Substitution Dynamical Systems: Spectral Analysis 3540186921, 9783540186922

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Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann

1294

Martine Queffelec

Substitution Dynamical SystemsSpectral Analysis

Springer­Verlag Berlin Heidelberg New York London Paris Tokyo

Author

Martine Oueffelec Universite Paris XIII, Departernent de Mathematiques 93430 Villetaneuse, France

Mathematics Subject Classification (1980): 11 K28, D43 ISBN 3-540-18692-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18692-1 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

PREFACE

The subject of this book is the spectral analysis of substitution dynamical systems.

i have tried to bring the study up to date, and in view to deserve a large audience, I have specially detailed the background, giving sometimes elementary rather than short proofs. Substitution dynamical systems might be considered as a source of examples in ergodic theory and spectral theory. We present a rather different approach, consisting, given some substitution dynamical system, in a deep study (when possible) of its spectrum ­ which means spectrum of the unitary operator adjoint to this system. Many problems still subsist, in particular for substitution of non­constant length. I have tried to write this areount in English but I am aware of that I have not perfectly succeeded. I beg the indulgence of the persevering reader. Most of topics tackled in this book have been discussed in our "groupe de travail", mainly with

B. Host, J.F. Mt§la and F. Parreau. It is my

gn:atpleasure to thank them for their very efficient help and assistance. During the period of the work on my dissertation, I benefited from the visit of many specialists in the subject. For numerous useful conversations, I express my gratitude to M. Dekking and M. Keane who initiated us

to

substitutions, and also S. Ferenczi, T. Kamae,

J. Lacroix, M. Lemanczyk, P. Liardet, M. Mendes France, J. Peyriere, G. Rauzy, J.M. Strelcyn, J.P. Thouvenot ••. I have received many encouragements to write this book ; I hope their originators are not too much disappointed now T. Ramsey reread the first part of the manuscript and made useful remarks. I thank him very much, and I do not forget C. Simon for her typing.

Paris,

Juillet 1987.

INTRODUCTION

Our purpose is a complete and unified description of the spectrum of dynamical systems arising from substitutions of constant length (under mild hypotheses). The very attractive feature of this analysis is the link between several domains : combinatorics, ergodic theory and harmonic analysis of measures. The rather long story of these systems begins perhaps in 1906, with the construction by Thue [103J

of a sequence with certain non­repeti-

tion properties (rediscovered in 1921 by Morse [80J)

o 1 1 0 1 001 100 1 0 1 1 0 This sequence (called from now on the Morse sequence) can be obtained by an obvious iteration of the substitution else, as an infinite block product for any

0­1

deduced from

block

the sum of digits of

n

in the

e is the

±1

01 , 1

01 , or

01 x 01 x 01 x ••• , where

B, means : repeat

by exchanging 0

B

0

and

B

and then

1

Also, if

2­adic expansion , u

B x 01,

the block S2(n)

=

denotes

(un)

with

irrS (n) 2

Morse sequence.

The Morse sequence admits strictly ergodic

(=

minimal and uniquely

ergodic) orbit closure and a simple singular spectrum, as observed by Keane [54J. The various definitions of Morse sequence lead to various constructions of sequences, and thus dynamical systems ­ substitution sequences [40J, [45J, [48J, [67J ­ a class of

0­1

zed Morse sequences then

[35J,

[56J,

[12J, [18J,

[68J, [79J

I

[57J,

[86J

[90J

[19J, [37J, [78J then

[21J,

.•.

sequences introduced by Keane, called generali[54J

, admitting in turn extensions

[58J,

•.•

[69J, [7 OJ

[62J

­ q­multiplicative sequences, [14J then [65J,

I

q = (qn)

, qn

integer

2 •

VI

In this account, we restrict our attention to the first category of sequences, but in case of bijective substitution (chapter IX), we deal with particular G-Morse sequences and q-multiplicative sequences. Ergodic and topological properties of substitution dynamical systems have been extensively studied ; criterion for strict ergodicity ([19J,

[78J}

([19J , [91]),rational point spectrum ([19J, [67J r [68]), condition for presence of mixed spectrum ([19J) and various mixing pror

perties

entropy

([21J)

But except

0

are main investigations and results in these last years.

in some examples

([S3J

r

[48J .•. ) no descriptive spectral

analysis of the continuous part of the spectrum has been carried out. Indeed, not so many dynamical systems lend themselves to a comprehensive computation of spectral invariants. I mean,.mainly, maximal spectral type and spectral multiplicity (see [94J for a rather complete historical survey). Of course, transformations with purely discrete spectrum are quite well-known ([107J)

and in this case, spectrum is

simple. In the opposite direction, countable Lebesgue spectrum occurs in ergodic automorphisms of compact abelian groups as in

K-automor-

phisms (see [16J). A very important class of dynamical systems, with respect to spectral analysis, consists of Gauss dynamical systems. Guirsanov ([39J) proved a conjecture of Kolmogorov : the maximal spectral type of a Gauss dynamical system is equivalent to

eO , where

0

denotes the spectral measure of the process and its spectral multiplici ty has been shown by Vershik ([10SJ, [106J ), to be either 1 - with singular spectrum (see also

Whether finite multiplicity

[26J) - or infinite. Then arose the question of 2 (or

for Lebesgue spectrum) was

1

possible and the last results in multiplicity theory have been mostly constructions of suitable examples ; I just quote the last three im-

[94]

portant ones.: Robinson E.A. Jr. in

exhibits, for every m;::1

, a

measure - preserving transformation with singular spectrum and spectral multiplicity

[7l], [72J,

m. On the other hand, Mathew and N,adkarni in

construct for every

N

2 , a measure-preserving transformation with

Lebesgue spectrum of multiplicity

N

function). In the-

se examples, transformations are group extensions. Recently, M. Lemanczyk ([61J) obtained every even Lebesgue mUltiplicity. Turning back to substitution dynamical systems, we prove the following: for a substitution of length q over the alphabet A (or q-automaton [12J), the spectrum is generated by k:> IAj probability measures, strongly mixing with respect to the q-adic transformation on most examples, these measures are specific

T ; in

of Riesz

VII

products, which is not so surprising because of the self-similarity property inherent in this study.

(Note that such Riesz products play

a prominent part in distinguishing normal numbers to different bases [50J ; see also

0 1J,

[84J).

Earlier Ledrappier ([59]) and Y. Meyer ([77J I already realized classical Riesz products as maximal spectral type of some dynamical system • The generating measures of spectrum of some q-automaton are computable from a matrix of correlation measures, indeed a matrix Riesz product, whose rank gives rise to spectral multiplicity. For example, the continuous part of the Rudin-Shapiro dynamical system is Lebesgue with multiplicity 2, while, using mutual singularity of generalized Riesz products (analysed in chapter I), we get various singular spectra with multiplicity 1 or 2, as obtained by Kwiatkovski and Sikorski ([58J) (see also

[34J, [35J I.

For substitution of non-constant length. no

spectral description seems accessible at present but we state a recent characterization of eigenvalues established by Host ([40J) and list some problems. We have aimed for a self-contained text, accessible to non-specialists who are not familiar with the topic and its notations. For this reason, we have developed with all details, the properties of the main tools such that Riesz products, correlation measures, matrices of measures, non-negative matrices and even basic notions

of spectral theory of

unitary operators and dynamical systems, with examples or applications. More precisely, the text gets gradually more specialized, beginning in chapter I with generalities on the algebra trum

Mvr)

and its Gelfand spec-

We introduce generalized Riesz products and give a criterion

for mutual singularity. Chapter II is devoted to spectral analysis of unitary operators, where all fundamental definitions, notations and properties of spectral objects can be found. We prove the representation theorem and two versions of the spectral decomposition theorem. We restrict ourselves, in chapter III, to the unitary operator associated to some measure - preserving transformation and dedUce from the foregoing chapter, spectral characterization of ergodic and various mixing properties (strong, mild, weakl. As an application of D-ergodicity (ergodicity with respect to a group of translations

OOJ I,

we dis-

cuss spectral properties of some skew products over irrational rotation ([33J, [36] , [51J , [93J ).

VIII

In chapter IV, we investigate shift invariant subsets of shift space, such like orbit closure of sequences. Strict ergodicity can be read from the given sequence, if taking values in a finite alphabet. The correlation measure of some sequence - when unique - belongs to the spectral family; hence, from earlier results, we

derive spectral

properties of the sequence. We give a classical application to uniform distribution modulo 2n

(Van der Corput's lemma) and discuss results

around sets of recurrence ([5J, [8J , [29J ' [97J ) • From now on we are concerned with substitution sequences. All preViously quoted results regarding substitution dynamical systems are proved in chapters V - VI fied notations

, sometimes with a different point of view and uni-

(strict ergodicity, entropy 0, eigenvalues and mixing

properties). We are needing Perron-Frobenius' theorem and for sake of completness we give too a proof of it. Till the end of the account, the substitution is supposed to have constant length. We define in chapter VII the matrix of correlation measures

and show how to deduce maximal spectral type from it. Then we

prove elementary results about matrices of measures which shall be used later. In chapter VIII, we realize

as a matrix Riesz product and this

fact provides a quite simple way to compute it explicitely. Applying the .techniques immediately we treat the first examples : Morse sequence and Rudin-Shapiro sequence and a class of sequences, arising from commutative substitution,

(particular G-Morse sequences) admitting gene-

ralized Riesz products as generating measures.

An important class of substitutionsis studied in chapter IX without complete success. It would be interesting in this case to get a more precise estimate

of spectral multiplicity which is proved to be at

least 2 for substitution over a non-abelian group. Finally the main results, on spectral invariants in the general case are obtained in chapter X - XI, by using all the foregoing. We have to consider a bigger matrix of correlation measures involving occurences of pairs of given letters, instead of simple ones, which enjoys the fundamental strong mixing property, and provides maximal spectral type of the initial substitution. Spectral multiplicity can be read from the matrix

, as investigated

with Rudin-Shapiro sequence and some bijective substitution. We obtain in both cases a Lebesgue multiplicity equal to 2, while N-generalized

IX

Rudin-Shapiro sequences [J2] admit Lebesgue multiplicity

N¢(N)

[87J.

In an appendix, we suggest an extension to automatic sequences over a compact non-discrete alphabet. We give conditions ensuring strict ergodicity of the orbit closure. As explained above, we preferred to develop topics involving spectral properties of measures and for this reason, the reader will not find in this study a complete survey of substitutions. A lot of relevant contributions have been ignored or perhaps forgotten : we apologize to the mathematicians concerned.

CONTENTS

PREFACE INTRODUCTION CHAPTER I

THE ALGEBRA

M(T)

1. Basic definitions

1

2. Generalized characters

4

3. Generalized Riesz products 4. Idempotents in

5. Dirichlet measures CHAPTER II -

••••••••••.•.•.••••••

and decompositions of

M(r)

7

•.

11

....•.•.•••.••........•••••••

12

SPECTRAL THEORY OF UNITARY OPERATORS

15

1. Representation theorem of unitary operators

15

2. Simple spectrum

.•.......•.•.•..

