove a general lemma on cocycles. We use the notations un in particular denotes the nth convolution power of p.
1.3
;
LEMMA 2.3. s
a
Let
be a topological semigroup acting on a set
T
T x B
be an additive cocycle on
.
and
B
Consider a probability measure u
on T such that r(g) = Sup Is(g,x)I , there is some xEB is finite , exp T r(g)dp(g) (i) for
T > 0
such that
)T
(ii) for some positive integer
eas(g,x) dun(g)} < 0
lim I Log {Sup
n-*n Proof
Since
x
a
Sup 1 s(g,x)dp (g) < 0 x6B for a > 0 small enough.
.
Then
J
For each
:
p
a
in
(0,T)
set
an = Log{Sup ( eas(g,x) dun(g)}
is a cocycle we have for all positive integers
( eas(g,x) dun
(g) =
n, m
eas(g1g2,x) d,n(g1)dpm(g2)
I
as(g1.g2'x) a.s(g2'x)
r J
e
e
n du (g1) dim(g2)
105
as(g2,x)
(
{Sup
e
J
as(g1'y)
(
f
n
m
du (g1)} dp (g2)
e
y
< {Su p
f
as(g1,Y)
e
n
du (g1) }{SXp
as(g2,x)
f
m
dU (g2) }
e
y an = Log{Sup ( eas(g,x) dun(g)} X
so that the sequence This implies that
r eas(g,x) dun(g)} = inf n
Log{Sup
urn
n
n
(eas(g,x) dup(g)
Sup
<
is subadditive.
an
i
)
X
From the relation 2
ex < 1 + x + x elxl we deduce that
and
eas(g,x) dp (g)
Sup J
<-
+ a Sup
1
f s(g,x)dup(g)+ 2
I
Therefore the claim follows readily from assumption (ii).
PROPOSITION 2.3. that
T}1
eT R(M)
Let
Gl(d,]R) such
be a probability measure on
i
is strongly irreducible and contracting. If for some is finite then there exists
dp(M)
a0 > 0
T > 0
such that for each
1
a in
(0, a01
,
lim {
n
_
dpn(M)}1/n
Sup
x,yttp(]Rd)
J
<
1
6(x,y)
Xry
Proof that
:
B
Let is a
B = {(x,y)
;
x E P(]Rd)
y E P(]Rd), x , y}
.
It is clear
Gl(d,]R)-space under the action M E Gl(d,]R), (x,y) E B
The relation s(M,(x,Y)) = Log d(x,Y)
defines a cocycle on
Gl(d,]R) x B
.
By Lemma 2.2
106
IIMxAMyII
_
Sup s(M,(x,y)) _ _Syp Log (x,y)EB (x,y)EB <-
41C(M)
IIx1I
IIyII
IIxAyII IIMxIIIIMyII
.
Since we have seen (Proposition 111.6.4)
that for some
__
6(S x,S y)
Sua J s(M,(x,y))dp (M) = Su2 ]E {Log x,y x,y
the proposition
} < 0
p
p
p > 0
b(x,y)
follows from the preceding lemma.
We now introduce the space of Holder-continuous functions with which we shall work.
DEFINITION 2.4.
on
Given
we set for any continuous function
a > 0
f
P( ]Rd)
_ Sup
If I
I f (x) - f (Y) I and
If(x)I
ma(f)
Sup
d
d(x,y)a
x,YeP(IR)
xEP(IRd)
x#y
We define
which
1L (a)
as the set of continuous functions f
IIf IIa =
on
P(]Rd) for
is finite.
If I. + ma(f)
]L(a), with the norm
It is readily seen that
is a Banach
space.
The following theorem, due to Le Page [49]
,
is basic. Note that,
with its notations, we have for any continuous function ]E
f
on
P(]R
d
Pnf(x)
J
Therefore it shows that, in some sense, the distribution of converges to
v
THEOREM 2.5.
Let
Tu
exponentially fast.
1i
be a probability measure on
Gl(d,]R) such that
is strongly irreducible and contracting, and let
p-invariant probability measure on exp T k,(M)dpi(M)
v
be the
P(]Rd). Suppose that for some
if finite. Then there exists
a0 > 0
such that for
1
0 < a < a0
,
the operators
P
et
N
Pf(x) = J
Nf(x) = J f(y) dv(y)
defined on
'[ > Q.
]L(a) by
107
are bounded and satisfy
lim IIPn - N11 11n a
<
1
n4-
Proof : We have, using Lemma 2.2
If I.
<
Pf(x) I
,
and
IPf(x) - Pf(Y)
6(M.x , M.y)a
_ _ 6(xY)a
<
6(x,y)a
ma (f)
<
so that
P
is bounded on
2.3 that for each
for
IL(a)
IL(a). For
is bounded on each a
ao
a
e4aR(M) du (M)
I
0 < a < It is clear that N 4 we know by Proposition small enough .
there exist
(0, ao]
in
(
m (f)dp(M)
C > 0
and
p
such that {6 (M x
r
,
C pn
dun(M)
6(x,Y)
This implies that for
f
in
IL(a)
I (Pn- N)f (x) - (Pn- N)f (Y) I
IPnf (x) - Pnf (Y) I
6(x,ry)a
6(x,Y)a dun(M)
J
6(x,y)
C pn ma(f)
On the other hand, since
v
p-invariant.
is
Jf
(Pn-N)f (x) I=
.
ff
(y) I
dun(M) dv(y)
< J
mX(f)
6(M x ,
dun(M) dv(y)
J
and, since
6(,y) _<
1
,
I (Pn-N) f I,o = Sup I (Pn-N)f C.) j x
ma(f)
I
Sup
x,Y < C Pn ma(f) J
d}1n(M)] dV(Y)
f )
6(x,Y)
, O< p< 1
108
This proves that ma{(Pn-N)(f)}
II(Pn -N)flla = <
+ I(Pn-N)fI_
2CPnm(f) 2C Pn IIf IIa
so that
IIPn-NIIa<2CPn Pn-N = (P-N)n
Noting that
PN = NP = N
since
and
N2 = N , the limit
IPn-NII11n exists by subadditivity. This completes the proof.
of
Things are simpler when
has a density. In that case we may
p
work with continuous functions and we don't need any assumption on the moments of
11
The following result due to Tutubalin ([66]), holds.
.
Suppose that
PROPOSITION 2.6.
is a probability measure on
p
G1(d,]R)
Sl(d,]R)) which has a density with respect to the Haar measure. If
(or
C denotes the space of continuous functions on for
conclusions of the theorem hold Proof : We first prove that {Pf
k
and
P(IRd), the
acting on e
N
is a compact operator on
'
,
i.e. that
is a conditionally compact subset of ° . For any
fI_ < 1} and
in
P
P
f
SO(d), put
in
Tf(k) = Since {Tf
,
(k-e1
,
it is sufficient to check that
k e SO(d)} = P(]Rd)
Ifi_ < 1}
is a conditionally compact subset of
space of continuous functions on
Ascoli (see e.g. Dunford and Schwartz IV.6.7
in
SO(d)
Gl(d,]R)
=
(or
Sl(d,)R)). For
p1
k1, k2
dMI
J
< J If(N'e1
tends to
k1
,
I dM
I
< J I$(Mk1 1) k2
[17]) we have only to
we have
Tf(k1)-Tf(k2)
When
,
denote the density of
show that this set is equicontinuous. Let
w.r.t. the Haar measure dM on
'(SO(d)), the
SO(d). By the theorem of Arzela-
(Mk21) I dM
the right hand side converges to zero (this is
109
is bounded and continuous ; in general approximate
obvious:when in
by such functions). Since it does not depend upon
L1
{Tf
,
lfl
Q = P- N
< 1}
is conditionally compact. This yields that
and
, we have
lim (P - N)nf
=
n-*
lim (Pn- N)f = 0
n--
Therefore the modulus of each eigenvalue of Its spectral radius, namely
:
,
P
are compact operators.
By Theorem 111.4.3
Remark
f
lim
IIPn
is smaller than one.
Q
- NII1mn
is thus smaller than one.
,
It is not difficult to see that when Xn
Markov chain
(Y
=
on
n+1.
p1
has a density, the
Gl(d,]R) X P(]Rd) satisfies
the Doeblin condition (see e.g. Revuz [61], chapter VI, for the definition).
Complements 2.8.
(A first proof of the central limit theorem)
We shall show how Theorem 2.5
enables us to use the general
result of Gordin and Lifschitz ([28]) for proving a central limit theorem. In Section 5 we shall give stronger results under the same assumptions without recourse to the theory of Markov chains. Therefore the reader may skip these complements and we just sketch the arguments. Let
(Xn)
be the Markov chain on
E = Gl(d,7R) x P(]Rd) defined by
Xo = (Y1,xo),...,Xn = (Yn+1' Yn...Y1*xo),... i.i.d. matrices whose distribution
11
Theorem 2.5 . We suppose also that
x0
independent of
probability
R
{Y.1, n _>-
1}
is a random variable on
, with distribution
E
It is easily seen that
C101). Let
are
v
.
d P(]R ),
The transition
(Xn)
A
,
R(M,x),A) _ ]E ((Y1,
so that
Y1,Y2....
associated with this Markov chain is defined by, if
is a Borel set in
(Xn)
where
satisfies the hypotheses of
p ® v
a A)
is the unique invariant distribution for
is a stationary ergodic sequence (see Breiman 7.3,
0(M,x) = Log
. We want to show that Ilx lI
Log 11Snxoll - Log IIxo1I
n-1 =
E
a(Xi)
i=o
satisfies a central limit theorem. We may always suppose that the upper
110
is equal to zero (by looking at
y
Lyapunov exponent
(x) = J a(M,x)dp(M), then
case, if
a
nSn).
In this
f (x)dV(x) = 0 . By the theorem
of Gordin and Lifschitz it suffices to check that the equation
(I-R)V=a has a solution in
L2(11 ® V)
But, for any
.
n >=
1
,
Rn a(M,x) = ]E (a(Yn, 1E
Pn-1
For
a > 0
is in
small enough,
IL(a)
(see Lemma 4.2), and
Su2 IRn a(M,x)I < IIPn-1 01, M,x
Using Theorem 2.5 and the fact that N =
dV = 0
we see that this
J
quantity converges to zero exponentially fast.
= E Rn a(M,x) converges in
This proves that (I-R)t = a
course
n=o By [28]
.
N(0,a2)
to the gaussian law Suppose now that
- Log IISn xoll Vn r , with a2 = ,
a2 = 0
.
Then
and of
converges in distribution IIRIPII2
IIRvII=IIPII, which implies that
(X1)}2) = 0
1E
Rip(Xo) _ V(X1 )
and But
L2(p 0 v)
)(M, x) = a(M,x) +
n=o
VM,x) = a(M,x) +
a. a.
Hence, for
Pn
E
]P
and
f(x) = E Pn 4(x), n=o We thus obtain
f(Y1 xo) = a(Y2, Y1 xo) + f(Y2 Y1 xo) Since is
and
f
are continuous and since the distribution of
a
this yields
la ® V
a(M,x) = f (x-) - f
11 0 V
a. s.
a(M,x) = f (-x) - f
,n 0 V
a. s.
and more generally
Since the support of and since for all
is not confined a proper projective subspace
is bounded this implies that for some
f
M
V
in
Tu
.
c > 0
,
1 < IIMII< c,
We shall see in Lemma 8.2 that this cannot be true
under our assumptions, proving
a2 # 0
.
111
Exercise 2.9.
Show that if
lim IIPn-NII1In <
Exercise 2.10.
is a probability measure on the
Sp(d,1R) which has a density w.r.t. the Haar measure,
symplectic group then
p
, where
1
P
e(P(]R
acts on
)).
Show that the conclusion of Proposition 2.6 holds if
for some integer
p
pP
is not singular with respect to the Haar
,
measure. (Hint : Use Lemma 2, Section VII.8.1 (p. 709) of Dunford and
Schwartz [17]
).
Exercise 2.11.
Let
M in
matrices
_ Let
be a probability measure on the set of positive
c Mi'J
"k j
c
M.
p
M(d,]R) which satisfy, for some fixed ,
c > 0
< i,j k,k <_ d
1
x
d
Bc = {x S P(]Rd)
x =
(x1,...,xd)
,
E
c < xl < Cl
X. = 1
i=1
Show that
Define (BC,e)
j
is a bounded set for the metric
B c
]La(Bc)
as the Banach space of order
introduced in 111.4.7.
0
a
Holder functions on
.
Show that the formula (
Pf (x) =
I
f
defines a bounded operatorl on
(M)
]La(Bc) and that if
Nf(x) = J f(y)dv(y) where
v
is the unique
p-invariant distribution on
B c
then
lim IIPn - N 111 /n < 1 (Use Exercise I1I.6.5).
V.3. A lemma of perturbation theory
We recall some elementary facts about the spectral theory of bounded operators. Although the lemma we will be needing is well known we present its proof for the sake of completeness.
We refer to Dunford and Schwartz ([17], VII.3) for the proof of the following. Let E
.
T
:
E -> E
The spectrum of
be a bounded operator on a complex Banach space T
is
112
a(T) = {z c Outside
be the set of complex valued functions which are
F(T)
a(T)
bounded open set whose boundary is contained in
a(T)
of
U , we set
U
f(T)
Let
.
and if
and
f
T
be a
U
and
f E F(T)
is a finite union of
aU
C1
curves.
is analytic in a neighborhood
f
f au
It follows from the Cauchy integral theorem that on
.
is analytic.
R(z,T) = (zI-T)-1
analytic in some neighborhood of
If
has no bounded inverse}
z I - T
;
the resolvent
a(T)
Let
IE
depends only
f(T)
.
We shall use the facts that for
f, g
F(T)
in
(of + tg)(T) = af(T) + (ig(T) (fg)(T) = f(T) g(T)
f(a(T)) = o(f(T))
LEMMA 3.1.
Let
and
U1
boundary such that
the union of U1
N1
be two open sets in
U2
and such that
U1 fl U2 = 0
and
U2 .
I
a(T)
with
C1
is contained in
Define
R(z,T)dz
J
2rti
.
R(z,T)dz
N2
,
21
aU 1
Jau2
Then (i) N1 and N2
N1 + N2 = I (ii) For
are two projections such that N1N2 = N2N1
and
T(Ni(E))
i = 1,2,
spectrum of the restriction of
Proof and
Then
Let
:
U2
.
V2
and
V1
fi is in Ni
F(T) 2Tri
is contained in Ni(E)
to
T
is
Ni(E)
and the
a(T)fl Ui
.
be two non-intersecting neighborhoods of
i = 1,2,
For
N1T = TN1
.
let
fi
be the indicator function of
U1
V.
and
fauf1(z) R(z,T)dz = YT) .
Since
f1 + f2 =
Moreover
f
= f1
1
on a neighborhood of and
f1f2 = f2f1
For any complex a set ga(T) = TN1 + a N2
a(T)
,
f1(T) + f2(T) = I
so that (i) is clear.
ga(z) = zf1(z) + af2(z). The relation
yields that, if
T1
is the restriction of
T
to
113
N1 (E) , a(ga(T)) = 0(T1) U {a}
.
o(ga(T)) = ga(o(T)) _ (a(T) (1 U1) U {a}
But
{Q(T) n U1} U {a}
contained in
for any
a
0(T) U {a}
so that
is
This proves (ii).
.
We consider now an analytic family of bounded operators V
E E VI , where
{T(C)
every
a
is an open set in
V we can find
e
This means that for
.
CC
and a sequence
E > 0
(A
n
)
of bounded
operators such that
(1)
En IAn1I<
E
n?o
(ii) for
lE-Eol < c
and
C
T(E) = E (,-,o)n n .
V
in
n=o We will use the following (see Dunford and Schwartz VII.6 [17]).
LEMMA 3.2.
be a neighborhood of 0
V
Let
in
T
and
e V}
be an analytic family of bounded operators on the Banach space Suppose that for some rank-one operator
1im n-
P
Then we can find
e > 0
E
N
IIT(0)n-NII
<
such that for
1
It;l
< e ,
T(Q = A(E) N(E) + where (i) X(t;)
(ii) N(S)
is the unique eigenvalue of maximal modulus of is a rank-one projection such that N(1;)Q(E) =
and
(iii) The mappings 23P
(iv)
that for each
and for each
P
:
1) First
d p Qn(C) II S
are analytic.
p c IN there exists
c > 0
such
c{1 +2p).
notice that since
This implies that restriction of
0.
n e IN
Il
Proof
T( Q .
T(0) Ker N C Ker N
T(0)
to
N(E)
lim T(0)n = N , NT(0) = T(0)N = N. and
T(0)N(E) C N(E). The
is the identity and on
Ker N
T(0) = T(0) - N . Therefore the spectral radius of the restriction of T(0)
to
Ker N
is
lim II(T(0) - N)n[IIIn = lim IIT(0)n - NllhIn = P
.
114
This implies that
a(T(0)) G B(O,p) U {1}
.
is a small circle around 1, then for
in Ker N ,
x
Moreover if
y
R(z, T(O))x
is analytic in the disk whose boundary is
y , so that by
the Cauchy Theorem
2ii For
x
in
N(E)
, T(O))x dz = 0
R(z
J
Y
R(z, T(O))x = (z-1)-1x
,
and
R(z, T(0))x dz = x
1
2i fl(
Whence N = 27ri
R(z, T(0))dz
J
Y
2) When
is in
E
If
<
II
for the resolvent R(z,T(Q).
, write
V
0)]n IR(z,0) II-1 then E [{T(C) - T(0)} R(z n
is the
inverse operator of {zI -
T(0)} R(z,0)
R(z,0) = Id
Thus R(z,0)-1{zI
- T(C)}
1
= E [{T(E) - T(0)} R(z,o)]n n
R(z,Q =
and
R(z,O)[{T(E) - T(0)} R(z,0)]n
E
n=o
->
This shows in particular that 3) Let
D1
be the disk with center
disk with center B(0,p)
and
0
{1}
and radius
, z
6f U}
is contained in
132-
I
U
,
and radius
13p
, D2
the
an open set which contains
is contained in
D1 U D2
,
and
.
IT() - T(0)II
We shall suppose that
D1
U
such that
M = Sup{ IR(z,O) II o(T(E))
is analytic.
U
.
Let
y1
< M . In that case, by 2),
(resp. y2)
be the boundary of
(resp. D2). We define
N(E) = 27i J
R(z,E)dz Y1
Using 1), 2) and the preceding lemma we see that analytic, that
N(0) = N
and that
Id - N(Q = 271ri J
N(E)
R(z,t;)dz Y2
E -> N(C)
is
is a projection. Moreover
115
Let us show that if
N(E)x =x and
Suppose that
then the rank of N(Q
IN(S) - N(0)11< 1
one there exists some
N(E)y = y
in
A
I
Since the rank of
.
N(0)) (x- ay) II IIx - ay II = IN(Q (x - ay) II = x = ay when IIN(E) - N(0)11< 1 .
is
N(O)
N(O)(x - ay) = 0
such that
is one.
.
We have
<=
II IIx-XyjI
II
Thus
4) Choose IN(Q - N(O) II < Let
<x
°
o'o
exists
1
.
IIT(E)- T(0) II
<-1
1
c E
x
x'> =
c > 0 such that for IEI < E , and E -> N(C) is analytic.
If
A(E) C E
x' E E'
and
be such that
0
then the rank of
ICI < E
II = 1, Nx
IIx0
0
= x
0
and
is one and there
N(2;)
such that
The formula
x'>
X(V x'>
0
choosing
0
is analytic. We find the wanted decomposition
shows that
Q(C) = T(E)(Id - N(Q).
4) To prove (iv) first note that, for since it is an eigenvalue of T(E) 2+
continuous, thus
Now for each
a() n
in
>
.
A(0) =
But
is in
IEI < E, A(1;)
and
1
A
U
is
3P IN
T(,)n (Id-N(Q) = 21i
1
zn R(z, )dz .
Y2
Therefore
dP dEp
Qn(C) -
IldEd'ap
and
for c = 27f Sup {
11-!L
d
P
1
2711
II
Y2
zn d P R(z,t;)dz dip
< c {1 32P}n
z C Y2
II
IEI < E}
The proof of the lemma is complete.
Remark 3.3.
It is easy to modify the proof in order to show that if,
for some m e IN, the family
{T(Q, E c V}
is merely of class
Cm
then an analogous result holds. (In the statement one has to replace analytic by
Cm
and the integer
p
in (iv) has to verify
0 <_p < m).
116
V.4. The Fourier-Laplace transform near 0
Given a probability measure
where
z c G, x t P(IRd)
P(lRd). Note that
and
Gl(d,IR) we define formally
on
p
_ z Log (T(z)f)(x) = f e
_
=rµ IIx II
dp(M)
is a complex-valued function on
f
a
Our first task is to show that for
that for some
Let
be a probability measure on
p
G1(d,IR) such
is finite. For any
exp T R(M) dp(M)
T > 0
is an
small enough,
IL(a) near zero. Namely,
analytic family of bounded operators on
PROPOSITION 4.1.
studied in Section 2.
P
is the operator
T(0)
a with
J
0 < a < 2 T
there exists
such that
q > 0
analytic family of bounded operators on
{T(z),
IzI < p}
is an
IL(a).
The proof is tedious but straightforward. It will require the following lemma in which directions x
LEMMA 4.2.
y
and
For
that, for any
and
z e M
M in
denote two unit vectors with
0 < a < 1
, there exists
c1, c2 > 0
IMxMI
-
__
Log IMYII
ci exp{[t4a)IRe z1+2a4 1r(M)
I
c
6(x,y)a
x,ycP(IRd)
Proof of the lemma
_ ez Log I1 MYI I,
6(x,Y)
Log
(ii) _ Sup
I Mx I I
-a
x,ytP(IRd)
a) First notice that for
6(x,y)2=1-<x,y>2=IIx-Y Replacing, if needed,
y by -y
2
f(M) e
2ak(M)
IIx1I = Ilyll = 2 x
112{1-114Y
we may suppose that
Ix-YII
<
6(x,Y)
<
IIx-Y112 = 2
11x-y11-
1/2
b) We now prove (i). Let F(x) = exp(zLogIIMxII).
DFx(h) = z
F
:
be defined by
IRd -> E
Its differential jMxl1z-2 <Mh,Mx>
DF
x
satisfies
for
1
}.
this case 1-_
such
Gl(d,IR)
ez Log
Sup
y
and
x
.
h c IRd
-
In
117
and
<
IIDFXII
IIrxIIRe z-1
Iz I
IIMII
Therefore, if a = IRezI ,
IIx II =
1
IF(x) - F(y) I < IIx-yII
and
then
IIy II = 1
IIDFax+(1-X)yII
Sup
Izl IIx-y 11 e(a+2)L(M) IF(x)-F(y)I c e
and
(a+2)k(M)
c = 2/ Izi
for
6(x,Y)
M = c exp(a+2)k(M)
This implies that for
and
11x
N = 2 Sup IF(x)I 11=1
< M 6(x,y) A N
IIF(x)-F(y) II
Whence, if M6(x,y) <
1
< M6(x,Y) < Ma' 6(x,Y)a
IIF(x) - F (y) II
and if M 6 (x,y) ? 1 IIF(x)-F(y) II < N <_ N Ma' 6(x,Y)a This yields
IF(x)-F(y)
<_ 2{c exp(IRez I+2) k(M) } a e IRezI R(M)
_
6(x,y)a proving (i). c)
(ii) is proved along the same line, using the fact that if
G(x) = Log IIMxII
then
Proof of the proposition T(z)
<
IIDGxII
:
e2k(M) 11X II-1
a) We first show that for
is a bounded operator on
]L(a). If
f c ]L(a)
0 < IRezI < T-2a 1 +0t and x,y are unit
vectors, then
IT(z)f(x) I
ezLog IIMXII = Ij
I
c3 IfI-
IfI. j exp IRe zI P, (M) du(M)
for some c3 > 0
if
IRezI < T
.
On the other hand, by the previous lemma,
IT(z)f(x)_ T(z)f(Y) I
<
j Iez Log 1114x 111 If(M'x) -
6(x,Y)a
I
d(x,Y)a
_
+j
zLog IIMxII _ I
Ie
du (M)
zLog IIMYII
_e
6(x,y)a
dp(M)
118
ma(f)
(
- IReI i(M) dp(M)+cl Ifl- J
e((i+a) IRezI+2)2(d
(M)
< c4 (ma (f) + If Imo) for some constant
c4 > 0
IRezI (i+a.) + 2a < T
as soon as
,
IRezI < T-2a
Hence, if
1+a
IIT(z)f IIa = IT(z)fl. + ma(T(z)f) <_ (c4+c3) llf IIa and
is bounded.
T(z)
{Log
is a unit vector with direction
x
where
Tnf(x) = J
n E IN let
b) For each
dp(M)
II Mx II}n
x
Taking into account the relation
I(LogIIMxII)n - (LogIIMyII)nI
IL°gIIMxII -L°glIMyIII n-1
<
IL°gIlMx ll b(x,Y)a
I
1
1L°gllMylll`rri
i=o
(x.Y)a
exp 2a i(M)
c2 n i(M)n+1
we see that
T f(x) - T f (y) I
Logn
I M. II
_
(x,y) a
l
(x,Y) (
+ J f(My)
dp (M)
IL.gnllMll - L°gnIIMyll I
_
dp (M)
d(x,y)a
< ma(f) J IL°gnIIMxII Idp(M) + I fl-c2n f
i(Mjn+ic2a.Q()dp(M)
On the other hand
i(M)n
ITnfI. < If I_ Sp f IL.gnIIMXII I dp(M) < Ifl.
dp(M)
J
This shows that
IITnIIa
< c2 n f i(M)n+l
E
n
nI
IITnIIa< c2
E
n
InII
Therefore
is finite if
IITn IIa {T(z)
,
IzI
< n}
n+1
In
n(M)
n
J
z n
so that
d11 (M) + 2 J R(M)n dp(M)
( e2ai(M) E nlz
In
and
e2a i(M)
-p (M) + 21 eIzli(M)dp(M)
2a + IzI < T
is analytic when
p = T-2a 1+a
Theorem 2.5, together with Lemma 3.2 and Proposition 4.1, yields
119
the following result of Le Page [49].
Let
THEOREM 4.3. T
be a probability measure on
p
Gl(d,IR) such that
is strongly irreducible and contracting. Suppose that for some T> 0,
-u
is finite. Then there exists
1 exp T i(M) du(M)
0 < a < a0 , then for some
e > 0
,
the operators
a0 > 0
such that if
T(z),
IzI < E , are
bounded on IL (a) . Moreover, for
IzI < c
T(z) = X(z) N(z) + Q(z)
where
1(0) = 1
(i)
N(0)f(x)
,
p-invariant distribution on
f dv
v
is the unique
P(]Rd
(ii) p = lim IIT(0)n - N(0) Ila'n < (iii)
i
1
is the unique eigenvalue of maximum modulus of
),(z)
is a rank-one projection and N(z)Q(z) = Q(z)N(z) = 0
T(z), N(z)
and
The mappings
(iv)
IX(z)I ?
(v)
such that for each
23p
and for each
p e IN
are analytic ;
there exists
c > 0
n E IN
aP II
1+2P
Qn(z) II < c
dzp
The following will be used for studying the speed of convergence in the central limit theorem. Under the hypotheses of the theorem, if IL
is the function identically equal to one,
COROLLARY 4.4.
Suppose that for some
X"(0) = a2 > 0 . Then there exist that for
n E ]N and
a e (0,ao], A'(0) = 0
13 > 0
,
c > 0
0 < r <
and 1
such
f
Itl < 2
IITn() 1L- e t /2
af
and
2
1L 110t 5
Proof : We have, by Theorem 4.3, for
f
{(It3I+ItI)e t /4 + Itlrn}
ItI < e
Tra(it) = an(it) N(it) + Qn(it)
120
If we write
II
instead of
II
II
it
T" (a f) g - eII- t /2 T
11 II
, this implies for It I < at rn
IIa
it) - e t
Ilan(-ita
!5
/2N(O)II+IIQn(-)-4n(O)II
a
(we here use the relations N(0)IL = IL and Q(O)11 = Q(O)N(0)1L = 0) For notational convenience we write the proof for a = 1 . c1 > 0
On the one hand, by (v) of Theorem 4.2, there exist 0 < r <
1
f
and
such that
IIQn(
<
) - Qn(O) II
It I
IldQn(z) II
Sup
,n
IzI<E
It! rn
.
(3)
rn On the other hand IIa
n(itf)N(Lt) it - e -t -t /2
< la n(itf)
II
- e -t /2
+ x--but,
N( )
k > 0 (
Sup
<
II az N(z) II
- e-t /2
N(0 )M <
Making use of the relations
/n-
tk n it f la (f) I + l an it
X(0) = 1, A'(0) = 0
and
e--t
/2
(4)
I
X"(0) =
1
obtain 2
a(it) =
2
2 + t30(t) = exp{- 2 + t3 0(t)}
1
t +t 0 2
and
kn( t )
= e
3
2
n
Yn
This implies (since
Ieu - 11 < Jul elul)
e
I
t/2. ItI3
e 13
that
t3 0(t)
tz /2
tz
Hence, for
tk
and
,
it)N(it) IIa f
t
v Izi<E
Yrn
for some
IIN(0) II
IlNclt) - N(O) II
IN(lt) - N(0) II
being analytic,
I
Yn
/2
z
e
t
e 2
(t)I explt 0(t)I f vn
c2 > 0
small enough there exists
It! < 13 n
3
10
t z/4
such that, for Itl3
Vn'
we
121
n(it)I
and
c2
f
2 e /4
This yields, by (4),
2 t
t2
IIxn(it) N(it) - e
v
/2
c
N(0)II
2
/4
e
(kltl+It13)
(5)
rn
Vn_
The lemma follows from (3) and (5).
Remark that the statements proved above are valid for matrices with complex entries.
V.5. Central limit theorem
Y1,Y2,...
Given a sequence
of independent random matrices in
Gl(d,]R) with common distribution
11
and
prove under various hypotheses that for
Sn = Yn...Y1
central limit theorem. We shall always assume that irreducible and that
Theorem 5.1 that if in addition S
Tu
is strongly
has an exponential moment. We first show in
p
and the direction of
we intend to satisfies a
x c ]Rd, Snx
n
is contracting, then both the norm
T11
can be normalized in such a way that they
x
converge in distribution to a non degenerate limit. We also give the speed of convergence. We then prove, in Theorem 5.2, that the central limit theorem for
is also true when
Log IS n x11
Tu
is not contracting.
This is useful because it is often difficult to check whether
T
is
u
contracting or not.
The following important result is due to Le Page (491), the only improvement being
a2 > 0
.
distribution with variance
N(O,a2)
Let
be the centered normal
a2
THEOREM 5.1. (Central limit theorem, first form) where
{Yn, n ? 1}
Suppose that some
T > 0
: Let
are i.i.d. random matrices in
Sn = Yn...Y1
Gl(d,]R) with law u.
Tur is strongly irreducible and contracting and that, for exp T R(M) du(M)
,
is finite. Let
y
denote the upper
J
Lyapunov exponent associated with u- invariant distribution on
Then there exists (i) for any
a > 0
x # 0 , the
(Sn)
and
v
be the unique
P(]Rd).
such that ]R x P(1Rd)-valued random variables
122
-
1
(- {Log IISn x I I
- n ' } ,
converge in distribution to the product
N(O,a2) ®v
measure
(ii) The random variables ( - I - {Log IISnII -n y}
distribution to
N(O,a2) ® V
such that, for any
C > 0
(iii) There exists
Log IISnxII-nY Sup 11P (
< t) -
of
IIxII=1
of
A
ezLog
of Theorem 4.3, and
S.-x)
converge in
J
T(z)
t c lit,
t exp - u2 Z du I < C- .
-
is defined by
JIM. 11
P(1Rd) and
is a complex valued function on
f
notation
1
f2Tr
We first prove a lemma. Recall that
T(z) f(x) =
,
Yn-
.
IIxII = 1
.
,
For each positive integer n and unit vector
LEMMA 5.2.
x
Tn(z) f(x) = 1E{ezLog If
A(z)
is the eigenvaZue of T(z)
(6)
a'(0)
0 < a < ao
In particular
Y)21
a = 0
converges as
- n a2
and for
and
IzI < e)
a2= X"(0)- y2,
n -> - to a finite limit.
if and only if the sequence
{]E(Log 1ISn xII- n Y)2'
is bounded.
n >= 11 Proof
y
is the upper, Lyapunov exponent
1E { (Log II Sn x II - n
IL(a) with maximum
acting on
modulus which appears in Theorem 4.3 (for then
Using the
for the first two derivatives
A', A"
:
It is worth noting that (6) is only a consequence of the cocycle
property of
a(M,x) = Log
IIMXII
We proceed by induction. If Tn-1(z)f(x)
z Q(S = ]E {e
n-1
,x)
then
T"(z)f(x) = Tn-1(z){T(z)f}(x) z 6($ = 1E {e
,X) n-1
_ (T(z)f)(Sn_1x)}
123
= 1E {e
za(S n-1 ,x) zct(Yn ,S n-1 x) e
z U(Sn',x)
= IE {e Sn = Yn Sn-1
since
and
f (Y ( S
n
n-1 'X)) }
f(Sn-x) l
U(Sn,x) = U(Yn, Sn-1'x) + U(Sn-1,x)
This proves (6). We now prove the rest of the lemma. If
t
is a
small real number we can write by Theorem 4.2 T(t) = X(t)N(t) + Q(t) Tn(t) _ Xn(t)N(t) + Qn(t)
and
But for
IIx1I =
1
tLog IISnxII
Tn (t) 11(x)
IE (e We thus have
tLog Ilsnxll)
(dt n(t))N(t) IL x)+an(t)
dt1E(e
dt.(N(t) IL (x))
+dt {Qn(t) 1I(x)} so that, using Theorem 4.2
,
tLo g 1 d Y = lim n1E(Log IISnxII) = lim nl dt1E(e
IISn xII
)(0) = A'(0).
Now
a2 (e
tLog IIS xII -
2
dt
Notice
_
{e
2
e n") (0) = n(X '(0)-Y2)N(0) IL (x) + a 2{N (t) 11(x)}(0) dt 2 + 2 {e nt Y Q"(t) Il6o)(0) dt
d
that, by (v) of Theorem 4.2
nt Y
dt2
n
,
Qn(t) IL(X)1(0) = n2y2 Qn(0) 11(x) - 2n y dt {Qn(t) 11(x) } (0)
_
2
+ d 2 {Qn(t) Il(x)}(0) dt
converges to zero. Therefore, since
N(O) 1 = IL
_ IE{(Log IISnxII -nY)2} - n (U(0) - Y2) - a2 {N(t) 11(x))(0) dt
converges to zero. This proves the lemma.
We postpone the proof of the following lemma until section 8 (cf. Corollary 8.6).
LEMMA 5.3.
Under the hypothesesof Theorem 5.1 the sequence
IE{(Log IISnxII -ny)2) is not bounded.
1 24
Proof of Theorem 5.1.
