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LUND UNI VERSI TY PO Box 117

The Family of Belykh Maps

Persson, Tomas

2005

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Citation for published version (APA):

Persson, T. (2005). The Family of Belykh Maps. [Licentiate Thesis, Mathematics (Faculty of Engineering)].

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1

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(2)

T h e fa m ily o f B e ly k h m a p s To m a s P e rs so n 2 0

T HE FAMILY OF

B ELYKH MAPS

T OMAS P ERSSON

Centre for Mathematical Sciences Mathematics

C E N T R U M S C IE N T IA R U M M A T H E M A T IC A R U M

(3)

T HE F AMILY OF B ELYKH M APS

T OMAS P ERSSON

Centre for Mathematical Sciences

Mathematics

(4)

Mathematics

Centre for Mathematical Sciences Lund University

Box 118

SE-221 00 Lund Sweden

http://www.maths.lth.se/

Licentiate Theses in Mathematical Sciences 2005:1 ISSN 1404-028X

ISBN 91-631-6468-X LUTFMA-2014-2005

Tomas Persson, 2005 c

Printed in Sweden by KFS, Lund 2005

(5)

Inthisthesiswestudya lassofnon-invertiblepie ewiseanehyperboli sys-

temswith dis ontinuitiesintwodimensions. This isaspe ial lassofsystems

but itree ts manypropertiesof moregeneralnon-invertiblehyperboli sys-

tems.

Foraspe ialsubset ofparametersthesystemis espe ially simple. Inthis

asethesystem redu es to aone-dimensionalsystemand methods from one-

dimensional dynami s an be applied. We lassify the ergodi properties in

termsof theasso iatedsubshift and thenumber-theoreti alpropertiesof the

parameter.

WeshowthatforanopensetofparameterstheSinai-Bowen-Ruellemeasure

isabsolutely ontinuouswithrespe ttoLebesguemeasureandthe orrelations

ofHölder ontinuousfun tionsde ayexponentially.

(6)
(7)

I want to thank the sharp-minded Jörg S hmeling, my supervisor, for great

support and en ouragement. He hasmade my days interestingand joyful. I

alsothankmy o-supervisorSergeiSilvestrov.

Finally,I wantto thankmyfather,HansPersson. Hetaughtmethemost

importantthingaboutmathemati sitisfun.

(8)
(9)

1 Introdu tion 9

2 Restri tionto one dimension 15

2.1 Denition ofthesystem . . . 15

2.2 Theone-dimensional ase . . . 15

2.2.1 Asubshift withtwokneadingsequen es . . . 15

2.2.2 Classi ationofthesubshifts . . . 17

2.2.3 Invariantmeasures . . . 21

2.2.4 Ergodi properties . . . 27

2.2.5 Conne tiontoalgebrai numbers . . . 29

2.2.6 Distan eto thesingularityandreturntimes. . . 35

2.2.7 De ayof orrelations. . . 39

3 Absolutely ontinuous invariant measure for a lass of pie e- wise anehyperboli endomorphisms 43 3.1 Introdu tion. . . 43

3.2 The lassofendomorphisms . . . 44

3.3 Absolutely ontinuousinvariantmeasure . . . 47

3.4 Proofofthetheorem . . . 47

3.5 De ayof orrelations . . . 55

(10)
(11)

Introdu tion

Thisthesisisastudyofa lassofnon-invertiblehyperboli mapsonthesquare,

alledtheBelykhsystems. Thesesystemsarepie ewiseaneand hyperboli .

ThesimpleformoftheBelykhmapsmakethem easierto workwithandthey

arehopedtosharemanypropertieswithmoregeneral lassesofnon-invertible

hyperboli systems.WehopethatthestudyoftheBelykhmapswill ontribute

to a better understanding of non-invertiblehyperboli systems and that the

methods anbegeneralisedtoabroader lassofsystems.

In [27℄, Pesin studied a general lass of pie ewise dieomorphisms with

a hyperboli attra tor. He showed the existen e of the Sinai-Bowen-Ruelle

measure,or

SBR

-measureforshort,andstudied theergodi propertiesof this

measure. If

f : M → M

is thesysteminquestionthen the

SBR

-measure isa

weaklimitpointofthesequen eofmeasures

µ n =

n−1 X

k=0

ν ◦ f −k ,

where

ν

denotestheLebesguemeasure. Thismeasureisthephysi allyrelevant

measureasit apturesthebehaviouroftheorbitsofpointsfromasetofpositive

Lebesguemeasure.Pesinshowedthatthe

SBR

-measurehasatmost ountably

manyergodi omponents. Foramorerestri ted lass,Sataev[29℄showedthat

thereareonlynitelymanyergodi omponents. In[30℄heusedthisresultto

provethat under a onditionon the parameters, the Belykhmap is ergodi .

S hmeling and Troubetzkoy studied in [33℄ a moregeneral lass than Pesin's

andprovedtheexisten eofthe

SBR

-measure. Theirmethod todealwiththe

non-invertibility of the system was to lift the system to a higher dimension

and get an invertible system on whi h the al ulations was made. In this

waymethodsfrom invertiblesystems ouldbeused. Theresult ouldthenbe

proje tedba kto theoriginalsystem.

Amongtheabovementioned lassesaretheBelykhsystems. Thesesystems

wererststudiedin[5℄asamodelofthePoin arémapofasystemofdierential

equations omingfrom the study of phase syn hronisation. In [33℄ and [32℄,

S hmeling and Troubetzkoy studied the Belykh systems for a wider lass of

parameters. These systems are espe ially simple but it is hoped that they

ree t many interesting properties of Pesins lass and that the method used

forBelykhsystems anbegeneralisedtoinvestigateabroader lassofsystems.