20 22

••••.•...•••••

30

•••••••••..•••••••.•.•.•...•••..

3. Spectral decomposition theorems 4. Eigenvalues and discrete spectrum

CHAPTER III - SPECTRAL THEORY OF DYNAMICAL SYSTEMS 1. Ergodici ty

•••..•••.•.•••••.••••••••••.•.•••••••

2. Ergodic theorems 3. Purity laws and

...•.....•.•••.••.......•..••..

6. Example CHAPTER IV -

35

38

.•••...•.••••••••.

40

•.•••.• , ••..•.••••.••.•

44

.••.•.•••..•.•.••..•.•.••...••

47 55

D-ergodicity

4. Ergodic discrete systems 5. Mixing properties

35

•••.••.•••••••••••••••••••••••••.•••••••

DYNAMICAL SYSTEMS ASSOCIATED TO SEQUENCES

1. Topological dynamical systems

.••••.•••.••.•••••

60 60

2. Systems associated to sequences taking values in a finite set •••••••.•••••••••••••.••••••..•.••••• 69 •.•.•.•••..•••.

73

DYNAMICAL SYSTEMS ARISING FROM SUBSTITUTIONS .

87

3. Spectral properties of sequences CHAPTER V

1. Definitions and notations

..••..••••.•.••••.••••

2. Minimality of the topological system

•••••••••••

87 89

XII

90

3. s-matrix. Positive matrices .•••...•.....••....••.....•..••.•.

95 100

6. Complexity and topological entropy •••••••••••• 7. Recogni zabili ty property ..•.•...•...••••..••..

104 108

4. Unique ergodicity of the system 5. Matrices

CHAPTER VI

-

MJ1,

(X(s) ,T)

EIGENVALUES OF SUBSTITUTION DYNAMICAL SYSTEM. .•..•.•••.••.•••••.

110

2. Eigenvalues of an admissible substitution of constant length ....•.•....•.•••••....•••••.•..

116

3. Eigenvalues of an admissible substitution of non-constant length ..•.........•.••.••••...•..

125

1. s(X)-induced transformation

4. Discrete substitution of constant length

132

5. Open problems

136

•••...••.•.•••••••••••..••..•••••

CHAPTER VII - MATRICES OF MEASURES

144

1. Reduction to the matrix 2. Matrices of measures

144 151

3. Characters on a matrix of measures

156 160

CHAPTER VIII - MATRIX RIESZ PRODUCTS 1. E

160

as matrix Riesz product

2. Examples of maximal spectral type

. . • . . • • . . • . • • . • • • . • . • . • • .. 169

3. Commutative automata CHAPTER IX

-

.•.•••..••••. 164

BIJECTIVE AUTOMATA OR SUBSTITUTIONS

174

1. Structure of bijective substitution systems

175

2. Spectral study of bijective substitutions

180

CHAPTER X

MAXIMAL SPECTRAL TYPE OF GENERAL AUTOMATA

190

1. Coicidence matrix 2. Projection operator

190

P

•.•...•..•.•......•••.•• 195

3. Matrix of correlation measures

Z

.•.•......... 200

4. Main theorem ••.•....•.••..•..•.•••••••••..••••• 205 CHAPTER XI -

SPECTRAL MULTIPLICITY OF GENERAL AUTOMATA •.. 210

1. The convex set

K

.••..••••.••.••..••••...•.••• 210

2. Spectral multiplicity of substitutions of constant length ••••••••••••.••••.••••••.•••••••••• 216 3. More about spectral multiplicity

...•.•.•..•... 224

XIII

APPENDIX - COMPACT AUTOMATA

225

BIBLIOGRAPHY

•••••••••••••• ••••••••••••••• ••• •••••••••••••

231

•• •• •• •• •••••••• •• •• •••• ••••• •• •••••••• ••• ••••••• ••

239

INDEX

CHAPTER

I

THE ALGEBRA M ('F)

This first chapter is devoted to the study of the algebra

M ('ll') •

This study will be brief because we need only little about

and

M('ll')

there exist excellent books on the subject in which all the proofs will

([41 J ' [52J , [96J , [02J ).

be found,

We introduce the technics of genera-

lized characters to precise the spectral properties of measures such that generalized Riesz products, which will pleasantly appear as maximal spectral type of certain dynamical systems.

1.1. Basic definitions

We denote by

1.1.

'lr

the set of complex numbers of modulus equal to

'lr to

1, and we identify

complex measures on

M ('lr)

, equipped with the convolution of measures

u .. v (E)

f­or M('ll')

)1,

V

£

M ('ll'),

E

is the algebra of the borelian

=

lJ

(E­t)dv (t)

any measurable set.

is a Banach algebra for the norm

The elements of the character group

r

= 'lr

,

isomorphic to

, will

be considered sometimes as integers, with addition, sometimes as mul­ tiplicative functions on instead of

T , and in this i nt t __ > e

case

we write

n , the element

The Fourier coefficients of

lJ

£

are by definition ,

M('lr)

n

(n)

1.2. If

u

the sequence

is

0 , that is is

lJ

(E)

Yn ,

0

£

ZI

for every measurable set

positive definite

, that means

E ,

2

z

1 Si, jsn

2:0

J

for any complex sequence Conversely, the measure

]J

asserts that a positive

Bochner - HergZotz theorem

definite sequence

, is the Fourier transform of a positive

M(T) .

E:

1.3. We recall that ]J is a discrete measure if ]J = l: a j Otj (Ot the unit mass at t) and ]J is a continuous measure if ]J{t}= 0

for all

t

T.

E:

Md(T)

designs the sub-algebra of

M(T)

consisting

in discrete measures, and res. Every]J

where

]Jd

E:

E:

M (T) the ideal of all continuous measuc can be uniquely decomposed to a sum

M(T)

and

Md(T)

]Jc

E:

M (T) c

There is a necessary and sufficient condition for a measure continuous, which uses only the Fourier transform of Lemma 1.1. (Wiener)

]J

Let

E:

1.4. Let

]J,\!

E f

M(T)

E:

\!

1 2N+l

lim

>

N+oo

; we say that

and we write

jJ«

if

(n)

l:

-N

L

1(v)

is refered as the density

]J(E) = 0 of

So we are allowed to identify the ideal ]J «

Iv!

write

jJ

- with v

if

L

1(\!).

The measures

jJ « v

and

and we write

]J

o •

]J

as soon as ]J

f.v

vIE)

where

with respect to

\!

L(v) - of the measures

]J, v

are equivalent, and we

V«]J

In the opposite direction, we say that ]J

Zar

12

ab e o l u t e l u continuous with

is

u

v

N

measurable set. By the Radon-Nikodym property, E:

to be

M(T)

­oo

m). Of cour-

0 .

for the ideal of all measures

whose Fourier transform varushes at infinity. 1. 5. M('lr)

is identified with the dual space

functions on

'lr. Let

converges to

and

E:

C(T)*

M('lr)

of the continuous

It is clear that M('lr), a (M(T),

in the weak­star topology of

C(T», if

and only if -->

y

for every

E:

r •

A very important example of measure is given as weak­star limit point of a sequence of absolutely continuous measures. Example : Consider a sequence of real numbers j is z 0 , lajl $. 1 , so that 1 + a. cos 3 t J has at most one representation of the form n ­1,0,1 • It follows that N­l JI

j=O

(1+ a . cos J

, satisfying j) lR • Every integer j where E: • l: E: • 3

(a t

E:

J

n

J

3j t )

satisfies

n

l:

joo J

weak-Dirichlet set, if, for every positive mea-

E, there exists

supported by ( and J h j -

r

borelian set

i f there exists

uniformly on

Yj - > 1

It is said to be sure

a

We recall that

Dirichlet set

Let

11 du

(y.) s r J

-> 0 •

be a probability measure in

1.I

such that

M(T)

The following assertions are equivalent: a) b) c)

d)

is supported by

1.I

lim I0 (y) I = 1 y+oo 1 is limit in

1

weak-Dirichlet set

r

There exists an idempotent

h

0(h)

Let

h s f\ r

lim 10(y)

y+oo

sf, h f

such that h 1.I

I

1.I.

r

with

x

e

be a

1

=

cluster point of

constant,

lei

such that

h 1.I

r

1.I.

In particular ,

(Yj)

of characters which

and satisfies

lim 10(y . ) J j+oo Let

1 ,

of characters of

and d) implies b).

1

Now suppose b). There exists a sequence tends to infinity in

(y j)

of a sequence

(1.I)

which tends to infinity in

Proof: 1 =

L

a

=

1

•

1 . (y.)

J

in

r .

Necessarily

It is easy to see that

e

13

J Iy·Jk for

a

that

sub-sequence y,

Ild p ----> 0

that we choose in such a way of (YJ') Jk tends to infinity in r and to 1 in Hence

y.

Jk+l Jk c) is proved.

Assume c) and let

k

(y.)

be an integer

1 • We can extract, from

sequence of characters satisfying c), formly on E k+ l

a compact set

E k

of

E

k is supported by

and

measure, supported by

E , v(E)

chlet set • Finally, if

in

r,

r

and to XP

sub-sequence converging uni-

u-rmea s ur-e E

U

k ;:1 lim V(E k ) k+oo

1 - 11k E

. Let

k

and

E

We can choose v

=

p

1

in

L

E r

(y.) J

1

If

such that

X

YJ'

be a positive

is a weak-Diri-

E, weak-

is a probability measure, supported by

Dirichlet set, there exists ty in

a

tends to infini-

is any cluster point of

and

h

lim n+oo

is an idempotent in

(Y j)

Ix ,2n

r . Moreover

X

E

r \r so that

h f 1 , and

o

h u = u , which completes the proof

Definition I.4.

a

A probability measure enjoying one of the equivalent

properties of the proposition I.5 is called a DMeb."let paob ab i l-i t u measure. Remark

The set of the Dirichlet measures of

M('ll')

, whose orthogonal is an L-ideal, denoted only if

h u

=

0

for every idempotent

since continuous measures

u

y.... oo

Iv(y) I

< 1

••.

11

E 'I

for every probability

It can be found in Mc('ll') , ' I '

[41J

u E 'I i f and Clearly hEr , h f 1

are described by

It is not difficult to see that

'I

is an L-space of

h

11 = 0 d if and only if

v«

examples and new characterizations of

with the aid of the arithmetical properties of the

Fourier spectrum of measures ; more precisely those ideals are charac-

14

terized by properties of the sets

= > 0

and

{n

in the ideal •

Z,

10(n) I >

}

CHAPTER

II

SPECTRAL THEORY OF UNITARY OPERATORS

We introduce in this chapter the different notions of the spectral theory of unitary operators such as spectral measure, type, spectral multiplicity, multiplicity function,

maximal spectral

••• ; we establish

two formulations of the decomposition theorem for these operators, with our familiar notations. The results and notations would be used later, when studying the spectral properties of automaton sequences, and in the next chapter where we restrict ourselves to dynamical systems.