:
Although (i) follows readily from (iii) we
shall prove it directly. By Theorem 4.3 and Lemma 5.2, for
small enough,
1E{e
f
in 1L(a)
and
II x I
(Log IISnxII-nY)f(Snx)}=e-ir ty{a(
whenever
is real and
t
But
lim
e-if t y
a > 0
i = 1 , we have )0N(
it )f(x)+Q°( -
)f(x)} VIE
sufficiently large.
n
2 2 )n _ exp - t2 a
X(
n-*
a'(0) = y
since
a"(0) = a2 + y2
and
a # 0 since, by Lemma 5.3
,
by Lemma 5.2 . Notice that
IE (Log II S. x I I
-n
Y)2
is not bounded. On the
other hand
lim N()f(x) = N(0)f(x) =
f dv J
n-
lim Qn(7) f(x) = 0
and
n4 (use (v) of Theorem 4.2 with
p = 0). Therefore for any
i f(Log IISnxIl -ny)
_
lim1E(e
in IL(a)
f
t 2 a2
f dv
n--
Taking into account that each continuous function on a uniform limit of a sequence of elements of IL(a)
P(3Rd)
is
we immediately see
that
(7n (Log II Sn x II - ny) , Sn-x) converges in distribution to
N(0, a2) ® v
.
To prove (ii) it suffices to use (i) and the fact that for any
x
0
,
is a. s. bounded (see (a) of Proposition
Log IIS n x lI - Log II SnI I
111.2.2). y = 0
We now prove (iii). We may suppose a'(0) = 0
and
X"(0) = a2 > 0
.
(see Lemma XVI.3.2 of Feller [18])
all A > 0
and real
M > 0
there exists
:
t
1
In this case
.
Recall the Berry-Esseen inequality
VF_ 1
(t
J
u2
e
2
1
gn(u)-e
A
duI _T J -
U
A
where
¢n(u) =1E{exp a,r Log IISnxII} = Tn( a/n=) 1(x)
By Corollary 4.4 there are when
Jul
< S
such that for
2
> 0
,
c > 0
and
2
Idu+AM
.
0 < r < 1
such that
125
Sup IIx1I=1 Choosing
<.
(u)-a-u2/2I
(t
< at Vn) -
IISn x
I
{(IuI3+Iu1)e-u2/4 + Iulrn}
we obtain
A = B vn
Sup IIx
In
Jte-u2/2
1
<
svc
S+
c
f-
((Iu2I+1)e-u /4 + rn)du
Sn
F
2L
duI 2
2
Srn
(Iu12+1) eu/4du+
+Vn-c
from which (iii) follows.
In the next theorem (announced in [7]) we do not suppose that
T11
is contracting. In this case we cannot study the convergence in distribution of
S
n x , we do not even know if there is a unique
p-invariant distribution on closed subgroup of
THEOREM 5.4. where
P(1Rd). Recall that
(Central limit theorem, second form)
{Yn , n
distribution
are i.i.d. random matrices in
1}
p
.
is the smallest
: Let
p
.
Sn = Yn...Y1
Gl(d,]R) with common
Suppose that the following hold
(a) For some
T > 0
,
J
exp T R(M) dp(M)
is strongly irreducible
(b) G11
G11
G1(d,]R) which contains the support of
{Idet
(c) The set
MI-1/d
is finite
.
is not contained in a compact
M , M e G
subgroup of Gl(d,]R) .
Then there exists
1 {Log I I Sn xII - ny} and to
N(O,a2)
a > 0 1
such that for any
(Log II S n
ny)
x # 0
in
]Rd
converge in distribution
.
We shall be needing the following algebraic lemma which is proved in Chevalley, (45, ch. IV of [13]).
LEMMA 5.5. 1
<_ p < d
Let
be an irreducible subgroup of Gl(d,]R). For any
G
, we can find a direct-sum decomposition of
Ap]Rd = W1 ® W2 ® ... ® Wk such that for any
(a) if M E G
i e {1,...,k} then
:
(APM)(Wi) = Wi
Ap]Rd
p.
126
(b) there is no proper subspace W of W. M
for aZZ
in
G
such that (APM)(W) = W
.
This amounts saying that the action of
G
APIRd
on
is
completely reducible.
Proof of Theorem 5.4. since
First recall (see Proposition 111.3.2) that
:
is strongly irreducible, the sequence
Gu
, n >_ 11 is almost surely bounded for each x # 0 in 1Rd . Whence An- {Log I Sn x 11 - n Y} and {Log II Sn I I -n Y} converge to the same distribution (if one of them does converge). Now let p denote the index of Tu . By Lemmas 111.1.4 and 111.5.3 the sequence {Log I I Sn I I p - Log II AP Sn 111 , n > 11 is also bounded so that 7 {Log II Sn 11 - n y} and P {Log II AP Sn I I - n y p} have the same limit
{Log 11S,111 - Log I Sn x II
I
distribution. I t is therefore enough to show that
{Log I I AP Sn II - n p Y}
converges in distribution to some gaussian law N(O,b2), b > 0 {IdetMI-11d
whenever
M ;
is not in a compact group, i.e. when
M F_ G
p < d (see Proposition 111.7.1).
y = y1 ? Y2 = ... > Yd
Let
with
p
.
be the Lyapunov exponents associated
We know (see Proposition 111.6.2) that
but that
Yp+l < yp associated with (APS ) n
X
Y i=1
for and
Al
1 <-i:-5p
A2
are
p
1
yi = yp
The two top Lyapunov exponents
.
1
2
p-1 + E Y = Y P+1 i=1 1
(see the proof of Proposition IV.1.2). Therefore, here, Al = p Y
and
a1 > X2
Consider now the direct-sum decomposition
AP]Rd=W1 ®W2 ® ... ®Wk given by the preceding lemma, applied to
G = Gu
.
It is easy to see
that the set of all Lyapunov exponents associated with
(APS )
union of the sets of all Lyapunov exponents associated
with the
restriction of choose
W1
(APSn)
among the
restriction of
to each
Wi's
APSn to Wl
W2 9 ... 0 Wk , then
W.
,
Bn
is the
1 < i < k . Therefore we may
in such a way that, if and
n
An
the restriction of
is the
APSn to
127
lim -1 nLogIIAn1I n4co
lim n Log II Bn II < A
and
2
n-*
A2 < X1
Since
and
prove that N(O,b2)
.
I
I Ap Sn I
IA
n II
, J I Bn II)
, we are done if we
converges in distribution to some
{Log IIAnII- n A1}
Let us check that we may apply Theorem 5.1 to the sequence r = dim W1
(An). If
= Sup(
I
random matrices on
we may view
r > 1
first show that sends all of
Ap]Rd
decomposable
p-vector
r = 1
If
.
n
as a product of
An
i.i.d.
with a common distribution, say
Gl(r,]R)
each limit point of
into the line
p
. We
IIAPSnII-1 APSn
. Therefore there is a
W1
v1 A ... A v
which spans
W1
But
.
p
(AP M) (W1) = W1
for any
M
1Rd spanned by
v1..... vp
,
Gu
in
,
M(V) = V
hence if for all
contradiction with the irreducibility of
G11
is the subspace of
V
M
in
G11
in
,
. Knowing that
there are at least two Lyapunov exponents associated with largest one is
with
(APSn)
,
.
The
and the next one, being also an exponent associated
Al
is not bigger than
exponents associated with irreducible on
r > 2 p
A2
.
Therefore the two top Tp
are distinct. Since
p
is clearly
by construction (see (a) of the above lemma) we
W1
know, by the converse statement in Theorem 111.6.1, that
Tp
is
actually a contracting and strongly irreducible subsemigroup of Gl(r,1K) . We may thus apply Theorem 5.1 to the probability measure and
with
(Log IIAnII -n p y)
b # 0
Remark 5.6.
.
converges in distribution to some
This completes the proof of the theorem.
Suppose that
compact subgroup of
{Idet
MI-1/dM,
is contained in a
M E Gp}
. We know (see Remark 111.1.8) that there
Gl(d,]R)
exists a scalar product on ]Rd
for which all these matrices are
orthogonal. If we work with the associated norm, then for any unit
vector x in ]Rd n
Log I I Sn I I= Log IISnxII = a Log I det sn I
aE i=1
Log Idet Yi I
is a sum of i.i.d. real random variables. Therefore, if
Y = ]E{Log I det YiI } ,
then {
converge to N(O,a2)
where
Note that subgroup.
p
N(O,b2)
a = 0
vin
Log I I Sn I I - n y} and a2
if and only if
rn {Log I I Sn x1 1 - n y)
is the variance of Log Idet YiI e-Y
Y1
is contained in a compact
128 r
In these theorems the assumption that
J
exp T R(M) dpi(M)
is finite
-"
is unpleasant. The condition
seems more natural.
j i(M)2 d}i(M) <
In this direction we have the following result of Tutubalin [66],[68.
PROPOSITION 5.7.
Suppose that the probability measure
p
density with respect to the Haar measure of Gl(d,1R) (or Sp(d,1R)). Then, if (
i(M)2 dp(M) < m
J
Sl(d,IR), or
the distribution of
converges to N(O,a2) ®v , for some
(Log II Sn .11 -ny) ,
a>0
,
has a
.
Sketch of proof : We define by (2) a family of operators
acting on the space e of continuous functions on
{T(it), tclR}
P(1Rd). By
Proposition 2.6 (see also Exercise 2.9) we have a decomposition analogous to the one given in Theorem 4.3
For
.
t
small enough,
T(it) = X(it)N(it) + Q(it) where X , N
and
have the same properties as in this theorem except
Q
that they are only of class
.
To verify this, note that if
t -> T(it)
is finite, then
J'i2(M) dp(M)
remark 3.3
C2
is of class
C2
and use
The proposition is thus proved as was (i) of Theorem 5.1
.
.
Making use of the arguments developed for proving Lemma 111.7.1 it is not difficult to show the following.
PROPOSITION 5.8.
random matrices Theorem 5.4
.
Assume that the common distribution
If
p
of the i.i.d.
satisfies the assumptions (a) and (c) of
(Yn)
GP
is merely irreducible
1Rd
=
then there exist a direct
sum decomposition
and a real number
x
in Vi ,
Vn-
a > 0
V
V
such that for any
Log { II S. x l I - n y)
1
< i < k and any nonzero
converges in distribution to
N (0, a2) .
But it is worthwhile to remark that it may happen that for some
nonzero
x
in IItd
,
1 {Log IISnx II -ny}
does not converge to N(O,a2).
Consider for instance the case where for some sequence i.i.d. random vectors of ]R2- (0,0)
,
(an,bn)
of
1 29
a n
or
a-1
0
b-1 n
0
0
n
=
Y
b
n
0
n
b11
0
If 1P(Y1 = (b
then
0 )) # 0
is irreducible. It follows from
G11
1
Proposition 111.7.4
T Log
y = 0
that
For
.
x = (i)
and
IISn xII ?1
so that this latter sequence cannot converge to a
S. x11 > 0 ,
centered gaussian distribution. Notice that the same remark holds for
7 Log IISn 11
.
In this particular case
N(0,a2), if
some
x = (0) or (I
Log I I Sn x II
. And
when
a # 0
does converge to Y1
is not a.s.
orthogonal.
In the following exercise we propose to prove, using the arguments developed in this chapter, the central limit theorem for positive matrices due to Furstenberg and Kesten [25]
Exercise 5.9.
of matrices
Let
c > 0
M e M(d,IR)
and
p
.
be a probability measure on the set
whose entries satisfy, for
1
< i,j,k,i < d
c Mi1i < Mk,R < c Mi.j Using Exercise 2.11 and the same route as above show that if 2
J
Log IMIIdu(M)
is finite, then the distribution of
converges to some product measure
{Log IISn x11 - n y} 2
N(O,a ) 0 V
on ]R x P(TRd)
for each nonzero x with nonnegative
coordinates. (This is rather long but requires only the former arguments, working on ILa(Bc)
instead of
L(ot)).
V.6. Large deviations
Consider a probability measure
p
on
Gl(d,IR)
such that
strongly irreducible and contracting and such that for some 1 exp T R(M) dp(M)
Tu
is
T > 0
is finite. By Theorem 4.3 we know that if we
consider the operators
T(z)
as acting on IL(a)
,
then for
Izi
small
enough
T(z) = X(z)N(z) + Q(z)
.
In particular for all real small enough
t
and unit vectors
x
130
tLog IIS xll n
IE{e
_
_ T(t)n 11(x) = A(t)°1(t) ]L(x) + Qm(t) 11(x)
}=
Since the left hand side is positive and since faster than
la(t)In
goes to zero
llQn(t)ll
is positive and
a(t)
,
tLog lISnxll
1
lim
LoglE{e
} = Log a(t)
n-*W
But
is analytic
a(t)
a'(0) = y
,
and
with
X"(0) = a2 + Y2
a
2
> 0
(see Lemmas 5.2 and 5.3). This implies the following result due to Le Page [49]
]P (Log
(a similar result holds of course also for
,
llSnxll-ny<-nE)) Let
THEOREM 6.1.
.
hypotheses of Theorem 5.1
any unit vector
where
Sn = Yn...Y1
x
and
.
{Y., n > 1}
Then there exist
satisfies the
such that, for
A,B > 0
0 < e < B
lim n LogIP(Log IISnxll-ny>ne) = (c) n-_ where
q(E)
{t e - Log a(t) + ty} < 0
Sup 0
This is a direct consequence of the following lemma applied to
Zn = Log Il Sn x II - n y . This lemma is now well known (see Freidlin and Ventsel [19], Theorem 1.1 and 1.2, Ch. V), we just sketch its proof.
LEMMA 6.2.
Let
{Z
such that for some
be a sequence of real random variables
n ? 1}
11
> 0
Log ]E(exp t Zn) = c(t)
lim n-
,
n
exists on an open set containing
[-n, n]. If
differentiable and strictly convex on then for
[-n,r1]
c
is continuously
and if
c'(o) = 0
0 < e < c(n) TI
lim n Log ]P(Zn > n e) n-'O°
Proof
:
a. We have for any
IP(Zn > nE)
Sup O
[te- c(t)] < 0
t >_ 0
tZ -nt ) < e-tEn1E(etZ n
S 1E (e
'
so that
lim
Log ]P(Zn > n E) <_ -
n
Sup O
[tc - c(t)]
.
131
It is clear that the right hand side is negative since b. We define, for
c'(0) = 0.
0 < t < n , new probability measures
1Pn,t by
tZ tZ rn,t(A) = IE {e n}-1 1E {e n 1A} For
0 < E < z < c pn)
0 < a < z - E , the inequality
and
Z
1P(Zn > nE) >_ IP{In - zI < a}
-tZ tZ 2 IE(e n) Pn t (e n 1 IZn IIz I
tZ
1 Loglp (Zn>nE) a n Log]E(e ntz-na+n LoglPn,tZn nzI
in
tz
relation
lim
have
n
such that
[0,n]
c'(tz) = z. Making use of the
s Zn
1
Log 1En t (e
) = c(tz + s) - c(s)
-and of part a we
z
< n a) = 0
Log lPn tz (1Z n nz
Urn
Therefore
Log 1P(Zn> ne) a c(tz) - tzz = c(tz) - tzc' (tz)
lim
n-, n When z> E then c(tz) - tc'(tz)->c(tE)-tEC'(tE)=inf c(t)-tE, O
Sup n, n Log IP(Zn> nE) a - O
lim
EtE - c(t)]
it is easy to check that the convergence stated in the theorem holds uniformly in
x
if
IxII =
1
.
The following result is maybe more useful since it does not require any information about
THEOREM 6.2.
for some a > 0
Assume only that
T > 0
,
T11
.
is strongly irreducible and that
G11
f exp{T 2(M)} dp(M)
such that for any E> 0
and
lim n Log 1P(ILogIISnx n,
is finite. Then there exists x A 0
-nyI >nE) < - a
and
lim n Log ]P(ILogIISnII - n yI > nE) < - a n,_
132
Proof
:
First, replacing each
loss of generality that Let
Yn
y = 0
be the index of
p
-Y Yn , we may assume without
by
e
If
p = d
.
T
.
then the theorem follows
from the usual estimate on large deviations for sums of i.i.d. real valued random variables (see Proposition 111.1.7 and Remark 5.6). Henceforth we shall assume that in the proof of Theorem 5.4
p < d
Recall some facts we have seen
.
There exists a direct-sum decomposition
.
Ap]Rd=W®V M
such that for any An
(resp. V)
a) If
in
Gu
,
(Ap M) (W) = W
,
and, if
(Ap M) (V) = V
is a matrix which represents the action of
(resp. Bn)
ApSn on W
then the following holds.
is the distribution of
p
Al
,
then
is a strongly
T P
irreducible and contracting semigroup of Gl(r,]R)
Moreover
, where
r = dim W.
lim n Log IIAnII = Y = 0
n,
b) The upper Lyapunov exponent associated with Theorem 6.1
If we apply
and each w
W , w # 0
in
to
(A
n
)
(B n)
is negative.
we know that for some
(x> 0
< -a
(7)
,
UE -1 n Log]P(ILog IIAnwilI >n E:)
.
n
If we write this relation for a basis of
W we readily see that
1
Tim n Log ]P(Log IIAnil > n e) < -a .
(8)
n--
Since the upper Lyapunov exponent associated with m
B
n
is negative, for
large enough ]E(Log IIBmII) < 0 . Making use of the relation
Log IIBnII
km < n
that, for
,
we deduce from the usual estimate on large deviations small enough ,
a
1-
imnLog]P(Log1IBnII >n e) < - a
.
(9)
n4co
Since we may choose
IIApSnII = Sup(IIAnII IIBnII)
,
(8) and (9 ) yield
Tim n Log]P(Log IIApS.II>n e) < - a and since (see e.g. Lemma 111.1.4) the index of
(Log I I Ap Sn 11 - p Log 11 Sn 11)
Log]P(Log IIS.II > n e) < -a
lim n
T11
being
p
is uniformly bounded we have (10)
133
This implies that for each
x # 0
in IRd
-a
limn Log]P(Log IISnxII > ne) <
is more intricate.
The study of IP(Log IISnxII < -n F-) We fix an
x # 0
such that
x A x2 A ... A xP
.
is not in
V
It follows from Mostow
.
that we may suppose that the scalar product on ]Rd and G
are orthogonal in
V of
x2,... x
and first prove that there exist
in litd
APIRd
(sketch of proof
the algebraic hull
:
is irreducible. Therefore, for some scalar product,
GP
self adjoint
.
W
Since
Thus its orthogonal is
If for any
W
V
,
as the
GP-invariant). We have seen in the proof of
p-vector, say yl A...Ay
contains a decomposable
x2,...,xP
is
G
G-invariant.
is also
G-invariant and we may choose
orthogonal of W . It is Theorem 5.4 that
Gu-invariant, W
is
54]
W
is such that
x A x2 A
A xP
P.
V , then yI A...Ay
is in
p
is orthogonal to
x A x2 A ... A xp
seen that this implies that in
that
GV
,
My1
Myl A ... A Myp
for any
is orthogonal to
yl
is also in
is orthogonal to
x
x,....,xp
W
x
.
.
It is easily
But for any
and we obtain, in the same way,
This contradicts the fact that
.
M
Gu
irreducible.
We thus can choose
x A x2 A ...
A xp
on W
x2,... xp
such that the component
w
of
is not zero. By (7)
limn Log IP(Log IIAnwII < -n e) < - a
and, since IIAPSn (x A x2 A ... A xP)Il Tim-
n
>_
IIAnwII
LogIP(Log1IAPSn (xAx2A...Ax
P
,
)11<-ne) <-a
(12)
.
Now, using the inequality
IIAPSn(xA... A x
)
P
< IISnxllllsnx2ll ... lISnxpll
II
we have
Log IISnxII>Log IIAPSn(xA... AxP)ll -
p E
i=2
Log IiSnxill
Therefore IP(Log IISnxII<-ne) _<-P (Log IIAPSn(xA...AxP) II p +
E
IP (Log
IIs n
xi
11
24 n p
i=2
and, by (11) and (12)
limn Log IP (Log II S. x II
n-_
< - n e) < - a
=-n
p
is
134
Since this implies that
Log IP (Log II S. II < - n c) < - a ,
lim
the proof of the ntheorem is complete.
V.7. Convergence to
y p
We shall see, on a typical example, how the arguments developed
Ilsnxll for studying
o(S
n
x) = Log
can be generalized in order to
lxii
study other cocycles.
PROPOSITION 7.1.
Let
random elements of for some
1
< p
-< d
Sn = Yn...Y1
where
{Yn, n ? 1}
G1(d,]R) with common distribution ,
Tu
is
and
p
irreducible and that for some
(p-1)
T > 0
are i.i.d.
p . Assume that,
contracting and strongly
exp T 9,(M) dp(M) is finite.
, 1
Then for each nonzero v in AP-tJRdd and each nonzero w in AP ]Rd 1
7
{Log
IA p Sn w II - Log hI AP-1 Sn v11 - n yp }
converges in distribution to
a gaussian law N(O,a2) Proof
: Let
be the natural angular metric on
(resp. d2)
6 1
(resp.
P(AP-11Rd) metric
6
and put on B = P(AP- tlRd) x P(Ad]Rd) the
P(AP]Rd))
defined by, if
(v1,w1) e B
,
(v2,w2) e B
d((v1,w1),(v2,w2)) = 6 (v1,v2) + 62(w1,w2)
If we apply Proposition 2.3 to the random products we see that for
a
AP-1 S
6 (AP-1S v Ap-IS v ) 1 im n4
Sup v1,v2eP(AP-lJRd)
L L
{
1
r
and
lim
Sup n-- w1,w2eP(AP]Rd)
So that if we set for
then, for
a
APS
and
n
small enough
M
small enough
in
]B {
Gl(d,IR) and
n
1
-
2 }a
n
1/n <1
1(v1 v2
_
a-11/n
<
}
d2(w1,w2)
b = (v,w)
in
B
1
n
135
lim
Therefore if and if
}
IB { Sup b1,b2eB L
n-
ILa(B)
is the space of Holder
is the additive cocycle on
s
s(M,b) = Log M E: Gl(d,IR) and
IIAp MWII II
II
- Log
IzI
B
functions on
a
defined by
IIAp-1Mv1I IIvII
b = (v,w)), then the formulas
T(z)f(b) _ ( e s(M,b) define, for
1
Gl(d,IR) X B
w
(where
<
6(b1,b2)
d}1(M)
ILa(B) which admit a
small/ enough, operators on
decomposition similar to the one given in Theorem 4.3. Working with this decomposition one easily shows the proposition (as in section 5).
In the same way one proves a large deviation result. Remark 7.2.
Let
B = P(IRd) X P(A2IRd) x
x P(Ad-1
..
]Rd),
(we may also
work with the flag manifold). Under suitable hypotheses on
one can
T.
as soon as
p(Sn,b)
in fact study
p(M,b) = (a 1(M,b),...,(Yd-1(M,b))
where each a.(K,b) = 0
a, i
is an additive cocycle on
for all
K
in
0(d)
and
which satisfies
Gl(d,IR) X B
in
b
B
V.8. Convergence in distribution without normalization
We shall prove, under weak hypotheses, that converge in distribution unless it is to
0
.
Sn
or
Sn x cannot
By the same token we
shall show that the variance of the limit law which appears in the central limit theorem is not zero. A fact we have taken for granted above.
We first study invariant probability measures on IRd that
IRd is a
.
It is clear
Gl(d,IR)-space for the natural action
M e Gl(d,IR), x e IRd. Therefore we may define, given a probability measure
p
on
Gl(d,IR), a
p-invariant probability measure
as a distribution which satisfies
v
on IRd
136 (
f(Mx) dp(M) dv(x) =
JJ
11-invariant. Actually,
PROPOSITION 8.1.
Let
zero is the only
Proof
:
be a probability measure on
p
Gl(d,IR) such
is irreducible and not compact. Then the Dirac measure at
GP
that
f(x) dv(x)
]Rd. Notice that the Dirac measure
for any bounded Borel function on at zero is always
I
11-invariant probability measure on
Suppose that there exists a
Iltd.
p-invariant probability measure V
]Rd which is not the Dirac measure at zero. Let us prove that in
on
this case there is a real
V
in
Tu
(12)
IIMII < c
1
Let
such that for each M
c > 0 <
be the linear subspace of
]Rd spanned by the support of V.
Then
v(V) = IJ IV(Mx) dp(M) dv(x) = 1 so that
v(M 1V) = v(V)
support of
p-almost all
for
is contained in
p
M .
This entails that. the
{M C Gl(d,]R); MV = V}
is a closed subgroup. It must therefore contain account the fact that ..
Let
distribution almost all
Gu
{Xn, n ? 1) p
w
Gu
.
is irreducible we see that
.
But this set
Taking into V = ]Rd
be a sequence of independent random matrices with Mn = X1...Xn . By Lemma 11.2.1 there exists for
and
vu on
a probability measure 2-n-1
vw
converges weakly to
for
E
II1d
such that
Mn(w)Mv
n p -almost all M . Furthermore
n=o ( J
f dv = ]E {J f dv
for each bounded Borel function We now show that some
u
lim
on
]Rd
is almost surely bounded. If for
we can find a sequence
Sup IM1(w)II = n
IMn(i)(u)II-1 Mn(i)(u) that
f
{Mn(u), n >_ 1)
(13)
converges to a nonzero matrix
IIMn(i)(w)II- m .
But, if
Mn(w)v
such that
n(i)
A(w)
converges to
Vu , then
.
for any continuous
f
on
f dvu = lim J
iJ
]R d
with compact support we have
f( n(i)(u)x) dv(x)
and such
137
= lim J
i-
Ker A(w)
2
E
n-1 pn-almost
be a limit
M(w)
{M (w), n >_ 11.
point of the bounded sequence For
(x) f(Mn(i) (w)x) dv(x)
, a contradiction. Let now
V(KerA(w)) = 1
so that
1
n
M , we have
all
n=o
M(w)MV
vw
(14)
.
in the closure
This equality holds also for each M
in
Tp
of
T
n -n-1 2 E if we view this p n=o distribution as a probability measure on M(d,]R)). If there were some
M(d,]R), (because
M in
is the support of
T
such that
T11
1
IIMII<
then the zero matrix would be in
,
and (14) would imply that for each
T
is the Dirac mass at zero.
w , Vw
In view of (13), this would contradict our assumption. We have thus proved the L.H.S. of (12). It remains to show that a sequence
{A n,
n
in
1}
is bounded. Suppose that there exists
T11
Tp
such that
We may
lim IIA nII = - .
1
A
assume that
A
n n (14), for almost all w
in
Tp
II
f
in
T
,
n > .
as
By
A .
the fact that MA
is
n
= Vw .
W = {x a ]Rd; M(w) M Ax = 0}
and set
w
W , IIM(w)MAnxdI >
function
and every M
M(w)M An V
implies that
Fix such an
converges to some nonzero matrix, say
.
If
is not in
x
Therefore, for any continuous
]Rd with compact support
on
1 f(x) dvW(x) = J f(M(w)M An x)dv(x) = lim J 1W(x) f(M(w)M An x)dv(x)
but this implies that Choose some
V(W) = 1
Ax
such that
x
,
so that
W = ]Rd
is not zero. For all
M
in
Tp
M(w) MAx = 0 and the subspace H in the null space of and
M(w) = 0
a.s.
of
]Rd spanned by
M(w)
.
Since H
{MA x , M e T
is invariant under
, which implies that
We have thus proved that if
V
V
is contained Gp , H =
is not the Dirac measure at
then (12) holds. The proof of the proposition is completed by the following lemma.
IItd
is the Dirac measure at 0. 0
1 38
LEMMA 8.2.
Let
be an irreducible semigroup contained in
T
such that for some
<
I
for each M in
Gl(d,]R)
c > 0
T
Then
.
<
IMII
c
is contained in a compact subgroup of
T
Gl(d,]R) . Proof
Let
:
be the closure of
S
M(d,IR) and
in
T
p = inf {rank(M) ; M e s} . (i) We first suppose that Jordan decomposition that
p = 1
.
It is easily seen using a
contains a rank-one projection
S
Without loss of generality we may assume that a unit vector
in the range of
y
Px
.
For any x
.
<x,Y>Y
=
and if M1 ,M2 and M are in
P
P
is orthogonal. Choose
P
S
PM1 MM2Px = <M1 M M2 Y, Y> <x,y>y . But the spectral radius of each matrix in
S
is one. Therefore this
relation leads to
<M1 M M2 y , Since {u 1
T
=
Y>
M E T)
1
is irreducible we can find
= MI y , a .. , ud = Md y } and
basis of
t
]R
.
{v1
(15)
.
M1, M2,..
, M2d
... ,
= Md+1 y '
in
By (15) we obtain that the set {<M u. , v,>
is finite, so that
is closed and so invertible and
S = T
T
;
1
In particular the rank-one projection
.
are two
<_ i,j < d,
itself is finite. But in that case
must be equal to one. When
d
such that
T
vd = M2d y }
d = 1
T
P
is
the lemma is
obvious.
(ii) We now consider the general case. First remark that <
IJAP MII
1
for each M
in
there exists M
(Mn, n >=
1}
S
with
S
Let
Q
p
N
I
then any limit point
Ap N = 0
is smaller than
.
p
N
Q
matrix in .
S
The range of
of
But this latter relation ,
in contradiction with
.
be a rank -p
basis of the range of
IIAP MII <
and satisfies
S
entails that the rank of the definition of
(16)
The right hand side inequality is obvious and if
.
in
is in
< cP
and
{u1'u2,...,uP }
AP Q
is spanned by
be a
139
wo = u1 A u2 A ... A u
By (16) the spectral radius of
.
AP Q
is one
p
P = Q2,
so that, for
is a projection on
APP
]R wo
.
Consider now the
direct sum decomposition
...
AP ]Rd = W1 ® given by Lemma 5.5 (where we take Since for any w0
is in
i
W1
.
as the group generated by
G
(AP P)(w.) is contained in
,
As in (15), for some
M1, M2, M
for any
®Wk
in
on
=
±
.
1
is irreducible this implies, as above,
W1
AP M, M e T
that the set whose elements are the restrictions of W1
is finite. Since
in
T
is in the closure of
P
AP]Rd , and
we then have
S
S
we may assume that
W.
scalar product on
<(AP M1) (AP M) (AP M2) w0 , wo> Since the action of
T1)
to
,
it must exist some M
T
such that
APP w= AP Mw for all w c W1
.
to the fact that span of
But
ui,...,uP
irreducibility of
Finally when
AP M is invertible, whence this relation leads is one-dimensional. This implies that the linear
W1
is invariant under , we see that
T
p = d
is a compact semigroup contained in
S
,
T . Taking into account the
p = d
Gl(d,]R). By (9.16) of Hewitt and Ross [36] or Exercise 11.2.4, contained in a compact subgroup of
Remark 8.3.
S
is
G1(d,]R).
The above proof is the only one which makes explicit use
of the fact that the considered matrices are real. But the lemma holds also for matrices in Gl(d,O)
as
(To verify this write any matrix M
Gl(d,G).
with
M = A + iB
A, B
in
in
Gl(d,]R) and work with the
matrices
A
B
of
Gl(2d,]R). See the details in Remark 2.4 of
COROLLARY 8.4.
matrices in
Let
Sn = Yn...Y1
, where
181).
Y1, Y2,
Gl(d,]R) with common distribution
p .
"'
are i.i.d.
Assume that
GP
is
irreducible and not compact. If for some zero.
x
in
]Rd
,
Sn x converges in distribution, its limit is
140
Proof
It is clear that the limit distribution of
:
Snx is }i-invariant.
Therefore the claim follows from the proposition.
If we consider the distribution measure on
of
}in
M(d,IR), we see as above that when
Sn as a probability pn converges weakly
then its limit distribution is the Dirac measure at the zero matrix.
One may find in
[8 ]
Under the hypotheses of Corollary 8.4, for any non zero
COROLLARY 8.5. x
in
1Rd the sequence of the distributions of LogllSn xll
is not tight on ]R Proof
generalizations of this result.
:
Suppose that this sequence is tight. Then, if n
distribution of
, n= 1,2,...
.
Six , the sequence
{n
E
uX , n > 1}
denotes the
1il
x
is tight on
i=1
]Rd. But it is easily seen (look at the proof of Lemma 1.3.5) that each
limit point of this sequence is a IRd. The proposition yields that
}i-invariant probability measure on n converges to 60. Therefore E U x n 1 i=1
for any
c > 0
,
n
lim n4-
1P(Log IIS, xll < - c) = 1
E
i=1
1
which entails that the sequence
Log llSn xll
is not tight.
In particular, applying the above corollary to the sequence {e-Y Yn, n >_ 1}
we obtain the following which proves Lemma 5.3
Suppose that
COROLLARY 8.6.
Gli
is irreducible and that, for some
the sequence 1E{(Log Ids xII- n Y)2} , n = 1,2,... is bounded. n Then there exists a compact subgroup K of G1(d,]R) such that e Y Y1 x ¢ 0
,
is a.s. in K
V.9.
.
Complements
:
linear stochastic differential equations
We now consider briefly the continuous time analogue of the random products
Sn = Yn ... Y1
.
Namely
141
A random process
DEFINITION 9.1.
with values in
{St, t ? o}
Gl(d,1R)
is called a Process with Stationary Multiplicative Independent Increments (P.S.M.I.I) if the following hold is continuous in probability.
(i) (St)
(ii) For any S-11
St
... ,
tI < t2 < ... < to , the random matrices
0 Stl
St
are independent.
St
1
t
2
n n-1 Stl (iii) The distribution of St+s 1
In particular, if t ? 0, s ? 0, ut+s and by
t ? 0}
is the distribution of
ut
We denote by
- Pt * us (resp. G
T11
depends only on
s >_ 0
St , then for any
the collection
p
,
{ut
the smallest closed subsemigroup (resp.
for all G1(d,IR) which contains the support of ut It is straightforward to check that all the previous results,
subgroup) of
t ? 0.
which were proved for the discrete time model, are in fact valid for P.S.M.I.I. with these definitions of
and
T
G
u
this
moment, then for any
x # 0
{Log JISt xll, t > 0}
,
For instance if
.
P St
is strongly irreducible, and if some
Gu
has an exponential satisfies a central
limit theorem, and a large deviation estimate (one may use Exercise 9.4 in order to prove this).
An important application (see e.g. Arnold and Kliemann [1])
is
given by the so-called Linear Stochastic Differential Equations on IRd Namely let
Ao, All
{(Bt ,...,Bt)
,
t
..
,
be arbitrary matrices of order
Ar
be the usual
__>_ O}
d
Consider the Stratonovitch stochastic differential equation on
r
dxt = Ao xt dt +
and
r-dimensional Brownian motion.
E
d
IIt
Aixt o d Bt
(17)
i=1 or, equivalently, the Ito equation
r
dxt = A'' xt dt + E A. xt d Bt
A'o = Ao
i=1
Let
from
x
{xt(x)
is a
d x d
x
t ? 0}
x0(x) = x)
(i.e.
function of
,
.
r +
E
2
A.
i=1
denote the solution of (17) which starts .
It is clear that
xt(x)
is a linear
xt(x) = Stx
In other words we can write
matrix which satisfies the following S.D.E. on
r dSt = AoSt dt + E i=1
A
i S t odBt
,
So = Id
.
where
St
M(d,IR)
(18)
142
is the stochastic flow of diffeomorphisms
{St, t > 0}
In fact
associated with (17). As such Gl(d,]R)
(see e.g. Kunita [44] or Ikeda and Watanabe [37]). Let
the Lie subgroup of by
Ao, All
..
the manifold
}t = {Pt,
Ar
,
G
.
t ? 0) ,
.
The S.D.E. (18) can be considered as a S.D.E. on
contains an open set in
This implies that
.
Since
G
{Ao, All
is the closure
Gp
..