The Belykh map is dened as follows. Let

Q = [ −1, 1] 2

and dene the

(12)

Belykhmap

f : Q → Q

by

f (x, y) =

 (λx + (1 − λ), γy − (γ − 1)),

if

y > kx, (λx − (1 − λ), γy + (γ − 1)),

if

y < kx,

wheretheparametersare

0 < λ ≤ 1

,

−1 < k < 1

and

1 < γ ≤ 1+|k| 2

. SeeFigure

1.1. Inthisworkwewillstudyasimilarmapwiththeonlydieren ethat the

singularity set is the set

([ −1, 0] × {−k}) ∪ ({0} × [−|k|, |k|]) ∪ ([0, 1] × {k})

insteadof

{y = kx}

,thatisweapproximatetheline

{y = kx}

withapie ewise

onstant urve.

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111

00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000

11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111

00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000

11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111

Figure1.1: TheBelykhmapfor

γ = 3 2

,

λ = 3 8

and

k = 1 4

.

Weshowthat whenthemapexpandsarea(

γλ > 1

)then thereisanopen

set

P

ofparameterssu h thatthe

SBR

-measureisabsolutely ontinuouswith

respe tto Lebesguemeasurealmost surelyiftheparametersarein

P

.

There are similar results in the literature. In the ase when

k = 0

and

γ = 2

thesystemisthefatbaker'stransformation, studiedbyAlexanderand Yorkein[1℄. Inthis asethemapistheprodu t ofitsproje tionsto therst

and the se ond oordinate. The proje tion on the se ond oordinate is the

two-shift. Thissimpliesthe al ulationsandthe

SBR

-measureistheprodu t

of the one-dimensional Lebesgue measure and a Bernoulli onvolution. The

resultofSolomyakin[35℄impliesthatforLebesguealmosteveryparameterthe

fatbaker'stransformationhasan

SBR

-measurewhi hisabsolutely ontinuous

withrespe ttotheLebesguemeasure. AlexanderandYorkeshowedthatif

λ −1

isaPisot numberthenthe

SBR

-measureissingularto theLebesguemeasure,

sin ethentheFouriertransformationoftheBernoulli onvolutiondoesnottend

(13)

Lebesguemeasure,[11℄.

Inthe aseofexpandingmaps,thatismapsthateventuallyareexpanding

ineverydire tion,mu hisknown. Buzzi,[3℄,andTsujii,[36℄showedindepen-

dentlythatanyexpandingpie ewiseanalyti mapoftheplanehasanabsolutely

ontinuousinvariantmeasure. InhigherdimensionsTsujii[37℄showedthatany

expandingmapwhi hispie ewiseaneonnitelymanypolyhedralpie eshas

anabsolutely ontinuous invariantmeasure. Buzzi showedin [2℄that almost

anyexpandingmap whi h ispie ewiseaneonamoregeneraltypeof pie es

hasanabsolutely ontinuousinvariantmeasure.

Let

A

be anite set and allit analphabet. A wordis anelement ofthe

set

A = {a 0 a 1 · · · a n−1 | a i ∈ A, n ≥ 0}

and

A

is alledthelanguageof

A

. Alanguage

L

on

A

is asubsetof

A

.

Let

A N

betheset ofallinnitesequen esofelementsin

A

. Wedenethe

map

σ : A N → A N

by

σ : {a k } k∈N 7→ {a k+1 } k∈N

. A ylinderisasetoftheform

k [a k · · · a k+l ] k+l = {b 0 b 1 · · · ∈ A N | b i = a i , ∀i = k, . . . , k + l}.

Asubset

S ⊆ A N

issaidtobeasubshiftifitisinvariantunder

σ

and losed

inthetopologygeneratedbythe olle tionofall ylinders. Wesaythataword

a 0 a 1 · · · a n−1 ∈ A

is allowed if there is a sequen e

{i k } ∈ S

and an integer

m ≥ 0

su hthat

a k = i m+k

for

k = 0, 1, . . . , n − 1

. Thesetofallowedwordsis

alledthelanguageof

S

.

InChapter2we onsiderthespe ial aseofthe Belykhmaps when

k = 0

and

γ

and

λ

are arbitrary. In this ase the dynami s depends only on the

se ond oordinate and we therefore study the dynami s of the proje tion to

these ond oordinate. Themap

T : [ −(γ − 1), (γ − 1)] → [−(γ − 1), (γ − 1)]

is

thenthefollowing.

T (x) =

 γx − (γ − 1)

if

x > 0, γx + (γ − 1)

if

x < 0.

Thegraph of

T

is in Figure 1.2. Bya hange of variables

T

anbewritten

in the form

x 7→ γx + α (mod 1)

were

α = 1 − γ/2

. This is similar to the

β

-expansion,

f β : [0, 1] → [0, 1)

,

f β : x 7→ βx (mod 1)

, introdu ed by Rényi

[28℄in the ontext ofexpanding numbersin non-integerbases, seegure1.3.

The theory was further developed by Parry in [22℄, where he des ribes the

asso iatedsubshift the

β

-shift,dened belowin termsoftheorbitof

1

.

Healsoprovedtheexisten eofanabsolutely ontinuousinvariantmeasureand

al ulatedthetopologi alentropy.

Let

[x]

denote the integer part of the number

x

and let

{x}

denote the

fra tionalpartof

x

. Let

β > 1

. Forany

x ∈ [0, 1]

weasso iatethe sequen e

(14)

−(γ−1)

(γ−1)

−(γ−1)

(γ−1)

Figure1.2: Thegraphof

T

.

0 1

1

0

Figure 1.3: Thegraphofthemap

f β : x 7→ βx (mod 1)

,for

β = 3 4

.

d(x, β) ∈ {0, 1, · · · , [β]} N

denedasfollows. If

d(x, β) = {i k } k=0

thenforea h

k ∈ N

wedene

i k = [βf β k (x)] = [β {β{β · · · {β

| {z }

k

x }}}].

The losure ofthe set of all su h sequen es isdenoted by

S β

andit is alled

the

β

-shift. Itisinvariantundertheleft-shift

σ : {i k } k=0 7→ {i k+1 } k=0

andthe

map

d( ·, β)

satises

σ n (d(x, β)) = d(f β (x), β)

. Ifweorder

S β

with thelexi o-

(15)

graphi alorderingthenthemap

d( ·, β)

isone-to-oneandmonotonein reasing.