11.1. Representation theorem of unitary operators

Let

U

be an unitary operator on the separable Hilbert space

endowed with the inner be

and

product

. In the third chapter, H

will

U the unitary operator associated to an autoT morphism of the probability space . We mean that U is defined by

Uf(x)

=

U

H,

f(Tx)

=

•

1.1. Construction of spectral measures

For each

f

E;

H

t

definite since ZJ' t.1.­J.

n

L

z. z.

L

1. J

i,j i,j

1.

is positive

J

By the Bochner­Herglotz theorem (1.1.8), we associate to the element f

E;

H , a positive measure on

spectral measure

of

f;

of

T , denoted by

of' that we call the

is characterized by n

and its total mass

I

.

E;

2l:

16

Now, if

f

and

so that

g

are two elements of

--

a_

=

n

H, we consider

. The elementary identity

(1)

proves that (a n + a--) is the Fourier transform of a real measure -n on T ; consider then b = ia . The sequence (b n + b'" -n) n n is also the Fourier transform of a measure on T. We deduce that (a + a--) and (a - a--) are Fourier transforms of measures and the-

n

-n

reby

n

-n

= Gf,g(n) and (1) yields

an

Gf,f

where

Gf,g

is a complex measure on

(2)

Definition II.l. The family family of the operator U.

(Gf,g)f,gsH

is refered as the

spectral

1.2. Properties of the spectral family Definition II.2. If f E H , we write [Uf] for the cyclic subspace generated by f , which is the closure of the linear span of {Unf, n s:.z}. More generally, [U,f1, ... ,fkl should denote the cyclic subspace generated by f , ••• ,f E H • k 1 1.2.1.

Let

R

Z R(k)e i k t

be

a trigonometric polynomial on :Jr ,

then

I RI[ Proof

IIR(U)fll

L

2

(G

0f,g

from

HxH

into

M{T)

is a

bilinear continuous application.

Proof : We shall prove the continuity by showing the inequality For

' the bilinearity being evident by construction. R, trigonometric polynomial on T , If R(t)dOf,g{t)

I

I1 :;.

by 1.2.2.

IlgII H II

RII

I RII

L 00

2

•

(of) II r]

I gliH

by 1.2.1.

H II gliH

since (J fll = II fll • We derive from this inequality, the promised one by taking the sup on the trigonometric polynomials R with norm IIRll oo

:;'

1.

0

18

1.2.5.

f,g

For all

£

H ,

0f,g

the measure

is absolutely continu-

0g; more precisely

ous with respect to

IOf ,g I (B) :s.

B

for every borelian set

/olBT g T.

of

Proof : Applying the Schwarz inequality to the positive bilinear form (f,g) ---> of ,g (B)

, we obtain

• 1cr=lBT g Now, for any fixed B

£

> 0 ,

there exists a finite partition

(B n)

such that Llo

n f ,g (B)1

by definition of IOf ,g I lity gives rise to L 'of

10f ,g I(B) -

L 10f (B ) • 10 (B ) ,g (B)':s. n n g n (L

Of(B

n»1/2

10f,gl (B) :s. 10f(B)

and the result follows. Remarks

£

One more application of the Schwarz inequa-

s so that, endly,

a (B

(L

g

+

n

»1/

2

£

0

- This property, in fact, is equivalent to the property 1.2.4. - We can prove the following result: if

of

and

0g

absolutely continuous with respect to a same positive measure is

of

0f,g

are

w , so

and we have

. t;i dw 0

1.2.6.

M(T)

If

converges to

f

for we have the inequalities

in

H, of

n

converges to

in

19

1.2.7. in

For every

[U, f] ,

f

E

H

2

¢ E L (of)' we can define an element

and

¢(U)f , satisfying

denoted by

H

g in particular

Proof:

When

¢

is a trigonometric polynomial

R, R(U)f

is quite

well defined and this is the property 1.2.2. Let now

2 L (Of)

¢

nuals with

h­Rnll 2 L

and

R be a sequence of trigonometric polynon going to zero. We define ¢(U)f to be the

(of) [U,f]

unique element of associated to ¢ by the isometry W (that 1¢), is W­ as well as the limit in [U, f] of the sequence R (U) f by n 1.2.1 and 1.2.3 (i). It follows that oR (U)f,g converges to O¢(U)f,g in

(1.2.4)

M(T)

and

absolute continuity

n

Rn.Of,g

to ¢'Of,g

0f,g« Of

in

(1.2.5). The measures

Rn .of ,g

being identical, the proof is complete.

Remark

If

h

is given in

because of the

M(T)

[U, fJ

the property (iii) of the isometry

0Rn(U)f,g

and

0

we should prefer, instead of 1.2.7 W

(iii)

1.2.8.

If

operator on

Proof 1.2.7

¢

is a bounded borelian function on

T,

¢(U)

is an

H, bounded by

¢(U)f

is well defined whatever

we check that

f , and, as a consequence of

II¢(U) (f+g) ­ ¢(U)f ­

110¢(u) (f+g)

'"

= 0 ; in the same way,

110¢(u) (At) ­A¢(U)fll = 0 • Endly

!I¢(U)fI1

2

11¢(U)(Af)­A¢(U)fII H=

1I0¢(U)fl =

::.

0

gives the bound

1.2.9. Let B a borelian set of T , IB its indicator function and is the orthogonal ll.B(U) , the operator defined in 1. 2.8. Then, liB (U) c projection on the subspace of H consisting in the f with 0f(B )=0. Proof

{f

H , 0f(B

c)

=

O}

is a subspace of

H

because of

20 Cl f + Cl g + Clf,g + Clg,f and the absolute conCl f + g Denote by P the tinuity of Clf,g with respect to Cl and o g f is clear(1.2.7) operator 1 (U) • For every f E: H , ClP(f) = lB' B ly supported by B Now, i f f,g E: H , by 1.2.7 , the basic relation

.

Of ,g (B)

so that P = 1

B

p

2

(U)

= P • Combined with the fact it is an I-normed operator, is the orthogonal projection on

0

H

B

We have proved, in particular, the so-called spectral representation theorem of the unitary operators Theorem II.l. space that,

U

Let

be an unitary operator on the separable Hilbert

H. There exists a family of measures on for every

q(U)f,g> =

(3 )

or,

bounded borelian function on

f

dO

Y

f ,g

(of,g)f,gE:H

such

Y

(t)

II.2. Simple spectrum Definition II.3. trexe exists In this case,

The operator

h E: H H

such that

U

is said to have simple spectrum, if

[U,h]

unitarily equivalent to the operator (V¢)(t) =eit¢(t) Proposition II.2.

H.

(1.2.3) Suppose

U

V

2

(oh) and u 2 defined on L (Cl by h)

is isometrically isomorphic to

L

is

f,g E: H.

with simple spectrum, and let

Then 0=

Proof as

In any case, it is obvious from 1.2.5 that

Clf,g

o

as soon

Cl But, in general, the converse is false, and this fact g gives rise to multiplicity. When U has simple spectrum, denote by W 2(Oh)' the isometry of H onto L so that, by (iii),

21

0

and the opposite implication is quite evident. Remarks

- We sall see in the next section, that in fact, the follo-

wing properties are equivalent : (a)

U

(b)

For every

noticing that [U,f]

and

has simple spectrum

of ,g

[U,g]

0

f,g £ H, 0f,g = 0 implies of 0g' by means the orthogonality of the cyclic spaces

.

We sbould possibly prefer the notation rather than

0f,g

- When

0 , or U

[U,f] J..

"f

and

g

are

U-orthogonal"

[u,gj

has no more simple spectrum, the proposition 11.2

takes the following form If

°9

of 1..

then

f,g

(next lemma II. 5) •

[f+g, UJ

£

f,g c [h,U] , 0 implies f,g = proposition to the cyclic space [h,u1 )

°

Now i f

H = L 2(T,m)

Examples: 1. Consider to the

1T

-irrational rotation , n

£

Z}

and

x - - > x+8

(restricting the

g

U

the operator associated

U

has simple spectrum. It

H , for which the linear span of is dense in L 2(T,m) . It is well-known that

suffices to exhibit a function {x ---> f(x+n8)

o

Of

f

£

the candidates are exactly those functions

f

whose Fourier transform

never vanishes. Let us sketch the proof rapidly: suppose

¢

£

H , ¢

[u, f] this means, ¢ '" f (n8) = 0 for every n e Z 2-functions 2 the symmetricalof f). By continuity of the L '" L , ":::" v ¢ :: 0 unless f vanishes. Later, ¢ '" f :: 0 and ¢. f :: 0 so that 2(T,m) we shall construct explicitely the isometry W, between L and 2 L (of)' 2 2. Consider H = L ([0, 1J ,m) and ¢ any continuous one-toorthogonal to

(f

=

one function on tion by rated by

dm ,

, with

I¢I = 1 .

U, the pointwise multiplica-

¢ , has a simple spectrum. The sub-algebra of ¢

is dense, because

theorem) so that

J ¢n

[O,lJ

H

[U,f]

in other words,

¢

where Of

f = 1 • We deduce that

is the image of

m

under

3. Consider now, the precedent example with ¢

is

C([O,l]), gene-

is one-to-one (Stone-Weierstrass af(n) ¢ ¢(x) = e

4'ITix

1/2-periodic. A new application of the Stone-Weierstrass theo2([0,1/2J,m) rem shows that [u, 1[0,1/2J] = L and [U,1I.[1/2,1]] =

22 2([1/2,1],m), 2([O,1],m) hence L = [U,f] e [u,g] where f,g are L u-orthogonal functions with of = 0g = m . U has no more simple spectrum. Simple Lebesgue spectrum It is easy to characterize the pairs (H,U) with H isometrically isomorphic to L 2(T,m) and U unitarily equivalent to the multiplicat i on operator

V

) L2 T,m) ( , Vf(t

on

pIe Lebesgue spectrum on Proposition 11.3. h E: H

exists

Proof :

If

the isometry

U

We say

(Unh)

z nE:

h ' as S1m-

U=

there

forms an orthonormal basis of

H.

H , we construct W(Unh) = e int

forms an orthonormal basis of 2 by setting H onto L (T,m)

W of

U

H •

has simple Lebesgue spectrum on

such that

(Unh)

itf(t ). =e

W exchange two orthonormal bases, and W U w-1(e i n t) = ei(n+l)t = eit.eint so that

U

is unitarily equivalent to the operator

V •

°

f with [U,f] = H and 'C m . f 1 d E: L (T,m). , > where the density and d is 0 m. a.e. df·m Of f 2 f In addition, if we denote by W the isometry of H onto L (T,of)

On the other hand, there exists

=

W(Ug) = e i t W(g) Thus

h

=

W-1( __1 _)

!CIT

orthonormal basis of

satisfies, H, as

Unh int

is an

and

o

(_e_)

Idt

11.3. Spectral decomposition theorems In this section we shall introduce the concepts of maximal spectral type, spectral multiplicity and multiplicity function of an unitary operator. Notation:

The class of measures equivalent to a fixed measure

E: M(T), is called the type of

denoted by

Theorem 11.4. (first formulation of the spectral deconposition theorem) Let

U be an unitary operator on a separable Hilbert space

exists a sequence

of elements of

H

>

such that

H. There

23

[U,h.].L [U,h.]

and

(a)

and for any other sequence and (b)

, we have

Remark:

vh.