,
G . Moreover if
Ar}
,
then it is
G
(see Lemma 7.1
G
of
is connected it is easy to see that
irreducible if and only if that
takes its values in
St
ut is the distribution of St
where T11
be
G
whose Lie algebra is the one generated
Gl(d,iR)
Therefore
easy to see that
[8])
is a P.S.M.I.I. with values in
(S t)
of
G G11
is
strongly
is irreducible, which amounts to say
G
acts irreducibly on ]Rd
Besides, note that
St)-1/d
Mt = (det
satisfies the equation
St
r dMt = Ao Mt dt +
Ai Mt o d Bt
E
1=1
Ai = Ai - d (trace Ai) Id
where
,
and that r
d(det Mt) _ (trace Ao) det Mt dt + E (trace A.) det Mt o d Bt i=1 r
det Mt= Exp{(trace Ao)t + E
so that
(trace Ai) Bt} d
i=1
and
This gives that ]E(Log det Mt) = ttrace Ao
yi = trace Ao
E
i=1
Finally using well knownestimates on solutionsof S.D.E. (see e.g.
of [ ) one obtains that
Theorem IV.2.4 for each t
]E(exp2k(St))
is finite
.
Getting these results together we may apply the theorems 111.6.3, V.5.4 and V.6.2
to the process
Let
PROPOSITION 9.2.
(S t)
. We obtain
be the solution of
xt(x)
dxt = Ao xt dt +
r
A. xt
E
o d Bt
i=1 starting from
at time
x
0 . Suppose that the following hold.
(i) There is no proper linear subspace
A
i
(V)
is contained in V for all
(ii) There is no matrix Q(Ai -
1
(trace A ) Id)Q i
Q
-1 ,
i = 0,
of IRd
V 1,
..
,
r
such that
.
such that all the matrices
i = 0,...,r
are skew-symmetric.
143
Then
x # 0
(a) For all
lim
,
Log llxt(x)ll
= Y
t and y > d trace A. 02
(b) For some
{7 (Log ll xt (x) II - t y) gaussian measure
,
and all
> 0
x # 0
t > 0} converges in distribution to the
N(0, a2)
.
a > 0 and all x # 0
(c) For some
,
,
c>0
,
lim t LogiP(lLog 11xt(x)11 - tyl > tc) < -a
t--
Note that when (ii) is not satisfied then all the matrices
(det St) 1/d St lie in a compact subgroup of G1(d,IIt) Therefore we may find a norm on IRd such that Ix(x) 11 = (det St 1/d llxll for any lRd and t >_ 0 X In this case (a) holds but and only if
trace Ai = 0
(b) holds but
trace A ,
y = ,
d i = 1,...,r
,
a = 0
if
and (c) always holds.
We have of course much better results when
T11
is contracting.
To check this condition on examples it is worth noting that we have a concrete description of subsemigroup of t > 0}
and
T11
{Exp t Si
.
Namely,
TV
is the smallest closed
which contains all the matrices
Gl(d,]R)
t e ]R, i = 1....,r}
,
.
{Expt So,
This follows either
from Hormander's hypoellipticity theorem (see Siebert [65] and especially Mc Crudden and Wood [53]
,
Proposition 3) or from the
support theorem of Stroock and Varadhan (see e.g. Arnold and Kliemann
Suppose that
Example 9.3.
that trace
(A1.) = 0
generated by Then the
d
,
Ao, All
Ao, All --I Ar
0 < i <- r ..
,
Ar
,
are
d x d
matrices
such
and such that the Lie algebra
is the set of all trace-zero matrices.
Lyapunov exponents associated with the solution of (17)
the full flag manifold is a (Y1 > Y2 > ... > Yd) ut-boundary for each t > 0 (see IV.4) and we have a complete are distinct
,
description of the behaviour of the solutions of (17) at infinity.
This follows readily from the fact that in this case from Proposition IV.2.3 since
Exercise 9.4.
GZ(d,1R)
Let
such that
Tu
{St, t > 0}
]E (11Sl 11 T)
and
Gu = SR.(d,1R)
contains an open subset of
be a P.S.M.I.I. with values in
is finite for some T > 0
.
G11
.
144
a. Show that there exists
c > 0
such that for any
0 < t <
1
,
]P( IS-t1 11 5 c) >- 3/4 b. Deduce that for each
G1(d,]R)
such that
IIA
1
I
<_ c
0
<_
and
1E(IISS AIIT)
so that
r > 0
< 3 ]E(IIS, IIT) 3 cT ]E(IIS1 IIT)
]E(JISs IIT)
c. Prove that for any
there exists a matrix A in
s < 1
,
Sup
0tr
]E(IIStIIT)
is finite.
CHAPTER VI
PROPERTIES OF THE INVARIANT MEASURE AND APPLICATIONS
The main result to be proved in this chapter is that under suitable hypotheses, the satisfies for some
}1-invariant distribution
d
P(IR
)
_
a
<X,y>
I
Since for any nonzero is close to
on
a > 0 11-11
S
v
x
dv(x) < W .
(1)
]Rd the distribution of the direction
in
V , this relation indicates that in somme sense
IISnX II is not too big. More precisely we shall show that for any <Snx,y> r > 0 1
lim - Log n-' nr
IISnxII
= 0
a.s.
.
I<Snx,y>I
This will imply that the previous limit theorems about actually valid for the coefficients of
IISnx11
are
Sn.
All the main results of this chapter are due to Guivarc'h and Raugi.
To get an idea of the way we shall prove (1) we first consider a degenerate situation. The i.i.d. matrices subset of
Gl(2,IR) and of the form
Yi b, Cal
Oi c
145
Y1,Y2,...
are in a compact
146
In that case
is of course not irreducible. We suppose that Y1 > Y2
Tu
y2 = ]E(LogIc1I).
Y1 = IE(LogIa1l) and
where
it is readily seen that
Tn = Yn...Y1
Setting
Tn e1
+ un e 2
e1
where c
ct
un = at1b1 +
lim sup (Ia
c ...c
lb
1
n a ..,a
n
1
-Y n-11)1/n < e Y 2_Y
0 < a <
]E{
a
n=o
the series
,
u
n
um
n-1
1 .
.
a} ]E{Ian
. a n-1
b
nIa}
1
E {]E Iat
<
<
1
<
u. a)
I
1
(2)
.
n-1
converges a.s. to a random variable Moreover for any
n
n-1
1
1
Since
.c
(a21 b2)+...+ a1...an-1 (a-1bn)
a
no
la}n-1
IN
{Iai1 b1 Ia}
1
c
The negativity of
]E (Logl-1I) implies that for
small enough
a > 0
a1
]E {
lat Ia}
<
,
1
1
is finite.
so that ]E (Iu.l') Now notice that
converges a.s. to v
implies that the distribution
of
e1 + u- e2
e1 + u - e2 , which is
p-invariant (in
p-invariant distribution can be written as
fact every
Xv +
e2 0
A
1
,
as is easily deduced from Proposition 11.3.3).
Therefore
a
IIX 11 J
2 a/2 } < dv(x) = IN {(1+u)
I<x,e1> I and, in this non-irreducible situation, (1) holds for Consider now a sequence in
y = e1
of independent random matrices
Y11Y2,...
Gl(d,IR) whose common distribution
p
is such that
T11
is strongly
irreducible and contracting. To get an analogy with the preceding example we write
(where
an
M
= T
T
=
n n
n
K
n
, where
a
0
Bn
Cn
In
is a real number,
Bn
K
n
is orthogonal and
a (d-1)-dimensional column vector and
147
C
n
a matrix of order d- 1). We shall see that
_
that if
V
n
converges to a random direction
to (2) and that
distribution
.
Z
with
By the same type of argument as above we shall deduce
has an exponential moment, then
p
1
a > 0
.
_
a
IIxI1
for some
can be written as a series similar
>
<x,e1>
dv(x) <
In this irreducible situation
has no particular
e1
role and (1) will follow.
The first four sections of this chapter are devoted to the proof of these results and to their applications to limit theorems. In the following we study the regularity properties of the invariant measure v
and give some examples.
6.1. Convergence in the Iwasawa decomposition
As said above we shall write Kn
is orthogonal and
As before, {e1,...,ed}
Tn
Mn = Y1...Yn
satisfies
Mn = Tn Kn where
as
= 0
,
i = 2,...,d
stands for the canonical basis of
They are many ways to express
M
n
]Rd.
as such a product. As far as we are
only concerned with the behaviour of
Tn e1
, any of these will do. Just
for convenience we choose a particular one, sometimes called the Iwasawa decomposition, given by
LEMMA 1.1.
Any matrix M in
Gl(d,]R) can be written as
M = s(M) k(M)
where
k(M)
is orthogonal and
s(M)
is lower triangular with positive
diagonal entries. Such a decomposition is unique. Proof : We apply the Gram-Schmidt orthogonalization procedure to the
vectors
M e1, ..
, M ed . We can write *
M e1
= all u1
*
M e2
= a21 u1 + a22 u2
148
M ed fui,...,ud}
where
i = 1,...,d
.
= adl ut
+ ... + add ud
is an orthonormal basis of
matrix
k(M)
by the relations
matrix
s(M)
by
k(M)ui = ei
i
Then M
*
]Rd. We choose
aii > 0,
This decomposition is then unique. Define the orthogonal
= k(M)
*
*
a.i
if
i ? j
0
if
i<j
M = s(M) k(M)
so that
s(M)
and the lower triangular
Using Theorem 111.3.1 we may introduce the following
Notation 1.2.
If
When
Tu
unique
are independent random matrices
Y1,Y2....
distributed according to
pi
Mn = Y1..Yn
we set
and
is strongly irreducible and contracting,
p-invariant distribution on
Z
Sn = Yn...Y1
denotes the
any non zero random
lim M V = 6_ .
vector such that a.s.
n
n Recall that for any
PROPOSITION 1.3.
P(]Rd) and
v
x # 0
Suppose that
contracting and write
Z
7P( = 0) = 0
,
v
since
is proper.
is strongly irreducible and
Tu
according to the Iwasawa
Mn = s(Mn) k(Mn)
decomposition. Then, a.s. s(Mn)e1 lim <s(Mn)ei,ei> -
Z
Proof : First notice that s(Mn) e1 = <s(Mn)e1,e1>e1
so that
k<s(M1)e1,el>I
=
IIs(Mn)* e1 II = IIk(Mn)*s(Mn)*el II = JIMne1II
and M n Mn e1
s (M
n
)
s ( M )
e1
* 11Mn
e1
I2
<s(Mn)el,e1>2 s(Mn)e1
<s (n)e1 ,e1> Now write a polar decomposition Un
are orthogonal matrices and
* = K A U n n n n
M
of
M
*
n
(K
n
and
An = diag(a1(n),...,ad(n)), with
149
a1(n) > a2(n) ? ... > ad(n) > 0). Recall that the index of since
Tu
(applied to
is one
111
is contracting. It thus follows from Proposition 111.3.2
M
that
)
n
Jim a2(n)
a. s.
a1 (n) = 0 converges a.s.
and that
.
It is clear, at least from the proof
of this proposition, that the limit of Since for any
y
in
,
is actually
Z
]Rd 2
<Mn Mn et
,
y> _ = a1(n)2<e1,Un y> + O(a
(n) a
(n) )
2
1
it follows that
<s(Mn)e1,y>
n- <s(n)e1 ,e1>
<M n e1
= lim
, Y>
n-
= Jim
*
n-
_
y>
,
This proves the proposition.
For any M
in
Gl(d,]R) let
satisfying
a(M)i,i = s(M)i i
The matrix
n(M)
a(M)
denote the diagonal matrix
i = 1,...,d
,
,
and
n(M) = s(M) a(M)-1.
,
COROLLARY 1.4.
is lower triangular with 1 on the diagonal.
Suppose that for any
p = 1,2,...,d-1
is
,
T11
p-contracting and
p-strongly irreducible. Then if Y1...Yn = Mn = n(Mn) a(Mn) k(Mn)
converges a.s. to a random matrix
n(Mn)
If, moreover, ]E {k(Y1)} to
N- .
is finite then
a(Mn)1/n
converges a.s.
diag(e Y1 ,...,e Yd).
Proof
:
Since
s(M ) = n(M ) a(M
n
n)
we have
s(Mn)e1
<s(n)e1,e1> = n(Mn)e1 Therefore the proposition yields that
n(n)e1
converges a.s.
150
In the same way, for any
p = 1,...,d-1
Ap Mn = {Ap rl(Mn)}{AP a(M)}{AP k(Mn)} is an Iwasawa decomposition of basis
{eil A ... A eid
(when we consider on Ap1Rd
AP M
n it < i2 < ...< id}
;
the
in the lexicographic
order). The proposition applied to these matrices gives that, for any p
,
Ap n(Mn) (e1 A e2 A ... A e
. This implies that
converges a.s.
)
p r1(Mn)
converges since for
i > j
<>1(Mn)e.,ei> = 1(Mn)(e1Ae2A...Ae. ] J- 1Aei),e1Ae2A...Ae.
Ae
j-1 j
On the other hand if
a(Mn) = diag(al(n),...,ad(n))
al(n)a2(n)...ap(n) = Therefore when
IIA)M*n
(elAe2A ...Aep)II
is finite and
1E(i(Y1))
then
T11
p-contracting and
strongly irreducible, then
lim--Log al(n)...a (n)=limn Log IIApM*(eiA...Ae )II=Y1+...+Y which implies that
Remark 1.5. If
v
a(Mn)1/n
The process
P
P
P
converges a.s. to
diag(eY",...,e1d)
is a Markov chain on
k(Mn)
0(d)
is the unique distribution on the flag manifold F
which is
*
invariant under the image
of
u
by transposition (see Proposition
p
IV.4.1) then the distribution of el A e2,
converges to
v
.
..., el A ... A ed-1)
In order to show this, write
Sn = r)(Sn ) a(S11) k(Sn)
.
it's also the case for
Since k(Sn)
So and
M
and k(Mn)
n .
Sn = Yn...Y1
as
have the same distribution,
Thus it is enough to notice
that
k(Sn)*'(el,
,e1A...Aed-1)
=
S*.(el,
converges in probability to a random variable with law
le1A...Aed-1) v
(this
follows readily from (ii) of Theorem 111.4.3).
We now suppose that
p
two upper Lyapunov exponents
has a first order moment. In this case the yl
and
Y2
are distinct when
strongly irreducible and contracting (see Theorem 111.6.1)
.
T11
is
1 51
THEOREM 1.6.
If
is strongly irreducible and contracting and if
Tu
1 i(M) dp(M) < W , then a.s. lim
Proof
:
For any
M
t(M) =
We have, for
in
s(M )e n 1
II<s(Mn)el,e1>
Lo g
n
s(M)
=
<s(M)el,el>
M and
M'
in
Z
II - Y2 - Y1
set
Gl(d,]R) s(M)
-
x(M) = t(M)el - el
and
.
(3)
IIM*el11 Gl(d,]R) ,
MM' = s(M) k(M)M' = s(M) s(k(M)M') k(k(M)M') MM' = s(MM') k(MM')
and
.
Therefore by the uniqueness of the Iwasawa decomposition s(MM') = s(M) s(k(M)M') Since
<s(MM')el,el> _ <s(M)el,el> <s(k(M)M')el,el>
this yields
immediately
x(MM') = x(M) + t(M) x(k(M)M')
.
We thus get the following formula, analogous to (2) x(Mn) = x(Y1...Yn) = x(Y1)+t(M1)x(k(MI)Y2)t ... + t(Mn-1)x(k(Mn-1)Yn).
To study this series we shall use the relations
IIx(M) II
< exp(2k(M))
(4)
IIs(M)xii
IIA2 M*II ; x orthogonal to e }
sup {
and
IIxII Since
x(M)
positive integers
is always orthogonal to
m
(5)
IIM* ells
1
and
el
these imply that for all
n ,
IIt(Mn+m)e1-t(Mn)e1II = IIx(Mn+m )e1
-
x(Mn)e1!I
n+m-1 E
IIt(Mj.) x(k(Mj.)Yj+1)II
J=n II A2 M* II2i(Y j
E ]-n JIM* ellI2
Now, the exponents associated with same, we know that a.s.
e
(Yn)
+d +1
and
(6)
(Y*)
being the
1 52
I A2 M. J *
lim 1 Log
j-
JIM* e1
II
2 = lim1 Log IIA2Y*...YIII - 2 lim ]
11 2
J
1 Log 11 Y* ...Yl el II J J
_ (Yl + Y2) - 2 Y1
=Y2 -Yl . Therefore for each
we can find
e > 0
A2 Mi
=
is finite
]P(Log R(Yj+1) > j e)
<
so that, using the Borel-Cantelli lemma, there exists a.s.
that for
,
e
On the other hand since ]E(91(Y1))
jIo
j > Ni
(Y2-Y1+e)
<
JIM* e1112
such that, if
N1
N2 > 0
such
j> N2
exp 2R(Yj+l) < exp 2j e . Hence we obtain for
and any m > 0
n > Sup(N1,N2)
j(y2-yl+3c)
IIt(Mn+m)el - t(Mn)el II z
and, since
is the limit of
Proposition 1.3)1
e
E
J=n
t(Mm+n)el
as
m
(see
n(Y2-Y1+3e)
IIt(Mn )e 1 -
Z
II
e
=
(Y2-Y1+3e)
1-e This yields that for each
e > 0
a.s.
,
s(M )e
n Log II<s(Mn)el,el>
n
II
Y2-Yl+3e
The theorem will follow once we have proved the relations
(4) and (5).
On the one hand we have (see the beginning of the proof of Proposition 1.3)
IIs(M)e 111 IIx(M) II
<
IIMM
e1 II
<_
I<s(M)e1,e1>I
On the other hand, since orthogonal to
el
< IIMII JIM*-' II < ezR(M)
JIM*e1II2 s(M)
leaves invariant the subspace
,
IlelAs(M)*yII
Iis(M)xII sup{
;
IIxII
x orthogonal to e } = sup d 1 YeIR
IIyII
IIs(M)*e1As(M)*ylI
- syp
I<s(M)el,el>I Ilyll
1 53
IIM*eI A M*y I I
= sup y IIM*elll IIytl IIA2 M*II
IIM*eill The proof is now complete.
We have already noticed that it is in general not true that for almost all x
Y = YI
converges to
w , n LogIISn(w)xII
for every nonzero
(see in particular the end of section 11.3). In fact we have,
working for convenience with transpose matrices,
COROLLARY 1.7.
If
Tu
is strongly irreducible and contracting and if
is finite, then for almost all
lE (i(Y1))
(i) If x
is not orthogonal to
w
Z(w)
then
limn Log IIY*(w)...Y*(W)xll = Y1 n-+m
(ii) If x
is orthogonal to
then
Z(w)
ni n Log IIY*(w)...Y*(w)xII < Y2 < Yl Proof
:
Making use of the relation IIuII2IIvIl2_2
IIuAvil2 = we have, if
Mn = Y1...Yn ,
e I 1I2 IIMxll2=IIM n
<x
*
n
Since, a.s.
II Mn eill
IIMn eI AM* x112
2
*
2
IIMn eill
2
,
lim n-1 Log
n
lim -nl Log
and
MnMn eI
n,.
elll=Yl JIM* el A M*xIl IIMn e l
2 11
the claim follows from the fact that, because
MnMn el IIMn elIl 2
M M* e
limn Log IIMneIIII<x, n*n 12 n-_ For
1114, e111 2x 2
Z
=
s(Mn)el <s(Mn)el,el>
>I < Y1 + Y2 - Yl = Y2
.
matrices, this corollary is Osseledec's theorem. To see
this, note that since the upper exponent associated with
(Yn)-I
is
1 54
we always have in this case
-Y2
lim inf n Log So that, if
I
is a nonzero vector orthogonal to
x
his
n
We also have
Let
Y1,Y2....
distributed according to Tut = {M ; M 1 E T
u
be independent random matrices Sn = Yn...Y1 . Suppose that
and
is strongly irreducible and contracting, and that
is finite. Then the two lower Lyapunov exponents
]E (9(Y1))
Yd
are distinct. Moreover let
v'
be the distribution on
which is invariant under the law of Y11 S
-1
Then, for almost all
n with
Z(w)
Log 11Yn...Y* x11 = YZ
n-
COROLLARY 1.8.
-> lim inf - n Log Il (Yn)-1... (Yi)-1 II =y 2*
IIY*...Y* x II
n-
Z(w)
and
Yd-l
and
P(]Rd)
be the limit of
Z
w and every vector
x not colinear
,
1im inf - Log 1ISn(w)x11 > Yd-1 n nProof
We know, applying the preceding corollary to the sequence
:
Y11, Y21,
...
that a.s., for any
y
orthogonal to
Z(w)
lim sup n Log II (Y1)* n ..(Y-1)* Y11 > -Y d_1 n*m are the > -Yd-1 > ... > -yl (Y-=Yd )). Let now x be a vector such that n . We can find a vector y such that <x,y> # 0 and = 0.
(We have used the fact that
exponents associated with x # Z(w)
Since
I<x,y) I
=
1<(Snl)*y
,
Snx>1 < II (Snl)* YII IisnxII
we have
lim inf - Log 11 S. x II > - lim sup - Log 11 (Snl)* Y11 > Yd-1 n n-
Remark 1.9.
vector
z(w)
n-
Since Osseledec's theorem states that there exists a such that
Ledrappier [46])
lim - Log IISn(w)z(w)II= Yd
we see that actually
lim -1 n Log 115n(w)Z(w) II =
n
Yd
(see e.g.
155
Suppose now that for any
p = 1,..,d-1,
TV
is
p-contracting and
p-strongly irreducible. Then one can show that with the notation Corolarry 1.4, a.s.
lim n i = 1,...d
of
,
n
Log IIM*(N*)-leill = Yi for a direct proof). The
(see Guivarc'h and Raugi [34]
Osseledec's subspaces are thus described by the random matrix N- .
Exercise 1.10.
Prove that if
Exercise 1.11.
x
in
.
is strongly irreducible and
Tu
contracting then there exists a random vector nonzero
is, for
Show that the first part of Corolarry 1.4
matrices, a direct consequence of Corollary 11.7.1
2 x 2
such that for any
Z
Iltd
S
S
'n
1im
*
n
Z Z a. s.
2
n-
Find the limit of
VI.2. Limit theorems for the coefficients
We first prove the following important result of Guivarc'h and Raugi.
THEOREM 2.1. Tu
Let
p
Gl(d,1R) such that
be a probability measure on
is strongly irreducible and contracting. If for some
1 exp T R(M) dp(M)
is finite then there exists
p-invariant distribution
on
v
{ II YIIP1
Proof
:
Let
f
such that the
P(1Rd) satisfies
dv(x) <
"XII <x,Y>I
B = P(1Rd) x P(A21Rd)
and
p
:
p(M,(x,w)) = Log
.
Gl(d,]R) x B -> 1R be the
cocycle defined by
_
a > 0
T > 0 ,
IIA2MwIIIIx(I2 IIwII IIMx1I2
156 for
M
in
Gl(d,]R)
x
,
Sup x,w
w
IRd and
in
in
. We have
A2 IRd
<_ 22(M)
p(M,(x,w))I
Mn = Y1 Y2 ... Yn
Moreover, for
Sup1E{p(M*,(x,w))) < ]E{Log IIA2M*II} - 2 inf]E(Log n n x,w x is negative for
n
large enough since
Hence there exist unit vector
IIA2 MwIIIIxII2 n
Log Sup x,w
n
n-
C > 0
n
)
IIxII
(see Corollary III.3.4).
yl > Y2
It thus follows from Lemma V.2.3 that there exists
lim
IIM*xII
a > 0
<0
}
]E{
such that
IIMnxII2 IIwIi and
S
0 < S < 1
,
such that for any
,
x
IIA2 Mnll ]E
)aa <
CS
n
.
(7)
IIMnxII2 We may of course suppose that
a < 1
In this case we have, taking
.
into account Proposition 1.3 and (6)
lim ]E
}
"Z11
1E{
II
n-r°°
s(Mn)el II
<s(Mn)el,el>
{IIA2 M II
j=o
lE{e2a 2(Y j+1) 2
IIM3 e
II
}
}a
2a 2(Y ) {e
<
This is finite for in IRd
by K-1Z
by
i
Y .
}
1-
small enough. Now for any unit vector
a
choose an orthogonal matrix K
replace each Z
l
K 1Y K , then M
i
such that
n
Ke1 = y
x
IIK-1
Z II
I
But
-1 Z,e1>I
2(K-l
2(Y1) =
}a' ]E
Y1K)
,
I
IK-
2a 2(K lY K) {e l } 1CS
1 ZII
=IIZII
and
.
< K ' Z,e1>=
Therefore a
1
-I<x,y>Ia
_ 2a 2(Y dv(x) = 3E{--Z-i_)a <]E {e 1 } II
The proof is complete.
C
11-a
n
K
and
we obtain as
above that 1E{
y
If we
is replaced by K- 1M
Since the formula (7) is valid for any
.
.
157
As a consequence let us show
PROPOSITION 2.2.
Let
theorem. Then for any
and any nonzero
are independent
x, y
IRd
in
Sn x, y>
1
r Log
n
a.s.
= 0
ISnxli
Proof : We shall show that for
large enough
S
IISnxli
1
= 0
lim
n-co
Y11 Y2
satisfies the hypotheses of the above
u
r > 0
lim
n-.
where
Sn = Yn...Y1
matrices whose distribution
a.s.
,
,
(8)
<Snx,y>
n
which implies the proposition. By the Borel-Cantelli lemma it suffices to prove that for any
c > 0
S x
n
IP(I<
£
n=1
,
y>I < c
n_S) <
11Snx11
We shall first use the exponential convergence of the law of
S x
n IISnxII
to v
.
Define
fn
[0, 1] > IR
:
by
if 0< t < E n
1
if E < t < 2e n -a if 2c n-S< t <1
2- t ns E-1 0
It is easy to see that for all t, t' 'fn(t) - fn(t')I = It-t'I
Therefore if we define
On
:
P(JRd) -> IR
n (x) = fn (I<
s
n-S
0,11
in
ns Cl
by
x , y>1) IIx11
we have for 11x11 = IIx' II = 1 (I<x,y>I- 1<x',y>I)ns
,n (X-) - n(x') I
E
S
2n
a(X,X')
(see the beginning of the proof of Lemma V.4.2). This entails that, with the notations of Chapter V, 110,111a =
ma(cn) 6
2e
S
+1
1 58
for any
a
a
for
0 < a < 1
,
Sn E
Taking into account Theorem V.2.5
.
we obtain,
small enough,
x
y>I < en-S) < E
1P(I<
n=1
]E{
n
IISn x11 <
lE {
n
n (Sn x)}
n(Sn x)} - J
n dvl
+ n J) n
dv
IIPn-NIIa II0nIIa + n S On dv n
C+E J for some
But
C > 0
.
E J dv
<
n
n=1
dv
n
n
I< X y> I< 2e n-S}
E v{x ;
IIxII'
n
< E v{X, { n
IIXII
}1/0 >
n1/s}
(2E)
I<x,y>I
is finite as soon as
11-11I 1/S
d v (x) <
I<x.Y>I This holds for
sufficiently large by Theorem 2.1
S
.
For such
8
(8)
holds.
COROLLARY 2.3.
Let
So = Yn...Yl
where
random matrices with common distribution
Yl,Y2,... 1
.
are independent
Suppose that
strongly irreducible and contracting and that for some 1E{exp T9,(YI))
is finite. Then there exists
(i) For any nonzero vectors lim
a > 0
T11
is
r > 0
such that
x, y
Log I<Sn x, Y>I = Y
n,+_ n and - {Logl<Sn x, y>I- n y} law
converges in distribution to the gaussian
N(O,a2) (ii) If
On) 2
i,j
,
1 < i, j < d
are the coefficients of the matrix
Sn, then the 1Rd -valued random vector
{I (Log
I (Sn)i'j I - nY) , 1 5 i, j S d}
converges in distribution to a random vector = Xk,R
Xi j
for any
1 < i,j,k,R <_ d
X which satisfies
and such that
X11
is
159
distributed according to Proof
:
N(O,a2)
.
The statements in (i) are direct consequences of Proposition
2.2, the law of large numbers for 1 Log IIS
(see Corollary 111.3.4) n n xII and the central limit theorem (Theorem V.5.1). In order to prove (ii),
taking into account the convergence in distribution of
n (LogI(Sn)111I- ny)
to
N(O,a2), it suffices to show
that, for any
1 < i, j < d Log I(Sn)111I
n Log I(Sn)i,jl
converges to zero in probability. It follows from Proposition 2.2 that a. a.
,
lim n (Logl(Sn)1
Logl(Sn)1,jI) _ 1
I<S e
n 1'e1
lim n (Log
>I
e>I
I<S e
n
- Log
1
'
= 0
]
IIsn e1II
IIsn e111 On the other hand I<S e
I<S ee>I
e,>I
LogI(Sn)1,jI - LogI(S.)i'jl = Log
Log
n s*
IIf*le JII
n
II
's*
e
n
= Log i<e
>I-LogI<e,n
,
1
i
II snej II
J
>I
I I Snej II
converges in probability to an a.s. finite limit (this follows from the fact, seen in Theorem 111.4.3, that a random direction
Z
satisfying
converges in probability to
n
j
IP( = 0) = 0
for any nonzero x).
Therefore (Lo g
n Exercise 2.4.
I
n 1,1I- Lo g
(S )
i
Suppose that
contracting and that for some Show that for any nonzero to
-yd
(S
n
Tu'
)
i1jl)
converges to 0 in
is strongly irreducible and
T > 0
,
J exp T i(M) d}i(M)
x , 1 Log IIS-lx II
n
probability.
n
is finite.
converges almost surely
.
VI.3. Behaviour of the rows
As the preceding results, the following is due to Guivarc'h and
1 60
Raugi (see [59]). We use the notation
Suppose that
THEOREM 3.1.
and that x
is finite for some
converges to
,
.
is strongly irreducible and contracting
Tu
exp T Z(M) d}i(M)
f P(]Rd)
in
1.2
T > 0
Then for any
.
Using the
a.s.
Z
converges to
Proof : We first show that
.
Z
notations of the first section we have d
= M el
Y ...Y e
n
1
n
1
s(M ) k(M )e
n
1
n
= 1
E
i_1
n
e.> s(M )e. n 1 1' 1
Hence
s(n)e1 <s(M )e1'e n
M1 e1 <M
n e 1' e I >
s(n)ei
d
+
> 1
E
n e1'e1>
<M
i =2
Taking into account (5) and (i) of Corollary 2.3 we see that, for
i = 2, lim
,d l n
,
I s (Mn)eill Log
n-*
I<Mnei,el>I
<
lim -nLog n,
IIs(Mn)eill I<
ne1'e1>
II A2 Mn II <
lim - Log nn II n el II I<e1 ,M*el> I
<
(Y1 + y2) - 2y1 < 0
.
Therefore a.s.
n (see Proposition 1.3) and Now for any that
e1
x
in
n M.-e1
P(]Rd)
Z
<s(Mn)el,el>
converges to
Z
choose an orthogonal matrix K
K 1 Mn Z
K -'--Z
so that
.
When we apply this theorem to the sequence that if
such
K 1Yi K
Applying the latter result to the matrices
.
we deduce that converges to
s(M)e1
Mn e1 <Mle1,e1>
lim
Sn = Yn...Y1
then
Y11Y2....
we find
converges a.s. to some direction
(this improves Theorem 111.4.3). Choosing
x
equal to some basis
Z
1 61
we see that the directions of the rows of e. 1 a.s. to the same limit. vector
Sn
converge
VI.4. Regularity of the invariant measure
We shall now see, following Guivarc'h and Raugi, that although the p-invariant distribution
is often singular, it always has some
v
regularity.
Suppose that
PROPOSITION 4.1.
T11
Then there exists
on
is contracting, strongly
T > 0
irreducible and that, for some
is finite.
f exp T R(M) dp(M)
,
p-invariant distribution
such that the
a > 0
v
P(]Rd) satisfies -
Sup d
and
u
dv(x) <
f
ycP(7R )
Proof any
:
If
y
are two orthogonal unit vectors, we have, for
x
Ix A y I
2
1
=
I
12
IIx
- <x,y>2 2 <x,u>2
.
Hence 11-11
Il
I<x,u>l
llxAy11 Since, by Theorem 1.2, there exists some
Sup
f(
lxII {
the proposition is clear
-
}a dv(x) <
.
Under the above hypotheses, there exist
such that for any
C > 0
such that
I<x,u>
Ilu 11=1
COROLLARY 4.2.
a > 0
x
in
P(]Rd) and
v({Y ; 6(x,Y) < E}) < C
Ea
x
and radius
v(B) = J
a(x,Y)
B 6(x,y)a
e d(x,Y)-a
dv(Y) < E:(' 1
and
.
Proof : This follows readily from the fact that if center
a > 0
e > 0
dv(Y)
B
is the ball with
1 62
As a consequence let us prove that the dimension of
V
is
positive. First recall the definition of the Hausdorff dimension of a
A
subset
(P(1Rd),6). The reader is referred to
of the metric space
Billingsley [5]
for details. For any
I > 0
and
E > 0
let
ks(A,E) = inf E (diam B.) i
where the infinimum extends over all countable coverings of A by closed balls
of diameter less than
B,
i
E
,
and let
R(A) = lim R,(A,E) E4O
A
The Hausdorff dimension of
dim A = sup{(
is defined by
i (A) = W} = inf{ : R(A) = 01
:
13
dim V = inf{dim A ; A is a Borel subset such that V(A) J01. Now if
v(A) # 0
E v(Bi)
V(A) Za(A) >_ v(A)
so that
and if
and
(B
<
)
i
is such a covering we have
C E (diam Bi)a
dim A ? a . This proves that
dim v2_! a > 0.
According to Ledrappier [46] one may conjecture that, for
2 x 2
matrices,
dim v = - 2y f
y) (x) dv(x) du(g)
Log d(B
(9)
(This is actually proved in [46] but for a weaker notion of dimension).
We have seen (see 11.7.2) that plane
H = {z E
IC
;
Imz > 0}
g.z when
g = [ a
d
]
.
is just the action of 1R U {-} p
, then
If
.
V
GP
Sl(2,1R) acts on the upper half
by the formula
az + b cz + d
Also that the extended action on the real axis Sl(2,]R) on
P(1R2)
if we identify
P(1R2) with
is the closed subgroup generated by the support of
is carried by the so-called limit set
L = {X E ]RU {oo} ; ](gn) C G11
L
of
Gu
,
i.e.
, his g n i = X} n_ a_
(This follows from the fact that
L
is closed and
G
- invariant or
u
1 63
converges
directly from Corollary 11.7.2 which asserts that to a random variable with law
V)
.
L may be arbitrarily small
It is known that the dimension of
V may be (and is often)
(see e.g. Beardon F3]). In particular singular.
It is worth rephrasing the above corollary in the context of projective transformations.
Let
COROLLARY 4.3.
G1(2,1R) such that
be a probability on
1i
strongly irreducible and contracting. If for some eT £(M)
Tu
is
,
is the probability measure on
is finite and if v
dp(M)
T > 0
]R
f
which satisfies
J
f(ax+b) dp(a,b,c,d) dV(x) = J f(x) dv(x) cx +d
for any bounded continuous function
a > 0
such that
C > 0
and
]R, then there exist
on
f
(i) For any
in
y
]R
J Ix-yl-a dv(x) < C (
(ii)
J
(1+x2)a/2 dv(x) < C
.
In particular the distribution function
F(x) = v(--,x]
is H6Zder
continuous.