Parry [22℄ proved that the map

β 7→ d(1, β)

is monotone in reasing and

inje tive. Forasequen e

{i k } k=0

thereisa

β > 0

su hthat

{i k } k=0 = d(1, β)

if and only if

σ n ( {i k }) ≤ {i k }

for every

n ≥ 0

. The number

β

is then the

uniquepositivesolutionoftheequation

1 = X ∞ k=0

i k x −k .

The subshift

S β

is theset of sequen es

{i k }

su h that

σ n ( {i k }) ≤ d(1, β)

forevery

n ≥ 0

. If

x ∈ [0, 1]

then

x =

X ∞ k=0

d(x, β) k β −k .

Anysubshift

S

anbe lassiedin thefollowingway.

Denition1.0.0.1. A subshift

S

is saidto be asubshift of nite type, SFT,

if the setof forbidden wordsis nite. Equivalently,a subshiftis of nite type

ifitisasso iatedwithanitedire tedgraphwithlabelledverti es,thatisthere

isanitedire tedgraph

G

withlabelledverti essu hthatasequen eisin

S

if

andonly if there is apath in

G

whi h yields the same sequen e by readingof

thelabelsof the verti es alongthe path.

Asubshift

S

issaidtobeso ifitisasso iatedwithanitedire tedgraph

withlabellededges.

A subshift

S

is said to be spe ied, or have the spe i ation property, if there exists a number

k

su h that if

a

and

b

are words in the language of

S

thenthereisaword

c

oflength

k

su hthat theword

acb

isallowed.

Notethatif

S

isof nitetypethenitisso . Therearesubshiftsthat are

notso and subshiftsthat arenotspe ied.

It is possible to hara terise the dierent types of subshifts

S β

in terms

ofthepropertiesofthesequen e

d(1, β)

andmake onne tionstothenumber- theoreti alpropertiesof

β

,seeforexample[7℄and[31℄fora olle tionofresults

onthesubje t. Amongotherresultsare

S β

is of nite type if andonly if

d(1, β)

either terminates with zeros or

isperiodi . [22℄

S β

isso ifandonlyif

d(1, β)

iseventually periodi , thatistheorbitof

1

under

f β

isnite. [6℄

S β

isspe ied if andonly if thereis an

n

su hthat there are no

n

on-

se utive zerosin

d(1, β)

. [6℄

(16)

If

β

isaPisot numberthen

S β

isso . [22℄

If

S β

isso then

β

isaPerronnumber. [20℄, [9℄

Themethodsfromthe

β

-expansion anbeappliedtothemap

T

withsmall

hanges. Wewilldes ribetheasso iatedsubshiftandgiveanalogousresultsto

thosementionedaboveforthe

β

-expansion. ThisisdoneinChapter 2.

InChapter3westudy a lassof mapssimilar totheBelykhsystems. We

prove theexisten e of an absolutely ontinuous invariantmeasure and prove

exponentialde ayof orrelationforHölder ontinuousfun tion.

(17)

Restri tion to one dimension

2.1 Denition of the system

Put

Q = [ −1, 1] 2

and

S = ([ −1, 0] × {−k}) ∪ ({0} × [−|k|, |k|]) ∪ ([0, 1] × {k})

.

Let

Q 1

and

Q −1

betheupperrespe tivelythelower onne ted omponentof theset

Q \ S

.

Considerthe lassofmaps

f : Q \ S → Q

denedby

f (x, y) =

 (λx + (1 − λ), γy − (γ − 1)),

if

(x, y) ∈ Q 1 , (λx − (1 − λ), γy + (γ − 1)),

if

(x, y) ∈ Q −1 ,

wherethe parametersare

0 < λ ≤ 1

,

−1 < k < 1

and

1 < γ ≤ 1+|k| 2

. These

aretheBelykhmaps.

2.2 The one-dimensional ase

Here we onsider the ase

k = 0

. In this ase the dynami s in the se ond

oordinate do not depend on the rst oordinate and the dynami s in the

rst oordinate are ompletely determined by that of the se ond. Hen e the

interestingdynami stakepla einthese ond oordinateandwethereforestudy

the proje tion of

f

to this oordinate. Let the map

T : I γ → I γ

, where

I γ = [ −(γ − 1); (γ − 1)]

bedened by

T (x) =

 γx − (γ − 1)

if

x > 0, γx + (γ − 1)

if

x ≤ 0.

Wehavedened

T

tobe

(γ − 1)

at

0

for onvenien e,butwe ouldjust aswell havedeneditto be

−(γ − 1)

.

2.2.1 A subshift with two kneading sequen es

Let

I −1 = [ −(γ − 1); 0)

and

I 1 = [0; (γ − 1)]

. Forany

x ∈ I γ

we asso iatea

sequen e

i = {i k } k=0 ∈ {−1, 1} N

dened by

T k (x) ∈ I i k

for any

k ∈ N

. Then

x

and

i

satisfy

x = γ − 1 γ

X ∞ k=0

i k

γ k .

(18)

We let

Σ γ

denote the losure of the set of all su h sequen es and dene the

map

π γ : Σ γ → I γ

by

π γ (i) = γ − 1 γ

X ∞ k=0

i k

γ k .

Theleft-shift

σ

isdenedby

σ( {i k } k=0 ) = {i k+1 } k=0

. Itiseasytoseethat

π γ (σ n (i)) = T n (π γ (i))

. Theset

Σ γ

isinvariantunder

σ

andishen easubshift.

Weendowthesubshift

Σ γ

withthelexi ographi alordering,denotedby

.

Be ause

T

ispie ewisemonotonein reasingthismakesthemap

π γ

monotone

in reasing.

Wedenote by

γ = {γ k } k=0 ∈ Σ γ

thesequen ethatsatises

γ − 1 = π γ (γ)

.