V

[vh

Proof

i

J

n

=

H

F

i

j

satisfying (a)

1

= V(h 1,h 2",,),

V ( f 1 ' f 2' ... ) (t1 ' t 2' ... ) quence

of elements of

Vh!

This formulation means that

the operator

for

)

U

is unitarily equivalent to

•

2

co

on the space L (T,vh ) by n it it 1 2 (e f 1 (t 1 ) ,e f 2 ( t 2) , ... ), t i £ T, the s e-

being uniquely determined by

We shall contruct such a sequence

U , (h

n).

First step:

We construct, by induction, a sequence (e j) of elements of H such that the spaces Hj = [U, ejl satisfy rn . We start with an orthonormal basis of the separable H, (£j)' and we put e 1 = £1' HI = [u, e 1 ' Suppose e ,e are so constructed that the spaces 1 1",. j H1".H). are orthogonal and £1' ... ,£). belong to@Hi,weproceed i$.j this way suppose n j is the first n so that £n is not in the sum

@ Hi' if possible, Hence i$.j e)'+l

(PHI

£

j+1

and therefore,

H.

)

(£) nj

H n

is isomorphic to

Second step: va

Suppose

H')

@ H. i$.j is not already in

is orthogonal to £j+1 £j+1

WU f (t )

then

@

i$.j

(e

The so-constructed sequence and

j+1 , and we put

j

is the orthogonal projection on

Hj+1 = [u,e j+1] £j+1 e HI @,.,@ Hj+1 , I f )

P

-

nj

clearly,

n.

n

L2 ( v

=

e j+ 1

n

)

HI @..

o

.e

,

Hj

belongs to

ei,e j by the isometry W

eitWf(t)

a £ H

We show endly that

P@ H. (£j+1) i$.j a

satisfies

j) e

+

.

o

for

i f

j

satisfying

,

is such that

v

«va en is maximal for the absolute continuity.

for every

n

24

x

Every 0", l..

j::;N

x, J

H

3xj

can be decomposed to a sum

o

since

L o J' L (Oh') 1

1

1

(with the notation of equivalent) and

1.1.2.3). That means

isometry Q from the following

1

"V h,

(= unitarily

1

J "U/[u,gJ by 1.1.2.3 again. The converse 1 f, g E: Hand UI , f] c: ul [u, g]. There exists an 2(Of) 2(Og) L onto L such that QV = VgQ ; notice f

U/[U,h

is true : suppose

V h

2(0) L 2(T) is an operatoX' fX'om L into it eitQ(v) Then O,T positive measuX'es in M(T) , suoh that Q(e v) 00 Q is the muZtipZication opeX'ator by some funotion

the linear span of all the eigenvec-

H

If

H denotes the closed subspace of elements with disd crete spectral measure, and H theLclosed subspace of elements with c continuous spectral measure, H = H by lemma 11.15 , so that c d H Hd $ Hc . If there exists at least one element in Hd (or in Hc)' U

is said to possess

a discrete

U

possesses discrete

as well as continuous components,

(or a continuous) component, and if

U

has a

mixed spectrum

Examples : 1) Turn back to the operator

U

defined on

by the 1l'­irrational rotation x - - > x + 8 . (§.1. 3). U has a discrete spectrum and om L 2­ I k l 0k8' Indeed, it is obvious that e ikt is ksZ ik8 an eigenvector of U corresponding to the eigenvalue e As 2(T,m) H L is generated by the (e i k t) , k s , om is discrete (corollary 11.18). U

has no other eigenvalue, hence the type of

om'

Let now f s L 2 , f(k) f 0 whatever k s Z so that Of om and 2 2 L (T , m) = [U,f] z: L (Of) by an isometry W (1.2.3 and 1.3. ) . We 2(Of) shall prove that Wg=h is defined on L by h(k8)=g(k)/f(k} k s By the totality of the translated t --> f(t+n6) ,gsL2(T) admits a decomposition of the form

and If now, we put

g(t)

L a

g(k}

L a

h(k6}

n n

f (t + ns ) f(k}e i k n 8

g(k) /t(k)

h s L 2 (Of)

= L jh(k8) 1 2 1f (k) 12 k

J jhj2dOf

since

Llg(k)1

2

< +

00

It remains to check that W : g --> h is an isometry of L 2(T) onto 2(Of} f (x + n8) = Unf (x) ; Wg = h = e int and W L . Consider gn(x) 2 n n extends to an isometry of L (T) because of the totality of the (Unf) •

H = L 2(T,m)

2) Let on

f

s H

and

U

by Uf(x) = ¢(x) f(x+8)

where

I¢I

1,

6

S

[0, 21T].

the unitary operator defined

33

Then,

om

2-

L

!k l

kt::lit

°*

°

0ke ' where

is the spectral measure of

the function 1 Denote by

\!

IT

af(n) =

,

the measure

q,(n) (x) f(x+n6)

0ke . If f(x) dx

q, (x + (n­1) e). It follows that «

o

where

q,

(n)

and

.

1.6.

0f«

2(T)

(x) =q,(x).q,(x+6) .••

lx fe om . Conversely, approaching every

lynomials, we establish easily

f t L

\! ,

therefore is

by trigonometric poand, combining the both

v

om"" v ,

We close the general study of the Hilbert operators by a classi-

cal theorem. Theorem 11.19.

(Von Neumann)

H, then for all

spaoe

1 N where

P

f

If

is an isometry on the Hilbert

U

H ,

t

N­1 Z Unf ­­­> Pf , n=o

is the orthogonal projection onto the subspace of

U-inva-

riant vectors.

Remark:

When the isometry

is no more an unitary operator on

U

U

the spectral family associated to If we put and for any

Of (n)

n :::. 0

af(n) = crf(­n)

n :S 0

f t H , the sequence

one and the spectral measure

(af(n»ntZ of

H

is still a positive definite

so defined. Most of the properties of

the spectral family are valid. In particular, arising from the Von Neumann's theorem, we have, for any

A t T

where

P is the orthogonal projection onto the proper subspace corA iA responding to the eigenvalue e Indeed, applying the dominated

convergence' theorem to the sequence of polynomials

1:.

L

N n­oo

1 N L

n­oo

lim N­>­co

0f {n ) e ­inA

Z

n N n rx(mod 1)

for every

r

E:

on

[O,lJ

1N, and the system

is ergodic too.

3) Consider the direct product of two dynamical systems (X (Xl X X , B lJ 0lJ T) where 1 2, I,B I,]1I,T1), 2,B 2,lJ 2,T 2) 10B 2, 2 T(x A direct product of ergodic dynamical sys= (T (X 1 1),T 2(x 2 1,x2) tems need not be ergodic: suppose for example, there exists A 0, (X

f1

».

E:

where

2

L (X1,]11) , f 2

f

1 Obviously

, f

2 £1

E:

2

L (X

2,]12)

such that

are non-constant normalized eigenfunctions. L . £2 is a T-invariant function inJl

°

37 4) tem,

G

and

f

(Group extension)

Let

be a dynamical sys-

an abelian compact group equipped with the Haar measure X ­­­> G

m G,

a measurable map. We define on the product space

(X x G, B 0 B ' u 0 m G G)

the transformation

Tf(X,g)

=

T

f

by

(Tx , g.f(x»

It is easy to check that

T is an automorphism of the product space. f (X x G, B 0 B P 0 m , T is called a group extension f) G G, of the initial system.

The system

Anzai

[4J

proved the following .

Theorem III.3.

Assume the system

not ergodic i f and only i f there exist

X ­­­> T

measurable map

(l) Proof:

>

y

ergodic. Then> T f > Y I 1 > and ¢

G

is >

a

such that

y(f{x}) = ¢(x) • ¢{Tx}

-1

­ a.e.

y G , Y I 1 , satisfies (l) h(x,g) = ¢(x)y{g). It is easy to check that h is Tf­in-

We just sketch the proof : if

consider

variant but

h

cannot be a constant

unless

C

C

0

because

J

Y dmG = 0 which is incompatible with Ihl 1 Conversely, supG pose T non­ergodic and h: X x G ­­­> a non­constant invariant f function. By a straightforward calculation, we see that ¢y , defined on

X

by

¢

y

(x) ­ ­

JG

h(x,g} y(g) dv{g)

, has a

T­invariant, and thus

y , by Fourier unicity, h should be independent of g and by hypothesis on h , a T­invariant function. Or h is a non­constant function which is impossible since T is ergodic. Therefore, there exists I¢yl a non­zero constant constant, modulus. If

for some

y

Remark:

T

¢

Y

= 0

for every

G , and the function ¢

measurable from :

realizes (l).

is a particular case of what we call a skew product : f is a dynamical system and (Sx)X X a family of endomor-

phisms of another measure space T

=

(Y,C,v). If,

(X x Y , B 0 C)

X x Y --> X x Y

defined by

into T

(x,y)

(x,y) ---> Sx Y

is

(Y ,C), the transformation (Tx, Sx y)

is an endomor­

phism of the product measure space, called a skew product.

38 111.2. Ergodic theorems We suppose

p

a probability measure.

Theorem III. 4. T

and

(Van Neumann mean ergodic theorem). Le t

be a measure-preserving transformation of

II.!

N-l L:

N n=o

f

for every of

£ L

foT

n-EJ(f)11

2 (X, jJ) , where

EJ(f)

H= L

2

(X, B, u )

(X,B,jJ). Then

N->-oo> 0

2

is the conditional expectation

f . given J • the sub o-algebra of

T-invariant sets of

It follows from the proposition I1.20, that

EJ(f)

Jf

B

•

, i f the

du

system is ergodic. This theorem is the formulation of the thereom II.19 for U = U . T As a consequence of it, we have the following formulation of the theorem II.20. Theorem IlLS. every

B £ B

(Khintchin). Let and every

£

> 0 ,

(X,B,jJ,T)

a dynamical system.

For

the set

contains a relatively dense sequence of integers.

Proof

This theorem precises the recurrence theorem of Poincare

which asserts, under these hypotheses, that almost all points of return infinitely often to

B

B . This is an immediate consequence of

the theorem II.20 since

B

and Theorem IIL6. cal system. If

0li

B

(n)

II

=

= u (B) 2 •

(Birkhoff's ergodic theorem). Let £ (X,p)

(X,B,p,T)

a dynami-

f

1

N

N-l L:

j1

n=o

If the system is ergodic, the a. e.

limit is

ff

-

du

We admit this famous theorem, and give two corollaries of it.

a.e.

39

The system

Corollary III.7.

for aLL

1

lim N+oo

which means Proof or 1

is ergodic if and onLy if.