Proof and
:
The mapping
'Y(°°)
= e1
]R U {oo} > P(]R2) defined by P(x) =xe1 +e2
T :
sends
v
onto the unique
P(]R2). Therefore, by Proposition 4.1 such that for any real
,
p-invariant measure on
there exist
y d(T(y),T(x))-a
dv(x)
<
C
dv(x)
=
C
J>R
and
( J
]R
But
d(1(y),Y(x))
II (x e1+e2) A (yet +e2) II Ix e1 +e 211
Ily et + e 211
C > 0
and
a > 0
1 64
Ix-y I /1+J
and
6(1'('),T(x)) =
1+yZ
I(Xe1 +e2) A e1 II
1
1+xZ
1+xZ The proposition follows immediatly.
satisfies a law
v
Finally we notice that the probability measure
of pure type with respect to the natural measure md . Recall that
]R U [-I, m2
On the projective line
and
d
P(]R ),(see II.5.1).
is the unique rotation-invariant distribution on
can be identified with the Cauchy
distribution and is equivalent to the Lebesgue measure.
PROPOSITION 4.4.
Let
be a probability measure on
p
has only one p-invariant distribution v
- or
V
and md
- or
v
is purely discrete.
Proof : Write
component of
v
on
P(IR)d). Then
is absolutely continuous w.r.t.
-either
for the discrete (resp. continuous) P * V2
It is readily seen that
.
and
are mutually singular ,
(resp. v2)
V1
V
G1(d,]R) which
is continuous.
Therefore the equality *
V = V entails that thus
.
But
v1 + u
V2
and
11 * v2
have the same mass,
By uniqueness either
V = V2
or
v2 > }1 * v2
V2 = 11 * V2
*
V = u
Suppose now that
V
.
v2
is continuous and write
V = V1
Va.c
.
(resp. V ) for s
its absolutely continuous (resp. singular) component w.r.t.
for any M
Gl(d,]R)
in
Mmd is equivalent to
and
m
.
d
Since
(see 11.5.1)
(
u * Va.c =
I
M Va.c dp(M)
is absolutely continuous, entailing as above that
11 * Va. c
uniqueness of the continuous or
Example 4.5.
V
p-invariant measure either
V
= V
a.c
By
is absolutely
is singular.
It was already noticed by Kaijser in [38] that is it easy
to exhibit singular invariant measures. Consider for instance two non negative
2 x 2
matrices
1 65
[ai
bt
c,
dt
M1
Idet Mil = 1
such that
show that if
a2
1 d2 J
L c2
M2
ci > 0
,
b2
and
> 2
d
i = 1,2
for
.
Let us
is the Bernoulli measure defined by
P
u({M1 }) = P({M2}) = 1/2 then the
11-invariant distribution
We shall consider
V
on
prove that the Lebesgue measure of the support For
i = 1,2
is singular.
P(1R2)
]R U {-} and
as a measure on the projective line
V
and
x
in IF, U {m}
V
of
S
is zero.
set
a.x + b. i
q.(x) =
i
c.x + d.
i
a2
a = Max(a1,
It is readily seen that if
.([O,a]) under
is contained in
T.
,
[O,a]
implying that
Therefore
.
(x) I =
.
Now, since for
<
2
1
12
di
(cix + d
we see that if
p = Min(d1,d2)
then
o Oi o
(Oi
, i )
...
is an interval of length smaller than
A = U {(c 1
,
p n a . Hence if
... , 4i ) [O,a] n
A
then the Lebesgue measure of
V(A)=(11n * V) so that
S
[o,a]
n
2
1
;
,in = 1,2}
i1
is smaller than
=_L_
2n p n a
.
But
,...,¢i )-1(A)) = 1 n
E
2n it,...
is contained in A . This shows that for any
n > 0
<_ 2np-na . Since p > 2 , m(A) = 0
m(A)
Exercise 4.6.
Let
11
be a probability measure on
a unique invariant distribution v
V
is the smallest closed subset
M
Exercise 4.7. (or
[O,a]
x ? 0
any
of
then
[0,a]- is invariant
is contained in
S
bt
b2) c2' d1' d2
c1
in
Tu
Gl(d,]R) which has
P(]Rd). Show that the support
on S
of
P(]Rd) which satisfies
.
(a) Suppose that
P
is a probability measure on
G1(d,]R)
S1(d,]R)) which has a bounded density with compact support w.r.t.
166
the Haar measure. Show that (b) Deduce that if
has a bounded density.
V
is absolutely
V
is not singular then
P
continuous.
Show that the Cantor distribution is
Exercise 4.8.
some non-irreducible probability measure
VI.5. An example
Let
(5.1)
:
p
p-invariant for
Gl(2,IR).
on
Random continued fractions
{An, n
1}
>_
be a sequence of independent random
positive integers with a common distribution
p , defined on some
1P). Consider the 2 x 2 matrices
(0, A
0
[1
Yn and call
p
A
n
their distribution. For any
in the projective line
x
]R U {co} , 11 Al + A2+ 1
+An+x Notice that Z
with values in
converges a.s. to a real random variable [0,1]
The distribution
.
V
of
is the
Z
p-invariant distribution on the projective line. This follows readily from the fact that the
A n's
are positive integers and from the
elementary properties of continued fractions (see e.g. Billingsley [5]). (Much less obvious is the fact that this also holds for arbitrary real i.i.d.
A
n
under an integrability assumption, which is an immediate
consequence of Theorem 3.1). Let us show that
V
is singular as soon as the
a.s. constant (in this case
Tu
A,'s i
are not
is strongly irreducible and
contracting, cf. Exercise 5.4). This result is due to Chatterji C1?. First recall that any real number
t
in
(0,1)
may be expanded
in a continued fraction, i.e. t = lim (a1 (t)
n-
where we define inductively
;
a2(t)
...
;
an(t))
1 67
1
(a 1) =
When
a1+(a2;...;an)
is irrational this expansion is unique. In particular since
t
A1(w),...,An(w)
law of
V
V-a.s. unique. As a consequence the
is continuous, the expansion is
under
1
(at;...;an) =
a1
under
is the law of
]P
a1(t),...,an(t)
V
Consider now the mapping
T
(0,1) -> (0,1)
:
defined by
T(t) = t - [t] Since T
(a1(T(t))
;
a2(T(t))
;
a3(T(t))
;
...) = (a2(t)
;
a3(t)
;
...)
,
is equivalent to the shift on the sequences of integers. The 0-1
law thus implies that
is invariant and ergodic under
V
(1+x)-1
well known that the measure under
T
dx
on
(0,1)
(see e.g. Billingsley). Therefore either
w.r.t. the Lebesgue measure or a1(t), a2(t), ...
v = 1
(0,1)
(Log 2) -1
.
But it is
is also ergodic V
is singular
(1+x)-1 dx
.
But
are not independent under this latter distribution
(elementary computations). This proves that
(5.2)
T
V
is singular.
Let us outline a proof of Ledrappier's formula (9) in this
simple situation, which is in this case due to Kinney and Pitcher [43]. Let
Jn(w) = Y1 (w)...Yn(w) 0,1] It is readily seen that we may write
Mn = Y1...Yn =
where
qn = Angn-1 + qn-2
[Pn_i
Pn
L qn_1
qn
The length .
IJn(w)I
of
Jn(w)
IJn(w)I = IMn(w)'0 - Mn(w)'1I =
But, since
q
n
+
qn2 Therefore, if
'
2 qn-1
]E(Log A1)
associated with
n,. n
{gn_I(w)(gn(w)+qn-1(w))}-
5 q n-1
2
lim
satisfies
(Yn)
15
gn-1(gn
+q n-1)
2 + 2 - qn-1 qn
is finite and if
y
is the Lyapunov exponent
then
Log IJn(w)I = - lim n Log IIMn(w)e2112 = -2y
On the other hand
IF a.s.
168
V(Jn(W)) = V({t e (0,1)
n II
p({A()})
i=1
i
(Recall that under
V
a1(t),...,an(t)
,
distributed according to if
p(i) = pi
a1 (t) = A1(u),...an(t) = An(W)})
;
are independent r.v.
p). Making use of the law of large numbers,
,
n
n Log V(Jn(W)) = 7E (Log p(A1)) E
pi Log pi
a.s.
i=1 (Note that since
Log x < x-1
E pi Log i = ]E(Log A1)
and since
is
finite
IEpiLogpil =EpiLog Z + 2EpiLogi < (E 2)-1 + 2 1E (Log A1) i pi i is also finite).
Therefore
Ep Logp
Log V(Jn (W) )
i
lim
n- Log J. (W) = In other words if for any
in
t
In(t) = is a (0,1)
;
2y
i
]P
a. s.
we set
(0,1)
a1 (s) = a1 (t) ,...,an(t) = an(s) }
then
Log V(In (t) )
him n-
Ep. Logp. 2Y
Log IIn(t)
1
V-a. s.
Onecan deduce from this formula (see Kenney and Pitcher [43]) that
dimV= -
E pi Log pi 2y
In order to recover Ledrappier's formula (9) it suffices to note that if
M = (0
1)
then n
\)(M-In(t)) = p(a)
p(ai(t)) = p(a) V(In(t))
ll
i=1
(use the formula
V({s ; ai(s) = a, a2(s) = al(t),
an+1 (s) = an(t)}))
Thus
d (M
N))
= p (a)
J Log d(M dN)
on
(0,I)
and
1V) dV dp(M) = E pi Log pi
169
An interesting example of such a measure is the Minkowski
(5.3)
measure denoted by distribution on
(0,1). We may define it as the
on
?
]R U {oo} where
(0
n)
=
11-invariant
is defined by
p
n
I
n
,
2
In order to describe it let us notice that if
is the Bernoulli
p
measure which verifies i(A) = 11(B) = 1 /2 for
B=( then
0)
p-invariant. (This is proved as follows
is also
?
B=(
11)
ti
:
if
V
is
ti
p-invariant then
Z(6A+6B) * V =v
v=26A *v+2 6B * (26A+Z dB)
and
V
2 6A * v+ 4 6BA * v+ 4 6B 2* v By induction we have for any
n > 0
,
n
V= E - 6 m=1
and, going to the limit
v But
m=1
*v
d
Bn
2n
,
m=1 2m
E L 6
p =
* V+
Bm-1A
2m
V
dBm-1A
, hence
V = ?
by uniqueness of the invariant
Bm-1A
2m
measure).
We now define inductively the Brocot sequence of order p1(n) p2 ( )
q1 (n)< - .. -
po(n) qo(n)
po(0)
For
n
0
p1(0) 1 q1(0) = 1 . The sequence of order
n = 0 , qo(0) = 1 and
is obtained by adding to the sequence of order
pi(n) + pi+1 (n) qi(n) + qi+1 (n)
n
the fractions
i = 0, ..., 2n-1 '
For instance the first three Brocot sequences are order 0
0
1
1
order 1
order 2
1
0
0+1
1
1+1
=
0
0+1
1
1
1
1 +2
3
2
1
1
2
1
1+1 2+1
__
1
1
3
1
n + 1
170
Ifs M
2 x 2
is a
matrix such that
Idet MI = 1
q and
,
it follows from Bezout's theorem that q
M = (P -P q '-q
P) q
Whence, since and
x
B'x =
x+1
11
MB-0 =.P MA-1 = MB-1 = + . Making use of this q q P +q result one sees immediately that if Y1,...,Yn are in {A,B} then
MA-0 =
,
Y ...Y n
Pi (n)
10,11 _
for some (with
0
_<_
i < 2n
Y. = A
or
.
B)
(n) Ci
Pi+1(n)
4 . (n)
1
i + 1(n)
Since there exists only one sequence
Y11 ...,Yn
which satisfies this relation we conclude that
Pi+1 (n)] _
i(n) ' qi+1 (n)
rpi(n)
(In
pi+1 (n)]
)( qi(n)
U
' qi+1(n)
Obviously this formula defines
?
1
2n
)
uniquely.
Related examples may be found in Chassaing, Letac and Mora D,] Exercise 5.5
.
is also taken from this paper (see also Letac and
Seshadri [51]). It gives a non-trivial example where the Lyapunov exponent is explicitely computed.
Let
Exercise 5.4. p
A be a not a.s. constant real random variable, and
be the distribution of
Y= (0
1
A)
1
a. Show that b. Let
Tu
be the
v
is strongly irreducible and contracting.
p-invariant distribution on the projective line
1R U {oo}. Prove that the Lyapunov exponent equal to
y
associated with
f Log x dv(x) , when there exists some
T > 0
]E{IAIT} is finite. Show that this is also true when A integer-valued, as soon as
Exercise 5.5.
[11]
distributions on
[51]. For
]E(LogIAI)
a > 0
,
11
is positive-
is finite.
let
and
as
]R+ defined by
aa(dx) = 2 exp - 2 1 0
dx
va
is
such that
be the two
171
2 (x + X) 1(0,co) dx
va(dx) =
where
j ur-1 exp -
Kr (a) =
a
(u + u) du
0 2
a. Consider two independent random variables the law of
X
(resp. A)
is
va
(resp. Aa)
.
X
and
A
such that
Show that the law of
A)- 1
(X +
is
Va
.
(Hint
:
compute the Laplace transforms).
b. Show that the Lyapunov exponent distribution of
0 (1
1
A)
y
2Ko(a) is equal to
aK
1 (a)
associated with the
SUGGESTIONS FOR FURTHER READINGS
We provide some recent references on limit theorems for random matrices and related topics. We make no claim for completeness but the quoted papers often contain a large bibliography.
(1) Lyapunov exponent for stationary sequences
The main properties of Lyapunov exponents in the stationary case can be found in Ledrappier [46] . Guivarc'h [32] (see also Royer [62] Virtser [69],[70]) gives a criterion ensuring that two given exponents are distinct for markovian products. See Ledrappier [48] for an assumption implying that the exponents are not all equal, in the general stationary setting.
(2) Boundary theory
After the fundamental work of Furstenberg (see [20] , the set of bounded harmonic functions was determined
E221, [23] )
- for absolutely continuous distributions on connected groups by
Raugi [59]
(see also Guivarc'h E311),
- for distributions on discrete groups of matrices by Ledrappier
[47]
.
(3) Limit theorems (3.1)
Onecan find a proof, under our usual irreducibility
assumptions, of - the functional central limit theorem, - the law of iterated logarithm, - the renewal theorem,
173
174
- the local limit theorem,
for the sequence
Log II Sn x II
in Le Page [49], [50]
properties are studied in Guivarc'h [33] The central limit theorem for
Sn
.
Recurrence
(see also Bougerol [9]) written iri the polar and the
Iwasawa decomposition is proved in Raugi [59]
.
References to earlier
proofs and applications can be found in Tutubalin [68]. (3.2)
Properties of the solutions of the difference equation on
d IR
Xn+1 = Yn Xn + Bn Bn
(where
[41] d = 1
is in
IRd
and
and in Le Page [50] ,
Grincevicius [34]
(3.3)
.
Yn
in
G1(d,1R))
are studied in Kesten
Stationary solutions are given in [8]
.
For
proves a central limit theorem.
Without irreducibility assumptions, the central limit
theorem is not yet fully understood. The latest reference is Raugi [60].
(4) Positive matrices
The reader will find in Cohen [14] a nice account of the applications of products of positive random matrices to demography and an extensive bibliography. Kesten and Spitzer [42] study the convergence in distribution of such products.
(5) Linear stochastic differential equation
A goog introduction to this subject is the survey of Arnold and Kliemann [1]
. A nice application is given in Pardoux and Pignol [58].
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PART
B
RANDOM SCHRODINGER OPERATORS
INTRODUCTION
Numerous equations arising from one dimensional discrete physical systems lead to the analysis of a linear second order difference operator H, acting on a complex sequence 'Y
n
, n e 7L , by
(H'1' )n = b -1 ((-AY ')n + an 'fn)
In this formula, A is the discrete Laplacian (AY')n ='n+1 +T n-1 and an bn , are two fixed sequences of real numbers with bn > 0,
Tn
'
representing the physical properties of the medium. Generally such operators are associated to "time dependent" equations and we give some typical examples
:
(i) A solution of the Schrodinger equation i
-i) t satisfies a
0(n,t) = Y'n
a
at
= H of the form
HT = a'Y where H is the classical Schrodinger
operator, that is the operator associated to b
n
= 1, Vn E 2Z, and a
n
is
the potential at site n. 2
(ii) A solution of the wave equation bn
iat satisfies a
0(n,t) = 'n
a
2 - _
of the form
at
2
HT = a 'Y where H is the "Helmotz operator" that
is the operator associated to an = 0, Vn E Z Z, and bn is the diffusion coefficient at site n.
(iii) A solution of the heat equation bn (A )n of the form at = ¢(n,t) = Tn a XtsatisfiesH'Y =a4' where H is the Helmotz operator.
Similar equations and operators also appear in quasi-one dimensional systems associated to an infinite wire of finite cross section with R sites. We have only to replace the sites n by (i,n) where i e (1,
..
,
R), (the integer k is called the width of the strip) and
the real sequences an,bn by matrices sequences. It is well known that the spectral properties of the operator 11
183
184
viewed as a self adjoint operator on an Hilbert space, govern the asymptotic behavior of the solutions of the associated time dependent equation. In quantum mechanicsthe number I$(t,n)12 (normalized in a itH
of the time f) n dependent Schrodinger equation, represents the probability of presence of
suitable way) associated to a solution (t,n) = (e
a particle at the site n at time t. Roughly speaking, when T is associated to the continuous spectrum of H then we have the "diffusion" T
behavior
lim
1
T}+ 2T -T
I$(t,n) 2dt = 0, and if T is associated to the
point spectrum of H then we have the "localization" property lim
Sup t
E
I4(t,n)12 = 0.
(See D. Ruelle [53] for more details). In
n,N
deterministic systemswith periodic structure, it is known that only diffusion behavior occurs. But it has been remarked that localization appears when this regular structure is perturbed by impurities or inhomogeneity in the medium. Thus a "metallic" wire suddenly becomes an insulator. We first give below a brief historical survey of the mathematical approaches to this subject.
(1958) that for the
It was first announced by P.W. Anderson [1]
classical multi-dimensional Schrodinger operator with an independent identically distributed family of random potentials, the spectrum has to be pure point for a "typical sample" assuming the disorder "large enough". It was later conjectured by N. Mott and W.D. Twose [46]
(1961)
that in the one dimensional case, this should be true at any disorder. The works of H. Furstenberg, H. Kesten [20]
(1960), H. Furstenberg [19]
(1963), V. Osseledec [49] (1968) provided the essential mathematical
background used in the first rigorous approaches of the subject. It was first proved by H. Matsuda, K. Ishii [44]
(1970), A. Casher
J.L. Lebowitz 110] (1971), L.A. Pastur [50]
(1973), Y. Yoshioka [64]
(1973) that there does not exist an absolutely continuous component in the spectrum of H. An essential step was achieved in 1973 when I. Ja. Goldsheid, S.A. Molcanov, and L.A. Pastur [24] gave the first proof to the conjecture of Mott and Twose (they actually dealed with the continuous case). Their original proof has been later considerably simplified and extended by R. Carmona [7] [52]
(1982),
[8]
(1983), G. Royer
(1983), J. Brossard (1983). In the "discrete" case the same result
has been obtained by H. Kunz, B. Souillard [37]
(1980), J. Lacroix [38]
(1982), F. Delyon, H. Kunz, B. Souillard [13] (1983). Moreover Goldsheid
gave a similar announcement in a strip [23](1981) and the proof can be
185
found in J. Lacroix [39]
(1983) [40]
[41]
(1984).'All these previous
proofs of localization are rather technical and at times the essential guiding principles are not easily understood. Fortunately, in the late of 1984, S. Kotani [36] clarified the situation, giving a rigorous statement to an earlier claim of R.E. Borland [5]
(1963).
Our essential goal is to provide a direct and unified treatment to the foregoing problems, in the general setting of operators H introduced at the begining of this discussion here in after called Schrodinger operators. The essential tool will be the theory of products of i.i.d. random matrices developed in the first
part of this
book. We are mainly concerned with the independent case but a lot of definitions and properties are given in the general ergodic case. The "almost periodic case" is also of great physical and theoretical interest but most of the proofs have nothing to do with random matrices.
Interested readers have to look at the survey of B. Simon [57] where they can also find an extensive bibliography. Since it seems that the theory of random matrices can hardly be used in the multidimensional case (up to now,limiting procedures in strips whose width goes to infinity have not been successful) we restrict ourselves to the one dimensional case and strips.
In chapter I the essential definitions and properties related to the spectral analysis of the deterministic operator H are given. In particular we construct a sequence of "good approximations" of the spectral measure of H and establish the existence of "slowly" increasing generalized eigenfunctions. Moreover the links between the singularity of the spectrum and the fundamental notion of hyperbolic behavior of a product of matrices are pointed out.
In chapter II we define an ergodic family of Schrodinger operators which contains as essential examples the classical Schrodinger operator and the Helmotz operator. Some weak properties of the spectrum of H considered as a subset of JR are given, before introducing the essential
concept of Lyapunov exponent. Positivity of this exponent is carefuly studied since this property is crucial in order to obtain absence of absolutely continuous spectrum. The distribution of states describing
186
the asymptotic behavior of the eigenvalues distribution for the operator restricted to "boxes" is of great physical interest and we discuss in detail its regularity properties together with the links with the Lyapunov exponent (Thouless formula). Kotani's criterion insuring localization property is then introduced in the general ergodic case but its main application to the independent case is discussed in the following chapter. Finaly we give a straightforward application of the central limit theorem on SL(2,1R)
to the asymptotic
behavior of the conductance.
Chapter III is devoted to the proof of the conjecture of Mott and Twose both in classical Schrodinger and Helmotz case. In the first model, Kotani's criterion gives immediately the solution, but in the general case the proof is more involved and requires some Laplace analysis on SL(2,IR) . As a consequence extra properties of the
distribution of states are obtained.
All these foregoing results are generalized in the chapter IV to the case of a strip. Most of the previous proofs in the one dimensional case
can be translated with some care. But some problems
are much more involved, especialy positivity of Lyapunov exponents. General results given in the first part of the book are then very useful. The proof of localization in the general case requires also much more work since Laplace analysis on symplectic groups needs some knowledge about symplectic geometry.
Numerous related topics, non-stationary processes for instance, are not tackled when they don't appear as direct applications of products of i.i.d. matrices, thus we don't intend to provide a complete survey in the theory of random Schrodinger operators. Moreover we are aware of that a lot of pioneer and connected works are not cited since we have focused our attention to a precise mathematical aspect of the subject.
CHAPTER I
THE DETERMINISTIC SCHRODINGER OPERATOR
I.1 The difference equation. Hyperbolic structures
be the linear space of complex sequences T = (Tn) where n
Let
runs through the set of integers FL . The operator H is associated to two
given real sequences a and b with b
n
# 0
Vn c 2Z, and acts on o+by the
formula
(HT )n = bn
1 [T1
- Tn+l + anTJ
For a complex number A every solution of the difference equation HT= A lies in a two dimensional subspace of.ice spanned by the solutions p(A)
and q(A) constructed from the initial values po(A) = q-1(A) = 1, p-1(A) = qo(A) = 0, such that
:
Tn(A) = pn(A) To(a) + qn(A) T-1(a) From now in order to avoid too complicated notations we don't write the variable A in the solutions of the difference equation. A solution Y' of
the difference equation is constructed from initial values To and W-1 by a product of "transfer matrices" Yn defined by
Y
=
ra
n IL
n - Ab n 1
-1
0
187
:
188
n S=YYn-1 n
if n >, 0
Yo
Sn = Yn1 Yn+1 ...
Y
Thus
=S
n+1 J
-1
if if n
oI
n
if n >, 0
n
,
T_
n-1
1
lyn
=S
o
if
n
-1
The transfer matrices Yn and therefore the products So belong to the group SL(2,O) of two by two matrices with complex entries and of determinant one. If A is real then Y. and Sn belong to the subgroup SL(2,]R) with real entries. The construction of the solutions of the difference equation by such products of matrices is the essential link between the two parts of this book.
of SL(2,O) is said "hyperbolic" if the eigenvalues of
A matrix 0
have distinct moduli. If V is an eigenvector associated to the eigenvalue of modulus stricly less than one, then the sequence IlanVll
goes to zero when n goes to + - and for any vector W non proportional
to V the sequence TJII goes to +- when n goes to + -. (The same situation occurs when n goes to -
,
replacing V by the eigenvector
associated to the eigenvalue of modulus strictly greater than one). Let now (sn ) n e 7l ,
that s
n
be a sequence of matrices of SL(2,¢). We say
has an hyperbolic structure in the positive direction if there
exists a non zero vector V of Q2 (called the contractive vector) such = 0. It is readily seen
that lim IIsnVII
n- +-
to V then lim Is
n->+W
WII =+ W
n
that for any W non proportional
(from the determinant property). We define in
the same way an hyperbolic structure in the negative direction. The existence (for some A) of hyperbolic structures for the sequence Sn is the most important question discussed in the following chapters. In particular for a real A the existence of hyperbolic in both directions with the same contractive vector n implies the existence of a real non zero solution Y' of the difference structures for S
lim
equation with
4'
Inl _+W
n
= 0. The next "Osseledec's theorem" is a basic
tool to prove the existence of hyperbolic structures
PROPOSITION 1.1
Let (C.), h
ri
E
i
be a sequence of matrices of SL(2,O)
such that :
(i) lim 1 LogjjC n++. n
n
-1
..
1 II
=Y
189
(ii) n lim-
Log
0
n
Then there exists a non zero vector V such that
lim
n Log
n,+-
n ... 1
VI
I
and for any vector W non proportional to V
lim
...
Log 11 C'
n
1
WII =Y
The proof of this proposition can be found in [49]. The euclidian norm of a matrix of SL(2,¢) is no less than one, thus y is non negative.
When y is strictly positive 0sseledec's theorem asserts the existence of an hyperbolic structure for the sequence sn moreover we see that the sequences 1IsnVII
and 9' n 'n-1 ... Q1 or IlsnWII converge to 0 or
+ - with an exponential rate.
We now give two obvious
(but useful) lemmas related to the
solutions of the difference equation. The Wronskian of $
..yy
and T in obis
the sequence W ( ' ) Wn(T'$)
= '4 $n =
LEMMA 1.2
$n+1 Tn
Let $ and T be two solutions of the difference equation
Hu = Au, then W(T,$) is constant and this constant is zero if and only if $ and T are proportional.
Proof
in
$n
:
S
n
for n < -1
and
N-1 $n-1J ITn+1
$n+1 S
Tn
$n
n
for n,0
Since the determinant of Sn is equal to one the result is obvious.
LEMMA 1.3
Let $ and T be in £, m and n two integers with m S n, then
we have the "Green formula" n E
bk ((HT) k $k - Tk(H$)k) = Wm-1(T,$) - Wn(T,$)
k=m Proof
:
Straightforward computation.
190
Exercise 1.4
:
Let P be a solution of HP = AP, m and n two integers with
0 < n. Prove that if bk is positive for m . k
m < -1
ITn Yn+lI +
Tm Pm-ll =
n then
2
ty
I3 m iI (boI'YoI
+ b-1IT-1I2)
1.2 Self-adjointness of H. Spectral properties
The Green formula of the lemma 1.3 suggests that H should be a
symmetric operator on a suitable Hilbert subspace of ,. From now we assume in all this chapter that the sequence b is strictly positive and bounded away from zero (that is the sequence b-1 is bounded)
We denote by YO the subspace of ou defined by
{P ce/ E bn
ITnI2 < +m}
(when the symbol E has no index of summation it's understood that this index should run through ZL). Endowed with the scalar product E b
subset of
d is an Hilbert space. If D is a dense linear n 4n Tn such that H(D) C `% we denote by (H,D) the linear operator
with domain D. Let Do be the dense linear subset of % of sequences with a finite number of terms different from zero. From the Green formula we know that (H,Do) is a symmetric operator and it's not difficult to see that its adjoint (H,Do)
is equal to (H,D1) where
D1 = {P E 4U/ HY e & _ (P e :%/ E a2 bn1 lPn12 < +m } LEMMA 2.1
If the sequence a is bounded then H is a bounded self adjoint
operator.
Proof
Let a,
HY II = (E bn-1 then by the (E
and
R2 b-1
n
be the bounds of the sequences a, b-1 and P e dam:
pn-1 + Pn-1 - an triangle inequality
Pn12)1/2 :
I2)1/2 + (E b1IY 12)1/2 + (E b-1 a2 IP I2)i/2 n n n n-1 n IT n+1
:
IIHPII. II Y11 (2a + aR) Thus H is a bounded operator on'. The assumption on the sequence b
191
implies that ki G Q2 (2Z) and hence
lim
Inl
= 0 for P e r . By the
P
---
Green formula we see that H is a symmetric operator on
and hence
self adjoint. In many applications the sequence
LEMMA 2.2
is not bounded, so we prove :
a
The operator (H,D1) is self adjoint.
Proof : As remarked in the lemma 2.1, H is a symmetric operator on D1, z
hence (H,D1) is a symmetric extension of (H,Do) with (H,D1) _ (H,Do) the result follows from the theory of self adjoint extensions of symmetric operators 19 1.
From now, when we speak of the operator H, it will be always understood that we actually deal with the self adjoint operator (H,D1). As we shall see in the exercise 2.3 below, the sequence b has not to be bounded away from zero in order that (H,D0) possesses a self adjoint extension, but such an assumption seems reasonable since a solution of HP = AP which is non zero at infinity does not have any physical meaning in general.
Exercise 2.3 (
For n e N let C (b
) 1/2
b
n
be the sequence 1/2
If we assume that E Cn = + n=0 prove that the operator (H,D0) is essentially self adjoint (and (H,D1) Cn = min (bn bn+1
'
-n -n-1)
)
is its self adjoint extension). Hint
:
Use the result of 1.4 to prove that when
a # 0 the equation
HP = AP has no non zero solution in dV2 (see [4] VII theorem 1.3).
We can apply to H the general theory of self adjoint operators
which we may find in [9]. There exists a resolution of the identity E which associates to each borel subset d of 1R the projection Ed of d& and for P, P in QID,oO,T(d) = <Ed4,P> is a bounded complex radon measure on JR
(We denote the positive measurelo
number with
m A
0, R1 = (H - XI)
-
,
a complex
by
01
is a bounded operator on '% and
the spectral theorem asserts that < RO ,P> =
J(t-A)- I
do can be written as the orthogonal sum of three closed subspaces invariant under H (Lebesgue decomposition of p) defined by
:
192
you
r
{ c
db/ °
is absolutely continuous}
16P = {$ c
Y./ °
is pure point}
dl' =
V
e
=
p / °
is singular continuous}
The spectrum Q of H is the union of
G-a,
(]-p, s which are the
spectra of the restrictions of H to the invariant subspaces
Apa, 6 p
A=
We say that the spectrum of H is pure point if we have
%s
in
pp
this case 49 has an orthogonal basis of eigenvectors. In our case this Lebesgue decomposition of 'X can be obtained by the decomposition of a single measure on ]R as it is shown below. Let ek be the orthogonal basis
defined by (ek)n = bnl do (we remark that
of
II ek 112
P e Ob, = 4`n). We denote by °m,n the measure
=
bkl and for
°em,en , by on the
measure an n and a the "spectral measure" a = °° + a-1. For each n in 7L,
pn(X) and qn(A) defined in the first section are polynomial functions in A and thus the operators pn (H) and q (H) are well defined. n
pn (H) e° + qn (H) e 1 = en
PROPOSITION 2.4
The equality is obvious for n = 0 and n = -1 and is proved by -ek+l - ek-1 + ak ek induction for all n from the identity bk H ek = Proof
:
COROLLARY 2.5
For each T in QU the positive bounded measure a
is
absolutely continuous with respect to a. Proof
:
Let 4 be a Borel subset of ]R with °(d) = 0. Then
IIEAe°1I2 = Go (A) = 0 and IIEe 1112 = a-1(d) = 0, thus by the proposition 2.4
:
E,en = pn(H) Ede° + qn(H) E,e 1 = 0. This implies that Ed = 0 and hence
aT(A) = IIEAT112 = 0.
This corollary has a direct consequence for the Lebesgue decomposition of
.
Let aa, °p, as be the three parts of the Lebesgue
decomposition of a.
PROPOSITION 2.6
With the above notations,
, q a,
(T p, (3-s are the
topological supports of the positive bounded measures
°,
°a,
°p,
moreover x is an eigenvaZue of H if and only if °(a) > 0. Proof
:
By Weyl's criterion we know that A eG-if and only if there
a
193
exists a sequence ('n).30 in D1 with lim
= 0 and
11 (H - AI)'Yn11
11 Y.11 = I.
n ->+ Taking in account that for A c IR and 'Y e D1 we have the relation
II (H- AI) Y'112= 1 (t-a)2 daY (t) the result follows for W a
other parts of the spectrum we remark that if Y =
"Lebesgue decomposition" of Y e I then oa = aTa
+
y,P
and a. For the + Ys is the
ay = ors
,
We associate to H the measure valued matrix S given by
:
,
up = aYp.
a-1,0
r
a0,-1
a-1
The spectral measures a
LEMMA 2.7
can be constructed from the
m,n
spectral matrix $ by the formula IPn
pmq )
am,n
$I
qr. Proof
:
Using the expressions of en and em given by the proposition 2.4
it's enough to compute am,n(t) = <E,, em , en >.
Exercise 2.8
Prove that {EAe°, E,e 1 / A is a Borel subset of 7R} spans
a dense subspace of I. Exercice 2.9
Prove that (an + ari+l) and a have the same nul sets.
Some authors work with the operator H restricted to the half axis
N denoted by +H
.
This operator is defined on the sequences `Y = (Tn)n,0
by
(+HP)o =
bo1
(-Pl + ao'Yo)
(+HY)n = (H'Y)n if n , 1 The Hilbert space + db , the spectral measures +am,n , the resolvent +R
are defined in the same way than for H and we set +a = +a o
Exercise 2.10
Prove that
(i) pn (+H) e° = en (ii) For each Y' in
respect to
(n a IS) + ,the
(iv)
+a Y,
is absolutely continuous with
a
(iii) +am,n = Pn pm +a +
measure
o
{ EDe
(m,n a 1V)
/ 4 is a Borel subset of IIt} spans a dense subspace of
+,{
ov
194
For m,n E IN prove the following relations
Exercise 2.11
M
Pn Pm d+a =
n
-2
+
A pn pm d a = bn
(ii)
:
m do
b- 1
n
-1
an dm - bn
-1
bm
The next exercise shows that when
n+1 ({dm
n-1
+ Sm
J mA # 0 then there exists an
hyperbolic structure in the two directions for S , with distinct n contractive vectors. Let +m(A) = J(t-a)-1 d+a(t) for
Exercise 2.12
(i) We define the sequence ( P
for n
m(A) Pn + qn
Prove that H
+T
= A
n e 2Z, by
for n , 0
+'Yn = <+R, eo, en> Tn =
),
m X# 0
+T
-1
+T-1
= 1
(ii) Deduce from (i) that for n 3 0
<+RA e o, en> _ +m (X) pn + qn
(iii) Prove that
+ b n=0
+T.1
2
r
- J m(+m(A))
n
6ma +
Hint
RA -
<+Ra +RA
+
R
= (a-a)
+
R
+
RA and compute
e o, e o>
(iv) The same property holds for the negative half axis and a complex number A with
m a# 0 is not an eigenvalue of H. This provesthe above
assertion about hyperbolic structures. Exercise 2.13
Prove that it's possible to recover the sequences (an)n>,0
and (bn)n>,0 from +a.
Hint
use 2.11.