If we let

−γ = {i k }

denotethe sequen e su h that

i k = −γ k

for ea h

k

then

−(γ − 1) = π γ ( −γ)

and

Σ γ

istheset

Σ γ = {i | −γ ≤ σ k (i) ≤ γ, ∀k ∈ N}.

(2.1)

We will all

γ

the upper kneading sequen e and

−γ

the lower kneading se-

quen e.

Sin ewehavedened

T (0) = (γ − 1)

there isno

n

su hthat

σ n (γ) = −γ

.

It ishoweverpossiblethat

σ n (γ) = γ

. Ifwehad dened

T (0)

to be

−(γ − 1)

thenwewouldhavetheopposite ase.

Welet

Ξ : (1, 2) → {−1, 1} N

denote themap that maps

γ ∈ (1, 2)

to the

upperkneadingsequen eof

Σ γ

. Themap

Ξ

satises

π γ (Ξ(γ)) = γ − 1

.

Let

K

denotethesetofkneadingsequen es. Dene

P n = {γ ∈ K | γ = (γ 1 γ 2 · · · γ n )

forsome

γ 1 γ 2 · · · γ n }

andlet

P = S ∞

n=1 P n

. Forany

γ ∈ K

dene

L(γ) = sup {n ∈ Z | ∃k : γ k γ k+1 · · · γ k+n−1 = ( −γ 0 )( −γ 1 ) · · · (−γ n−1 ) }.

Let

K n = {γ ∈ K | L(γ) = n}.

Foranyword

A = a 0 a 1 · · · a l

and

k, l ∈ N

wedenoteby

k [A] k+l

the ylinder

setdened by

k [A] k+l = {i ∈ Σ γ | i m+k = a m , m = 0, 1, . . . l }

=

k+l \

m=k

σ −m ( {i ∈ Σ γ | i 0 = a k }).

Wewillusethenotation

[A] = 0 [A] l

.

(19)

2.2.2 Classi ation of the subshifts

Webeginbydening somedierenttypesofsubshifts.

Denition2.2.2.1. A subshift issaid tobe of nite type, SFT, if it isasso-

iatedwithanite dire tedgraphwith labelledverti es.

A subshift issaid tobe so if the language is asso iatedwith a nite au-

tomaton,thatis anitedire tedgraphwith labellededges.

Asubshift

S

issaidtobespe ied,orhavethespe i ationpropertyifthere existsanumber

k

su hthatif

a

and

b

arewordsinthelanguageof

S

thenthere

isaword

c

of length

k

su hthatthe word

acb

isallowed.

Inthis se tion wewill provethe followingtheorem, that hara terisesthe

threetypesofsubshiftsinDenition2.2.2.1intermsofthekneadingsequen e.

Theorem2.2.2.1. Thesubshift

Σ γ

isofnitetypeifandonlyif

γ

isperiodi .

The subshift

Σ γ

isso ifandonly if

γ

iseventually periodi .

If

γ > √

2

thenthe subshift

Σ γ

isspe iedif andonly if

γ ∈ K n

,for some

integer

n

.

This anbeformulatedequivalentlyin termsoftheorbitof

(γ − 1)

.

Theorem2.2.2.2. Thesubshift

Σ γ

isof nite typeif andonlyif the orbitof

(γ − 1)

isperiodi .

The subshift

Σ γ

isso ifandonly ifthe orbit of

(γ − 1)

isnite

If

γ > √

2

thenthesubshift

Σ γ

isspe iedifandonlyifthe orbitof

(γ − 1)

isboundedaway from

−(γ − 1)

.

Considerthe ase

γ = √ 2

. Let

A = h

− γ − 1 γ + 1 , γ − 1

γ + 1

i , B = h

−(γ − 1), − γ − 1 γ + 1

i ∪ h γ − 1

γ + 1 , (γ − 1) i .

Then

T −1 (A) = B

and

T −1 (B) = A

. Hen e

Σ γ

is notspe iedeventhough

γ = 11( −11) ∈ K 1

.

If

γ < √

2

then

f (A) ⊂ B

and

f (B) ⊂ A

and

Σ γ

an not be spe ied.

Howeverthereisno

γ ∈ (1, √

2)

su hthat

γ

isperiodi oreventuallyperiodi .

Let

A

be an alphabet and

L ⊆ A = {a 1 a 2 · · · a n | a k ∈ A, n ∈ N}

a

language. For

x, y ∈ L

wedenetherelation

by

x ∼ y ⇐⇒ (axb ∈ L

ifandonlyif

ayb ∈ L, ∀a, b ∈ A ).

Denition2.2.2.2. Thelanguage

L

issaidtoberationalifthequotientgroup

withrespe t totherelation

isnite.

(20)

Thefollowingtheorem anbefoundin[10℄. Itwillbeusedintheproofof

Theorem2.2.2.1.

Theorem 2.2.2.3 (Kleene). A subshift isasso iated with anite automaton

if andonlyif it'slanguage isrational.

We annowproveTheorem2.2.2.1.

Proof. Werstprovethatif

Σ γ

isso then

σ n (γ)

isperiodi forsome

n

.

Assume that

Σ γ

is so and

σ n (γ)

is not periodi for any

n ∈ N

. Then

thereexists aninnitesequen e

i 1 < i 2 < · · ·

su hthatthesequen es

γ i k γ i k +1 γ i k +2 · · · , k = 1, 2, 3, . . .

areunique. Considertwosequen es

γ i k γ i k +1 γ i k +2 · · · , γ i l γ i l +1 γ i l +2 · · · .

Withoutlossofgeneralitywemayassumethatthereisa

j

su hthat

γ i k = γ i l , . . . , γ i k +j = γ i l +j

and

γ i k +j+1 > γ i l +j+1 .

Thesequen e

γ i k γ i k +1 γ i k +2 · · ·

prolongtheword

γ 0 γ 1 · · · γ i k −1

butitdoesnot

prolongtheword

γ 0 γ 1 · · · γ i l −1

. Hen e thequotientgroupisnotniteand

Σ γ

isnotso . Thisprovesthat

σ n (γ)

isperiodi forsome

n

if

Σ γ

isso .