(X,B,p,T)

A,B E B ,

1

A' B

P (A) 0p (B)

N P (A) 0p (B)

{a}

If A is a T-invariant set, choosing B = A, we get P (A) = 0 and the system is ergodic. In the opposite direction, we apply f = n

the ergodic theorem with inner product with

liB •

and the result arises taking the

A

An approximation argument entails the second corollary. Corollary IILB. lim N+oo f, g

for an

E

lim N+oo

The system 1

N

l:

n 0 Consider B U T-nA. B is an inva-

riant set which contains -n

=

ll2(A)

i = 1,2

\1

or

0

2

II = all

belong to 1

1

+ Sll2

be a convex

fiT' a > 0, S > 0 ,

and therefore

III (A)

1 \1 and ll2 are also T-ergodic. If \1 t \1 ' \1 J.. P 2 1 1 1 2 the preceding theorem, and we can find A J with III (A) = 1

\1 2 (A)

or

\1 (A) = 0 , which implies

2 ergodicity of

\1 (A) = a

\1 ; thereby

PI

= P2

t

0 and

or P

=0

using and

1 . This contredicts the is extremal.

J such that ffiT is not ergodic, and let A 1 1 • :D. and ll2 p(A) = a, a t 0 or 1 We define \1 = I-a IJ • RCIJ A 1 A are easily seen to belong Since A is an invariant set r 1J and \1 1 2 to ffiT now II a ll + (1 - a) \1 2 and II is not extremal 0 1 Conversely, suppose

\1

a

Consider, in particular, the probability space unique probability measure on

T

(T,B,m).

m

is the

, ergodic with respect to any irra-

tional rotation (corollary 111.11). G. Brown and W. Moran, in introduct in this case, the following generalization: countable subgroup of probability measure on Defini tion III. 3. for every

d

£

T

, acting by translation on

T

D

designs a

• Let

II

be a

T .

II

is said to be

D-quasi- invarian t ,

\1

is said to be

D-ergodic

if

£

is a correlation function of

C

I N

C(k) = lim j+oo

k £:N

for every

u n +k un

l:

n c l

c

would contain a difference set

B- B

with d (B) > 0,

A

for some

£

> 0 . In this case,

and thus

(h) >

for some

£

intersective would intersect

11 h

£

11 • Since

0

A

11 E

A, ptA) > 0 ,

1A

£

d(B) > 0

e < u (A) 2 , and a difference set

possesses the same property

£

p(A)2 > 0

{a}

theorem (111.5)

would be correlative. Now, from the Kintchin's has bounded gaps if

E

B- B

,

with

this can be deduced from the

well-known property of measures on Lemma IV.21.

Ec

Let

Fourier spectrum of

p

be a set with arbitrary Zarge gaps. If the £

Mcrr)

is incZuded in

E , then

p

is conti-

nuous

Proof

We give a short proof of this fact, using that the discrete

(1.4) is the smallest element in f+, the positive elements of r. Now let (n with lim n = + be such that j j) C n.+jJc E , and consider X a cluster point of idempotent

h

d

00

,

J

X

y

rc £

E

C

r ,

since

o Let

and X f c

hlxI 2 and

Bc:N*

p

C

Putting

h . But h Xy £ f is continuous

with

of the sequence

E

h

instead of

and

d*(B) > 0 , and let u

(1

B

(n)

(h

o From (3)

o

(h

Xy

XY •

X)

h

d,

we get for every

• X)

o , therefore

be a correlation measure

82

cr

and

is not continuous. Since

cr

the spectrum of

RB(n+k) RB(n) = 0

is included in

B-B

if

i

k

B-B ,

and we conclude with the

0

lenuna.

Nevertheless J. Bourgain constructed an intersective but not correla-

[8J .

tive set

2) Examples of correlative sets can be found in

. Using a

useful criterion the authors prove in particular that {n - n k j and

{n

2

,

n

r

j

> n

where

k}

(n )

is infinite ,

j

n e :N}

are correlative sets.

3.3. Examples of correlation measures 1. Recall the example considered in chapter III, section 6 : We defined the operator

a

11

U

is irrational and

on

L

2(T)

Uf(x) = 4>(x)·f(x+a)

by

T

the function defined on

4> (x)

by

a

-1

if

0

x
(x)4>(x+a) ••• 4>(x+(n-1)a)

J211

d

()

4> n (x)

0

is the correlation measure of the sequence x

(mod 211)

£

:

d

4>(n) (x), for

this is the consequence of the uniform distribution

of the sequence

{x + nc l

, for every

x

£

,

and the ob-

vious relation 4> (n+k) (x)

4> (n) (x)

• 4> (k) (x + nc )

•

Thus we get

t

4>(n+k) (x) 4> (n) (x) =

n (k) (x + na )

n

(k)

dx (x) 211 = o(k)

, for every

x

£

T

(theorem

83

2. Let q £ , q 2 and one complex numbers, satisfying u

for every

n, a

£

u

n

aq +b

a

u =

• u

a sequence of modulus

u

b

I

o

b < qn • Such a sequence is said to be

,

q-muZ-

tipZieative . We shall prove that a

q-multiplicative sequence

admits a unique correlation measure

cr , which is the generalized , ... ,u _ u ' that means o,u 1 q 1

Riesz product (1.3)

, constructed on

w*

o = lim N

I P ( t) = q

where

II

+ u

1e

u

it + ••. + u

q_ I

e i (q-I) t I 2

Proof: We write u(n) instead of un' for sake of clearness . For the unicity of the correlation measure, we observe, following

[7 ] ,

that, for any fixed

a ,

k

in arithmetical progressions

U Pm m

u(n+k) u when

n

runs over

I'

m

,

e

N-I

such that

De m

the progression

(We omit the details). Now, i f am

there exists a partition of

,

constant a

N 2: I , Pm

m

+ q

n

r

with

CO,N-I] l' ¢

a

m

1 is a Cauchy sequence in verges to a sequence u E: AJifl- satisfying u = 1;k (u), and sequence

u

1;

=

(i)

lim n-+""

(ii)

there exists a letter

for every

+

begins in

E:

1;k

on the

(later denoted by 0) so that

a

shall design the fixed point of u, and thus

A

A

a.

1; , according to the preceding pro-

position. We shall only consider the letters of appear in

o

and con= a • The

is assumed to fulfill the following hypotheses (H)

1; (a)

n

U

JN

0

From now on,

u

A

may be obtained by iterating the substitution

a

letter

A

For an arbitrary

a, 1;kn(x) begins by 1;kn(a) so that by the same word 1;kn(a), whose length

X

E:

k

a.

A

which actually

now, is the set of all letters in

I;;n(O),

O.

Examples

1)

1;, defined on

two fixed points, exchanging in

u

u

1;""(0)

the letters

{0,1}

by

0 ---> 01, 1 ---> 10

011010011001

o

and

1

and u

v

oo

1; (1)

admits when

is the Morse sequenae

2) 1;, defined on {O,l} by 0 --> 01, 1 ---> 0 leads to the fixed point u = 1;""(0) 0100101001001 ••• called the Fibonaaai e e que n c e

[90J.

89

3) Consider 1 ---) 30, 2 ---> 21,

on

{O,1,2,3}

defined by

=

a

3 ---> 00. Choosing

fixed points. If U = 0 o 2 and identifying in

0

to

we may reduce A 0, 2 with 1

0 ---) 01 ,

or 2, we get two {O,1,3J. Starting with

and

1, 3 with -1

we get

the Baum and Sweet sequence.

V.2. Minimality of the topological system Let

u

where

fulfills the conditions (H). Following the

preceding chapter, we associate to where sed

T

is the

T-orbi t of

Theorem V.2. a c A.

in

u

the topological system

one-sided shift on u

in

A

lN,

and

X

= O(u)

(X,T),

is the clo-

AIN •

The system

there exists

Proof

u

(X,T)

k 2: 0

Suppose the system

is minimal i f and only if, for every

eucb that

(X,T)

oon t a i ne

(a)

O.

minimal: every word of

u

occurs

with bounded gaps (theorem IV.12) ; in particular, the letter 0 with bounded gaps. Every a £ A is a letter in for every k 1 , u contains the words

occurs in u since u = k 2: 1

Now

o

k

for

I

goes to infinity with

k

and

u;

contains

large enough.

Conversely the system is minimal if

0

occurs in

u

with bounded

n 2: 1 , will occur in

gaps for, in this case, every word

u

with bounded gaps and, if B is any word in u , the same will hold for B, since B C for n large enough. Suppose that contains O. (a) contains and thus O. Putting (1)

K

sup a£A

contains we see that taining

u

0

inf k2:1

for every

{k, a

£

contains

O}

A . Using the identity

is the juxtaposition of words

O. The minimality is established.

u

=

all of them con-

0

If u = without any other hypothesis on s , the con"u o occurs in u with bounded gaps" is sufficient to o ensure the minimality of the system (X,T) .

Remark

dition

Examples

Consider on

{O,1,2}

1 : 0 ---> 0 1,

the substitutions 1 ---> 20,

2 ---> 1 1

90

o -->

01,

o --> 010, The systems arisin:; from

and

--> 22,

2 ---> 11

--> 02,

2 --> 1 •

3

are minimal. Minimality is fai-

ling for We introduce now a terminology issued from the theory of positive matrices and Markov chains. Definition V.2.

A substitution is said to be irreducible on A if for every pair a, B of letters of A, we can find k = k(a,a) k such that (a) contains a . is said to be primitive if there

exists

k

is,

can be chosen independent of a, B)

k

such that

Remarks

means that

In fact, since

ter of

K

A

A, for

£

A

(that

being assumed to satis-

2) If

a

£

thus every let-

A.

is primitive ,X

u , but only on

u,

large enough. Letting N= J+K

J

contains

and this for any

fixed point

a, a

•

is exactly the set of all letters in

is defined by (1),

A

for every

is primitive,

contains all the letters of where

a

The condition ensuring the minimality of the system, in

1)

theorem V.2 fy (H).

contains

does not depend on the chosen

(since every letter occurs in u), and

gives rise to exactly one minimal system that we sometimes denote by

(X

,T).

V.3. Let

­ Positive matrices be a substitution defined on the alphabet

Notations:

If

Band

the occurrence number is the number of Definition V.3.

i

e

in

B

occurring in

B

is a positive

e

We call

matrix whose entries are M

are two words in

of

s

x

s

A

A*, we denote by

In particular, if

and we denote by t

ij

matrix

i,j

=

(

£

i

M

£

M

Le(B)

A , L tB) i

the

A

with non­negative entries, not all

equal to zero) whose entries are integers

0 .

91

For every q

, L

j E A , q

.. 1J

i

L L.(s(j» i 1

for every

!s(j)!. I f

=

j E A ,and

tic matrix. Note also the identity If

B

A*

is a word in

ponents are

i

M(sn)

q.S

=

=

where

S

is a stochas-

(M(s»n . lR s whose com-

designs the vector in

L (B)

Li(B), 0

M

is of constant length

s

s-1 • It is clear that

L(s(B»

= M.L(B),

L : A* --> lRs ,

in particular, L (s (j» iEA . We shall call the composition function, and sometimes, M the composition matrix. Most of the properties of a substitution composition matrix

s

arise from those of its

M(s) . We are thus led to describe more generally

the properties of positive matrices arrl prove the Perron-Frobenius I theorem (which also can be found in [99J). Recall that

s

is said to be

primitive if its composition matrix M = M(s) is primitive which k means M is strictly positive for some k (= all its entries are positive), since = L. (sk(j». 1J

Theorem V.3.