:
Exercise 2.14
The sequences +T(a), -'V(A) are constructed for the two
half axis in the same way than in 2.12. Prove that the symmetric matrix
n
,
em>
= a
m X O 0) is given for n < m by
(
+`1(m)
`Y (n)
W( T, +y')
(where W is the wronskian)
Exercise 2.15 that n k=0
2
Let A be a real number and n a positive integer. Prove d
d
bk Pk = Pn a Pn+l - pn+1 dl pn
195
1.3 Slowly increasing generalized eigenfunctions
We say that Y'
is a generalized eigenfunction if H'Y = AY for a
E
real A and 'Y is non zero. (When ' E
'Y
is an eigenfunction). A fairly
general theorem about Carleman operators asserts that such sequences
are slowly increasing [4]. Actually it's possible in the simpler case of difference operators to give a direct and easy proof of this result. As we have seen in 2.5, °m
,n
is absolutely continuous with respect to °
(thank's to the inequality I°m,nl ` 2 (Qm + °n)). Let S be a matrix of
density of S with respect to a.
For a almost all real A, the matrix S(A) is symmetric and
LEMMA 3.1 positive.
Proof
:
For each borel subset S of ]R, the real matrix $(5) is symmetric,
positive since ao,-1 = °-1,0' and for real x,y
(x,y) $ (4) xy1 1= 0x e0 + y e1 (5) The conclusion follows easily.
>, 0.
\\1
For a almost all real A there exists a generalized
PROPOSITION 3.2
eigenfunction Y' such that for each E > 0
-T (n) lim InI ±m Ii -+E
=0 ra
For ° almost all X we can write
Proof
O1
K* where K is an
S = K ILO S
orthogonal matrix, a and 6 are two positive numbers with a , S, a +a =1. By virtue of the lemma 2.7 0
nnn =
( P,
4 )K
Pn
[0
SJ
K ( qn
°
/'Y
If we denote by
see that on >, a
the firstcolumnof K and 'n
°
= pn'o +
we
bn' and that the
2 `Yn 0 . We know that on(IR) =
0
qn'Y-1
f2
1+E
sequence bnlis bounded then for any e > 0 we have
E
n#0 n
n
E
n#0 n
,f2 thus
(
< + oo
1+E
o a.e. 'P2
0
This implies that lim
InI >m
n
1+E
° a.e.
do < + m
196
The same result for the operator
Exercise 3.3
H on the half positive
2
axis
for +a almost all A, lim
:
Pn
n->+- n l+e
= 0
Prove that the Proposition 3.2 could be
Exercise 3.4
For a almost
:
all real A there exists a generalized eigenfunction 'Y such that for every
c
we have
t2(TL)
c 22(7L) .
1.4 Approximations of the spectral measures
The spectral measures a are generally obtained as weak limits m,n of spectral measures of the operator H restricted to "boxes". These
limits are independent from the boundary conditions on the boxes and this allows us the construct new approximations which are absolutely continuous with respect to the Lebesgue measure on]R.
A box A is a finite subset [M , N] of 7L , and we say that a sequence of boxes is going to 7L if the associated sequences of M and N are going
to - - and + - respectively.
For a box A anda real number x we define the operator n Hx on the Hilbert space of sequences ('Yn) , n e [M , N], (with the scalar ,gyp
product inherited from 7) by (AHxY)M = bMl (-i'1
(AHx'Y)n = (HT
N=
1 + aM'M)
for M+l
)n
bN1(-T
n
N-1
N-1 + (aN x) Y'N)
Let ^p and "q be the solutions of HW = A'Y with initial values
^qM = 0. It's easy to see that Hx is a self FPM qM-1 = 1' PM-1 adjoint matrix on ^90, its eigenvalues are the roots of the polynom '
^ PN+1
nam X
they are simple and if A is an eigenvalue we have
(A) _
N
APn
Apm(kEM bk Pk)
Let x c1R, m,n a 7L be fixed, then the sequence
PROPOSITION 4.1
converges weakly to a
Proof
:
m,n when A goes to 7l.
Fix A with 3m A
0, x c ]R, n e 7l . A function
n
4'
iA
be extended to
by ^'n = 0 if n 0 A ; with this convention
in
n ax m,n
nt At can
197
( (nH-a
I)n'Y)
(H-aI)nY'
n'YM
+
e
M-1
+
n
TN eN+l
- xn Y' N eN
When
kl - + the sequence ek converges weakly to zero, thus if the Al is bounded when A goes to 2Z (and hence weakly compact) the sequence eM-1 + ATNeN+l sequence AO = ATM x A T eN converges weakly to zero N n A 4'
is bounded by
RA en the sequence
when A goes to 7L . Taking A T =
b-lJIm al-1 . With this choice we have en = (H-A)AP + A and
n
R. e n =
A Y
RA"
If T is any weak cluster point of AT we have
.
'Y = R. en and thus A R. en converges weakly to % en when A goes to TL
Taking in account that I(t-A)_1
_
en a
em> =
d
AOx
n(t)
(t-A)-1 d om,n(t)
I
we obtain the expected result since it is well known that convergence of integrals of the continuous bounded functions t -i
t1X
for mA # 0
implies weak convergence.
LEMMA 4.2
For
m X
Aq eM>_- n N+1
o we have xAqN
eM
A
PN+1 - x PN Proof
C
Let Y` e
n
with
TN+1/- YN-1 "
YM+1
(A Hx
- aI)`Y = e
M
this equation can be written
TM+11 M
1
TM+1 = (aM - AbM) TM
TN-1 = (aN - AbN) TN - x`YN
and we obtain the relation
YNNJ = YN YN-1 ... YM
:
[fr]
L
taking in account that ' N ... YM
from
= `YM.
LEMMA 4.3 Proof
:
If
If
PN+1
L PN
qN+1l
the statement follows
qN
m A > 0, m z . 0 then APn+l - zAPN # 0
APN+1 -
zA PN = 0, z is an eigenvalue of the operator A Hz
(defined in the same way as AHx). For z = x + iy the adjoint operator (AHz)* is given by
:
198
M:n
((^HZ)* `Y)n = (A11zp)n
N-1
((A Hz)" `Y)N = (AHz'Y)N + 2iybN1 TN For an eigenfunction P associated to z
`Y, `Y> = X11TI12 = = aIITII2- 2iylTN12
thus
(3
mA)
IT I
'
'
N12 and this is impossible, the right member of
'N
this equation being strictly positive. We recall without proof some basic facts about the Poisson kernel of the upper half plane. These results are proven for the unit disc in [32]. Let C. be the Cauchy distribution with density 1 II
(t-a)S2+S 2
dt,
z = a + i6 with S > 0. Let f (z) be an analytic function on m z > 0 such that f can be continuously extended to IR U{m}. Then this extension satisfies the Poisson formula
f(z) =
f(t) dCz(t)
:
(Sm z > 0)
J
Furthermore let F(z) be the function associated on m z > 0 to a positive ua+us+up is the bounded measure p by :F(z) _ J(t-z)-1 dp(t). If u =
Lebesgue decomposition of p, then (i) lim 6m F(x+ic) exists for Lebesgue almost all x of IR and is equal t 0
to the density of
a.
(ii) (µs+pP) is supported by the subset {x a IR/lim 5m F(x+iE) _ +m} E+0 In particular if lim im F (x+it) exists and is finite for all real x, Ey0 then p is absolutely continuous with respect to the Lebesgue measure on
IR and its density is given by this limit. Its also useful to remark (
that
dC
J Z dp(t), that is,
m F(z)
m F(z)
is the Poisson integral of
dt
p and the above statements are nothing else than Fatou's theorems. %
We define Aam n as the average of A am n with respect to the Cauchy measure e by the formula A
aM'. (A) =
1am,n(A) dCi(x)
PROPOSITION 4.4
The measure
am
n
has a density with respect to the Apnnpm
Lebesque measure on IR given by
2 II
A2
pN+1+ pN
199
din
In particular we have the .formula 1
Proof
:
A 2 PN +
1
II
nax
Thank's to the relation
Ap np Aax n m M
=
m,n
=
1
2
PN+1
it's enough to
prove 4.4 for m = n = M. By the lemma 4.3, for 3 m i> 0 the function z
nqN+1
z -* -
qN
is continuous on Ci m z3 0 and has obviously a limit _ ZAPN pN+l when IzI -> + m . From the lemma 4.2 and the properties of the Poisson integral
nq
- iA q
N
N+1
(
F(A) = II1 J(t-A)-1 do(t) M
A
A
pN+1 - 1 pN {{
As it is readily seen that for A real limb m F(A + it) _ 64.0
1
A 2 pN+l
+ n Pi2
the result follows from the above remarks.
The sequence
nconverges weakly to a by the proposition 4.1 m,n m,n (this is true for am,n and hence for the averaged Cr
when A goes to 7L
measures). One of our essential goals is to obtain (under some hypothesis) that a
is pure point, and it seems surprising to work m'n n x am which is absolutely continuous (rather than with am,n n which is pure point). But we shall see in the chapter III that this nti
with
choice is well adapted to the proof of. localization in the random case.
Exercise 4.5 (HT)n
Hint
Compute
^
and am,n for the Laplace operator am,n
-fin-1 - Tn+l
pn(A) _ sh(n-M+1)w
A
(where ch w = - 2
sh w d am,n
and for A c ]-2,+2[,
da
Prescribing
Exercise 4.6
cos[(m-n) Arcos - 2 (1-A2) -1 /2 211
boundary values at each endpoint of the box AHy'x
A, we define for x,y a JR the operator
by
(nHy'x 1')M = bMl(-'FM+1 + (aM y)TM)
if M+1 : n < N-1
(AHY,x `F )n = (HT )n
(^Hy'xf)n =
bN1(-TN-l
Prove that for' m A # 0 <^ nRy,x
M
M
+ (aN-x) TN) A
A
qN+1 - x qN A
1 ,
e
e >
- xApN+y(AgN+1 - x'qN)
200
Let +Hy be the operator on the "box" 10, +-[ with the
Exercise 4.7
boundary condition y at the endpoint
0 (see 4.6).(Wh+n y = 0 this is
just the operator +H defined in 2.10. Let +m(A) = J
dto(t)
defined in
2.12.
m(A)
(i) Prove that <+Ry e o, e°> =
1-y+m(A)
X
Hint
use 4.6 with N = 0, x = 0
:
(ii) Prove that the measure
+ay
++'a'
= J
dCi(y) is absolutely continuous
with respect to the Lebesgue measure on R. + MM
Hint : J m
m(a)
is bounded on J m A> 0
1 - i+m(A)
(Remark that Exercise 4.8
mA
> 0 m +m(A) > 0) .
Compute
m(a) for the Laplace operator (exercises 2.12
and 4.5).
1.5 The pure point spectrum. A criterion.
Let A be an eigenvalue of H associated to a normalized eigenfunction f (the eigenspaces are one dimensional !), then
a m,n (X) =
en>
and the density matrix S(A) defined in 1.3 has the form 2 0 Y
1
2 Y
o
+Y
Yo -1
2
-1
TT
Y' 2 1
In this case S(A) is of rank one and each column vector is proportional to
0 1
Therefore, in order to prove that A is an eigenvalue it seems natural to
prove that each column vector of S(X) can be taken as initial values of
a solution of H'Y = A' which is in
a almost
everywhere so this is only an heuristic argument ...). If I is an open subset of 7R, we say that the spectrum is pure point
on I if as(I) = as(I) = 0 (or Gj afl I = (5s (lI = o).
201
Let I be an open subset of]R for which there exists two
THEOREM 5.1
positive constants C and p with p < 1 such that for each box A and ti Im-n
m,n a A,
r"' am
2p (II)
Cp
Then if E bn 1
.
< + -, the spectrum of H is pure point on I and
the associated eigenfunctions faZZ off exponentiaZZy at infinity. Proof
:
On the set of bounded measures on ]R endowed with the weak
topology (that is convergence of integrals of bounded continuous functions), for each positive lowersemicontinuous function f, the map p --' IpI(f) is lowersemicontinuous. Together with proposition 4.1 this
Cp (m-n). Choosing m = 0 and m = -1 in the
implies
lemma 2.7 this can be written :
fIP
n
I
+q
doo
doo,-1I do < Cp InI n
do
doo-1 + q
do
da-1 I
fI
Pn
n
do
do
C P I n+1 I
do
Observing that a sequence in k1(2Z) is necessarily in k2(ZZ) hypothesis E b112 p
nI
the
< +- implies that for a almost all A of I the
two column vectors of the matrix S(pa) can be taken as initial values of a solution of HT = A'Y with 4' e off. These column vectors can't be
simultaneously zero (the trace of S(A) is equal to 1, a a.e.). Hence a almost all A of I is an eigenvalue
and a is pure point on I. For an
eigenvalue AI letting To = ao(X) and W-1 =
ITnI ` Cp n
Go'- 1(A)
we obtain
j
5.2
(i) In the above proof,the hypothesis of 5.1 has only to
hold for m = 0 and m = -1,but as it's easily seen these two statements are equivalent. (ii)The same result holds when we replace ^a^'
mn
by ^ox m,n
(for a fixed x or a bounded sequence x^) in the hypothesis of theorem 5.1, but checking such a property
seems much more difficult,
the measures 'ax being not as tractable as the m,n have'a nice formula (4.4).
n
am,n
for which we
(iii) If the sequence b is bounded (in which case oV is
isomorphic to k2(7L) the assumption
Exercise 5.3
E bn /2
p
ni < +- is vacuous.
Let I be an open subset of It, then if E b /2I a I(I)<+. n n m,n
202
the spectrum of H is pure point on I.
The criterion 5.3 can be obtained as in [37] using the following "Ruelle formula" proved in [53] I I1
T I 12
lm
=
c
lm T-+.
A+7L
1
T
:
T
0
I
I
eitH
l A
c
T I I2 dt
where P is in, IIc is the projection from Pb onto the subspace
V + OVS and 1Ac is the projection operator [1Ae(4)] n = n if n ¢ A
=0 Exercise 5.4
if nrA
Let I be an open subset of R.
(i) Prove that if
for each integer m (or only for m = -1 and m = 0) we
have I IIc EI emll = O,then the spectrum of H is pure point on I. I
(ii)Prove that if for each m (or m = 0 and m = -1) E bn(Iam,nl(I))2 <+m
then the spectrum of H is pure point on I.
eitH'v 112
fT
T
Hint
IlAc J
dt `niAbn(Iam,nl (I))2
0
(iii) Observe that the criterion obtained by restriction of (ii) to
boxes is stronger than 5.3.
1.6 Singularity of the spectrum
Osseledec's theorem 1.1 together with proposition 3.2 about slowly increasing generalized eigenfunctions, are powerful tools to obtain singular properties of the spectrum. The two following lemmas and the
added remark
actually contain essential ideas developed in the sequel.
We assume in this section that lim 1 Log(1 + I a n+±m InI
n
I
+ b. ) = 0.
I
We say that the sequence Yn(a) has an hyperbolic behavior if there exist two strictly positive numbers y+(a) and y_(X) such that
lim 1 LogIISn(a) II = y, (X) n+±- I n I
If the sequence Yn(a) has an hyperbolic behavior, Osseledec's
theorem 1.1 insures the existence of two contractive vectors V +M and V (x)
203
LEMMA 6.1
Let m be a positive continuous measure and A a Borel subset
of IR, such that the sequence Yn(A) has an hyperbolic behavior m
a.e.
on A. Then the spectral measure o is orthogonal to m on A. Proof
:
Osseledec's theorem 1.1 implies existence of contractive
vectors V+(a) and V -(X) m a.e. on A. Such A are eigenvalues of H iff V+(a) and V -(X) are proportional b
n
.
from the hypothesis on the sequence
The measure m being continuous such countable set of A is m
negligible. This implies that each non zero solution of HY = aW is exponentially growing in at least one direction of 2Z for m almost all A
in A. Since for a almost A there exists a non zero solution of HW = aW which is slowly increasing, the conclusion follows.
LEMMA 6.2
Let A be a Borel subset of IR such that the sequence Yn(A)
has an hyperbolic behavior a
a.e. on A. Then the spectral measure a is
pure point on A. Proof
:
Osseledec's theorem 1.1 implies existence of contractive
vectors V+(a) and V -(X) a
a.e. on A. Moreover for a almost all A,
there exists a non zero slowly increasing function W solution of H' =a' and thus V -(X) and V+(a) have to be proportional a
a.e. on A. Hence,
on A, a is supported by the eigenvalues of H (if any).
REMARK 6.3
Lemmas 6.1 and 6.2 can hardly be directly applied to a
deterministic operator H. But we shall see in the sequel that in the "random case" and under broad conditions, for a given measure m, the hypothesis of 6.1 are satisfied for "almost all samples". Thus we could think that occurence of the hypothesis of 6.2 is highly improbable...
Nevertheless we shall see that, in the independent case, this hypothesis holds with probability one amazing fact in localization theory.
!
This is certainly the most
CHAPTER II
ERGODIC SCHRODINGER OPERATORS
II.1 Definition and examples
We now suppose that (a
bn), n E 2Z, is a stationary random process
n '
This means that the real random variables (an(w), bn(w)) are defined on some complete probability space (D, a , F) and that there exists an
invertible measurable transformation S of U leaving F invariant and In general we don't write the such that an+1 = an oS bn+l variable w and when a property depends only upon the common law of the o.e.
,
sequence (an,bn) we omit the index n and speak of the variables (a) and (b). We say that the family H(w) of associated operators on
A
2is
ergodic if a B invariant measurable subset of 0 is of zero or one F measure. It's easily seen that H o 6 = U 1 H U where U is the shift
operator on
2Z , (UT
)= T n-1, n-1'
The most important example in this book will be the "independent case", that is (an,bn) is an independent identically distributed
sequence of random variables with values in l2. Then (C,a,, F) is constructed as the 2Z power of a probability space (1R2, w = {(xn'yn)}, n E 27., we define 0(w) = ((x
For
and an(w) = xn,
n+1' yn+1)} bn(w) = yn. An other example is the "quasi periodic case" where (C,a.,F) is the one dimensional torus [0,211] endowed with the Lebesgue
probability, e is the "rotation" 6(w) = w + 2fl a with a an irrational
number. The sequences an = aegn, bn = b o0n are constructed from two
205
206
real continuous functions a and b on 0. A particular case is the "Mathieu operator" associated to the functions b(w) = 1, a(w) = 6 cosw where 6 is a real number. A degenerated example is given by the "periodic case" where (o,3.
1P)
is the finite set {0,1,2,
..
,
r-1} with
the uniform distribution. The "rotation" modulo r acts by 0(w) = w+l. More general examples of almost periodic or limit periodic cases are also of great interest [57]. Ergodicity does not require any disorder as shows the periodic case (we don't assume any mixing property) nevertheless that is a sufficient assumption to imply ]P a.e. properties
of the spectrum of H(w) as a set. If we want to obtain stronger properties, as absence of absolutely continuous part in the spectrum, some randomness is necessary (the spectrum is absolutely continuous in the periodic case). In some sense the Lyapunov exponent defined in the section 3 is a good indicator for the required disorder as it is shown in the section 5.
Throughout this chapter we assume that H(w) is an ergodic family of random operators.
11.2 General spectral properties
We assume in this section that the random variable (b) is strictly positive and bounded away from zero, then by 1.2.2 for ]Palmost all w the operator H(w) is self adjoint on the Hilbert space
dl-1)(w)
(note
that the scalar product is itself random). The shift operator U is a
unitary map from X-8 onto 'A thus H and H o0 are unitary equivalent.
For 4 , T in d0 o 6 we have the equality < (E6. 0 thus a n
o
LEMMA 2.1
T> = <E6 U¢ , UT> and
B = a n+1
Let A be a Borel subset of]R, m and n two integers, then the
function am,n(A) defined] a.e. on 0 is mesurable. Proof
:
By a monotone class argument it's enough to prove this property
for an open interval A = I
E. For such a A we have
((a-6
am,n(A) = lim lim
t+o 610
1
R J- W
m
cm,
n
e > dt
207
rb thank's to : lim 1 f e1o II
a
By virtue of 1.4.4 we obtain
A- ZZ
II
(am,n(a)+am,n(b))
:
em,en> = lim 1 f(A - (t+ie))
t+ie
am,n(]a,b[) +
m =
1
pn 2
pN +
pm 2
dA
pN+1
This last expression is obviously jointly measurable in (w,t) and this gives the result.
The following lemma due to L.A. Pastur [50] has a well known consequence in the "periodic case" since it implies that such operators have no eigenvalues.
LEMMA 2.2
Let x be a fixed real number. ForT' almost all w, a is not
an eigenvaZue of H(w).
Proof
:
Defining the subset 0, of H by 0A = {a(1`) > 01 the lemma 2.1
implies that 0X is measurable. The relation an o 0 = an+1 added to the fact that a and (an +a n+l) have the same null sets proves that 0X is 0 invariant thus ]P(HA) = 0 or 1. If we denote by tr EA
the trace of the
projection EA onto the eigenspace associated to A, then tr EA = 1 if
tr E. = O if w ¢ D,, and tr E. = 1 b n an(A). Taking expectation we obtain : P(QX) = E TE(bn an(A)). Stationarity implies that TE(bn an(A)) is a constant which has to be zero and thus P(Q X) = 0. w e DX ,
The two following lemmas are proven by H. Kunz and B. Souillard
in [37]. LEMMA 2.3
There exists a closed subset (1 ofiR such that forTP almost
all w, (F (w) _ ff. Proof
:
Following the lines of [37], let Rn be the countable set of
open intervals of TR with rational endpoints and length
.
For I e Rn
the subset D1 = {a(I) > 0} is 9 invariant (see the proof n of 2.2) and hence of zero or one]P measure. Let Sn = {I a Rn / P(QI) = 1} ci
= n U n=1
Q (w) =
I
;
it's easy to check that for P almost all w,
3IeSn (and hence is closed !).
and
208
In the quasi periodic case this result holds for every w in Q [3]
and in the independent case the set T has a more explicit description.
In the independent case, assume that b = 1 Vn c 27. Then n for]? almost all w,G-(w) = [-2, +2] + Sp where Sp is the topological LEMMA 2.4
support of the law p of (a). Proof
We denote by H and V the operators (HY)n
(V')n = an 'P
-Tn+1 - Tn-1 ' such that H = H +V. It's easily seen that the spectrum of
V is for]? a.e. equal to Sp
,
that IIHII = 2 and that the spectrum of H
is equal to [ 2, +2]. Applying the result of the exercise 2.5 below
(the proof can be found in [31]) we obtain G (w) = [ 2, +2] +Sp. For the converse we give the proof when Sp is a finite set, the general case handled in [37] being a straighforward extension of the main ideas given below. Let A e [-2, +2], by Weyl's criterion there exists a
sequence (4'k), k
e N, in %such that II1k1I = 1, lm II (H-xI)TkjI = 0. k- +-
H is a bounded operator so we can choose `1k to be zero outside of a box Ak.
If a E Sp and if N is a positive integer, the set Qo ={an = a/n=0,1,..,N)
is of positive]? measure and by "Poincare's recurrence lemma" we know 8-n(S2a))
that]? (lim
= 1. Then for]? almost all w, there exists for
n;+each positive integer N a box of length N such that an(w) = a for n in this box. This implies that for ]P almost all w there exists for each k ti v a box Ak of same length as Ak and such that an(w) = a for n e Ak. Let Uk be the translation of 2Z such that Uk(Ak) = Ak , taking in account
that H is translation invariant, that is UkH = H Uk, we obtain letting
$k = Uk 4'k
:
(H - (A+a) I) 0k = Uk (H - XI)
,,k
+ (V - aI)
k.
The last term of this equality is zero and a new application of Weyl's criterion using the sequence
k
gives the result.
An important consequence of the lemma 2.4 is that such operators have no discrete spectrum (but can have pure point spectrum !). Furthermore we see that their spectrum depend upon the law p only by its support Sp, so we can't expect any continuity property of the spectrum with respect to p without strong assumptions on the convergence of supports.
Exercise 2.5
Let A be a bounded operator, B a self adjoint operator
209
and H = A+B. Prove that if A is in the spectrum of H then d(A,o(B)),IIAII where d(A,o(B)) = inf{la - YI/Y a spectrum of B}. Hint
See [31] chapter V theorem 4.10.
:
Exercise 2.6
-Fn-1 - Tn+1 +
(HY)n
Hint
Compute the spectrum of the periodic operator (-1)n Tn
Perform a Fourier transform and for the general periodic case
:
see [55] theorem 1.
11.3 The Lyapunov exponent in the general ergodic case
In this section we assume that]E Log(1+Iaj +Ibl) < + -the random variable (b) taking any real value.
We recall in the particular case of Schrodinger matrices the definition of the Lyapunov exponent given in A.I.2 Let A be a complex
number and yn(A) _ ]E Log Il Snll
,
n e 2Z , (see I.1) .
It is readily seen
that for positive integers m,n (i)
Ym+n
Ym + Yn ]E Log(2 + (a-Ab)2) < +
(ii) Yo
(iii) Y-n = Yn-1 (n Thus yn
.
1)
is a positive subadditive sequence and the Lyapunov exponent
y(A) is defined by y(A) = lim
:
1 Yn(A)
InI ++m InI
= inf n Yn(A) n%1
A function f on ¢ with values in
is called subharmonic if
f is upper semicontinuous and if f(A) is less than the mean value of f over the circles of center A. Observing that the same submean property is true for discs and taking limits when the radius is going to zero,
we see that two subharmonic functions almost everywhere equal (with respect to the Lebesgue measure on ¢) are equal. The role of subharmonicity of the Lyapunov index has been first pointed out by M. Herman in [2:6] and used in the general ergodic case by W. Craig and
B. Simon in
111.
210
PROPOSITION 3.1 Proof
:
The function y(A) is subharmonic.
Sn(A) is an holomorphic matrix in the variable A, thus
LogjjSn(X)II
is subharmonic and
the same is true for yn(A). The
infinimum of a sequence of locally bounded subharmonic functions is itself subharmonic and this yields the result.
As it was shown in A.I for the independent case, y can be
recovered as a pointwise almost everywhere limit
PROPOSITION 3.2
:
Let A be a fixed complex number. Then the sequence
1 LogllSnl1 converges ]P a.e to y(A) when nl goes to +m . Inl Proof
:
This is just the subadditive ergodic theorem thank's to the
relation, for positive integers m,n
LogJJ Sn+mll
.
:
Loglj Snll + (Log lI Smll) o Bn .
See [42] for a proof.
It is readily seen that for each fixed A the sequence i LogJJY
n
In j
converges]P a.e to zero, thus the proposition 3.2 together
II
with
Osseledec's theorem (I.1.1) implies that for each fixed A such that y(A) > 0, there exists for]P almost all w, an hyperbolic structure in the two directions of a. If we think to theip a.e pure point spectrum, we see that we must obtain the a.e in the reverse order, namely
:
for
8 almost all w there exist hyperbolic structures in the two directions of Tl with the same contractive vector for o(w) almost all A. This
inversion of a.e properties is the most important problem discussed in the sequel.
As we are mainly concerned in this book with the independent case we give without proof a result of S. Kotani [35] asserting that y is "often positive". We say that the process (a
b n) is deterministic if n (ao,bo) is measurable with respect to the subsigma algebras generated '
by {(ak,bk)/k -11.
PROPOSITION 3.3
Assume that {A e]R/ y (X) = 0} is of positive Lebesgue
measure, then the process (an,bn) is deterministic.
The proof given by B. Simon [58] assume that (a) is bounded, but the general case is handled in [33]. A typical example of deterministic
211
process is given by the Mathieu operator and the next proposition shows that the converse of 3.3 is not true.
The very easy proof given here is due to to M. Herman [27].
PROPOSITION 3.4
Let H(w) be the Mathieu operator
:
[H(s) Y].
-Yn-1 - Yn+1 + S T. dos(211 .(% + w)
where a,s
are real numbers with a irrational.
Then for any complex nwnber A , y(a) >. Log2I-
For a complex number z we set z = z e2IIin a and we define the n matrix Zn by Proof
:
2 (zn + 1) - Azn Zn(z)
-Zn
= z
0
n
Observing that Z(e) = znYn(w,A) we have
...
IIZn (els)
and thus
Z.(eiw)II
= IISn(s,X)II
:
(211
Yn(X) =7E LogjISn(s,A)II = 1 J The function z -> Log1IZn(z) ... Yn(X)
LogJIZn(eis)
...
Zo(eiw)II dw
0
211
Zo(z)II
is subharmonic and this implies:
LogIIZn(0) ... Zo(0)II = n LogJ2I
Exercise 3.5
Compute the Lyapunov exponent for the operator
-fin-1 - Pn+1 + (1)n Tn and compare spectrum of H computed in 2.6 (Hf)n
{A/y(A) = 0} with the
.
11.4 The Lyapunov exponent in the independent case
In this section it is assumed that we are in the independent case and that1E Log(1+Ia! +Ibi) < + - the random variable (b) taking any real value
In the independent case the Lyapunov exponent is given by the
212
Furstenberg's formula involving an "invariant measure" and this allows us to prove extra properties of positivity or continuity for this exponent.
We recall there some definitions given in the first part of the book. For a non zero vector x of ]R2, x is the corresponding point of
the compact projective line denoted by X. A matrix g of SL(2,]R) acts on X by g.x = g and for two probability measures li, v on SL(2,]R) and
X respectively, we define the "convolution" p p * V(f) = J
v
by
f(g.x) du(g) dv(x) where f is a continuous function on X.
We say that V is p invariant if p 'F v = v. For a real A
,
p
a
is the
law of the matrix Y
I
=
a-Ab
-1
Let v be a probability measure on X such that
.
0
1
v(xo) = 0 for
xo=`O (11
ti
then we can define the image v of v on R by the u
application x -*
v
where x =
1v
We shall say in the sequel that we are in the "Helmotz case" when there exists a value a'F of A and a constant c such that the law of (a,b)
is supported by the line a -X*b = c. In this case the operator H can be
written
:
(HT )n = bn1[-fin-1 - 'yn+l + c'n] + a
and the matrix Y has the form Y
=
I
L
c-(A-a'F) b
-1
:
1
OJ
1
PROPOSITION 4.1
If the law of (a,b) is not concentrated on a single
point of ]R2 then for A E n (A # a'F (i)
'Fn
in the
Helmotz case. :
There exists a unique p1 invariant probability measure v, , and
VX(X) = O, vX E X.
(ii) Y M > 0 and y (X) = J JLog Imo- dpA (g) dv1 (x) (iii) If there exists a > 0 such that IE(lafa + lbla) < +- then Y(A) = ILog Itl dv1(t)
213
Proof :Looking at the theorem A.II.4.4 its enough in order to prove (i) and (ii), to verify that the closed subgroup G. of SL(2,]R)
generated by the support of uX leaves no probability measure fixed on X
r1
Under the hypothesis of 4.1 G. contains at least two matrices
gI =
g2 = and
IL
0
rl
Letting d = a-S we see
with a
JI
1
0
that G. contains the matrices u = g1 g21 = _
1
0
61
0
11 and
1
v = gl g2 = l
Ithen uxo = xo
If we set xo
.
d
1
-
1
and for every
0 /
x of X the sequence un x converges to x
Thus any fixed probability o measure under the action of G. must be concentrated on xo, but .
v.xo # xo and such a probability measure does not exist. The hypothesis of (iii) implies that IIgIIa is 1i1 integrable and we know from the theorem A. VI .4 . 3 that there exists
> 0 such that
sup Jd-S(x,y) dvx(x) < + - where d is the usual distance on X defined yEX
f .
IIIXIInIIyII
d(x,y)
by
Choosing y
2 S'2
2
IuZI
l
r
dv; (x) < +- and hence
v
I
1
=(11we obtain for x =I\v/ u f 2
2
Log (u Z ) dvx (x) < +v
J
ti v Let now m and t be the positive measures on X such that m and k are the Cauchy and Lebesgue measures on R. -1 -1 I_-II2 dg m dg lm (x) = = We have the relation dg R dm IgxII
dg
-1 R
.
di
dk
(x-)
(x)
dm
2
Letting f(x) = dR(x) = II
the above result implies that Log f(x)
2v 2 u +v
dg-lm
is vA integrable and remarking that -1 (x) = f(g.x) we obtain dg
-1
rr
II(Log
+
og a-) dud dv Log-4k
I
111111
t
= 0
dm
dg- R
Applying now Furstenberg's formula
Y(A)
(( 2 JJ
_-2 REMARK 4.2
r J
di-1
-1
m dud dvX
Log
2
dm 2
Log (v-2) dvX (x) = u
(
f
11 Log
Log I tI
dua dv1
di
dvAti(t)
The exponent y is easily computed in the Helmotz case at
the point A* as the spectral radius of the matrix
214
rc
1
1
0
and we find
:
L
if Icl < 2
y(A*) = 0 y(a*) = Log
2(IcI +
cV
`-4)
IcI 3 2
if
In the general ergodic case we have seen that y is
uppersemicontinuous (3.1). In the independent case y is actually continuous and we divide the proof in two parts (which is easy) and the case A = A
the case A # a*
be a sequence of real numbers distincts from as and
Let A
LEMMA 4.3
:
(which is more involved).
with limit A. Then for any subsequence extracted from vX
and weakly n
convergent to v the function y(A) has a limit along this subsequence equal to
:
I(ao,v) =
JJ
Log
iLg2cL
dud (g)
IIXII
°
Moreover v is uA
d\) (K)
invariant. 0
Proof
:
f (X) =
Define for each A E R the function I
Log
du (g). From ILog
I
X
. Log IIgII, it is readily
1
1
IIXII
a-ab
IIXiI
11 0 J then
seen that fX is continuous on X. If we set g, = C 1
Sup If X1 (x) -
fX2 (-X) I
IE Logllg, gall)
, JR
XEX
2
1
This yields the first result writing
Y(an) - I(Xo,v) = % (fX- fA 0
n
n
)
:
+ (VAAn - va o
)
fa
o
In order to see that V is uXinvariant we note that uX4 vA n
o
and taking in account that
= vX n
converges weakly to uXo on SL(2
'JR.)
n
and
pXn that X is compact the proof is complete.
COROLLARY 4.4
The function y(A) is continuous at each point ao
distinct from a4. Proof
Let an be a sequence with limit a0 # a
know that each limit point of vA n
From the lemma 4.3 we
invariant, hence equal to vA
is uX o
.
0
215
and y(on) converges to
converges weakly to v.
This implies that v.
n
o
I(ao,VA ) which is equal to y(ao) by 4.1. 0
In the Helmotz case y is continuous at the point A
COROLLARY 4.5
(Assuming that (b) is bounded when Icl Proof
:
> 2).
We know from 3.1 that y is an uppersemicontinuous non negative
function, hence each point A with y(A) = 0 is a continuity point of y and this gives the result for Icl . 2 in view of 4.2. In the hyperbolic c
1
1
O
> 2, the matrix
case Icl
has two real eigenvalues el and e2 with
and we denote by x1 1) the associated > 1 > l e2 l el x2 = C = eigenvectors. Each uX invariant probability measure v is a convex
(1)
I
l
combination of 6_
x
Thus continuity
and 6of y at
,
and we see from 4.2 that I(a6_ ) = y(a*).
,
the point A
follows from 4.3 if we prove that
with An -> A* is such that v(x2) = 0.
any limit point v of a sequence vA n
Let T be the closed subset of X defined by T = {x/u , v 3 O or u`< v < O} . It is readily seen that T is invariant under the action of any matrix
-1 1
Cd
with d > 2. Assuming c > 2 and the variable (b) bounded, T is rc-(A-X x)b
0
-ll
for A near A
invariant under the action of IC
A near a *
,
.