Wenowprovethatif

γ

isperiodi then

Σ γ

isofnitetype.

Thesubshift

Σ γ

isofnite typeifand onlyifit isasso iatedwithanite

graphwithlabelledverti es. Assumethat

γ

isperiodi . We onstru tagraph

in ordertoprovethatthesubshift isofnitetype.

Assumethat

γ

is

n

-periodi . Let

V = {a 0 a 1 · · · a n−1 | a 0 a 1 · · · a n−1

isanallowedword.

}

betheset ofverti esandlet

E = {(A, B) | A, B ∈ V, AB

isanallowedword.

}

bethe set of edges. Then asequen e is in

Σ γ

if and onlyif there is a orre-

sponding path in the graph

G = (V, E)

. Indeed, a sequen e

a = a 0 a 1 a 2 · · ·

is in

Σ γ

if and only if thewords

a k a k+1 · · · a k+n−1

,

k = 0, 1, . . .

are allowed.

This onditionis obviouslysatised foranypathin

G

. Furthermore, forany su h sequen e the words

a k+nm · · · a k+n(m+1)−1

and

a k+nm · · · a k+n(m+2)−1

,

k = 0, 1, . . .

,

m = 0, 1, 2, . . .

are allowedandthereforethereis a orresponding pathin

G

. Anexampleofthe onstru tionisin Example2.2.2.1below.

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Itisnowtimetoprovethatif

σ n (γ)

isperiodi forsome

n

then

Σ γ

isso .

Assumethat

σ n (γ)

isperiodi forsome

n

.

If

γ

isperiodi then

Σ γ

isofnite typeandhen eso . Assumethat

γ

is

notperiodi . Write

γ = α 0 α 1 · · · α m−1 (β 0 β 1 · · · β n−1 ) = γ 0 γ 1 · · ·

. Wemay

assumethat

α 0 · · · α m−1 > β 0 · · · β n−1

.

Foranyniteword

a 0 · · · a N −1

wedenethestate

(k, l)

;Put

k = max {j | a N−j · · · a N −1 = γ 0 · · · γ j−1 },

l = max {j | a N−j · · · a N −1 = ( −γ 0 ) · · · (−γ j−1 ) }

andlet

k = k

if

k ≤ m + n,

l = l

if

l ≤ m + n,

k = m + n + r

if

k = m + pn + r, p ≥ 1

and

l = m + n + r

if

l = m + pn + r, p ≥ 1.

Then

0 ≤ k, l ≤ m + 2n − 1

. Let

S

bethemap

(a 0 · · · a N−1 ) 7→ (k, l)

.

Let

V = {(k, l) = S(A) | A

isanallowedword.

}

be the set of verti es.

Denethesetofedges

E

by

E = 

(k 1 , l 1 ) → (k L 2 , l 2 ) | L ∈ {−1, 1}, a 0 · · · a s−1 L

isanallowedwordwith

S(a 0 · · · a s−1 ) = (k 1 , l 1 )

and

S(a 0 · · · a s−1 L) = (k 2 , l 2 ),

where

a 0 · · · a s−1 = γ 0 · · · γ k 1 −1

if

k 1 > l 1

and

a 0 · · · a s−1 = ( −γ 0 ) · · · (−γ l 1 −1 )

if

l 1 > k 1

.

Weprovethatthegraph

G = (V, E)

determinesthesubshift

Σ γ

. Observethat

theword

A(β 0 · · · β n−1 )B

isallowedifandonlyifthewords

A(β 0 · · · β n−1 ) i B

,

i = 1, 2, 3, . . .

are allowed. This implies that for any

a ∈ Σ γ

and any

j

the

state

S(a 0 · · · a j−1 )

is dened and from the vertex

S(a 0 · · · a j−1 )

there is a

uniqueedgelabelledwith

a j

goingtothevertex

S(a 0 · · · a j )

. Hen ethesubshift

determinedby

G

ontains

Σ γ

.

Conversely,let

a

be asequen edetermined byapathin

G

. Then forany

i = 1, 2, 3, . . .

the word

a i−s+1 · · · a i = ±(γ 0 · · · γ s−1 )

where

s = max {k, l}

,

(k, l) = S(a 0 · · · a i )

. Clearly

a 0

isanallowedword. Assume that

a 0 · · · a i

isan

allowedword. Theword

a i−s+1 · · · a i

isallowedandbythe onstru tionof

G

it

is learthattheword

a i−s+1 · · · a i a i+1

isallowed. Hen e

a 0 · · · a i+1

isallowed

andbyindu tion

a ∈ Σ γ

.

Thegraph

G

isobviouslynite so

Σ γ

is indeedso . SeeExample2.2.2.2

belowforanexampleofthe onstru tion.

We annowprovethat if

Σ γ

isofnitetypethen

γ

isperiodi .

(22)

If

Σ γ

is of nite type then it is so and

σ n (γ)

is periodi for some

n

.

Assume

γ

isnotperiodi . Thenwe anwrite

γ

as

γ = α 0 · · · α m−1 (β 0 · · · β n−1 ) ,

where

β 0 · · · β n−1 < α 0 · · · α m−1

. Thenthereisa

k < n

su hthat

β 0 · · · β k−1 1 >

β 0 · · · β n−1

and

(β 0 · · · β n−1 ) N β 1 · · · β k−1 1

isallowedforany

N

. Forany

N

the

word

α 0 · · · α m−1 (β 0 · · · β n−1 ) N β 0 · · · β k−1 1

is forbiddenbut it ontainsno smallerforbiddenword. Hen e thesubshift is

notofnitetype. We on ludethatif

Σ γ

isofnitetypethen

γ

isperiodi .

Ifthereisno

n

su hthat

γ ∈ K n

thenforany

n

thereisan

m

su hthat

[γ 0 · · · γ m ] = [γ 0 · · · γ m ( −γ 1 )( −γ 2 ) · · · (−γ n )].