1

(Perron-Frobenius)

M be a primitive positive matrix.

Let

Then a) M admits a strictly positive eigenvalue

A of

for any other eigenvalue

8,

such that

8 >

IA I

M,

b) There exists a strictly positive eigenvector corresponding to 8, c) 8 is a simple eigenvalue. Proof

a) Consider

,

an eigenvalue of

M

satisfying

A of

for any eigenvalue If and

Y

is an eigenvector associated to

1,1

IYi l

L m.. Iy·

j

1J

J

I

L

(admitting the value Define now, for

x

+co

0

,

mijlyjl 1Yi!

if

x

= 0) •

Yi

"

0

,

the function

L m.. x.

r(x)

, for every

so that

min j i

I, !

,

min j i

1J

x.

1

J

r

by

M.

92 The function

e

r

sup r (x) = xl"o x::::. a

z

sup rex)

e

{x e lR

s

{OJ, x

so that

O}

exists and is attained. Clearly,

II x] =1 mijx j

min j i

since

is u-s-c. on the set

for every

x

0 . We claim that

for some

y

0 , Ilyll

e

is an

eigenvalue too.

e

Write

min i

would be different from

1 . If

0 , using the primitivity of

z = My - ey

M, we should

have

o i

k

= M(M

k M > 0 , so that, putting

where eX

k M z


0

with

=

for any

IAI

=

e .

If

elvl

ek!YI = IMkyl

1.J

M, with

and for every

i

YJ.I

i The Yj must have identical arguments e ¢ and positive eigenvector corresponding to A _ A is

A

e::::. IAI

and the already used above arguments

Mlyl

Ii: j

and

M

Mlyl

Mklyl

ex < Mx

which provides a contradiction. There-

A , eigenvalue of

My = Ay , elyl = IMYI show that

It follows that

k y) - e M y

y

ei ¢

> 0

is a strictly necessarily and

e b) We already proved this assertion by proving the following.

Lemma V.4. M • If

x

Let

e

be a dominant eigenvalue of the primitive matrix

is an eigenvector corresponding to

positive eigenvector corresponding to

c) We begin to establish that the to

e , Ker (M - e I)

e> Ixl

is a strictly

e. eigen-subspace associated

, is one-dimensional. In the contrary, let

x

and

93 y

independent vectors in

Ker (M - e I). From the preceding lemma, xi xl are different from zero, for every i • z = x - - . Y is thus

and

Yl

another eigenvector corresponding to again the lemma,

z

We prove now that ci ty of MX

ex

Ker (M - eI) = Ker (M - 81) 2 x

o

o.

Applying

which implies the simpli-

Ker (M - e I) 2 \ Ker (M - e I) •

E:

+ x , where

o

zl =

must be identical to zero.

e . Suppose o

e , satisfying

x

E:

Ker (M - eI), and, for every

n;::. 1

n n-l e x +n8 x. o There exist

large enough, and a constant n

(2) Now

M

n

°lxo I

x0 I ;: . n0

0

C > 0

such that

n -1 8 0

n M0

n 8 o.

is a primitive matrix with positive dominant eigenvalue n n n n We deduce from (2) that M 01 x I = e 01 x I ans thus M Ox = 8 Ox o 0 0 0 in a). This contredicts our assumption and

Ker (M - 8 I) 2

=

as

Ker (M - 8 I) ,

0

which completes the proof.

Remark: A positive matrix is said to be irreducible if, for every i,j , there exists k 1 such that > 0 . In the irreducible hyJ.J

pothesis, the Perron-Frobenius'theorem is almost the same: b) and c) are still true, only a) must be changed into

8

JAI

a ') M admits a strictly positive eigenvalue for any other eigenvalue A of M

8 , such that

Nevertheless, it is possible to precise the form of the eigenvalues

A

8 , satisfying

is the

8 • We need the following definition:

The period

Definition V.4. M

IAI =

d

1

g cd, for every

The reader will find in

of the irreducible positive matrix

i , of the set

{k

(k)

1 , m..

J.J.

> O}

•

the description of irreducible matrix.

In particular Proposition V.5. An irreducible positive matrix is primitive if and only if its period d is 1 (= aperiodic matrix). Proposition V.6.

d > 1.

Let

M be an irreducible positive matrix with period

M admits exactly

d

eigenvalues

A

satisfying

IAI

=

8 ,

94

whieh are

e.e2TIik/d, k

O,l, .•. ,d-l.

Consequences of the Perron-Frobenius'theorem Let

any primitive substitution on

A ,

(satisfying the hypotheses

(H». We deduce from the Perron-Frobenius'theorem, that each letter of u

occurs in

u

with a positive frequency, which is the first step to

the unique ergodicity of the topological system

For every

Proposition V.7.

v ec t or e

n

(L

en

e.

We interpret the property of

e

to be a simple and dominant

eigenvalue. We may decompose the operator

into a sum e P e + N is some projection onto the one-dimensional eigen-subspace

where

P e Ker(M-eI), and

N

M

is an operator satisfying

ver, the eigenvalues,

Ai' of

decomposition , Mn = en P

e

PeN =NP

0 Moreoe suplAil < e , accor-

N, are such that

ding to the assertion a) of the theorem

L (a)

s-dimensionat

, c on o e r q e e to a e t r i c t l u positive e i q en o ec t o»

(a»)

corresponding to Proof:

the sequenee of

a e A

n + N

Mn lim -- = P • Let now e n-+co en

and

is the s-dimensional vector

v.3. We deduce from this

(0, ..• ,0,1,0,0) lX

n

M

and

=

L(a) tends to Pe(L(a». The vector v(a) Pe(L(a» en positive eigenvector corresponding to e (theorem V.3.b»

When

is of constant length

1

and

I.

I = qn

q,

) en is a strictly

o

for every

a ,

In the general case, we have (a) I

1

For every

Proposition V.8.

a e A

a e A ,

I

(a)

I

tends to

e

(n -+ 00) •

Proof

This is obvious with the preceding proposition, since, deno-

ting by

ll, the s-dimensional vector

Proposition V.9.

of rea'l numbers

Le t

L

j

I

n

a e A (a»

(a)

I

and

j

(1. •• 1),

eA. When

admits a 'limit

d

j

Isn(a)1 =

n -+ 00 • the sequence > 0

which is indepen-

dent of

a

Proof:

Applying again the proposition V.7, we observe that the se-

.

quence of s-dimensional vectors

n

I

(a»)

I

0

converges to the limit -

95

v(a) 00 tive,that we should denote by

(X,T)

of

1 • If

2 , and we write

of length

. If

=1

, the for

may be identified

to the set of all letters of some fixed point of a primitive substitution

defined on

, the proposition V.10 can be derived from

the proposition V.9.

w , a "letter" of the alphabet

Let

in the following way if

(w)

.

We define the substitution

96

we set '"

(2)

o

So defined, (w) by juxtaposition. Lemma V.11.

I =

)1-1"'YII;(w

II; (w o ) I , and we extend

to

and

admits as a fixed point, the sequence (uo ..•

=

•.•

...

whose "letters" are all the words of length

u, without repe-

in

u.

tition, oa aurri.n q in the same order as in Proof:

.

0

We iterate, for that, the substitution

in u of length , w '" uou1 ... u = I;(u) and from (2) ,

=

(uo ...

begins by

.•.

since

... (ull;(Uo)I-1 •..

wand this is sufficient to prove the existence of

a fixed point for

I;;(W)

denoted by

It is easy to check, for every

n

o

and thus

=

Lemma V.12.

is primitive if

Proof

on the first word

= u ou1.··uI G(w) 1-1

. G(w)

,

is primitive

the same conditions (H) as

, . It suffices to

prove then, the irreducibility .of I; on rl • Let wand B in rl • Using again the identity u '" r,n (u) for every n , B c ,P (CI.) for some CI. A and p 1 ; now, ,m(w contains CI. for m m beo o) cause of the primitivity of " and Be ,m+p(w I. Writing o n n n n , (w) "'" (w o ) ­ ' (w o ) ­ ' (wl .. = YOYl.. · Y )1_ C1. o Cl. 1 · .. 1 I,n(w o we get, iterating (2) , (3)

'0

n x,

(w) =

(Y ... Yo- I) (Y I•• ·Yo)··· (Y 0 '" '" I

,n

(to )

o

1-1

o"

Ct

.Ct o _

'"

2) •

97 contains as "letters", all the words of length

We notice that

,

1;;n (w

in

o

) ' Taking

contains

and the lemma is proved

B

that

n = m+p, we see

large enough, and

m

0

We achieve now the proof of the proposition V.10. Applying to the substitution

, the preceding proposition, we obtain readily L

lim n.... OO

n

B

(w)

I

But from (3), when W

o

n --->

(w))

00

I

> 0 , independent of

= dB

w.

=

Finally

•

' and the proof is complete.

Example 1;;(1)

Let

u

be the Morse sequence.

= 10 . M(1;;)

M =

possesses

values. We shall describe

1;;2

u = 1;;(U)

e

= 2

with

and

on the alphabet

A = 0

1;;(0) = 01, as eigen-

= {(OO),(Ol) ,(10),

(11 ) } 1;; (00)

= 0101

so that

1;; 2 ( (01) )

(01 ) (11 )

1;; 2 ( (10) )

(10) (00)

1;; 2 ( (11 ) )

(10)(01)

1;;2«00))= (01) (10)

Thus, the composition matrix

M 2

of

0

C

M 2

1

0

0

1

1

0

1;;2

and also,

has the form

:)

The eigenvalues of M are e 2, A = 0, 1, -1 . The normalized, 2 1 1 1 1 positive eigenvector, corresponding to e , is v = (6' 3' 3' 6) so that, the frequencies of the pairs are respectively d(OO) Remark:

=

1

6

= d(11)

We shall see later, that, in the general case, the dominant

eigenvalue of

is always

e

(the Perron-Frobenius eigenvalue of 1;;).

98

We shall derive from this observation, an algorithmic method to evaluate the frequency of any word occurring in If

B

is any word in

generated by

u.

u , we already defined

[B] to be the cylinder

B, that means

o

b.

J

jBj - 1J

j

if

If

M(u)

£

, is a

T­invariant probability measure,

is defined

lim 1 Card{n < N j j .... oo N:J

B}

by lJ.(

for some sequence

, and for every

(Nj l

u n···un+IBI­1

t

[BJ

The last proposition asserts that M(u)

if

a

£

belongs to

A .

We are now looking forward conditions on the

T­invariant measure

lJ.

on

X(s)

s

ensuring the unicity of

• The following has been proved

by P. Michel [78J Theorem V. 13.

the system

(X,T)

is minimaZ J

it is uniqueZy ergo-

die.