Thus for
0
1
JII
vA is supported by T and any limit point V is also supported
by T. Since we have el > 1 > e2 > 0 we see that x2
T. The same works
for c < -2 replacing v by -v in the definition of T. REMARK 4.6
(i) Corollaries 4.4 and 4.5 are direct consequences of
general theorems about perturbations of random product of matrices which
have been proven by H. Hennion [25], H. Furstenberg,.Y. Kifer [21]. In the case Icl > 2 and (b) bounded, D. Ruelle [54] has proven that y is
actually analytic at A = a*. But when lcl.= 2, analyticity can break down as shows the exercise 4.12 (iv). Such singularities have been
studied by B. Derrida [16] . (ii) In the case c = 2 (c = -2) the matrix
unique invariant probability measure 6- with
I
has an
xl
1
xl = l
with x-l =I
(6X /
-1
1
l)).
l
(iii) In the case IcI
< 2, the matrix
can have a lot 1
Cc
0
-1 1
216
v) does not
of invariant probability measures v but the integral I(A
depend on the choice of v and is equal to zero. In order to see this, we remark that such an integral does not depend on the choice of the norm on]R2 since if we set f(x) = Log
is any norm on
where IIjxIII II x
1R2we have
JJ (f(g.x) - f(x)) dv(x) duA*(g) = 0 c
1
Q 1 is a
Now there existsa real inversible matrix Q such that Q
l
0
rotation and choosing the norm IIIxjjI = IIQxII we obtain I(a*,v) = 0.
We now turn our attention to a result of E. Le Page [ 43 ] improving the corollary 4.4. We say that a function f defined on an open subset U of ]R is locally Holder continuous if for any compact T GU there exists
a strictly positive a = a(T) and a constant C(T) such that If(t) - f(s)I : C(T) It-sia
PROPOSITION 4.7
,
s,t e T
Assume that there exists a strictly positive constant s
such that]E Log(Ials + Ibis) < +- . Then the function y(A) is locally
Hd Zder continuous on JR \ {X *} . Proof of IIt2
:
We assume that the law of (a,b) is not concentrated on a point in which case the result is obvious.
The most important part of the work is to prove that the function A+ v
is, in some sens, locally Holder continuous. Using this property Furstenberg's formula will give the result for y(A). The proof uses
heavily the Holder spaces £ introduced in A.V
and the spectral
properties of the operator PXf(z) = Jf(g.R) du(g) on £ , for a small enough. Our essential goal is obtain some uniformity (with respect to A in a compact set) of the spectral decomposition of the operator PA.
Unfortunately this property is not easy to check since the function A - PX is in general not continuous from]R to the space of bounded
operators on £a. We give below the essential steps of the proof of 4.7. In all the lemmas,T is a fixed compact subset of ]R \ {a
LEMMA 4.8
There exists a(T) > 0 such that for a e 10, a(T)], A E T,
the spectrum of PA acting on %a is contained in a disc of radius strictly less than one, except the eigenvalue 1 which is simple.
Moreover for f r Y,a
217
f = VA(f) II + QA(f) where the spectral radius of QA is strictly less
P X
than one and Proof
:
1 (x)= 1
vx e X, vA(QAf) = 0, QX(II) = 0.
For each fixed A this is just the theorem A.V.4.3. The
uniformity with respect to A E T is a direct consequence of the uniform convergence of
]E LogjISn(A,w) XII
to y(A) on {A,%}
F-
T x X
(We remark that this result gives again the continuity of y(A) on T). n
In order to obtain this uniformity we remark that for a sequence 1 n-1 k (xn An) converging to (x0 ,a0) the sequence vn ,
converges veakly to vX
(each limit point is 0
1
1A
n JO ua 5x invariant)n Moreover
0
looking at the proof of 4.3 we obtain that -IF, LogjISn(Xn'w)XnII
converges to vA (fA 0
LEMMA 4.9
vn(fa
= y(A0) and this yields the result.
)
n
0
There exists R(T) > 0 and for a e 7 O,B(T)J a constant C(a)
such that for (s,t) e T, f E
IIPsf - Ptflla
. C(a)Ilfjja It-sla/2 /2
Proof
Using the notations of A.V.2.4 and 4.3 we have for f E ct a
s,t E T
(a >01)
:
(i) IPsf(X) - Ptf(X) I < ma (f) 1E[5a (gs.x, gt.x)] Since 5(gs.x, gtx)
IbI
IIgsiI
Is-tl we obtain from the
IIgtII
integrability condition in 4.7 the existence (for a small enough) of A(a) such that
:
IPsf(x) - Pt f(x)I . ma(f) A(a) I s-tia
(ii) IPS f(x) - P f(Y)I . ma S
5 (g Since
s
X, g
(f) 1E
5n( gs X, gs. Y) 5a (X, )')
Y) s
<
IIgs1I2 we obtain the existence (for a small
5 (x, y) enough) of B(a) such that
:
IPs f(x) - Psf(Y) I : ma (f) B(a) 6a (X,y) (iii) From (i) and (ii) we obtain that I(Ps-Pt)f(x) - (Ps-Pt)f(y)I < 2ma(f) maxLB(a), A(a)] min(6°(X,Y),It-sla)
and the result of 4.9 follows from
218
min(6a12(K,Y), Is-tIa 6-('12(R, Y)) 6
Is-tia/2
Remark that the lemma 4 implies that A - PX is continuous if we
think of PX as a bounded operator from Za to .9a/2* The following lemma is technical and the proof can be found in [43]. (Remark that the assertions of 4.10 would
if A -> PA
was continuous from ]R to the bounded operators on oU
a
LEMMA 4.10
(i)
Sup
I I
For a E
(QA - I)-1 I
0, a(T)]
).
(lemma 4.8) we have
<
I
AET < where K is a compact subset contained (ii) Sup (PX - zI)IIIa +AET /2 ZEK in all the resoZvent sets of P11 A E T II
LEMMA 4.11
There exists 6(T) > 0 and for a e ] 0, 6(T)] a constant D(a)
such that for f e `,P,a
,
(s,t) E T
Ivtf - vsfl < D(a)It-sIa/2 Ilflia Proof
From 4.10 (i) we know that there exists a circle r in ¢ with
center 1 such that intersection of the spectrum of P., A E T,with the associated disc is equal to the eigenvalue 1 (when a < a(T)). Moreover using the relation Rt(z) - Ra(z) = Rs(z) (Ps-P t) Rt(z) where Rt(z) _ (Pt-z I)-1 and the lemmas 4.10 (ii), 4.9, we see that for a < a(T) A S(T) = 6(T) and f EX,
a
(s,t) E T, there exists K(a) with
IIRt(Z)f-RS(z)flla . IIfII5 It-sIa/2K(a) /2 The "Cauchy" formula vt(f) Il = 1 f 211i r
dz gives the result.
The proof of the proposition 4.7 is now easy since with the notations of 4.3
Y(t) - Y(s) = vt(ft - fs) + (vs-vt)(fs) and we have only to check that the function (A,x) -
f1(x) has a
219
continuous derivative on IR x X.
(Example A.II.5.2
Exercise 4.12
revisited)
Suppose that (a) and (b) are independent random variables distributed according to the Cauchy laws C a
+i6
and Ca
+i6
with 6l > 0, 62 > 0,
61 +62 > 0 (We recall that a Cauchyllaw Ca+i6 with 6 > 0 has the density
6d2
and when 6 = 0, Ca is the Dirac distribution 6a)
2
(t-(x
+6
)
C. where z is the unique solution with OM z> 0 of
(i) Prove that
the equation
z+
1
z
:
= (al - A a2) +
(6l + H 62)
(In the Helmotz case (that is when 61 = 0) we assume d # 0) (ii) Prove that y(X) = Loglzj (iii)In the case a2 = 1, Ch y(X) = 4 (42+X )2 + 62 +
2
,
z being defined in (i).
= al = 0, 6l = 6 > 0
prove that
(2-a)2 + 62)
and therefore
y (A) = y(-A) >, y(0) = Arg sh (z) (iv) In the lIHeflmotz case a1 =2, 61 =a2 =0, 62 Chy(X) =6
> 0 prove that
1+6 -
+
16 and hence for A near zero y(X) has the singular behaviour Y(A) ti JAI1/2
6/2
In various problems of random walks in random environment
the
.corollary 4.5 needs to be stated in a more precise way. From 4.3 and Ax 4.6 (ii) we know that in the Helmotz case with c = 2, if A then -> the sequence vA
converges weakly to 6-
.
scale on the parameter A and the space 1 X
We may hope that with a new ,
it's possible to obtain a
less degenerated limit. We give below the essential features of a
theorem of W.R. Schneider [56] in the simplest case. In this purpose
/u\
it's more convenient to use the change of variable)
v 1 -
ru-v
on X and \\\ v
to change the sign of A in order to work with the matrix
Z=
rb+l
ab1
= [0
[1
1]
+Xb
01
1
11
If we assume that (b) and A are non negative, then the matrix Z leaves
220
invariant the subset X+ _ {x t X /x =(u/ , uv 0}. Denoting always by v the law of Z and vX the invariant measure (for A > 0) we see that v is supported by CO, + - C and v, converges weakly to 6o when A + 0. We
also assume that B = E(b) < + -. Exercise 4.13
With the above hypothesis and notations let e be the
probability measure on X+ defined by 6X(f) = If([0
ol-
x)
J
111
i) Prove that 6A is the invariant measure for the matrix
as
Alb A
1
This shows that the problem of finding the weak limit of
6X when A + 0
is not trivial since the limit matrix is the identity matrix and we are in the most degenerated case.
ii) For t > 0 we denote by eA(t) and p(t) the Laplace transforms
eX(t) = J e is d WA (s), P(t) =7ELe tb1. Prove that the invariance equation for e. yields ea (
t)
=
P
(
l exp (
)
t ) - i s+l
dW
p(S)
(s,t,A) dW,(s) where
( s)
111
and
(t) 1-p(
)1
=
1¢
[exP(_ 7s+l) - exp(-ts)J
O(s,t,a) _
iii) Prove the following relations
lim
(1-p(s)) = t
A+0 ts2 exp(-ts)I
C Z t
(where C is a numerical constant). ti
(iv) Prove that if
is a vague limit point of 6X when A i0 then the 2
Laplace transform V of v satisfies
0(t) = a s v(t) dt`
Conclude that
is proportional to S1.
(v) Let now V be a weak limit point of ea when A + 0. From (iv) we know
that V = a 6X + (1-a)6X where a c [0,il, x1 =\1/ and x, 1
(0)
221 u
For x e X' we define f(R) =
u+v to f and substracting the function 2
TX = (v+(1+a)u) 1 u [ u+v
-
(v+au)2
Applying the invariance equation .
v+(1+a)u
prove that 0
(FA) = 0 where
b R(v+(1+A)u) + db(u+Av)
Furthermore prove that
Sup IF (x) Sup+
u+v
converges to zero when X+ 0. I
This implies that a = 1, thus 0. converges weakly to 6x when X+ 0.
11.5 Absence of absolutely continuous spectrum
We assume in this section that (b) is strictly positive, bounded away from zero, and that It Log(1+lal+b) < + -.
The following results
are mainly due to L.A..Pastur [50] and K. Ishii [29].
Let A be a BoreZ subset of R such that the Lyapunov
THEOREM 5.1
exponent y(A) is strictly positive almost everywhere on A with respect to the Lebesgue measure.
Then a a (A) = 0 1P a. e. Proof
:
Define the measurable subset W of 0xR by
we
0,
A e A,
Inl I
1
n
:
LogIIYn(w,X) II = 0
A is not an eigenvalue of H(w)
W= )(w,A)
,mTnT I
Logllsn (w,A)II = Y(A)
I
The lemma 2.2 and proposition 3.2 implie that for each A e A, 1P(w/(w,A) a W) = 1 and thus using Fubini's theorem, for1P almost all w
( the set A. = I A e A S
(w,1) e W
Y(A)>0
is of full Lebesgue measure in A. For A e Aw
,
let V+ and V
be the contractive vectors given by I.1.1
applied in the two directions of 2Z. These vectors are not proportional (otherwise A would be an eigenvalue of H(w)) therefore each non zero
solution of HT = AT is growing with an exponential rate at least in one direction of Z Z. Looking at the proposition 1.3.2 we obtain
222
o(w)(A(w)) = 0 and thus oa(u) (A) = 0.
REMARK 5.2
The above theorem can be stated in the more general form :
"Let A be a Borel subset of ]R, m a positive measure on it, such that y is
strictly positive m almost everywhere on A. Then for F almost all w, o(w) is orthogonal to m on A". As a consequence, if we know that y is strictly positive everywhere, taking a fixed spectral measure a(wo) as a measure m, we see that a(w) is orthogonal to o(wo) for F almost all w. This means that the dependence of a upon w is in general extremely "chaotic", although the support of a is F a.e. constant (2.3).
REMARK 5.3
We have proven in 5.1 that for F almost all w, for almost
all A of A, there exists an hyperbolic structure in both directions of
M with distinct contractive vectors. This suggests that the existence of pure point spectrum is some what unexpected since in this case we ought to obtain that for F almost all w, for a(w) almost all A of A there exists an hyperbolic structure in both directions of 7l with the same
contractive vector
!
(But there is no contradiction since a is F
a.e. orthogonal to the Lebesgue measure).
COROLLARY 5.4
The absolutely continuous part of the spectrum vanishes
F a.e. in (at least)the three following cases : (i)
The independent case, assuming that the law of (a,b) is not
supported by a single point ofJR2. (ii) The Mathieu equation when IsJ
> 2.
(iii) In the general ergodic case assuming
that the process (an,bn) is
not deterministic.
REMARK 5.5
In the general ergodic case it has been proved in [58 J
that for each Borel subset A of positive Lebesgue measure and contained
in {AIy(X) = 01 then aa(A) > 0 F a.e. that
a
This result added to 5.1 shows
= {y = 0} 8 a.e. where the closure means essential closure
with respect to the Lebesgue measure.
On the half positive axis, the first proof of absence of absolutely
continuous spectrum was given by K. Ishii in [29] and we outline his method in the following exercise.
223
Exercise 5.6
Let +H be the operator H restricted to the positive half
axis as explained in the exercise 1.2.10. (i) Prove that for <
+
o
bm A # 0
0
Hint
:
d+o(t)
1
R), e , e > = J (t-a )
= lim -
9n =+Z
1
n=0 Pn pn+l
pn
n-f-
use the lemma 1.4.2
(ii) Let A be a Borel subset of F. such that y is strictly positive
almost everywhere on A and
/
(w,A)
W=
:
E Q, A E A lim Log!I Sn(a,A) II = y(a lil
n-
n
lim n (lanl + lbnl) = 0
n- +prove that for1P almost all a, the set Aw = (X E A /
(w,A) a W and y(A) > 0)
is of full Lebesgue measure in A. (iii) Prove that for A E Aw :
_ .{
0
E +O
Hint
+m
Mm( nE0
:
1
p (A +ic) pn(a +ia)
)
0
n+1
observe that
l p
( A )
l
: lp (A + ic)l and that if pn(A) = 0 we can
use the identity (I mz # 0)
Pn(z) Pn+1(z)
+ pn(z) pn-l(z)
- (a n -
zb n )
Pn+l(z)pn-1(z) l
Up to now we have only shown that the positivity of y implies absence of absolutely continuous spectrum and we may ask wether this positivity implies the pure point spectrum property. It is known that it is false in general, in example the Mathieu operator has no point spectrum when the rotation is given by a Liouville number a [57j. Hence even when y has a strictly positive lower bound on ¢ (when Isl
> 2 see
3.4) we can't assert the absence of singular continuous spectrum.
However S. Kotani [36J has obtained a weaker result on the positive half axis and we follow his proof in the next exercise.
Exercise 5.7
Let +Hy be the operator on the positive half axis with
boundary condition y at the point zero (see 1.4.7) and A be a Borel
224
subset of ]R such that y is strictly positive almost everywhere on A. We want to prove that for ]P almost all w spectrum of
for Lebesgue almost all y, the
Hy is pure point on A (if any) and that the eigenfunctions
are exponentially decaying.
(i) Defining W in the same way than in the exercise 5.6 and looking at the result (ii) of 1.4.7 prove that for IF almost all w, for Lebesgue
almost all y, for +ay almost all A of A we have
lim
LogjIS.(w,A)JI = y(a)
n- +oo
n
(ii) Observe that under the same almost sure assumptions than in (i) the sequence IISn (w,1)(1)II
is either exponentially growing or exponentially
decaying (with rate y(A)). Hint
:
use
Ossedelec's theorem.
(iii) Looking at the proposition 1.3.2 adapted to the operator Hy and ay (see exercise I.3.3) prove under the same almost sure assumptions that only the second alternative can take place and conclude.
11.6 Distribution of states. Thouless formula.
In this section we assume that (b) is strictly positive and bounded away from zero.
The distribution of states is the probability measure on ]R defined by k(A) _ ]E[bn
on(A)]. This expectation does not depend on the variable
n thank's to the relations(bnon),6 = bn+1 an+1 and k(IR) an(iR)
= 1(since
b- 1
=
n ). This probability measure k is sometimes called
"integrated density of states" but we don't use this expression since in many cases k is not absolutely continuous !.
We say that a sequence of boxes A goes to infinity if the boxes are increasing and if the length of A denoted by JAI goes to infinity (compare with the definition of a sequence going to 2Z in 1.4).
Generally we choose A = [O,N] with N going to infinity as sequence of boxes going to infinity.
For a box A we define the probability measure kA by kA
E
JAI neA
PROPOSITION 6.1
b a
n n
Let f be a k integrable function. Then for any sequence
225
of boxes going to infinity the random sequence kA(f) converges P a.e to
k(f) and JE[kA(f)] converges to k(f). Proof
:
This is just the Birkhoff ergodic theorem.
For any sequence of boxes going to infinity the random
COROLLARY 5.2
sequence kA converges weakly]P a.e to k, and7E[kA] converges weakly to k. Proof : We use 6.1 looking at a countable dense subset of the space of
continuous functions with compact support on1R.
PROPOSITION 6.3
The topological support of k is P a.e equal to the
spectrum of H(s). Moreover k is a continuous measure. Proof
:
k =
1E(boao + b_1o-1) and the support of (bo o + b-la-1) is
equal to the2 support of a. Then by the lemma 2.3 we obtain the first part of 6.3. For the last statement see the lemma 2.2.
Usually the distribution of states is defined as the limit of the distribution of eigenvalues of the operator H restricted to boxes and it's observed that this limit does not depend upon the boundary conditions given at the endpoints of the boxes. In order to prove this AHy,x result let be the operator H restricted to a box A with boundary conditions y and x as explained in 1.4.6. If we denote by ( ^An'x)nEA %H),x the eigenvalues of we define the eigenvalues distribution :
^ky,x = 1 E b Aay,x = 1 IAIncA n
PROPOSITION 6.4
n
E
8
JAI etA ^Xy'x n
Assuming that the random variable (a) is bounded, for
fixed x,y E it the random sequence
^ky'x converges weakly to k P a.e.
and1EE^ky' converges weakly to k, when A goes to infinity. Proof
n
:
Under the hypothesis on the random variable (a) the sequences
ky'x and kA are supported by the same compact subset of 1R. Then by
corollary 6.2 it's enough to prove that for each positive integer m the sequence 1tm(A ky'x - kA )(dt) converges to zero when A goes to infinity. This last integral is equal to FAT1
nE
bn
<((^HY,x)m - Hm) en, en>
226
In this sum the only non zero terms are those of index n with
M < n
M+m and N-m,< n. N (where A = [M,N]). Since the operators ^ Hy'X
H are uniformly bounded the proof is complete.
The definition given here of the distribution of states and the construction by the proposition 6.1 can be found in [ 3 J. Some other
authors [34] define the distribution of states as the weak limit of the distribution of eigenvalues of the operator H restricted to boxes by means of the subadditive and superadditive ergodic theorem. As usual it's possible to obtain the convergence in 6.1 with an exponential rate assuming 4, mixing properties of the dynamical system (Q,a ,F,6). We have
the same speed of convergence under this hypothesis for the distribution
of eigenvalues, but the proof is more involved [34]. We may ask what are the functions f for which the proposition 6.1 apply. Indeed bounded measurable function are available since k is a probability measure, and when the random variable a is bounded any measurable locally bounded function works since k has a compact support (by example a continuous
function). In the general case following [12] we prove
LEMMA 6.5
Assume thatlE Log(1+l al) < + - then for
:
m x # 0 the
function f(x) = Logla-xl is k integrable. Proof
It's enough to prove that Log(1+lxl) is k integrable. Let V be
:
the two diagonal (and hence self adjoint) operators (V Y) +
n
= b-
n
1
(a ±2)'Y
n
V and H are self adjoint with the same domain D1 (see 1.2) and it's readily seen that V
< H < Vt.
The distributions of states k- of V (defined in the same way than for H) are the laws of the random variables b-1(a t2). Thus for any increasing (decreasing) positive function g+ (g-) we have k(g+) < k+(g+)
k(g )
k (g )
Taking g+ = Log(1+lxl)
{no)
g
,
= Log(1+lxl) .
11
{x,<0}
we obtain the expected result since b-1 is bounded.
In the independent case the distribution of states is related to the invariant measure v. on the projective line X given by the proposition 4.1. Let X
be the subset of X defined by
n
227
x ={X/x=('),uv PROPOSITION 6.6
O}
In the independent case, assume that the law of (a,b)
is not concentrated on a single point of1R2, and that the random A*in the Helmotz variable (a) is bounded. Then for each real A (A # case)
k((--, A ) = )X(x ) From 4.1 we know that for a # ax there exists a unique u
Proof
invariant probability v. on X. Let xo be the vector ( 10) and for N > 0 N 1 k X where u is the kth power of convolution of uX N+1 k 0 ua va,N on SL(2,]R). Each weak °limit point of VA N when N } + - is uA invariant
and hence equal to vA. This implies that the sequence 'OX
,N
converges
weakly to v. when N - + If we denote by 9X_ the indicator function of the set X-, it is readily seen that
:
_
=
N
va,N(X )
N+1
IElkEO
where xk
]1X- (xk)]
(:_)
N B ut kEO 1X- (xk) is the number of change of sign of the sequence pk for 0 < k < N. Let ao, All
..., AN be the roots of the polynom pN+l then by
the Sturm theorem [ 2 ]
(see also [l3]) this number of change of sign
is equal to 0 if A < ao, r+l if ar < A < Ar+l
,
N if A > AN
.
Thus if
we note by NH the operator H restricted to the box [0,NJ with zero boundary conditions at each endpoint and Nk its distribution of states, we have
:
IVA,N(X-)
-JE[Nk(]_m,A
N+1
We know that vA N converges weakly to vA
,
that ]E[Nk] converges weakly
to k (by 6.4) and that v. and k are continuous measures thus
lim N->+-
va(X) = 1im ]E[ k(
k( -co,A])
N- +-
We now turn our attention to a remarkable relation between the Lyapunov exponent and the distribution of states first pointed out by D. Thouless
62] and hence called the "Thouless formula". We follow here
the proof given by J. Avron, B. Simon [3] and W. Craig, B. Simon [12].
228
PROPOSITION 6.7
Assume that the random variable a is bounded and
E(Log b) < +- , then for each complex number A y(A) _ ]E(Log b) +
Loglt-al dk(t) J
Proof
:
Let A be a complex number with 3 m A # 0 and N 1,l. The roots of
the polynom pN+l
(qN+1), are the eigenvalues of the operator H '
restricted to the box [O,N]
endpoint. Denoting by Nk
([1,N] ) with zero boundary condition at each
(Nk) the distribution of these eigenvalues it
is readily seen that N
r
Loglt-al dNk(t) + 1
J LogIt-Al
N+1 kEO Log bk
dNk(t) + N kE1 Log bk =
N+1
LogIgN+1(X)I
N LogIgN+1(A)I
The random variable (a) is bounded, thus the probability measures Nk, Nk are supported by the same compact set. The Birkhoff ergodic theorem
and 6.4 mplie that
Nm N+1
Log
:
pN+1(a)I
N
+
N LogIgN+1(X)I
= JLoglt- Al dk(t) +]E(Log b),]P a.e.
We have thus proved that each entry of the matrix S
= N
rpN+1 pN
qN+111
qN JI
has
the same asymptotic behaviour. Since we know from 3.2 that
lim N-+- N
LoglISnII = y(A)
we can conclude that for
]P a.e. 6m X0 0
y(X) _ ]E(Log b) + J Logl t-aI dk(t) Each function in the above formula is a subharmonic function on
for
y this is the proposition 3.1 and the subharmonicity of JLoglt-al dk(t) is straightforward. Since two subharmonic functions almost everywhere equal, with respect to the Lebesgue measure on ¢, are everywhere equal, we obtain the result.
REMARK 6.8
:
We have assumed that the random variable (a) is bounded in
the proof of the Thouless formula. But if we only assume that
]E Log(1 + lal +b) < +- then y(X)
]E Log b and JLogl t-al dk(t) are well defined (the latter for 6 m A 0 0 by 6.5). This suggests that such a ,
229
formula should be true under this weaker hypothesis...
We know from 6.3 that the distribution function K(A) = k(]-m,Xl) is continuous. We list below sow extra regularity properties of K. K is said locally Log Holder continuous if for all compact subset T of R there exists CT > 0 such that for s,t e T, Is-tI
we have
IK(s) - K(t)I < CT (Logls-tl-1)-1
PROPOSITION 6.9 (i) In the general ergodic case assume that (a) and (b) are bounded, then K is locally Log Holder continuous. (ii) In the independent case assume that (a) is bounded and that there
exists a > 0 with]E(ba) < + - then K is locally HOZder continuous on
R\A
.
(iii) In the independent case assume that bn = 1, Vn a 7L , ]E(a2) < +
and that the law of (a) has a density on R, then K has a continuous derivative.
(iv) In the independent case, assume that the conditional law of (a) given (b) has a bounded density $b on R, such that]E(bjkkbll) < +
Then
k has a bounded density with respect to the Lebesgue measure. Proof
(i) Since (b) is bounded there exists S such that Sb < 1 and then by the Thouless formula
:
Y(SA) -]E Log Eb =
Loglt-Al dk(t)
v where K(t) = K(lt).
Thus this latter integral is positive for all X e 0
and this implies that K and hence K must be locally Log Holder continuous by standart analysis arguments [12]. (ii) The result follows from 4.7 and the properties of the Hilbert transform [47]. For, P
E:
L2(]R)
we denote by 'T its Hilbert transform and
if ]A,B[ contains the support of k, we choose P(A) = K(A) if A e ]A,B[, T(A) = 0 if A i ]A,B[. By integration by parts and the Thouless formula: ti
1
(Y(A) - LogIB-AI) for almost all A with respect to the Lebesgue 4'(A) = II measure. Letting O(A)
(y(A) - LogIB-AI)
,
@ is locally Holder continuous on
IR%{B,A*}, hence 0 is also locally Holder continuous and 0 = T
a.e
hence 0 = P on R \{B,A }. Since B can be chosen outside of any compact
230
we obtain the result.
(iii) This result will be proven in 111.4.1.
Let a be a number with )m a> 0, N an integer. We denote by
(iv)
of the operator H restricted to the half axis [N, + m [ or
the resolvent
]-
N
,NI
Following the lines of 1.2.12 and 1.2.14 we define
Nm (A) =
± n = <+R
,
eN , en> N
N. The functions +'Y can be extended to the whole of
where n > N or n
7L in such a way that these extensions satisfy H'Y = A'Y, and we find N' _ (Nm)(Np)
+ (Nq)
where Np and Nq are solutions of H'V = AY' with the
,
boundary conditions N +pN
N +q
_ N
_ N
-pN
N-1
1
-qN+l
N
N
N
N
+'N-1
+qN
-pN+l
-qN = 0
Then it's easy to verify that
< RA e N ,
:
-1
eN >
1
(. + m) +
N-1 m) + AbN
(
- aN
The functions (N++m) and (N-1 m) are functions of the process (an,bn) for n >. N+1, n < N-1 respectively and have positive imaginary parts. Taking expectation in the above formula, first integrating with respect to
(aN,bN), we obtain
f JELb
This yields the expected result, using Lebesgue's convergence theorem in the formula
k(h)
_ o II
:
h(x) ( im f
dk(t) ) dx
J
where h is a continuous function with compact support.
Let L1'a be the space of functions f e L1(IR)
such that there
a
exists g e L1(IR) with f(t)(1+t2) /2 = g(t)
(where f is the Fourier
transform of f). B. Simon and M. Taylor [61] have recently proven that in the independent case with bn = 1, n e ZZ, if we assume that (a) is bounded with a density in L1'a for a strictly positive a ,
then K is a
231
function. A typical example is given by the uniform distribution on
C
a compact interval.
When the density of (a) has stronger regularity, namely when (eita)l . lIE
a e sltl for some positive constants a and a such that
a < 8, then K has an analytic continuation to a strip {z c
/l) mzl <6}
for some 6 > 0. This result is proven by R. Carmona in [91 using the "Molcanov formula". Further results about Lifschitz exponents can also be found in this course.
Exercise 6.10
In the general ergodic case assume that (a) is bounded. Nk ' Prove that the sequence of probability measures (N , 1) converges
weakly almost surely to k where dNk _
II(N+1)
N
1
1
da
P2 + P2 N
2
k=O bk Pk
N+ 1
The two following exercises show that the distribution of states can be the same in two models which have in general very different spectral properties ([59] and [18j).
Exercise 6.11
"The Maryland model", see [18]. Let H = H + V.
Laplace operator HT = -yn-1
- 'n+1
is the
and for a positive number 8 and
irrational a we define (VT)n = 8 Tn tg(IIn a + a) where
a e [O,IIJ \ {II/2 - IIn a}n e 2Z = Q. This is not exactly a quasi-periodic model since a is not bounded but a lot of explicit computations can be performed in this "solvable model". Let U be the operator (U'Y)n = 2 iw
e2iII n aT
n
and z be the complex number
e
(i) Prove that H = H + i8 (zU - I) (zU + I)
(ii) Prove that for
mX>0
(zU+I)(I+z CU)-1 R1+i6 where C = Rx+i8 (H - (A - i8) I ) (iii) Observing that
1Cll < 1 expand (I+z C U)
in power series and
prove that lE [R A]
= Ra+is
Conclude that k has a density and k = k '
Ci6 (Cis is the Cauchy law
232
and see I.4.5 for k). Hint
dk as
:
for A real. dk(t)
lim 1. c+o
J
Exercise 6.12
t-(A+ir)
dk(t)
=3m 1
t-A-i6
The "Lloyd model", see [60]. Let H = H + V as in the
exercise 6.11 but now (VP)n = a
n
Y'n where a
n is a sequence of real
independent random variables with a common law Ci6 (see 4.12) (i) Let I be a finite set of integers (perhaps not all distinct) and s
k
= 1.
k c I, a set of positive numbers with E s
keI k
Prove that IE [exp it E sk ak] = exp(-sltl) keI
(ii) "Dyson's formula". Let A and B be two squares matrices with complex entries and t a real number.
Prove that et(A+B) =
etB + ft es(A+B)A e(t-s)B °
Iterating this relation, expand e
t(A+B)
in a convergent series.
(iii) Applying (i) and (ii) to the operators prove that
and
A
H and A H
esltl eit H
I[.eit
F
ds
HI
= e -St eit H
(iv) Conclude from (iii) that k = k
i6
11.7 The pure point spectrum. Kotani's criterion
We assume in this section that (b) is strictly positive, bounded away from zero and that IE Log(1 + lal
+ b) < +
Let a +(a -) be the subsigma algebra of a generated by the
process (a n b n) for n >,N (n,<M) and integers 11,N. For a '
measurable positive function
defined on
we denote by
+i the conditional expectation with respect to a-+ and for each
Borel subset d of the real line we set a+(4) = IE [a(A) l a+] . a+(0) are a+ measurable functions defined IP almost everywhere. It's possible to construct a topology on 52 such that D becomes a Polish space
233
and a_ its Borel sigma algebra. Thus there exists a choice of a+(A) such that a+ are positive measures on ]R with a+(]R) = o(IR) .
By a
monotone class argument we see that if F+ are measurable functions on 0 x IR with respect to the product of
with the Borel sigma algebra
of IR then
IE IJ( F+ do I a+Q
F+ da+
=(
and
J
IE If F+ do] = IE If F+ do+] (take care that a+ have nothing to do with -a of 1.2.10) We are now able to prove Kotani's criterion [36].
Let A be a Borel subset of IR and m a positive
PROPOSITION 7.1.
measure on A. We assume that y(X) > 0 absolutely
m.a.e on A and that a+ are
continuous on A with respect to m,
Then for ]P almost all w the exponent
almost everywhere
IP almost everywhere.
y(A) is strictly positive a(w)
on A, the spectrum of H(w) is pure point on A (if
any) and the eigenfunctions decay with the exponential rate -y(A). Proof
:
Define the subsets W+ of 0 x IR by
w e 0, A E A, lim
InI Log IIYnII = 0
n->±°
w± = { (w,X)
/
lim 1 LogilsnII = y(A) > 0 n->±m
InI
a+ are absolutely continuous with respect to m
The sets W+ are measurable with respect to the products of a+ by the Borel sigma algebra of IR and we know from the ergodic theorem 3.2 and the hypothesis that W+ are of
full IP ®m measure.
Let F+ be the functions defined by F+(w,A) = 1
if
(w,A)
= 0
if
(w,A) e W.
W+
Using the preceding discussion and the absolute continuity of a+ with respect to m
IE If F+ do] = IE If F+ do+] = 0 Then by Fubini's theorem, for IP almost all w, for a(w) almost all A in A, each solution of H'1 = AP is, in each direction of 2Z, growing or
decaying exponentially fast (thank's to Osseledec's theorem I.1.1). Looking at the proposition 1.3.2, for o(w)
all A there exists a slowly
234
increasing solution of HP = AY' and thus this solution must be
exponentially decaying in each direction of a with rate -y(A). From the integrability condition Log(1+b) < +- we know that for IP almost all w,
lbn(w)I . n for n large enough. This implies that a(w) is
supported by the eigenvalues of H(w), hence a(w) is pure point and the eigenfunctions decay with the rate -y(X).
REMARK 7.2
The above matching method of two solutions of the
difference equation was employed by S.A. Molchanov L45] and
R. Carmona Is I . If the process (an,bn) is deterministic then the hypothesis of 7.1 does not hold because in this case a+ = a and we know from 5.2 that if y > 0
m
a.e. on A then for IP almost all w, a(w) is orthogonal to m.
Thus in order to check the hypothesis of 7.1 some randomness is inevitable. We shall see in III sufficient conditions in order to deal with criterion 7.1.
11.8 Asymptotic properties of the conductance in the disordered wire
In physical litterature there exist several approaches to the notion of conductance. We only give there a formal definition and we refer for instance to J.L. Pichard [51] for physical background. We call "plane waves" two sequences u and v of the form u = einB n vn = e in0 where e is a real number different from a multiple of R. Then any sequence Y of complex numbers can be "projected" on the plane waves u and v according to
n+1 Y
where T n Rn
It
=
vn+l
+ 8
n un
n
a 1n
u.+1
- a
n vn
un+l
vn+l
u
v
n
n >, 0, from
= T
an
n 8n compute
is an invertible matrix. In order to
n
a-1 8-1
we set Z n = T n l Y n Tn-1
,
Ti
n = Zn n
n-1
'
Z
o
is readily seen that Rn = Ti nSn T_1 and thus if Y< satisfies the
235
difference equation (HT)n = XT relation
n
, n c [0,
..