Thisimpliesthat

Σ γ

annotbespe ied.

If

γ ∈ K n

then for any allowed word

C

of length

m

there are

k, l

with

k + l < n

su hthateither

Cγ l γ l+1 · · · γ k+l ( −11) , Cγ l γ l+1 · · · γ k+l 1( −γ)

or

Cγ l γ l+1 · · · γ k+l (1 −1) , Cγ l γ l+1 · · · γ k+l − 1γ

aresequen esin

Σ γ

. Hen ewehaveeither

f m+k+1 (π γ ([C])) ⊃ h 0, γ − 1

γ + 1

i ,

or

f m+k+1 (π γ ([C])) ⊃ h

− γ − 1 γ + 1 , 0 i

.

For

γ > √

2

there isan

N

,dependingon

γ

,su hthat

f N h

0, γ − 1 γ + 1

i = f N h

− γ − 1 γ + 1 , 0 i

= [ −(γ − 1), (γ − 1)].

Thisimpliesthat

Σ γ

isspe ied.

Example 2.2.2.1. Let

γ = (11 −1)

. The orresponding graphis in gure 2.1. Ifwelet

1 = 11 −1

,

2 = 1 −11

,

3 = 1 −1−1

,

4 = −1−11

,

5 = −11−1

and

(23)

6 = −111

thenwegetthefollowingadja en ymatrix.

 

 

 

1 1 1 0 1 1

0 1 1 1 1 1

1 1 1 0 0 0

0 1 0 1 1 1

1 1 0 0 1 1

0 0 1 1 1 1

 

 

 

11−1 1−11 1−1−1

−11−1

−1−11 −111

Figure2.1: Thegraphasso iatedwiththesubshiftdeterminedbythekneading

sequen e

γ = (11 −1)

.

Example2.2.2.2. Let

γ = 1(1 −1)

. The orrespondinggraph, onstru ted asin theproofof theorem2.2.2.1,is in gure2.2. Thegraphforthe subshift

determinedby

γ = 11( −11−111−11)

isin gure2.3.

2.2.3 Invariant measures

Inthisse tion we onstru tan absolutely ontinuousinvariantmeasure. The

methodfollowsthatappliedforthe

β

-expansionbyParryin[22℄. Weestimate

thenumberofallowedwordsoflength

n

inthesubshift

Σ γ

andusethisestimate

toestimate the Lebesgue measure of pre-imagesof any ylinder. In this way

(24)

(1,3) (0,4) (3,1)

(2,0) (4,0)

1

−1

(0,1) (0,2) (1,0)

1

−1

−1 1

(0,0)

−1

1 −1

1

−1

1 1

−1

Figure2.2: Thegraphasso iatedwiththesubshiftdeterminedbythekneading

sequen e

γ = 1(1 −1)

.

(0,2)

(5,3) (6,4)

(4,6) (3,5)

(1,3) (3,1) (2,0)

(1,0)

(0,1) (0,0)

(2,4) (4,2)

−1

1

−1 1

1 −1

−1

1

1 1 −1

1

−1

−1

1

−1

−1 1

(8,0) (9,1)

(11,3)

(12,4) 1

−1

−1 (10,2)

(7,5) (13,5)

−1

1 1

1

−1

−1 1

1

−1

1

−1

−1 (0,8)

(5,7) (5,13) (4,12)

(3,11)

(2,10) (1,9)

1

Figure2.3: Thegraphasso iatedwiththesubshiftdeterminedbythekneading

sequen e

γ = 11( −11−111−11)

.

we an onstru ttheabsolutely ontinuousinvariantmeasureasaweak limit

(25)

pointof thesequen e

1 n

n−1 X

k=0

ν ◦ T −n ,

where

ν

denotes the Lebesgue measure. Thelimit measure isthe measure of

maximalentropy.

By onstru tingaMarkovpartitionHofbauershowedin[13℄thata lassof

pie ewisemonotone in reasingmaps of an intervalhas at mostnitely many

measuresofmaximalentropyandifthemapistransitivethenthereisaunique

measureofmaximalentropy. Hofbauerusedthismethodformoregeneralmaps

in[14,15,16,17℄.

Denethefollowingmetri s.

d Σ γ (a, b) = max n 1

γ n | a n 6= b n o

, a, b ∈ Σ γ ,

d γ (a, b) =

γ − 1 γ

X ∞ k=0

a k − b k γ k

, a, b ∈ Σ γ .

Themetri

d γ

is theLebesgue metri in the sen e that it satises

d γ (a, b) =

γ (a) − π γ (b) |

.

Theorem2.2.3.1. Let

a, b ∈ Σ γ

. Then

2d Σ γ (a, b) ≥ d γ (a, b)

.

Proof. Assume that

d Σ γ (a, b) = γ −n

. Then

d γ (a, b) =

γ − 1 γ

X ∞ k=0

a k − b k γ k

≤ γ − 1 γ n+1

X ∞ k=0

2 γ k = 2

γ n = 2d Σ γ (a, b).

Thefollowinglemmaisobvious.

Lemma2.2.3.1. Forany ylinder

[C]

of length

m

wehave

d Σ γ ([C]) ≤ γ −m−1 ,

d γ ([C]) ≤ (γ − 1)γ −m+1 .

Lemma2.2.3.2. Let

γ ∈ K n

. Then for any ylinder

[C]

oflength

m

wehave

d Σ γ ([C]) ≥ γ −n−m−1

.

Proof. Let

j > 0

besu h that there isno

a, b ∈ [C]

with

d Σ γ (a, b) ≥ γ −m−j

.

Therearenumbers

k

and

l

su hthatthelast

k

lettersof

C

are

γ 0 · · · γ k−1

and

the last

l

letters of

C

are

( −γ 0 ) · · · (−γ l−1 )

. Sin e

γ ∈ K n

any

a ∈ [C]

has

the property that one of these hains of

γ 0 · · · γ k−1

and

( −γ 0 ) · · · (−γ l−1 )

in

a

ends after at least

n

letters. This impliesthat we annd

a, b ∈ [C]

with

d(a, b) ≥ γ −m−n−1

.