Proof

We have to prove, more precisely, for any word

B

of

u,

that lim

N.... oo uniformly in to compare If

N»

where

n

B

0

So we get

(5)

k. In order to make use of the proposition V.10, we try uk .•. u k+ N

t

sn(w)

to a word

for some

w

£

A*

and

n

1.

we may decompose :

and

B 1

N + 1

are words of

IBol

+ /B 1/

, whose length is

u

+

j +£ L:

i=j

I sn (u i) I

s

SUPlsn(all=r. a n

99

and

L B (uk'" u j +R.

-

lJ, ([B])

,l:.

(N+1 hi,( [B] )

-

k+N)

I;;;n (u i )

I• L

(X,T)

Since the system lJ, ( [B] ), for every

is assumed to be minimal, lim n.... cc , and thus, for any fixed e > 0

i

n

B

(;;; (u ) ) i

I ;;;n(u i) I

(6)

for

n

large enough.

According to

(5), we deduce from (6)

,

On the other hand, LB(Uk···Uk+ N )

) +L B(B 1

j +R. +,l:,

where the last sum takes into account the overlaps of n

intersects

;;; IU j +R. +1 ) ;;;

n

n

1

• B

;;;

(u, 1)

in order to get (6) and then

composition (4) of

B

onto two

intersects

times, so finally, we obtain the majorization

(u + at most i 1),

We choose

B

n

Bo and Jthe words ;;;n(u and may intersect i)

consecutive words, admitting that

uk",u + k N

N»

n , to have the de-

as also the estimation

Inf I;;;n (a) I ae:A Applying (5),

I

IBI

e:(N+1)

(R.+1)

LB (uk' .. uk N) N+1 + - lJ,([B])

The theorem is thus proved. In this case, for every

I

, and

5. 2e:

4r +

uniformly in

k.

0

f e C (X)

,

Jf d u

lim N+'"

l: f(Tnu) n 1

requires the recogniza-

Z;;k. This can be easily deduced is assumed to be an homeomorphism

Z;;(X). Suppose

Z;;

is an homeomorphism from

is recognizable for every

We proceed by induction on

one­to­one for every

X

onto

r

since

k • We prove easily that

k > 1. Suppose now

z:;k

Z;;(X)

•

k > 1 .

z:;k

Z;;k

is

recognizable with index

Kk ' writing K instead of K . We can find K K 1 k k+1 for every x,y e: X, X[O,K] = y[O,KJ i f Z;;k(x) [O,Kk+1J Z;;k(y)[O,Kk+1J

X ,

£

i

may be interpreted in the following way

bility property for the substitution

Lemma VI.5.

n

0

and the proposition.

z:;(X)

i f and only i f

onto

so that, for

for every

and

The proof of the corollary for

X

Tniu

(T

j

n i + ). ,

Z:;(X)

= lim

K . x

and

property. This proves

from

X.

such that,

is an homeomorphism

Suppose now that u[n,n+Kk+1J u[m,m+Kk+1J and already because K and n e: E . Thus K k+1 k k

n e: Ek+1

• m

£

Ek

112 m

T u

> 0

for some

and (TPu) for some

>

p

0

. From above, £

[O,R] '"

E • We deduce

which means

m

[O,R]

'"

for some

k 1 .

r

> 0

and

K,

Tmu =

(Tru)

0

E +

£

and, by definition of

Induced transformation Let

be a dynamical system, and

> 0 . We define

with y

Y

a measurable set in B,

Ty ' the transformation induced by

T

on

, by

Ty (y) '" Tn (y) (y) where

n(y) = inf{j

This function for

> 0 , Tjy

y ---> n(y) every

If, in addition,

Jy where

y

T

Y

y}

£

Y

£

is called the return time in

Y .

By-measurable, By = B nY, and finite

is

Y (from the Poincare's recurrence theorem).

£

is an ergodic automorphism n(y)

By

is the probability measure defined on

by

We shall not need this result known as Rae's lemma. Let us turn back to the system (X,Il,T) associated to L; strict ergodicity of the system, u gives positive measure open set on X (proposition IV.5 (X)) > 0 and when cognizable. We denote by S the induced transformation by L; (X) • Proposition VI.6.

Y

If Sy '" T

Proof:

y = L; (x)

If

= L;(X)

I L; (xo)

r

T

£

,(X)

L;

is re-

T

on

for some

(y)

I L; (xo) I I

i

E (1) . If

to any

j

have just to prove that !L;(x o) n, Write x = lim T lU so that y £

From the

, since

•

=

(y)

L; (Tx)

belongs to

> 0 , T Y

m. lim T lU , where i

m. lim T l

E:

L; (X) . We

L; (X)} .

m.=jL;(u ... u n.- 1)1 l 0 l

u , by the corollary

113

VI. 4 ,

m. +R.

E

E:

1

for

m.+R.

large enough • Thus

i

I;; (uo ••• u n +p- 1) I

1

mi

I;; (u n

+

i

••• u n +p- 1) i

inf{R. > 0

and large

I;;(u n)

1=

p

I for sufficiently

1;;(xo)1

1

0

i.

for some

i

We deduce from the above proposition and the identity (1), the following commutative diagram T

x

S

;;(X)

We suppose now, that

;;

sense that the words

;;(a)

ters

distinguishes different letters of and

a, b • We should say that

;;(b)

are distinct for different let-

is one-to-one on

;;

A, in

vestigate the structure of the system

(X(;;),T)

(primitive - recognizable - one-to-one on

A

and we in-

for such substitutions

A) that we might call

admissible substitutions, if necessary Proposition VI.7. To;;

by

=;;

Sk (y)

0

=T

Sk

;;

If

homeomorphism from k k

where

I ;;k(xo ) I

(y)

;;k satisfying

is an admissible substitution.

X onto

, for

;;k(X)

k

1 •

Sk , the return time to

if

;;k(X)

is an

, is defined

y

Proof: If we prove that ;; is an homeomorphism from X onto ;;(X), ;;k will be recognizable and satisfy the proposftions VI. 1 - VI. 6, as remarked in lemma VI.4 . Let

x, x'

X

E:

such that

;:;(x)

=

;:;(x')

= y.

I;;(x) I = I;:;(x') I R. and o 0 • It follows from our assumption,

According to the preceding proposition, thus that

= YoYl ••• y ,Q, - l

i:;(x o) =

xo

= x'0

xi,···

'X n

and applying a similar argument to -:

and

x = x'.

The systems topologically isomorphic. denoting by re u under z,

Corollary VI.B.

TR. y

E: ;:; (X)

, we get

0 and

(;:;k(x),

Sk)

are

• the image of the measu-

114

Proposition VI.9.

e,

where

If

is the restriotion of

as usual, designs the

to

eigenvalue of

s.

is the unique S-invariant probability measure on sIX) is S-invariant too, necessarily v

Proof

by the corollary VI.8. If

C·v

(u)

A*

Let

Ul

But

T [Ul]

E:

I;;

0

C , which is easily seen to be

for some constant

and consider the cylinder

T [Ul]

=T

11; (Ul o ) I

)

I

0

in

s (X)

From this ,

I; ([Ul] ).

I s (to

([Ul])

1;([Ul]»

=

that is

v (S I; ([Ul] ) )

which was to establish It remains to compare Lemma VI.10. letter

Let

a

(s E:

k

and

and [a]

e. We shall need the the oylinder of

x

E:

X , with first

a. Then

Proof of the lemma: der y = (x) where Ty

A

(X»

= T k I; (x)

Suppose x E: [S]

begins by

a

(xo)k

=

with

I

k
.. E jn lim >..q (15) , for some = z and n-+oo

Writing

>..2nit

and

satisfies 1

.

z = e 2n i, , it follows from (15) that

>..

is

125

lim qjn t _ T(modulo 1) and the decomposition of

t

_a_ q

b

+

jn

=

m

0

T

=

( 16)

b

(qL 1 )

o

for some integer

h

A

£

, and

hT

=0

(modulo 1) implies

m < h • This leads to

_a_

t

q

and

with

jn . q o(qJ_ 1)

We deduce from that T

in the

...!.- - adic

expansion must be ultiqJ Hence we are allowed to decompose

mately stationary.

t

c

. +-h Jn o

Z(q) x Z/hZ , is a continuous eigenvalue (theorem VI.14)

We have thus proved the theorem and the corollary

•

0

Remark: This proof can be shortened, in particular it is possible to establish the convergence of (Aqjn) , when j satisfies only (12), by analysing properties of

h. This last improvement can be obtained

also as a consequence of the forthcoming theorem, established by B. Host for general substitutions, and gives rise to a new property

of the height

h.

Corollary VIo18.

h

divides

qj ­ 1

where

j

is the period of occur-

rences of the first letter in Proof:

We mean

the last theorem

h with

divides qj­l where j satisfies (12). Apply 2n i/h , continuous eigenvalue, and j A = e

as above qjn(qLl) lim A n+ oo and

qjn(qL1)

Since

£

hZ

gcd(h,q) = 1

for

,

h

n

lim e n+ oo

2ni qjn(qL1)/h

1

large enough.

divides

qL 1

.

IJ

VI.3. Eigenvalues of an admissible substitution of non­constant length Assume

is, what we called an admissible substitution, but no more

of constant length.

126

Definition VI.2.

The function

if, for every word of

u

h is a cobord of s if and only if there exists such that g(b) = g(a) .h(a) for every word ab of u.

Lemma VI.19. 9

: A ---> T

Proof

The sufficiency is evident. Now let

and put

g(u n) = h(u

pears in any

n

... h(u

o)

is a function

9

and

u

is called a cobord of

h: A ---> T

n_ 1

)

A ---> T

. If

un

If

ab

h

be a cobord of

urn' m

n , clearly

is a word of

with bounded gaps, from the minimality of

with

u

n-l

s

u,

ab

ap-

(X,T). For

= a

g(b)

=

g(u

n)

=

g(a)h(u

=

n)

g(a)h(b).

Note that every cobord is constant, equal to

0

1 , when the alphabet

A = {O,I}

B. Host proved in [40J the following result

Let

Theorem VI.20.

j

s

an admissible substitution over

a)

Every eigenvalue of

bJ

A

::: 1

(17)

8

T

s

A.

is continuous

is an eiganvalue of

s

if and only if there exists

such that h(a) = lim A1sjn(a)

I

n-+co

being a cobord of

h

Proof:

exists for every

a

8 A

,

s.

We give only the main steps of the proof which goes as fol-

lows : A.

if

A

satisfies (14) for some

j,

A

is a continuous eigen-

value. B.

if

A

is any eigenvalue,

A

First of all, observe that, for every

satisfies (14) for some precise j. a

£

A ,

127

where

tM

is the transpose of the composition matrix

M, and

e

the s-dimensional vector (1 ••. 1), so that

if

t

denotes the

s-dimensional vector

If necessary, we may consider (18)

I

sj

(t ••. t)

instead

converges if and only if

ZS

and this holds if and only i f

to

0

s

.

and omit the integer j.