,
N], we obtain the
:
=
B=1
11N
\
I
1
a
This means that if we "input" the wave of "coordinates" ( the disordered box A = 10, (coordinates
..
,
1)
into
NJ, then the "output" wave have
SN
aN The reflexion and transmission coefficients tN and r,, of the disordered box A = L0,
0
..
/
HN
,
NJ are defined by the equation
(N / 2
t
and the conductance of A is the ratio C,, = IrN I
Our problem is to
find the assymptotic behavior of CN when N -
Let S U (1,1) be the subgroup of matrices in SL(2,Q) of the form
Ia B
Tn1
a
Y
). It's an easy exercise to verify that if Y e SL(2,IR) then is in S U(1,1). Thus ZN and therefore RN are in S U (1,1). As
//Tn-1
a consequence we may write
CN = I
:
(
2
(IIlNxIl2 - 1) where x is the vector I
I nNXII =
T_ I
I
1
)Of .2. Since
T_
nl Sn T-1xl I
=
I I
nl Sn y II
where y = T_lx, the asymptotic
behavior of CN is obtained from the asymptotic behavior of the columns of SN which is well known in the independent case.
THEOREM 8.1
Assume that the sequence (a
n
b
n),
n >, 0, is an i.i.d
'
sequence with values in TR2 and that its law is not supported by a single point of IR2. Then for A c IR (A # a* in the Helmotz case) (i) The sequence 1 Log C converges IP a.e. to -2y(A) when n - + n n Moreover y(A) is strictly positive.
(ii)Assume that there exists a > 0 with IE(Iala + Ibla) < +
then
there exists a strictly positive number a2(A) such that the law of the
sequence N (Log C law with
Vill'
ariance
n
+ 2n y(X)) converges weakly to a centered Gaussian
a(X). 2
236
Proof
I
:
From the inequalities
I Tn I I - 1
:
IISnYII 1 IITn1 Sny II : IITn111 IISnYII
we can write Log I Tn1 Sn Y I = Log I I Sn y I + Un where the random sequence Un is bounded since I Tn II . 2, 11 Tn1 II . I sine I I
I
I
Thus we obtain V n
1n Log C n =
n
-
?n
Log IISnYII
where
Vn = [Log2 -2Log Un - Log (1 - (IISnYII + Un)-1)1 We know from 4.1 that y(A) > 0 and from A.II.3.6 that Log Its yII V converges IP a.e. to y(X). This implies that n converges IP a.e. to n n zero and the result (i) follows. In order to prove (ii) we write
1 (Log Cn + 2n y(A))
V
:
V
2
A
(Log Its nYII
- n y(A)) +
n
/
'
The result (ii) now is a direct application of A.V.4.3 and of the IP
a.e. convergence of °n
to zero.
REMARKS
(i) The same result was already obtained by A.J. O'Connor [48] and T. Verheggen [63] under stronger assumptions on the law of (a,b). Actually they suppose that this law has a density with respect to the Lebesgue measure on IR. (ii) Extensions to the strip, using results of chapter III and A.IV.
can be obtained in a similar way and this is what is done by R. Jhonston and H. Kunz in I30].
CHAPTER III
THE PURE POINT SPECTRUM
We now prove the strongest singularity property for the spectrum of Schrodinger operators, that is the pure point spectrum property. Until the end of the year 1984 the only known proofs were very close from the original one given by the Russian school (see the introduction). In most of these works it was not clear what was really necessary in order to obtain the pure point spectrum property, since they dealed essentially with the independent or Markov cases under strong assumptions on these processes. A very clarifying idea of Kotani introduced in 11.7 and applied to the independent case in the first section gives very easily the expected result, at least inthe classical Schrodinger case, that is when bn = 1, Vn e 2z. It's not clear that such a procedure can be easily applied to the Helmotz case. Furthermore Kotani's theory does not use any approximation of the spectral properties of H by means of "thermodynamic limits" along sequences of boxes going to 2Z. Thus if we want to obtain some information about the speed of convergence of these spectral measures computed in boxes, we have to use an other way. In this purpose the second section is devoted to the study of the Laplace transform on SL(2,IR) . The corresponding
operators were already introduced in A.V. In section 4, using the spectral properties of these operators,we prove again the pure point spectrum property in the independent case, thank'sto the "good" approximations of the spectral measures of H obtained in I. These results are also used in section 5, in order to obtain a representation
237
238
formula for the density of states.
III.1 The pure point spectrum, first proof.
In this section we suppose that (an'b ) is a random process defined a n on some complete probability space (O,¢.,1P) but we don't require
ergodicity or stationarity. We only assume that for IP almost w (i) The sequence bn(w) is strictly positive and bounded away from zero.
(ii) lim 1 Log(1+lan(w)l + bn(w)) = 0 n-*±
Ini
This last assumption is used to check the hypothesis (ii) in Osseledec's theorem I.1.1 and to establish that for 0 < r < 1 the series E b rn is n 8 a.e convergent.
Carefull readers notice that the proof in 11.7.1 has nothing to do ,with ergodicity if we assume the existence of y properly. Thus we can prove exactly in the same way as in 11. 7. 1
PROPOSITION 1.1
:
Let A be a BoreZ subset of IR and m a positive measure
on A. We assume tnat there exist IP x m
a.e
lim 1 Log lSnl = y+
strictly positive functions
Furthermore we a.e. n+± nl suppose that a+ (defined in 11.7) are absolutely continuous with respect y,(w,A) such that
IP O m
tom on A. Tnen for IP almost all w, y+(w,X) are strictly positive for a(w) almost
all A, the spectrum of H(w) is pure point on A (if any) and the eigenfunctions decay with the exponential rate -y+(w,X)
As remarked by S. Kotani [36] and F. Delyon, Y. Levy, B. Souillard [14], if we suppose that the process (an,bn) is an
independent sequence of random variables with values in JR2 (not
necessarily identically distributed), then the assumption on a+ is easily checked
COROLLARY 1.2
Let (an,bn) be an independent process such that there
239
two functions y+(a),
exist
each A, lim 1 Log1S n InI
I
m
= y+
a.e strictly positive and that for a.e. (where in is the Lebesgue
IP
n-*±
measure on IR). Furthermore we suppose that there exist
two consecutive indexes n =no,
no+1, such that the conditional law of an given bn has a bounded
density b with IE (I I b II ) < + n
n
Then the conclusions of 1.1 hold, replacing a by an 0
Proof
:
+ an
+1'
0
In view of proposition 1.1 and exercise 1.2.9 we have only to
prove that for n = no and n = no+1
IE[a Ia +7 is absolutely continuous n with respect to the Lebesgue measure m. Choosing N = n0+2 and M = n0-1
in the definition of
in 11.7 these conditional expectations are
nothing else than expectations with respect to the random variables (an,bn) for n = n0 and n
we know that for
= no+l. Looking at the proof of 11.6.9 (iv)
m A > 0 and any n
:
_ -L(n++m) + (n-lm) + abn - anj where
n+1
MM is a function of (ak,bk) for k . n+l,
n-1m
and
(A) is a function of (ak,bk) for k . n-1,
with strictly positive imaginary parts. Denoting by IE
n
the expectation with respect to the single random
variable (abn) we obtain r
rlEn (on)] (dt)
m
III
%
n
t-a
Thus IEn (an) has a bounded density with respect to the Lebesgue measure on IR and the result follows.
Note that for each A the limit of 1 LogflSo1I is IP a.e. constant and
Ini thus we can choose y+ as functions of A only.
COROLLARY 1.3
Let (an,bn) be an independent identically distributed
random process such that IE Log (1 + IaI + b) < + = and that the law of
(a) given (b) has a bounded density
b
with IE E II b I I ] < + = . Then the
conclusions of 1.1 noZd. Proof
:
From
11.4.1 we know that y(A) > 0
everywhere (the assumption
on the law of (a) excludes the Helmotz case).
Thus the ergodic theorem 11.3.2 and corollary 1.2 yield the result
240
(Remark that the hypothesis (ii) of the begining of this section is
satisfied since IE Log (1 + al + b) < +m). REMARK 1.4
(i) In the Helmotz case, in example for the operator b- 1
(Hi')n =
n
(-fin-1 - 'n+l),
we obtain in the proof of 1.2 1
(n++m) + (n-1m-) + abn11 j
we don't obtain n the absolute continuity of c1+ in general. (See however the remark of and if we integrate with respect to the variable b
S. KOTANI in the appendix).
(ii) In the independent case, it would be very tempting to choose as the tail sigma algebras generated by the process (an,bn). Thus o+ would be obtained as expectations rather than as conditional expectations . Unfortunately it's far from evidence that the
mesurability property for the sets W+ introduced in the proof of 11.7.1 can be easily checked....
111.2 The Laplace transform on SL(2,IR)
Let G be the unimodular group SL(2,IR) , X the one dimensional real
projective space, C(X) the Banach space of complex continuous functions on X endowed with the supremum norm. As explained in 11.4, G acts
continuously on X if we use the usual norm topology for the two by two
matrices and the distance 6 on X defined by 6 (x, y) =
JJ
x n Y11
since it is readily
,
seen that
llYll
11x11
6(gl.x ,g2.y) . 6(5,Y)
11g211
I1g111
+ I1g1-g211 I1g211
Moreover a sequence xn converges to xo in X iff there exists a sequence
gn in G such that
x = g n.xo. n converges to the identity in G and n
g
In order to check this last property we observe that for
(rcos0
x
\r sin 8)
,
r e IR \ {0}
,
0 e [O,n [ we can write x = g0. x- where g0
is the rotation of angle 0 and x, = (1).
241
Let now u be a probability measure on G and t a real number. We define the operator Tt by T
t f (x) = J (--x-,-) f (g.x) du (g)
,
f e C(X).
IIgxII
It is readily seen that if f Ilglln
du(g) < -- for n > O,then Tt is a
positive bounded operator on C(X) for t c [-n,+n] and the Markovian operator To is just the transition operator for the left random walk of law u on X. We denote by Tt the dual operator, acting on the space M(X) of complex measures on X by (Tt v)(f) = v(Ttf)
,
v e M(X), f e C(X). A
probability measure v such that T+ v = v is called a p invariant 0
probability measure, such measures always exist
from the Markov-
Kakutani theorem (see A.I.3.5) but are not unique in general. Most of the sequel is devoted to the study of spectral properties of the operators Tt. These operators have already been introduced in A.V but we have to notice that we use them on C(X) rather than on Holder spaces. (see remark 2.10.(ii)).
Let n be a strictly positive number such that
PROPOSITION 2.1
J IIgIIn du(g) < + - . Then the map t -> Tt is analytic from [-n,+nJ to the space of bounded operator on C(X). More precisely if
Dt f (x) = J
then
(Log) IIgx11
nn I( tt-
)
t f (g.x) du (g)
IIgxII
- DS)f(x)I: IIfII
(IITt+EII+IITt-EII+IITS+EII+IITS-EII) 2E
S
for f e C(X), n >,
1,
e > 0,
(s,t) a
L-n,+n]
.
Proof : We first remark that the n th power of the above operators are
the operators associated with the n th power of convolution un of u on SL(2,7R) and that
du (g)] n
JIIgJJ' dun(g)
[ JIIgIIf
<
Thus it's enough to prove the above formula for n = 1. As in the classical proof of analyticity of the Laplace transform, the result follows from the Taylor formula (where o is strictly positive,
r E ]s,tD
.
2
at - os = (t-s) (Log a)as + (t-s) 2
and the bounds
:
(Log o)2
or
242
a
r < os + a t (oe
ILog
+ o-e)
e
for any e > 0
Assume that :
PROPOSITION 2.2
(i) there exists a strictly positive number n such that
(ii) u has a unique invariant probability measure v
(iii) y = f f Log
l
du (g) dv (x) > 0
IIxH Then there exists a strictly positive number t
0
such that the spectral
radius of Tt is strictly less than 1 for t e J O,tj.
Proof
:
Define fn(x) = n f Log Jlx
du
n (g) .
Igx1I As in the proof of 11.4.8, the unicity of v implies the uniform
convergence, of the sequence fn to -y on the compact space X. Applying 2.1 to the function f(x) = 1, 8x a X, s = 0
,
t e 1 0,T2] , e =
2
yields
1 + t En fn (x) + 4n Kn
Tn fi (x)
where K = J IIgIIn du(g) Now we can choose an integer no such that
sup f
(x)
no XCX and then there exists to > 0 such that for t e ]O,t0]
Sup
n Tto
- Z < 0
n D (x)
=
Tto IJ
<
1.
XEX
REMARK2.3
The result 2.2 can be obtained from the general theory of
analytical perturbations of bounded operators, if we know that the spectrum of To except the eigenvalue 1, is contained in a disc of radius strictly less than one. In this case if r(t) denote the spectral radius of
Tt it can be proven that dt r(t) t=0
-y and the result follows
from r(O) = 1 and y > 0.
But such an assumption on the spectrum on To requires strong regularity properties on u as we shall see in the sequel (Remark 2.7).
In order to investigate the spectral properties of To we prove the following , where T(ii) is the closed subsemigroup of G generated by the
support of 11
:
243
PROPOSITION 2.4
The eigenvalue 1 is the only eigenvalue of modulus 1
:
for To and the associated eigenspace is one dimensional, in (at least) the two following cases : (i) The orbits of T(u) are dense in X. (ii) Tnere exists a unique u invariant probability measure v. Proof
:
Let f CC(X) with Tof = of
,
JAl = 1,
lf(x) l = 1. The invariance equation 0
llfll = 1 andxo C X with
Tn f = an f implies
that
0 for g C T(u). The proof is complete in the case (i).
f(g.xo) = f(xo)
Assume now that there exists a unique u invariant probability v of
1n
topological support S. As it has been often remarked, for any x in X, k a 6X converges weakly to v. Then if y is n kI1 u a point of S and U an open set containing y we have
the sequence vn,-
lim V
n,x0
-
v(U) > 0. This implies the existence of g in T(u) such
(U)
that g.x0 E U and hence f(y) = f(xo)
Moreover for x E X we have
n
V
_
Vy c S.
:
n
_
- (f) = n kEl Tokf(x) = n (kEl Ak) f(x)
and this sequence converges to v(f) = f(xo) # 0. This implies that A= 1 and f(x) = f(xo). We also remark that the assumption (i) implies (ii).
In the sequel m will be the "Cauchy law" on X, that is the invariant probability measure on X under the action of S 0(2) and we have already
de-lm ,-.
used in 11.4.1 the relation
jjxjj2
dm
llgxll
2
We denote by u the image measure of u by g -} g
v
1
and Tt stands for the
v
operator Tt associated to the measure p. If V is a subspace of C(X), we denote by Vm the subspace of M(X) defined by Vm = {fm /f c V} (Remark that f -+ fm is one to one from C(X) to M(X)).
PROPOSITION 2.5
Assume that Jllgll2 du(g) < + -, then :
(i) (Ttf)m = TZ-t (fm), t c [0,2J, f e C(X)
(ii) Vk E IN , VA C ¢ , Vt C [0,2] v
[Ker(Tt-XI) ] km
.
:
Ker(TZ-t - XI)k
244
Proof
:
Let f,h be in C(X)
:
2-t
rr
CT2-t(fm)]
(h) =
f(x)(
h(g.x) du(g) dm(x)
)
JJ
IIgxII
IJ f(x) h(g-1.x)(
t dg-lm(x) du (g)
)
IIg 1x11
111
h(x)(L-)
JJ f(g.x)
t
dm(x) du(g)
IIgxII
=
[(Ttf) m] (h)
The assertion (ii) is a simple consequence of (i) and we may remark that (i) implies that TZm = M.
PROPOSITION 2.6
Assume tnat u has a density with respect to the Haar
measure on G and that J I I g II
on C(X) for t e r0,2] Proof
:
2
du(g)
Tt
is a compact operator
.
Let dg be an Haar measure on G and du = ¢ dg, then for u in G (
Ttf(u.x)
f(g.x)( ux
t
(gu 1) dg
)
IIgxII and
<+-. Then
:
ITtf(u.x) - Ttf(x)I
IIfII J
(II
IIgxII Thus if
)t
II
I
dg
IIxII
is continuous with a compact support the result follows from
Ascoli's lemma. In the general case, there exists a sequence 1n of continuous function with compact support, which converges to $(g)IIgIIt in L1(u). Taking
n(g) = IIgII-t Tn(g) and denoting by nTt the operator associated to
un = n dg, we obtain
IInTt - TtII
.
J
IIgIIt - Yn(g)I dg
and Tt is compact as limit in norm of compact operators.
REMARK 2.7 pa
a is said "spread out" on G, if there exists n such that
is not orthogonal to the Haar measure on G. In such a situation
245
To is a quasi compact operator (See A. Brunel and D. Revuz [6] for more details). Hence To has a finite number of eigenvalues of modulus one and the remainder of its spectrum is contained in a disc of radius strictly less than one.
Looking at 2.4 and 2.3 this is enough to use the perturbation theory for To (If u is pread out, then v is unique from A.II.4.3). Unfortunately it's not true in general that
Tt (t > 0) has to be
quasi-compact, when p is spread out and since we need the spectral properties of T2 we have to assume the existence of a density for p.
COROLLARY 2.8
Assume that
1g112 du (g) < + - and that p has a density 1
with respect to the Haar measure on G. Then for k c IN, A E 0 \101 we have [Ker(vTt
- XI) k m = Ker(TZ-t -
(i)
vTt
(ii) dim(Ker
- XI)
k
AI)k
= dim Ker(T2-t - AI)
k
(iii) The unique p invariant probability measure v has a continuous
density p on x with respect to m and T2 p = p Proof
:
We apply 2.5 (ii) to the values t and 2-t replacing p by N for
the latter. Looking at 2.6 we have only to use the fact that for a compact operator T, the characteristic spaces have the same (finite)
dimension for T and T* and that the map f -> fm is one to one. If u has a density, then G
= G thus v is unique and (i) applied to t = 2 implies
that v = pm with T2p = p.
THEOREM 2.9
Assume that
jjgjI2 du (g) < + = and that u has a density
with respect to the Haar measure on G. Then there exist
unique p,
(u),
invariant probability measures v, (v), with continuous densities p,(p)with
respect to m, and operators
Q. and Q2
of spectral radius strictly
less than one, such that
Tm(f) = v(f) 11 + Qf(f) TZ(f) = m(f) p + QZ(f)
Moreover the spectral radius r(t) of Tt is such that r(o) = r(2) = 1
and r(t) < 1 for t c ]0,2[. Proof
:
To is compact and u has a unique invariant probability measure
246
v thus by 2.4, except the eigenvalue 1, the spectrum of To is contained in a disc of radius strictly less than one, and we obtain the first v
decomposition. This result is also true for T
0
and by 2.8 the spectrum
of T2 has the same structure. We already know that r(0) = r(2) = 1 and Holder's inequality implies that the function t --> Log r(t) is convex
on [0,2]. From 2.2 this function is strictly negative at a point in 0,2[ and hence is strictly negative on ]0,2[.
(i) Most of the results of this section are true in a much
REMARK 2.10
more general setting. Actually it's possible to traduce all of them to the case of the action of a locally compact group 0 acting continuously on a compact metric space X. If we replace
Log
11x11
by an additive
llgxll cocycle a introduced in A.I.3.2, the Laplace transform is given by
Tt f(x) =
ot(g,x)
dp(g)
J
See for instance [41] and chapter IV. (ii) In view of A.V.2.5 the operator To acting on
possesses the
same decomposition as in theorem 2.9, without any density assumption on p. By perturbation theory such a decomposition holds for Tt if t is small enough (A.V.4.3). But we need a decomposition of T2 and this is the reason why we don't use the Holder space Za and work with C(X).
Exercise 2.11
It's not always true that v has X as support, even when
p has a density. (Think of p supported by the matrices of SL(2,1R) with non negative entries, then v is supported by the invariant set X+ = {x / x = (UV) / uv >, 0}). Suppose that p is supported by the
matrices of the form L1
where t is a real random variable such
0]
that the topological support
of the law of t contains an open set of
-2, +2[. We know from the proof of 11.4.1 that v is unique. Prove that the support of v is equal to X.
Exercise 2.12
Without any assumption on p prove that T1 is a bounded
operator of L2(m) and is self adjoint if p = u Denoting by y the limit of the subadditive sequence JLog
llgll
n
j
Log Ilgll dpn(g) (assuming that
dp(g) < + m) prove that the logarithm of the spectral radius
of T1 acting on L2(m) is greater than -y.
247
Hint
1 Log II Tn I n 1
I
> 1n Log f
IIJl du
n
- 1n 1 Log I I g II dun (g)
(g) dm(x)
IIgxII
As a consequence if we know that this spectral radius is strictly less is not amenable, see Y. Guivarch
than one (this is the case when G u
and Y. Derrienic [171) then y is strictly positive.
Suppose that u is supported by the matrices
Exercise 2.13
where t is a real random variable distributed
Lt
1 0J according to the Cauchy
v
law Ci6 with
> 0. Compute the eigenfunctions p and 'v of T2 and T2
with m(p) = m(p) = 1.
111.3 The pure point spectrum, second proof.
In order to obtain the pure point spectrum property in the general independent equidistributed case,we apply the results of the preceding section together with the criterion given in 1.5.1.
We denote by uA the law of the matrix
Y =
ralba
ll
1
Tt,A the operator
Tt associated to the law u, and by v. a ua invariant probability measure.
PROPOSITION 3.1
Assume that IE (a2+b2) < + -. Let J be a compact subset
of IR and suppose that for A e J, the law of a - bA has a density T with respect to the Lebesgue measure of ]R such that A -> TX is continuous from J to LI(IR) . Then there exists positive constants AJ,
BJ, CJ with AJ < 1 and such that
(i)
(ii)
IITI X11 I
5 BJ AJ
I T2A II
Proof
:
CJ
,
,
n e IN, A e J
n e IN, X e J
An easy Jacobian computation shows that u3 has a density
with respect to the Haar measure
of G and that A ->
X is continuous
from J to L1(dg). Then following the lines of 2.6 it's readily seen that
A -> Tt
1
is continuous from J to the space of bounded operators
on C(X) for t e [0,2]. We now observe that it's enough to prove (i) and
(ii) for Tt
rather for Tt'A since X
IITt,AII : J IIgIIt dua = IE([2 + (a-Ab)23t/2)
248
Looking at 2.9 we know that the spectral radius of T3
is strictly
less than one and that T2 X(f) = m(f) pX + Qf(f) where pA is the
continuous density with respect to m of v , and the spectral radius of QX is strictly less than one.
The continuous perturbation theory of compact operators (see Kato [31]) asserts that the maping A --> QX is continuous from J to the space of bounded operators on C(X) and A --- pA is continuous from J to C(X). The
conclusion of 3.1 now follows from the next lemma
:
LEMMA 3.2 Let A -TX be a continuous mapping from a compact set J to the spaceof bounded operators of a Banach space, such that for
each A e J the
spectral radius of TA is strictly less than 1. Then there exists AJ <1 and no, such that for Proof
:
n >, no
,
A e J,
IIT'II
AJ
The Logarithm of the spectral radius r(A) of Tx is an
uppersemicontinuous function which attains its maximum value f on the compact set J, thus t < 0. Furthermore i = Sup(Z, Log r(A)) is the pointwise limit of the decreasing sequence Sup(f, inf k LogIIT,'II) of
1fk,
lemma.
THEOREM 3.3
Let H be the Schrodinger operator associated to an
independent
identically distributed process with values in 1R2
such
that the variable (b) is strictly positive, bounded away from zero and IE(a2 + b2)
< + -
.
Let I be an open subset of 1R with a compact closure
J such that for A c J the law of a - Ab has a density TX with respect to the Lebesgue measure on 1R and A -> TX is continuous from J to
L1(IR). Then for almost all a the spectrum of H(s) is pure point on I (if any) and the eigenfunctions fall off exponentially fast.
Proof : Let in, n be two integers and A = [M,N] a box with M <m
YM
Yn-1 Yn-2
Ym
YN YN-1 " ' Yn From the formula 1.4.4 we can write
:
249
(I)) s n J (J
nl
IE ("a
II Z3 xmI
11 Z1 z2 z3
1
(
where x°° =
IIZ2 Z32x-U
) as
11
x-
p
)
Integrating with respect to Z1 we obtain that the above expectation is less than
1
I
:
11Z3 x-11
I Z2 Z3 x
N-n+l 2,a
I
II
II
Integrating with respect to Z2 this latter expression is less than
n-m
11 T1
A
1
l
l
11 T2-a+111
Looking at the proposition 3.1 we obtain ti
lE from nI (I)) :
I"
AJn-ml
AJ < DJ < 1 we have for IP almost all w and a finite
Then for
FJ(w)
BJ CJ
:
am nI (I) < FJ(w)
DJIn-mI
The theorem I.5.1 gives the result since the integrability of b2 implies that
I bn12
is IP almost everywhere finite.
LEMMA 3.4
With the notations of 3.3 the law of a - Ab has a density PX
with
Y', continuous from J to L1(IR)
A --
following
in (at least) the two
cases :
(i) In the classical Schrddinger case (that is bn = 1
the Law of the random variable (a) has a density
Vn e 7L), when
on IR and J
is any closed interval.
(ii) In the Helmotz case when the law of the random variable (b) has a density on IR and J is any closed interval not containing a*.
Proof Let P be the density of (a) or (b) then the density 'Y, of a - Ab
is given by Pi(t) = 'Y(t+a) in the case M and T, (t) =
in the case (ii). Approximating P in L1(IR)
1
p(t +c )
A- A
A -A
by continuous functions with compact support,
the result is straightforward.
REMARKS 3.5 (i) In the classical Schr6dinger case we have obtained the pure point spectrum property assuming that the law of (a) has a density
250 and lE (a2) < + m . These hypothesis are slightly different from those of
corollary 1.3 in which it is assumed that the law of (a) has a bounded density.
(ii) It's easy to see that lemma 3.4 applies to more general situations,
in example if we suppose that (a) and (b) are independent random variables each law being absolutely continuous.
(iii) The proof given in 3.2 applies to random Schrodinger operators in a half axis (As well as the results of the section 1).
111.4 The density of states
We assume in this last section that (a ), n e 2Z, is an n independent identically distributed sequence of real random variables, that the law of (a) has a density and lE(a2) < +- . In this situation we
shall prove that the distribution of states introduced in 11.6 has a continuous density constructed from the invariant function of the operator T2 X defined in the sections 2 and 3.
Under the above hypothesis the distribution of states
PROPOSITION 4.1 of (HT)n = -Tn+l
- Pn-1 + Lebesgue measure given by
da
anPn has a density with respect to the
pX (x) dm(x) nn f pa (x)
where pX and pX are the unique invariant functions of T2,X and T2 X
with m(pX) = m(pX) = 1, and m is the Cauchy measure on X. Proof
:
Let 0 be a continuous function with compact support on 1R, A
the box [-N,N]. We know from 1.4.4 that
IE [% + al
k(O)
where xm = Ill
lE [T2,a
1I
.
(
2II
I
J
0(X) lim IE
N;+-
IIY°Y- 1'
(
Y-N x- 112 2) da
IIYNYN-1...Y-N xmll
The latter expectation is equal to
(\Yo/-1...Y-N .xm)] where L is the function f(x) = 1.
Looking at the decomposition of T2 given in 2.9 this latter expectation can be written
:
251
TO A pX (x.) + IE IQZ
A
I (Yo Y_1 ...
Y-N . xm)]
From the decomposition of To given in 2.9 we have To+X
:
pX (xm) = vA (pA) + Qo,A pX (x.) .
Taking in account that the spectral radius of Qo,X and Q2 , are strictly less than one we obtain
iii J $(a) vx(px) dA
From lemma 3.4 and the proof of 3.1 we know that A --+ pk and A -+ pX the norm topology of C(X) thus da is continuous.
are continuous in
In [43] we can find an other kind of formula for da
Exercise 4.2
namely if we denote by/ cos2(x) the continuous function on X defined by cos2(x) =
112
u +v
2
,
x =1 v \
then dk = II J cos2(x) po(x) po(x) dm(x)
Hint
:
Write k($) = IE [o0(O)] and follow the lines of the proof of 4.1.
Moreover using the invariance equation for vX it is readily seen that these two expressions of the density of states are identical.
Exercise 4.3
Using the result of exercise 2.13 compute da when the
law of (a) is the Cauchy law C.
and compare to exercise 11.6.12.
REMARK 4.4 The same computation of IE [oo(q)] can be performed in the
general case of an independent identically distributed process (a
n
but in this case we don't obtain the density of states since
k($) = IE [b0 GO(O)]
'
b n)
CHAPTER IV
SCHRODINGER OPERATORS IN A STRIP
Most of the results obtained for the one dimensional Schrodinger operator in the preceding chapters can be adapted to the case of the strip. Deterministic properties are easily generalized, replacing at time, the Poincare half plane by the generalized Siegel half plane. For random operators, the theory of Lyapunov exponents is more involved and we refer heavily to part A chapter IV for related results. In contrast with the one dimensional case, positivity of some exponents is not easily checked and needs more assumptions on the "disorder". As in chapter III we give two proofs of the pure point spectrum property. The first proof is a straightforward adaptation of Kotani's criterion. In order to treat the Helmotz case and to obtain some information about "thermodynamic limits" we need some Laplace analysis on symplectic groups. Such results, easily obtained on SL(2,7R) , require here much
more work since we have to deal with the Poisson kernels associated to the boundaries of symplectic groups.
IV.1 The deterministic Schrodinger operator in a strip
Let R be a positive integer (width of the strip) and , linear space of sequences `Y = (Y'n), n e 7L with `Yn a 0
.
be the
The Schrodinger
operator H is associated to two given sequences of real symmetric square
254
matrices of order R, A
n
(HT)n = BnI(-2n+1
and B
n
,
with B
n invertible, and acts on `,g by
- 2n-I + AnYn)
We say that H is,a classical Schrodinger operator when B
n
= I and A
is
n
a tridiagonal Jacobi matrix whose entries (i,j) with Ii-jl = 1 are
equal to -1. For a complex number A every solution of the difference equation HT = AY lies in a 2R dimensional subspace of o`Pj and
Tn = Pn 'o + Qn '-1 where Pn and Qn are solutions of the "matrix difference equation"
An M n = A Bn Mn with the initial - Mn+1 + values Po = Q-1 = I, P-1 = Qo = 0. A solution 4' of the difference -Mn-1
equation is constructed from initial values 'Yo and `Y-1 by a product Sn
of "transfer 'matrices" Y
Y
n
in - A Bn
-1]
I
0
n
with
=
in the same way as in chapter I. We remark that the transfer matrices Yn and therefore the products Sn belong to the symplectic group SP(t,¢) of( 2R b
where J =1 I
2R matrices g with complex entries, such that g'J g = J
0j and g' is the transpose of g (Remark that
SP(1,¢) = SL(2,¢)). As explained in A.IV we can consider the exterior powers Apg acting on AP(¢22.) and we have the following generalization of Osseledec's theorem [42]
PROPOSITION 1.1
Let (gn), n e IN be a sequence of matrices of SP(t,¢)
such that (i)
lim n++m
LogjAP(gn gn-1 ... gl)II = Y1 + ... + YP
n p
1
(ii) lim
n++ n Let al > A2 >
Log IIgnII
R
= 0
the strictly decreasing sequence of distinct
-> Ar
-y1) . Then there , .... Yk, -Yk , - yQ-1 exists a strictly increasing sequence {0} = Vr+l C Vr C ... C V1 =
elements of (yl,
Y21
,
.
2R
such that :
(i) If x e Vi\ Vi+1
,
lim n Log lgngn-1
glxI! = ai , 1 . i S r
n,+m
(ii) dim Vi+l - dim Vi = number of elements of the sequence (y1,y2,.
,-yl
255
whicn are equal to ai).
Actually this theorem tell us that if yR is strictly positive,
the sequence
gl V 11 has I gn gn-1 an exponential behavior and is exponentially decreasing iff V belongs then for each non zero V
Q2R
in
to a R dimensional subspace the sequence g
of 0
2R
I
called the contractive subspace of
n.
We now turn our attention to the Hilbertian operator induced by H. It's easily seen, following the lines offp I, that H is essentially self
adjoint on the Hilbert space '= {Y'c Y5/ E Y'n Bn `Yn < + -} assuming
that Bn is a sequence of positive definite matrices such that the set of eigenvalues of the sequence B
n
is bounded away from zero. This
hypothesis will be in force throughout this chapter. We denote by
k c I1, ..
R] ,
n e 7l
(ek'n)m = 0 if m # n,
by a e2'm
(1,n),(j,m) ,
and by a
the orthogonal basis of 00 defined by
(ek'n)n = k th column of the matrix Bl. We denote
the spectral measure associated (n,m)
ek,n
to the vectors
the R xR matrix of measures a
(i,n),(j,m)'
ei,n
moreover
a is the sum of the traces of the matrices ao o and a-I -1. We remark that <'Y,ei,n> is the i th entry of `Yn and with the help of some
algebraic computation we obtain the following formulas where Pn,(i 1j)
is the entry of index (i,j) of the matrix P
n
thus a polynomial function
of A
PROPOSITION 1.2 R
(z)
E
k=1
ek,o + Qn'(1'k)(H) ek,-1) = ei,n (Pn,(i,k)(H)
ao,o (ii) am,n
a-1,o
P.
= (Pm'Qn)
a0,-1 a-1,-1
Qn
(iii) For each Y' in djthe measure a,Y,' is absolutely continuous with
respect to a. (iv) The spectrum
of H is the topological support of a and A is an
eigenvaZue of H iff a(A) > 0.
256
Following the lines of 1.3.2 we can prove
:
For a almost all real A there exists a generalized
PROPOSITION 1.3
eigenfunction P such that for each e > 0
:
lim
1
Inl-'+m in i
/2+e
= 0'
The spectral matrices o can be constructed from approximations m,n computed in boxes and as in 1.4 we introduce varying boundary conditions in order to obtain smooth approximations of a For a box A = {(k,n)/k e
m,n
[M,N]} and a real symmetric n matrix X we define the operator A Hx on the Hilbert space Qk of [11
..
,
, n e
9j
sequences P _ (Fn), n e [M,N] with the scalar product inherited from 1j by
(Hx
F) M = BM1(-FM+1 + AMIM)
( Hx Y')N = (H'Y)n
(^Hx p)N = B N
I
for M+1 < n < N-1
(-' N-1 + (AN-X) TO
Thus we can prove (with the notations introduced in 1.4)
:
PROPOSITION 1.4 (i)
a
m,n
Let X, m,n be fixed, then the sequence
n
ax
m,n
converges weakly to
when the box A goes to ZZ x{1, ..., R}.
{
(ii) For J m A # 0
C^PN+1
-1 QN+1 -
- X^PN]
where is the matrix <"Rx
X^QN]
ei,M ej,M>
(iii) If X and Y are real symmetric matrices of order Az with Y non
negative then (^PN+1 - (X + iY)'PN) is invertible for J mA>0 (Take care that actuaZZy^p N is a function of A
!).
We now want to compute the mean value of ox with respect to a A m,n probability measure on the symmetric matrices X analogous to the Cauchy measure in the one dimensional case. For this we recall some facts about the Poisson formula of the upper Siegel half plane (see [28]).