(26)

Theorem2.2.3.2. Assumethat

γ ∈ K n

. Then forany ylinder

[C]

su hthat

d Σ γ ([C]) > 0

we have

γ −n+1 γ − 1

γ + 1 ≤ d γ ([C]) d Σ γ ([C]) ≤ 2.

Proof. Bytheorem 2.2.3.1wehavethat

d γ ([C]) d Σγ ([C]) ≤ 2

.

Arguingasinthe proofof Lemma2.2.3.2weseethat there areintegers

k

and

l

with

k ≤ n − 1

su hthat either

a 0 = Cγ l γ l+1 · · · γ l+k −1γ, a 1 = Cγ l γ l+1 · · · γ l+k ( −11) ,

or

a 0 = Cγ l γ l+1 · · · γ l+k 1( −γ), a 1 = Cγ l γ l+1 · · · γ l+k (1 −1) .

are in

[C]

. If

m

is the lengthof

C

then a dire t al ulation gives

d γ ([C]) ≥ γ −m−n γ−1 γ+1

andhen e

d γ ([C])

d Σ γ ([C]) ≥ γ −n+1 γ − 1 γ + 1 .

Theorem2.2.3.3. Let

N (k)

denote the numberof allowedwordsof length

k

.

Then

2

γ γ k ≤ N(k) ≤ 4 γ − 1 γ k .

Proof. Thereare

N (k + 1) − N(k)

allowedwords

W

oflength

k

su hthatboth

W 1

and

W −1

are allowed. For ea h su h word

W −1γ, W 1−γ ∈ Σ γ

. Sin e

π γ (W −1γ) = π γ (W 1 −γ)

wemaythink oftheword

W

asifitwasasequen e,

su hthat

π γ (W ) = π γ (W 1 −γ)

.

Orderthese

N (k + 1) − N(k)

words lexi ographi ally and onsider three onse utivewords

W 1 , W 2 , W 3

. It anhappenthat

W 1

and

W 2

or

W 2

and

W 3

arevery lose. However

d γ (W 1 , W 3 ) ≥ γ−1 γ k

. Hen e

N (k + 1) − N(k) 2

γ − 1

γ k ≤ 2(γ − 1)

and

N (k) ≤ 4

γ − 1 γ k .

(27)

Considertheallowedword

i 0 i 1 · · · i k−1

oflength

k

. The ylinder

[i 0 · · · i k−1 ]

hasthepropertythat

d γ ([i 0 · · · i k−1 ]) ≤ γ(γ−1) γ k

. Hen e

N (k) γ(γ − 1)

γ k ≥ 2(γ − 1).

Thisimpliesthat

N (k) ≥ 2 γ γ k .

Corollary2.2.3.1. The topologi al entropy of

Σ γ

is

log γ

.

Corollary 2.2.3.2. The map

Ξ : (1, 2) → {−1, 1} N

is monotone in reasing

andinje tive.

Proof. If

γ < δ

then

Σ γ ⊂ Σ δ

by(2.1). Thisimpliesthat

h top (Σ γ ) ≤ h top (Σ δ )

andsowemusthave

γ ≤ δ

.

Theorem2.2.3.4. Let

I ⊆ I γ

beanyinterval. Then for any

m 1

m

m−1 X

k=0

ν(T −k (I)) ≤ 4 γ − 1 ν(I),

where

ν

denotes theLebesguemeasure. If

γ ∈ K n

and

γ > √

2

thenthereexists

a onstant

c > 0

,dependingon

γ

,su hthat

1

m

m−1 X

k=0

ν(T −k (I)) ≥ cν(I).

Proof. Theset

T −k (I)

onsistsofatmost

N (k)

disjointintervalea hofmeasure

lessorequalto

ν(I)

γ k

. Thisimpliesthat

ν(T −k (I)) ≤ N(k) ν(I) γ k ≤ 4

γ − 1 ν(I)

andhen e

1 m

m−1 X

k=0

ν(T −k (I)) ≤ 4 γ − 1 ν(I).

Assume that

γ ∈ K n

and

γ > √

2

. Then

Σ γ

isspe i and thereexists an

N

su hthat foranyallowedword

i 0 i 1 · · · i n−1

thereexistsaword

W

oflength

N

su h that

i 0 i 1 · · · i n−1 W C

is anallowedword. So

T −k (π γ ([C])

onsists of

atleast

N (k − N)

intervalsandbyLemma2.2.3.2and Theorem2.2.3.2there

(28)

exists a onstant

c 0 > 0

su h that ea h of these intervalshas lengthnot less

than

c 0 ν(π γ ([C]))γ −k

. Hen e

ν(T −k (π γ ([C])) ≥ N(k − N)c 0 ν(π γ ([C]))γ −k ≥ cν(π γ ([C])),

where

c

dependson

γ

butnoton

[C]

. ThisimpliesthestatementoftheTheo-

rem.

Corollary2.2.3.3. Thereexistsaninvariantmeasure

µ

,absolutely ontinuous

with respe ttothe Lebesguemeasure

ν

su hthat

µ(A) = lim

n→∞

1 n

n−1 X

k=0

ν(T −k (A)) ≤ 4

γ − 1 ν(A).

If

γ ∈ K n

and

γ > √

2

then

µ

isequivalentto

ν

.

Parry[25℄ hasshownthatthemeasure

µ

hasthedensity

h(x) = D

X ∞ n=0

χ [−(γ−1),T n (γ−1)) (x) − χ [−(γ−1),T n (−(γ−1))) (x)  ,

where

D

isanormalising onstant,andif

γ > √

2

then

h(x) > D γ(γ−1) γ 2 −2

.

Theorem 2.2.3.5. The measure

µ

is the measureof maximal entropy. That

is

h µ (T ) = h top (T )

.