(tMn. t )

converges modulo

11: -

converges

11:)

modulo

Moreover the convergence is easily seen to be geometric If

(19)

then, from (18)

n+co

for some constant (20)

C

and

Suppose now that

0 < r < 1 , independent of h , defined by

s A

n+co

S, and let

is a cobord of to

h

r

g : A ---> T

be the function associated

by the lemma VI.19. We exhibit a continuous eigenfunction

corresponding to If

m.

in the following way

A

x s X , let

r

n

n

(x)

Fix

be the smallest integer

r

T x s s (x). Define now the continuous function f -r (x)

A

(the function

r

n

9 (a)

if

x s T

n

n

-r (x)

n

0

such that on

X

by

sn ( [a] )

being continuous) .

n

Applying (19) we get k Ifn(Tku) - A fn(u)

I

C

r

n

which ensures the uniform convergence of the sequence continuous function

f

satisfying

(f

n)

to a

f

128

(21)

To derive the similar property to the lemma VI.IS

and its

consequence (9), we need the following. k = {T r,;n([a}) , aEA, O;;;'k< !r,;n(a)!} n a-algebra generated by P

Recall the metric partition and consider

Bn

Lemma VI.2l.

a(

cyZinders and,

the

P

n

U Bn )

=

n

B , the

13

B E n f

Proof:

generated by the

f E LI(X,Jl)

for every E n f

where

X

o-aZgebra on

f

converges to

in

I L (X,)l)

designs the conditional expectation of

f,

given

B

n

This lemma shall be used later to give an estimation of the

spectral multiplicity of the system

(X(r,;),T), so we give the comple-

te proof. G:!l,ren

E > 0

u (B

such that Consider tant on

and !l

n

f = :n.

a word of

W

[w]

and

f

and

B

B E n f , n

n

I

• If

B

E l'

f

n

n

EP n

is cons-

n

with value

B

f

,n([a]) , a

E

)l(Bn

f du

A , with

[wJ)

)l (B)

B

In particular, writing T

0

n

[w]) ::. E .

JllB)

k

u , we shall find

fn f p_ 1 k:::. qn_ p • From this

W =

wOwl···w

on every

B

L:

aEA (p-l)e- n nius eigenvalue of Choose on which

where

e

is the Perron-Frobe-

r,;.

k Bn ' the union of the T r,;n 1 to

f , when

129

f = :n. [w]

. We deduce from above

II f

n

-fll 1 L

(ll)

:::. 211 fll oo ll{f

n

t- f }

:::. 21 wi 8- n

0

and the lemma is proved. (22)

Let

f

z;; by z;;j

Replace

where

satisfies (12)

j

f

Denote by dn(a) the constant value of From the preceding lemma I lim en -+ n 00

and thus

lim Idn(a) n-+ oo

(23)

Z;;(a)

If

A,

be the eigenfunction corresponding to

I

J

1f - f

Z;;n ([a] )

= 1

S,

B

= E n f

n

Idu

for every

I

begins by

n

but let us write

I

on

If I =1.

with Z;;

Z;; n ( [a])

.

0

a

A •

1;n+1([a]) c 1;n([S]) In::: 1

I

and as

a consequence of (22) 1

lim d n+ l (a) /d (S)

n+ oo

(24)

n

.

We already assumed the period of appearances of the first letter in Z;; to be 1 • If a k+ 1 = 1;(a k)o I a j+1 and then applying tWice (23) we obtain lim d (a) /d (a) n-+ oo n+ 1 n

w = as be a word of u I of length 2 • ¢n ([w] ) C and T k 1;n ([w]) C 1;n([S] ) i f k= Iz;;n(a) I • But

(25)

f and

f

n

n

= A l1;n (a)

d (S ) n

on

I

d (a)

on

n

1; n ( [S])

.

Combined with the lemma VI.21 , we deduce lim Idn(S) - A1Z;;n(a)I dn(a)1 n-+ oo also

for some

A .

a

1

aj

1

0

.

j

130

(26)

Finally

•

I =

A

using (24), and (18) implies It remains to prove that steps,

h

fore,

lim dn(a) /d (S)

n-+oo

example,

as

n

g(a)

=

h(a)g(a) = g(S)

lim dn(a) /d (0)

n-+oo

if

is a word in

as

u, and there-

a, S

A . Put for

, it is easily checked that

n

is a word of

The proof is achieved.

A ,

. From the two last

exists for every letters

n

a

heal •

n+ oo

is a cobord of

lim dn(a)/d (S) exists if

n+oo

1 , for every

u, and

h

is a cobord of

0

Applications to mixing property M. Keane and M. Dekking in [21J studied mixing properties of dynamical systems arising from substitutions. In particular they proved the following. Theorem VI.22.

The dynamical system arising from a primitive substi-

tution is not strongly mixing.

Proof: phabet

For sake of simplicity, we restrict our attention to the al{O,l} . We suppose that

([llJ )

is

> 0

X

is infinite and thus

p([OOJ)

or

r = 1.1 ([OOJ) > 0

for example, and put -s sn = I I. I f w is a word in u , consider Dn = [w] n T new] If the system were strongly mixing, we should have lim 1.1(D ) = 1.1 ([wJ)2.

1.1

Let

On the contrary, the authors establish

n+

oo

n

(27)

for some constant

e > 0 , independent of

w

strong mixing property, choosing Recall the notation Le(B) word

e

in the word

B

i

to design the occurrence number of the and let

N»

appears in (0) if the word D n word e , leI = sn • Clearly L

w

w, which contredicts the

long enough •

< L

-

w

C

w

n •

wCw

appears in

(0), for some

131

L

and

;::, L

w

w

C

w

«(;n(OO».L

(l;n (0»

.L

OO

(;n(OO)

«(;N(O»

(l;N-n (0»

which infers

eN where p is sN is asymptotically equal to p a positive constant and e , the Perron-Probenius eigenvalue of I;; s 1 L N-n (I;;N-n (0» Therefore , lim u ([OOJ ) l'l.m 00 sN N-+oo sN

As already seen,

.

r. and

lim j1(D n)

r lim

n+ oo

n+oo

::: rp.j1 which proves (27)

with

e-n

([w] ) 0

C = rp

In the same article the authors provide examples of weakly mixing system arising from substitutions. This will be easy as a consequence of the theorem VI.20. Corollary VI.23. (;(0) = (01010)

The substitution (; defined on {O,I} by and (;(1) = (011), gives rise to a weakly mixing dy-

namioal system. Proof:

I;;

is an admissible substitution, and the integer

ned by (12) equal to

is equal to

1. Now, if

I;; • Prom the theorem VI.20, A ll;;n (0)

lim n-+oo Setting

A

1 • We shall see that

r

n

ll;;n (1)

sn+I

3s

r n+ 1

s

which leads to

n

n

I

and

+ 2r + 2r

n n

h 1

is a cobord of

(;,

defi-

j

h

A is an eigenvalue if and only if

I = s

n

lim n-+co

is

is the unique eigenvalue of

A I (;n (1)

I (;n (0) I

I =

1

, we have

132

r

s

e

If

27Tit

i(2.4

n

n

+ 1)

l(4 n + 1 _ 1)

3

n

lim A

implies

1

n

4t t 3 - 3 (modulo 1) and

t

O.

What can we say more precisely about this system ?

Remark:

We do not know how to describe the maximal spectral type of such a substitution (which shall be possible in the constant length case) but, if

is the correlation measure of the occurrences of

0

the sequence

0

in

u , defined by o(k)

O}

lim Card{n+K < N , u n+ k N+oo

we can establish the identity lim n+ oo

o(s n + k)

1

3(20 (k)

which means that there exists Nevertheless, i t seems that 0

such that

,

X0= 3 +3 Y

[I (Chapter I).

VI.4. Discrete substitutions of We proved in the section V1.2

a(k+l»

is not a Dirichlet measure and even

0

could belong to the ideal

reo)

X s

+

length that a substitution of constant length

always possesses a discrete part in its spectrum, which only depends on

q, the length of

S, and

h , the height of

s . M. Dekking gave

a characterization of the substitQtions of constant length with (purely) discrete spectrum. Definition VI.3.

Let

A = {O,l, ... ,s-l} . eXlst

k

an d

s

be a substitution of length

We say that

J' < q k

s

admits a cotncidence, if there

sue h t h at

sk

(S-l)j

(that means,

sk

admits a column of identical values) .

Example:

on

{O,l,2}

s

If we superpose

q , defined on

defined by

s(O)

11,

sO)

21, s(2)=10.

133

1;2(0)

1 2 1 2

1;2(1)

0 1 1 2

1;2 (2)

1 2 1 1

the third column gives the coincidence We suppose

is discrete i f and only i f

I;

Suppose that

plicity. :e

j

(1), ••• ,1; n

I;

easy to establish, for

2)

=

q.

admits a cofncidence.

I;

admits a coIncidence with

I;

then b Y

(0),

blocks

and

is an admissible substitution of length

I;

Theorem VI.24. Proof:

(k = 2

k

=

1, for sim-

Cn ' the number of coincidences in the By our hypothesis,

n

1 ,

C

I

1 • It is

(28) since a coIncidence at order n+I

.

, leads to

n

coincidences at order

q

Resolving the recurrence equation Cn+I

by putting

C

q

n

+ (q­I)C

C

n

1

1

,

we get

n

1

q­l so that

and

Finally, under the hypothesis

C

qn _ (q­I) n

n

C

I

1 ,

(28) infers

We shall deduce from this estimation that the correlation measure of the sequence

u

is a discrete measure, and shall conclude with the

proposition rV.21. Definition VI. 4 .

We say that a sequence

dic if, for every

gers

£

(un)

is mean-almost perio-

> 0, we can find a relatively dense set of inte­

such that

(29) Suppose that

1

n+ oo

(un)

N

if

k £

admits a unique correlation measure. There is then

a simple characterization of the property (29).

134

Lemma VI.25.

u = (Un)

correlation measure

Proof

0

is mean-almost periodic i f and only i f the

u

of

is a discrete measure.

From the Dunkl-Ramirez inequality, regarding

it is readily proved that,

P E M(T)

is discrete if and only if

0

,

(o(n»nEZ

is an almost periodic sequence, or equivalently

(30)

E > 0 , we can find a relatively dense set of inte-

for every gers

EE ' such that sup 10 (n) - o (n+k) I nEZ

s

.

It remains to compare (29) and (30)

E

k E E

if

E

Since

2 IUn+k-UnI2 = l: (Iu +kI2+lu 1 ) - 2 Re l: u n+ k u n n : A --> 'll' } , which clearly contains : X --> G

defined by

xCF¢) = F¢CX)

D, and wonder when

, provides a metric isomor-

phism. For simplicity, suppose that the only cobords of

1;

are trivial ones

and consider additive generalized eigenfunctions. Recall that t M group of JRs H =

Lemma VI.28.

is the transpose of matrix

Iv e JRs , tMn v F¢

tends to

0

M. We define the sub-

modulo

:r;s} •

is an additive and aontinuous generaZized eigenfuna-

143

i.i on of 'longs to

s-dimensiona'l veator

if and on'ly if the

($ (a) )aE:A

be-

H

We omit the proof. Note that

H

to