257
For k = 0,1,
..
k, we set Sk = {X + iY/X and Y are real symmetric
,
matrices of order Z with Y non negative and of rank k}. Then there exists a family P(Z,dX) of probability measures on So
,
indexed by
Z e L) Sk called the "Poisson kernel". If f is an harmonic function k=1 (see [281 for the definition) in each Sk, k = 1, ..., R , which can be continuously extended to S
0
U {W} we have the representation theorem R
f(Z) = 1
f(X) P(Z,dX)
,
Z E U Sk k=1
PROPOSITION 1.5
The measure valuated matrix no
m ,n
=
I
1
wax
m,n
P(iI,dX)
nas a density with respect to the Lebesgue measure on 1R given by Ao
d om,n
= 1 Ap (At)-l R m
dl
where
AP'
n
A4 = ^pN+1 APN+1 + ^PN APN
AGMX then it's enough APn AUX,n APm to prove the above result for m = n = M. From proposition 1.4 we know
Proof
It's easily seen that
:
that the function LAPN+l
,
- ZN -1L^QN+l
- ZnQN] is analytic and hence
harmonic, on U Sk, and extends continuously to So U {W}. Thus from k=1
,t
the above discussion we obtain (d m l > 0 ) d
nti
j
(t)
=
_ LnPN+l - 1APN] EAQN+l - i^QN]
t
n
nP N+1
Taking in account that the matrix A P
N
that for each real A r
lim ' m
e-o since
QN+1
is symplectic, we see
AQN
:
d"am(t)
f t-(A+it)
=
(AA )
We remark that the matrix nt is invertible
:
PN+1 nPN
II
^QN+1
= YN YN-1 ...
A
QN
YM I0 L
0 II J
Following 1.5.1 we now establish a sufficient criterion implying
258
the pure point spectrum property for the operator H. Let S(A) be matrix a
of density of
0,o
a-1,ol
with respect to a. If A is an eigenvalue
Loo,-1 Q 1, of H the terms of S(A) are well defined and let T1,
..
,
'fr an
orthonormal basis of the associated eigenspace. We denote by To, (T-1) <ei,o,
the R x r matrix with entries
Ti,
,
(<e i,-1, Tj>)
It is readily
seen that ,T
S(A) =
1
(T', T'
I
To
GM T
l
The matrix( T
\
)
1
0
1
I
is of rank r thus S is also of rank r. Furthermore
1
from the relation S' J Sn = J and if we denote by Ln the 2R x r matrix
n
I1
,f2
+1
1+1
,
fr Tr+1I
we obtain Ln J Ln = Lo J Lo
.
Since the
n
n
eigenfunctions are going to zero at infinity, Lo ' J La = 0 and this
implies that the rank r of L
is no greater than R. Finally we obtain
0
that the rank of S(X) is equal to the rank r , R of the eigenspace associated to A and that each column of S(A) can be taken as initial values of an eigenvector.
THEOREM 1.6
Let I be an open subset of IR for which there exists two
positive constants C and p with p (m,n) E A, i,j e [I, ...,R
^ti I
(j,n)I
a
(I)
Then if EIIBnjj1/2 p1n1
C P
<
1, such that for each box A.
:
im-nj
< + -, the spectrum of H is pure point on I and
the associated eigenfunctions fall off exponentially at infinity. Proof
:
Following the lines of 1.5.1 we see that for a almost all A
each column of the matrix S(A) can be taken as initial values of an eigenvector. These column vectors can't be simultaneously zero (the trace of S(A) is equal to 1
a.a.e) Hence a is pure point and from the
preceding discussion we can assert
that all the eigenfunctions are
exponentially decaying at infinity.
Exercise 1.7
We can define the Wronskian of two sequences of R x R
259
matrices Un and Vn by Wn(U,V) = Un+1Vn - UnVn+1. If (U ) and (V ) are two solutions of the matrix difference equation, n n prove that W (U,V) is constant. n (A symplectic matrix g verifies g ' J g = g J g' = J and thus the
"second" wronskian Wn = Un+1Vn - UnVn+1 is also constant).
It's possible to define the resolvent of H by means of m
Exercise 1.8
functions, using the same method than in the one dimensional case. We define the matrices
N
NM(A) = 'NUn = <+ % eN, en> where n
N
or n . N. The sequences NU can be extended to the whole of 7L such n that these extensions are solutions of the difference equation and we verify that NUn = (NPn)(NM) + (NQn) , whereNP and NQ are solutions of the difference equation with boundary conditions N _ N N N +QN-1 +PN PN -QN+1 = I N +1'N-1
_ N +QN
_ N
__ N
-1'N+1
_
-QN -
Let N and n be two integers and define the sequence V Vm = (NU + - NU) + m )(NU') ' W 1(NU,
, m 3 n
, w 1(NU, NU)
, m : n
-n
Vm = (N U
m)(NU)
m
by
= Vm
IV.2 Ergodic Schrodinger operators in a strip
We now suppose that the sequence (An,Bn) is defined on some ergodic
dynamical system (13, 2.,6,IP) such that (An+1' Bn+1) =(An - 0 , Bnc B). Furthermorewe assume that the law of the variable (B) is supported by a set of non negative symmetric matrices whose eigenvalues are uniformly
bounded away from zero and that IELog(1+llAll +IIBII) < + - . The Lyapunov exponents 11(A), ..., yR(A) are defined for a complex number A as limits of the subadditive sequences
:
260
Y
+ ... + Y(X) = lim 1 IE Log IIAPS n II p
l1
n->±W
InI
p ; 2.
for 1
(The limits for n = t- are equal since IISnII = IISn1II)
The subadditive ergodic theorem implies IP almost everywhere convergence
of 1 LogjiApSnII to 11(A) + ... yp(A) (for a fixed A). Moreover the I nnl
integrability condition IE Lo5(1 + II All + IIBII) implies the IP almost everywhere convergence of 1 LogHIY n to zero. Thus for each fixed A, II
In j
Osseledec's theorem 1.1 applies IP
a.e..Looking at the proof of 11.3.1
we see that the sums yl(X) + ... + y (A) are subharmonic, but except p
for y1 we can't claim that the Lyapunov exponents are subharmonic. The measurability property 11.2.1 still holds and we obtain the Pastur's lemma about eigenvalues of H looking at the proof of 11.2.2. The only change is that now, the trace of the spectral projector E. on the eigenspace associated to the eigenvalue A is no greater than R. Then the Ishii's, Pastur's theorem becomes
THEOREM 2.1
:
Let A be a Borel subset of IR such that the Lyapunov
exponent y5(X) is strictly positive almost everywhere on A with respect to the Lebesgue measure. Then aa(A) = 0 ,IP.a.e. Proof
:
We define the measurable subset W of 0 x IR by
Jwe0 W
=
(w,a)
A t A, lim
I+ InI
I
LogflYn(w,A) II
=0
A is not an eigenvalue of H(w)
lim
InI- InI
LogIIAP Sn(w,X) II, = YI(A) +
for p = 1, ..
... +yp(a)
, Z
The proof is now the same than in 11.5.1 remarking that y5 > 0 implies that the two contractive subspaces V+ and V A is not an eigenvalue V+(1V
= {0). Thus
are 2 dimensional. Since
IR25
= V+ ® V
and each non
zero solution of HY = AT is growing with an exponential rate at least
in one direction of ZZ. REMARK 2.2
No converse of 2.1 is known for the strip. In the one
dimensional case Kotani's proof uses heavily the Thouless formula. Such
261
a formula has been proved in the strip by W. Craig, B. Simon [12] in the following form
:
Y1(A) + ... + Y(X) = k
Loglx-AJ dk(x) J
They assume Bn = I, Vn e 7l, and that A is a bounded matrix. But it
seems that no formula of this kind is available for the individual exponent Yf(A) since it is believed that this exponent is not subharmonic in general. A weaker form of Kotani's converse would be "Let A be a Borel subset of positive Lebesgue measure contained in {a/y1(a) = 0) then a
a
(A)
> 0,
IP
a.e.". Such a result certainly can be
proven using the above Thouless formula, the m functions introduced in exercise 1.8 and following the lines of [58].
Exercise 2.3
Ishii's proof of 2.1 (see exercise 11.5.6) extends to the
positive strip
:
(i) Prove that for 5m A > 0 Pn+1_
Pn+1)-1
+Ra e eo> =
-
n
(Pn
Qn+l
(ii) Prove the expected result, following the lines of 11.5.6
Exercise 2.4
Let Y be a real symmetric matrix of order i and +HY the
Schrodinger operator with boundary condition Y at the point zero. Prove
that for 3 m A i 0 and a box A= 10,N] Y (i) < R; eo , e o> _ - [PN+1 + QN+1 Y1 (ii)i+Ra e , eo> = M [I -YM]
(where M =
0M
1
QN+1
introduced in exercise 1.8)
(iii) Denoting by
+o Y
the trace of
+o Y
0 o prove that
+tio
= f
+oy
P(iI,dY)
has a bounded density with respect to the Lebesgue measure on IR. (Hint : M = U + iV with U and V symmetric, V positive definite).
(iv) Following the lines of exercise 11.5.7 prove the generalization of Kotani's result
:
"Let A be a Borel subset of IR such that 11(X) is strictly positive almost everywhere on A. Then for IP almost all w, for almost all Y (with respect to the natural Lebesgue measure R
d yi,3 on the
i>,j
symmetric matrices) the spectrum of +HY is pure point on A (if any) and the eigenfunctions are exponentially decaying".
262
Hint
:
P(iI, dY) is absolutely continuous with respect to II d y, l.J i%J
(see [28]).
We now adapt Kotani's criterion 11.7.1 to the strip
PROPOSITION 2.5
:
Let A be a Borel subset of IR and m a positive measure
on A. We assume that yt(A) > 0 m.a.e on A and that a+ are absolutely continuous on A with respect to m,IP a.e. Then for IP almost all w,
y1(X) is strictly positive for a(w) almost all A, the spectrum of H(w) is pure point on A (if any) and the eigenfunctions fall off exponentially with a rate given by a Lyapunov exponent.
Following the lines of 11.7.1 we obtain that for IP almost all w, for a(w) almost all A the contractive f dimensional subspaces V+ and V have a common non zero vector. But we don't know in which Osseledec's subspace of V+ and V
lies this common
vector, neither if this common
vector is unique. Furthermore given such a common vector the rate of decay of the associated eigenfunction has no reason to be the same in the two directions of 2Z
!
From a physical point of view it seem's
natural to think that this common vector has to be unique and that it does not belong to proper Osseledec's subspaces of V+ and V-. Thus the rate of decay would be given by -yk(A) in the two directions of 2Z, but we don't know any proof of this ....
IV.3 Lyapunov exponents in the independent case. The pure point spectrum (first proof).
As seen in section 2, positivity of yR is a crucial hypothesis to obtain absence of absolutely continuous spectrum or pure point spectrum.
In contrast with the one dimensional case the only known proofs of this positivity require strong assumptions on the law of (A,B). In this section we suppose that (An,Brl) is an i.i.d. sequence of random
variables. In order to avoid too complicated proofs, we also assume that (A) is a Jacobi matrix, (B) a diagonal matrix such that the I l of (A,B) are independent random variables with random entries (a,b) l
263
values in uR2 . We say that the "Helmotz case" occurs when, for some
such that
index i, there exists a constant cl and a a value ai
ai -A *i
b
i
=c
i
Moreover we suppose that each (b1) is positive, bounded away from zero
and that lE Log (1 + IIAII + IIBII) < +m . We denote by ua the law of the 0J1
-aB LAI
matrix
L
Assume that no (al,bl) is supported by a single point
PROPOSITION 3.1
(Assuming A ¢ X
of IR2 . Then y1(A) > 0
Proof
:
if the Helmotz case occurs).
Following 11.4.1 we see that the subgroup G. of SP(f,IR)
generated by the support of u, contains matrices of the form u = 10 v = LA
0]
I]
where 4 can be chosen diagonal with no nul diagonal entries.
El
projective subspace spanned by the I first 1' (E2 ) the 22 2R x we of IR (last) basis vectors of IR Then for a vector x 11
We denote by
ll
jx +n4x \
obtain unx
I` x2
2J
x
vnx =
x2
X1 2
+ n d xl
If x e E 1 then If x ¢ E1 then u
n
-
f 6x21
. x converges to ` O /which belongs to E1. This
implies that each fixed probability measure on P(IR
2k)
under the action
of GA is supported by E1. The same is true for E2 and as a consequence GX leaves no fixed probability measure on P(IR21)
thus the result
follows from A.III.7.5.
We have thus obtained under broad conditions positivity of y1. As seen in section 2 we actually need positivity of yR and this is certainly the case if all the Lyapunov exponents are distinct. Unfortunately we have to work with contractive properties on the exterior powers of IR2R ua
,
of the semi group generated by the support of
as explained in A.IV. Such a property can hardly be directly
checked for Schrodinger matrices and we are led to prescribe stronger assumptions the law of (A,B) in order to make use of a more tractable criterion given in A.IV. Let F be the application of IRS'
EA(x) F(x)
L
I
in G = SP(I,IR) defined by
-I 0
where A(x) is the Jacobi matrix whose diagonal
elements are given by the coordinates of the vector x. In the one
264
dimensional case (1 = 1) a straightforward computation proves that the F(x3) F(x2) F(x1) of IR3 in G is regular at
application
with x2 40 (d is then of rank 3, which is the
each point of IR
dimension When I
of the Lie algebra L(. of G).
2, such a computation becomes tremendous thus we have to work
>,
in a more intrinsic way. This is what is done in [40] and we only give here a sketch of the proofs, for readers familiar with Lie algebras.
Let U be an open set in IRQ, then the closed subgroup GU
LEMMA 3.2
generated by Proof
:
F(U) is equal to G.
As in the proof of 3.1 we see that GU contains matrices of the
form u = 1I
and v = L I
IJ
where A is a diagonal matrix whose
diagonalsr runs through an open set of IRQ. We can write u = exp 10
01
and thus by varying A we can construct a family of
v = explA 00]
(J
elements of the Lie algebra U of C. For R.= 1 this family generates "]gas a Lie algebra and the proof is complete. For R. > 1 we have to verify that conjugates of these elements with respect to some matrices
of F(U) generate { . Let U be an open set in IR(X1, ..
LEMMA 3 . 3
elements of rank of
Then there exists (al,
the family (where gk
{Ad(g11) Xk
,
=
Ad(g11g21) Xk,
..
F(ak))
,
,
Xr) some non zero
an ) in U
such that the
0
:
Ad(g11g21,
..., gnl) Xk / k= 1, ...,r} 0
is equal to the dimension of Proof
:
Consider the set of subspaces of `
families for all the finite subsets (al,
..
spanned by the above ,
an) of U. It is easily
seen that a subspace of maximum dimension in this set is invariant under Ad g-1 for g in
F(U)
and hence is an ideal of e{, from 3.2. 1 i
a simple Lie algebra thus this maximal subspace must be equal to
For each open set U in IR1
LEMMA 3.4 set (al, (D(xl'x2' (al,
..
there exists an integer no and a
..., an ) in U such that the map ..
,
, an)
xR ) = F(xn) ... F(x2) F(x1) is a submersion near 0
0
0
Proof
:
Choose as (X,,
10 ..
XR)
matrices of the form
0
A.1 pJ where
265
A. has zero entries except the term (i,i) which is equal to one. Let (al,
an ) be the subset of U given by lemma 3.3. For t cIRR and
..
o
R
F(a+t) = (exp T) F(a) and thus
t.X. we obtain
T = I
i 1
i=1
5(a1 +tl, ..
,
+ to
an 0
(exp Tn ) F (an ) ... exp(T1) F (a1) 0
0
0
The result follows from 3.3 and the computation of the differential of S at the point (al,
..
,
an ). 0
REMARK 3.5
It's certainly possible to choose explicitely (al,
and to see that n
= 2R +1, but the computation is tremendous
o
2R+1 is obvious since dim
lower bound n
,
!
an
(The °
= R(2R+1).
We now formulate the expected results of this algebraic manipulation
THEOREM 3.6
(i) Suppose that for i = 1,
..
,
R the supports of the
laws of (a.-Xb.) contain an open set of IR, then there exists an integer no such that the support of ubo contains an open set of SP (9.,IR).
(ii) Suppose that for i = 1,
..
,
R the laws of (a. -abi) have a n
density on iR, then there exists an integer no such that }1X° has a density with respect to be Haar measure of SP(R,IR). Proof
:
Let U. be an open set contained in the support of the law of i
no
R
(ai -Xbi) and U = R
Ui then by 3.3, S(U
) contains an open set.
i=1
Furthermore t is an analytical mapping and thus its Jacobian function is an analytical function which is not always zero. This implies that the zeroes of this function are contained in a countable union of lower In0
dimensional subspaces of IR
,
hence negligible with respect to the
Lebesgue measure.
COROLLARY 3 . 7
Suppose that for i =1, ..
,
z the supports of the Law of
(ai -abi) contain an open set of IR or that the Law of (ai - abi) have
a density with respect to the Lebesgue of al, then y9(A) > 0. Proof
:
direct application of 3.6 and A.IV.3.5.
Before proving the main theorem of this section we give a useful
266
deterministic lemma. If X is a symmetric real matrix of order n then each spectral measure is a function of the
n(2+1) entries of X. We
denote by ak the spectral measure associated to the vector ek of the orthonormal basis (e1,
..
,
en) and xk the diagonal term of order (k,k)
of X.
With the above notations
LEMMA 3.8
J ak(dt) dxk = II dt
Proof
It's possible to suppose k = 1, then X can be written
:
xI
X =
u'1
where u is a (n-1) dimensional vector. For m a> 0 we set
u
R
1
rT
P = (X -XI)-1el and from the decomposition P =1
1
we obtain
xIT1 + u' = A Y1 + 1 ti
T1u+X = a0
Let h = (X -XI)u ,
then
4 = -PIh and P1 = (x1 -u'h A)1 r
Taking in account that u'h =
au(dt)
J
rr aI(dt)dx1
Jm
=3M
t-1
THEOREM 3.9
f
we see that
3m(u'h) > 0, hence:
t-A
(x1 -u'h -x)- dx1 = II 1
Suppose that each random variable (a ), i = 1, i
..
,
1,
possesses a bounded density with respect to the Lebesgue measure. Then if H is the classical independent Schrodinger operator, the Lyapunov exponent yz(A) is strictly positive VA E IR. Moreover for IF almost all a the spectrum of H(w) is pure point and the eigenfunctions fall off exponentially with a rate given by a Lyapunov exponent. Proof
:
Looking at 3.7 and 2.5 we have only to prove that a+ are
absolutely continuous with respect to a fixed measure.
We choose a generated by
(An)n>l'and
box A = 1-N,N], denoting by IE(i,n)
random variable a
IE
(i,n)
(
i, n
we obtain from 3.8
a(i,n)(i,n))
II
a-_ generated by (An)n<-2. In a
the expectation with respect to the :
n E (0,-1), ie(1, ..., z)
267
where
i
i
is the density of the law of (a ). Then letting A going to 7L
we obtain that IEi,n (a(i,n)(i,n)) has a bounded density and thus the same is true for a+.
REMARK 3.10
(i) It's also possible to obtain absolute continuity of a+ using the m functions of exercise 1.8 and following the lines of 111.1.2.
(ii) The proof given here excludes the case of random Bn. In this case the operator H restricted to a box has off-diagonal disorder and lemma 3.8 can't be used.
(iii) It has been announced that all the Lyapunov exponents are distinct under the hypothesis of proposition 3.1 [23]. This result seems natural but we don't know any proof ....
IV.4 The Laplace transform on SP(f,1R)
Most of the results established in 111.2 extend to the case of a locally compact metric group G acting on a metric compact G-space B (see A.I.3). In the sequel, a cocycle on (G,B) means a multiplicative
cocycle (that is the exponential of an additive cocycle). We suppose that there exists a continuous map s from B to G and a fixed b
0
in B
such that s(b) . bo = b for all b in B, thus a sequence bn converges to
b in B iff there exists a sequence gn in G such that gn converges to the identity of G and bn = gn - b. Let p be a probability measure on G,
p a multiplicative cocycle, we define an operator on C(B) the Banach space of complex continuous functions on B by
Tpf(b) = J
p(g,b) f(g`b) dp(g)
We say that p is u integrable if is a bounded operator on C(B).
,
:
f c C(B).
sup p(g,b) e L1(p). In this case Tp bcB
The next propositions are identical to 111.2.2, 111.2.6 and the proofs are the same.
268
Assume that
PROPOSITION 4.1
(i) There exists a strictly positive t such that pt is p integrable. (ii) p has a unique invariant probability measure v on B. (iii) y = -JJ Log p(g,b) dp(g) dv(b) > 0.
Then there exists a strictly positive number to such that
the spectral
radius of the operator T associated to pt is strictly less than 1 for
t E ] O, t0] .
Assume that
PROPOSITION 4.2
(i) p is u integrable.
(ii) p has a density with respect to an Haar measure on G. Then Tp is a compact operator on C(B).
Suppose that there exists a probability measure m on B such that g- lm has the same null sets as m and that r(g,b) =
dg
1
m (b)
is a
dm continuous function of (g,b). Then r is a multiplicative cocycle called a Radon Nikodym cocycle (R.N. cocycle) and we can prove exactly in the same way as in 111.2.5 and 111.2.8.
Assume that p is u integrable and r is a R.N. cocycle
PROPOSITION 4.3 such that
rp-1
:
is p integrable. If p has a density on G with respect to
an Haar measure, then fork e IN and X c ¢ \{0} (i) (Ker(T
- XI)k)m = Ker(T P
rp 1 -
:
XI)k
(ii) dim(Ker(Tp - XI)k) = dim(Ker(Trp-1 - XI)k)
COROLLARY 4.4
Together with the hypothesis of 4.3 assume that the
sequence II'rPII
is bounded. Then the sequence
IITn _lII
has the same
rp
property.
Proof
:
It is readily seen from the Jordan form of a finite dimensional
matrix that for a compact operator T the sequence IITnII
is bounded iff
the spectral radius of T is no greater than one and for each eigenvalue A of modulus
1 one have Ker(T-AI) = Ker(T-AI)2. Thus we have only to
use (ii) of 4.3.
269
We now apply these results to the case of the action of G = SP(1,IR)
on some boundaries B . For k e (1,
..
,
1) a non zero
k-vector x = u1 A u2 A ... A uk of Ak(IR21) is said Lagrangian if u .
i
J u .
= 0 where i,j e
..
( 1 ,
,
k) and J is the element
of G.
I
0I
0
If x e Ak (IR2R) we denote by x its image in the projective space P(Ak(IR2R)) and the Lagrangian manifold Lk is the compact part of the latter projective space associated to Lagrangian k-vectors. We say that x c Lk is contained in y c Lk+l if there exists z e IR2R
such that
x = y A z. As seen in A.IV,G acts on Lk by g
uI A u2 A ... A uk = g ul A g u2 ... A g uk and if (e1,
...
,
e21) is
the usual basis of IR2R, Lk = G. e A ... A ek. I
We only consider two boundaries of G, namely X1 = LR-1 and X1,2 the partial "flag-manifold" defined by
1,2 = {(xl,x2) / xI e LR-1
X1
,
:
x2 e L1
,
xlC- x2}
We We define on (G,Lk) the multiplicative cocycle Pk(g,x) _
11Agxll where the norm is taken in
Ak(B229, ). Actually if we take
x = uI A ... A uk and if we denote by m the 21 x k matrix associated to the coordinates of ul, 11x112
..
,
uk in the usual basis of IR21 then
= det(M'M).
There exist
on X1 and X1,2 two probability measures ml and m12 ,
with the invariance property
kmi
= m1, km ,2= m1,2 for k in the
(K is a maximal compact subgroup of G).
subgroup K = SO(21) fl G
Furthermore the associated Radon Nikodym cocycles can be computed, using a method explained by H. Furstenberg and Tzkoni in [22]. We give below a sketch of the proof (see exercise 4.7, 4.8 or [41] for a detailed computation).
PROPOSITION 4.5 r1(g,xl) _
dg 1m1
(P1-1(g,xl))1+2
(x I) =
dm dg
-1
m 1. 2
r1,2(g,(x1,x2)) _ dm Proof
:
(x1,x2) = (p1-1(g,xl))1(P1(g,x2))2
1,2
The cocycles rI and r1,2 are K invariant (that is
r1(k,xI) = r1 2(k, (xlx2)) = 1
of the form r1 = (pt-1)a
,
Vk E K) and thus they must be (see [22])
r2 = (PR-1) S(P1)1 where a,s,y are three real
270
numbers we ought to compute. In this purpose we choose a point b0 in B bo. Then if R is a
(B = X1 or B = X1,2 ) and go in G with
positive measure on B with the same null sets as m, we can write
dg dm
m = dg m dg-IR
dg-1R .
dR
dm
dR
Constructing a parametrical representation of B and choosing R as the image of the Lebesgue measure on the parameters, the computation of - 1R dg0
(bo) reduces to a Jacobian computation. This is enough to obtain dR
a, and we have to use two pairs (oo'bo) in order to obtain y and 6. "
The interest of this computation lies in the fact that in our case -1
the important coc ycle is B = PR-1(pR) and from 4.5 we obtain 2 -1 e = r1 2(r1) The following theorem summarizes the essential results .
needed in the following section. We denote by Tt the operator associated to at on C(X1,2 ) and by r(t) the spectral radius of Tt.
THEOREM 4.6
Assume that a has a density with respect to the Haar
measure on SP (R,IR)and that f
IIgIIR(R+1)
dp(g) < + - . Then
r(0) = r(2) = 1, r(1) < 1, and the sequence IIT2II
is bounded.
Furthermore assume that there exists a map A -> uX such that A - Tt X is continuous from IR to the space of bounded operators on C(X1,2 ) for t c [0,2]. Then for a compact subset J of IR there exist constants AJ
(i) IITi
,
BJ
II
(ii) I I TZ jJ
,
positive
CJ with AJ < 1 and
BJ AJ
,
n e IN
,
AeJ
CJ
,
n e IN
,
AeJ
Moreover (i) and (ii) are true if we only suppose that there exists an n
integer no such that uA° has a density with respect to the Haar measure
on SP(R,]R) Proof
:
From corollary 4.4 we know that the sequence IITTII is bounded.
We may consider that Tr
n seen that the-norm of Tr
is an operator on C(X1 2) and it is readily is the same on C(X1) and C(X1,2 ). Thus applying
again corollary 4.4 welobtain that the sequence IIT2II is bounded (taking in account that
e2
= r1 2(r1)-1). From 4.3 (ii), the spectral
271
radius of T2 is equal to the spectral radius of Tr11 acting on C(X1,2)
and hence to the spectral radius of Tr
on C(X1) thank's to the above
remark. Applying again 4.3 (ii) this spectral radius is equal to 1 and thus r(o) = r(2) = 1. We know from A.IV. 4.2 . that there exists a
unique p invariant probability measure v112 on X1 2 and taking in account that 0 = pi-1(p,)-1 we obtain yR = -JJLog'0 dv1,2 du. Looking at A.IV we see that yQ > 0.
Proposition 4.1 implies that r(t) is strictly negative for t e 10,2 since we know that r(0) = r(2) = 1, that Log r(t) is a convex function on [0,2]
and that this function is strictly negative at some point of
] 0,2 [. If J is a compact set, conclusion (i) follows from lemma 111.3.2.
To obtain (ii) we write TZ'A = Pn + Qn , where PA is the projection operator on the sum of the eigenspaces associated to the eigenvalues of T2
A
of modulus one and QX is an operator of spectral radius strictly
less than one. From continuous perturbation theory (see Kato [31]) we know that A -> P, and A -} Q), are continuous. Looking at the proof of
corollary 4.4 we obtain that PX is a diagonalizable matrix, thus the sequence
II PaIl
is uniformly bounded on a compact set. Result (ii) now
follows from a new application of lemma 111.3.2 to the operator QA. The last assertion of the theorem is obvious.
Exercise 4.7
To a Lagrangian element x = uI A u2 A
... A uR-1 we
associate the 21 x f-1 matrix of coordinates of (ul, 2R
usual basis of IR
and this matrix has the form
.., u a
,
c
b
)
R
in the
where a and b '
d
are in IR9-1, c and d are square matrices of order 1-1. (i) Prove that the mapping II defined by 11(x) = (a'd
1,
b'd
1,
1
cd
-
d1-1
ab' d 1
is one to one from the elements of X1 with d invertible, onto the
Q (1-1) product IRR-1 x JR -1 x
is a symmetric
2
IR
(remark that the last coordinate of II
matrix of order 1-1). =
(ii) Let bo = eQ+2 A ... A elk and go matrix, diag A = (19 (iii)
11
..
,
(AO
A-1
0
/
where A is a diagonal
1, a) with a > 0. Prove that
go
b
0
= b
0
is well defined in a neighborhood U of b0 and for x e U we set
272
II(x) = (r',s',N). Prove that II(go.x) _ (r'6, s'6, 6N8) where 8 is the diagonal matrix of order f-1 defined by diag 6= (1, 1, (iv) Using the proof of 4.5, prove that rl = (pR-1)
x1 = ul A ... A uR-1 ..
,
,
1, a).
We associate to a pair of Lagrangian elements
Exercise 4.8
(ul,
..
R+2
,
x2 = xl A uR the matrix of coordinates of
uR) in the usual basis of IR21 of the form (b) where a and b
are square matrices of order R. When b is invertible we set t = (b-1)1,1 (i) Prove that the mapping II defined by
(II(x1),t) where II
II(x1,x2)
:
is defined in 4.7 is one to one from the elements of X1
1,2
R(1-1)
invertible onto the product IRR-1 xIRR-1 x IR (ii) Let bo = (e
Z+1
where A = diag(8, 1, (iii)
II
A ... A e21 ..
,
,
with b
x IR.
2
e1 A ... A e21) and g o
01 = I`(00
A
-1)
1, a) with a and 8 > 0. Prove that o .b 'o
0
= b
0
is well defined in a neighborhood U of b0,and for (x1,x2) e U
we set II(xl,x2) _ (r',s',N,t). Prove that
ti
_ (8r'6,
II(go.x1' go'x2)
1 s'8, ANA, 62 t) where 8 is the diagonal
matrix of order R-1 with d = diag(1,
..
1,a)
,
(iv) Prove, using 4.5, that r1,2(pR-1)R (p
R)2
IV.5 The pure point spectrum, second proof
If we want to apply the results of the preceding section together with criterion 1.6, we have to remark that the entries of the matrix of ti0
density of
%
can be constructed from the cocycle 0.
For a box A = LM,N7, n e A , we denote by "R 0
6- '/2 0
M
n
/CA/2 n
m < n < N
where ^A n = AP'n+l
A
P
n+1
the matrix
+ ^P'AP n n
.
of the form
If m,n are such that
we denote by Z3, Z2, Zl the products Z3 = YmYm-2' ...,YM
Z2 = YnYn-1 .. ' Ym+l
Z1 = YNYN-1 ... Yn+1. Furthermore if el, '
is the usual basis of 1R2R we set
ui = el A ... A ei-1 A ei+1 A ... A eR
..., e21
273
u = e1 A e2 A ... A eQ PROPOSITION 5.1
With the above notation we obtain A
E r^Pm('A)-1 APn)i
1,j
J
i,3
J
L
E
9,
(ui,u))
Rn
e(Z1'Z2Z3
e(Z1,Z2Z3 4Rm . (u.,u)) O(Z2,Z3 nRm . (uj,u))J Proof
:
If A, B, C are square matrices of order t with C = AB then t(tr A'A . tr B'B)1/2.
C,.
E
ij
1J
1/2
Choosing A = APm(AA)
/2
APn we obtain that the left
B = (AA)
,
member of the inequality in 5.1 is less than k(tr A
tr nAm(nA)-1) 1
An(nA)-1
/2
We now remark that
tr(AAn(AA)-1) = E 02(Z1 Z2 Z3 ARn , (ui,u)) i=1 The same is true for the index m and taking in account the cocycle property for B2
and
u)) = 1
B(Z2 Z3ARn, (ui,u)) = 6(Z3 ^Rm, (u J ai)1/2
we only have to use the inequality (E obtain
assume that the sequence (A
box A = [M,N] , M < m < n
:
Ibjl to
J
n' N
and a BoreZ subset I of IR we have _
^Q(i,m),(j,n)I(I) s n J IITI,a II I
_
IIT2,aII da
First integrating with respect to Z1 in 5.1, proposition 1.5
and the Schwarz inequality imply
that the above expectation is less
than 2 II
J
and
B n) is an independent process. Then for a
3
IE Z J
< E IaiI
J
Let Tt,X be the operator introduced in theorem 4.6
COROLLARY 5.2
Proof
(E b4)1/2
the announced result.
I T2 I
E
j
(u.u)) dX J
A new integration with respect to Z2 gives the result.
If the independent random process (An,Bn) has the structure
274
described at the beginning of the section 3, it's now clear that the following theorem holds
THEOREM 5.3
:
Let H be the Schrodinger operator associated to an
independent identically distributed process (A
Bn) with the structure n described in V.3. Let I be an open subset of IR with a compact closure '
J such that for A e J the law of a. - Ab. (i = 1,
..
,
1) has a density
Ti X with respect to the Lebesgue measure on IR and A ->
'F.
X is
k. continuous from J to L (IR). If we assume that for i = 1, .. 1 I1(1+1) + b1(1+1)) < + m then for almost all w the spectrum of ,
IE(Ia
H(w) is pure point on I (if any) and the eigenfunctions fall off exponentially fast. Proof
:
We only have to collect the results of 4.6, 3.6(ii) and 5.2. As
in the one dimensional case the hypothesis about T in general.
A
is easily checked
APPENDIX
The spectral theory of random Schrodinger operators still gives
rise to a lot of papers and some questions left without solution in the preceding chapters have received partial answers during the redaction of this book.
1
The one dimensional case
F. Ledrappier has given a simpler proof of proposition 11.3.3
avoiding utilization of the Thouless formula. This result now appears as a direct application of a general fact in ergodic theory (F. Ledrappier "Positivity of the exponent for stationnary sequences of matrices" preprint).
S. Kotani has pointed out that in corollary 111.1.2, it is possible to handle the Helmotz case assuming now that there exists two consecutive indexes n = no, no+l such that (i)
bn is bounded
(ii) for each A r A, the law of an Abn has a bounded density
that sup
"on, X II
such
<+
AEA The proof is the same as in 111.1.2 and Corollary 111.1.3 is obtained
under the same hypothesis. (S. Kotani, private communication).
In [61], B. Simon and M. Taylor prove that for the classical independant Schrodinger operator, for any a > 0 there exists a
Bernouilli distribution of the potential such that the distribution
275
276
function K(A) of the distribution of states is not Holder of order a. This shows that the result (ii) of 1.6.9 cannot be improved in general.
2. The strip
In their paper "Anderson localization for one and quasi one dimensional systems" (preprint) F. Delyon, Y. Levy, B. Souillard precise the theorem IV.3.9, proving that the eigenspaces of H(w) are IP
almost surely one-dimensional. (See also the discussion following proposition IV.2.5).
3. The multidimensional case
At the beginning of 1985 the first proofs of the Anderson's conjecture (see the introduction) appear almost simultaneously in three preprints
:
- J. Frolich, F. Martinelli, E. Scopolla and T. Spencer "Anderson localization for large disorder or low energy"
- F. Delyon, Y. Levy, B. Souillard "Anderson localization for multidimensional systems at large disorder or low energy" - B. Simon, M. Taylor, T. Wolff "Some rigorous results for the Anderson model"
In these three papers, the localization property is proved in any dimension for the classical independent Schrodinger operator, assuming the disorder "large enough" and with a bounded density. The Kotani's criterion and Frolich and Spencer estimates (replacing the positivity of Lyapunov exponents in this context) are the essential tools.
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