Proof. It su es to show that

h µ (T ) ≥ log γ

sin e

h µ (T ) ≤ h top (T )

for any

measureand

h top (T ) = log γ

. Theorem2.2.3.4andLemma2.2.3.1impliesthat

forany ylinder

[C]

oflength

n

wehave

µ([C]) ≤ 4γ 1−n

. If

C n

isthepartition

of

I γ

into ylindersoflength

n

wehave

1

n X

[C]∈C n

−µ([C]) log µ([C]) ≥ 1 n

X

[C]∈C n

µ([C])(n log γ − log(4γ))

= log γ − 1

n log(4γ) → log γ,

as

n → ∞

. This showsthat

h µ (T ) ≥ log γ

.

Example2.2.3.1. If

Σ γ

isspe iedthenthereexistsasequen e

a ∈ Σ γ

su h

that

n (a) } n∈N

isdensein

Σ γ

withrespe ttothemetri

d γ

.

Let

{a n } n∈N = {a n,1 a n,2 · · · } n∈N

bedensein

Σ γ

. Dene

A n = a n,1 a n,2 · · · a n,n .

(29)

The ylindersets

[A n ]

are allnon-emptysin e

a n ∈ [A n ]

. Sin e

Σ γ

has nite

memorythere existswords

A T,n

oflength

T n ≤ T

su hthat

b = A 1 A T,1 A 2 A T,2 A 3 · · · ∈ Σ γ .

Let

k n = X n i=1

(i + T i ).

Thenby Lemma 2.2.3.1and Theorem 2.2.3.2theset

k n (b) } n∈N

is densein

Σ γ

withrespe tto

d γ

.

2.2.4 Ergodi properties

Thetheory ofpie ewise monotonemaps of anintervalis well developed. We

use this theory to on lude that the measure

µ

onstru ted in the previous se tionistheuniquemeasureofmaximalentropyand thatitisBernoulli.

Theorem2.2.4.1. Let

γ > √

2

. If

γ ∈ K N

then

T

ismixing.

Proof. Bowen[8℄hasshownthatanypie ewise

C 2

fun tiononanintervalwith

onedis ontinuityandderivativelargerthan

√ 2

isweaklymixing.If

T

isweakly

mixingthen

T

ismixingifandonlyifthere existsa onstant

K

su hthat

lim sup

n→∞ ν(A ∩ T −n B) ≤ Kν(A)ν(B),

foranymeasurablesets

A

,

B

.

Takeanytwo ylinders

[A]

and

[B]

. Weprovethat

d γ ([A] ∩ T −n [B]) ≤ Kd γ ([A])d γ ([B]),

(2.2)

for

n

large enough. Let

n A

be the length of the word

A

. Assume that

d Σ γ ([A]) = γ −N A

and

d Σ γ ([B]) = γ −N B

.

Let

n > n A

. Then

[A] ∩ T −n [B] = {a ∈ Σ γ | a = A · · · | {z }

n

letters

B · · · } = [

k

[C k ],

where

[C k ]

are disjoint ylinders of the form

[Ai n A · · · i j−1 B]

. There are at

most

N (n − n A )

su hnon-empty ylinders.

ByLemma 2.2.3.2wehave

d Σ γ ([A]) ≥ γ −n A −N −1

. Weget

d Σ γ ([A] ∩ T −n [B]) = X

k

d Σ γ ([C k ]) ≤ N(n − n A )γ −n−N B

≤ 4

γ − 1 γ n−n A γ −n−N B ≤ c 1 d Σ γ ([A])d Σ γ ([B]).

(30)

ByTheorem2.2.3.2

d γ ([A] ∩ T −n [B]) ≤ 2d Σ γ ([A] ∩ T −n [B])

≤ 2c 1 d Σ γ ([A])d Σ γ ([B]) ≤ c 2 d γ ([A])d γ ([B]).

Hen e

T

ismixing.

Theorem2.2.4.2. If

γ > √

2

then

T

istopologi ally mixing.

Proof. Weshowthatforanynon-empty ylinders

[A]

thereexists anumber

n

su hthat

σ n ([A]) = Σ γ

. Itthenfollowsthat

T

istopologi allymixing.

Let

n A

bethelengthoftheword

A

. Let

n γ

bethelargestnumbersu hthat

( −γ 0 ) · · · (−γ n γ −1 )

is awordin

γ 0 · · · γ n A −1

. Clearly,

n γ < n A

. There exists

k, l < n γ

su hthateither

Aγ l · · · γ l+k−1 (1 −1) ∈ Σ γ , Aγ l · · · γ l+k−1 1( −γ) ∈ Σ γ ,

or

Aγ l · · · γ l+k−1 ( −11) ∈ Σ γ , Aγ l · · · γ l+k−1 −1γ ∈ Σ γ .

Hen e

f n A +k (π γ ([A])) ⊃ [0, γ−1 γ+1 ]

or

f n A +k (π γ ([A])) ⊃ [− γ−1 γ+1 , 0]

. For any

γ > √

2

thereisanumber

N

su hthat

f N h

− γ − 1 γ + 1 , 0 i

= f N h 0, γ − 1

γ + 1

i = [ −(γ − 1), (γ − 1)].

Thisimpliesthat

T

istopologi allymixing.

Corollary 2.2.4.1. If

γ > √

2

then the measure

µ

is the unique measure of

maximal entropy.

Proof. In[13℄, Hofbauershowed that provided

T

is transitivethen there is a

uniquemeasureofmaximalentropy.

Theorem2.2.4.3.

T

isergodi withrespe ttothe measure

µ

.

Proof. In [19℄ it is shown that that any transformation of a ertain lass of

transformations with

n

dis ontinuities has at most

n

ergodi measures ea h

absolutely ontinuous. Thus

µ

isergodi .

Theorem2.2.4.4. If

γ > √

2

then

µ

isBernoulli.

Proof. Bowen [8℄ has shown that provided

T

is weakly mixing, the measure

µ

is Bernoulli. As mentioned in the proof of Theorem 2.2.4.1,

T

is weakly

mixing.

References

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