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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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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
T HE F AMILY OF B ELYKH M APS
T OMAS P ERSSON
Centre for Mathematical Sciences
Mathematics
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
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.
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.
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
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 thismeasure. If
f : M → M
is thesysteminquestionthen theSBR
-measure isaweaklimitpointofthesequen eofmeasures
µ n =
n−1 X
k=0
ν ◦ f −k ,
where
ν
denotestheLebesguemeasure. Thismeasureisthephysi allyrelevantmeasureasit apturesthebehaviouroftheorbitsofpointsfromasetofpositive
Lebesguemeasure.Pesinshowedthatthe
SBR
-measurehasatmost ountablymanyergodi 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 todealwiththenon-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 theBelykhmap
f : Q → Q
byf (x, y) =
(λx + (1 − λ), γy − (γ − 1)),
ify > kx, (λx − (1 − λ), γy + (γ − 1)),
ify < kx,
wheretheparametersare
0 < λ ≤ 1
,−1 < k < 1
and1 < γ ≤ 1+|k| 2
. SeeFigure1.1. Inthisworkwewillstudyasimilarmapwiththeonlydieren ethat the
singularity set is the set
([ −1, 0] × {−k}) ∪ ({0} × [−|k|, |k|]) ∪ ([0, 1] × {k})
insteadof
{y = kx}
,thatisweapproximatetheline{y = kx}
withapie ewiseonstant 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
andk = 1 4
.Weshowthat whenthemapexpandsarea(
γλ > 1
)then thereisanopenset
P
ofparameterssu h thattheSBR
-measureisabsolutely ontinuouswithrespe tto Lebesguemeasurealmost surelyiftheparametersarein
P
.There are similar results in the literature. In the ase when
k = 0
andγ = 2
thesystemisthefatbaker'stransformation, studiedbyAlexanderand Yorkein[1℄. Inthis asethemapistheprodu t ofitsproje tionsto therstand the se ond oordinate. The proje tion on the se ond oordinate is the
two-shift. Thissimpliesthe al ulationsandthe
SBR
-measureistheprodu tof the one-dimensional Lebesgue measure and a Bernoulli onvolution. The
resultofSolomyakin[35℄impliesthatforLebesguealmosteveryparameterthe
fatbaker'stransformationhasan
SBR
-measurewhi hisabsolutely ontinuouswithrespe ttotheLebesguemeasure. AlexanderandYorkeshowedthatif
λ −1
isaPisot numberthenthe
SBR
-measureissingularto theLebesguemeasure,sin ethentheFouriertransformationoftheBernoulli onvolutiondoesnottend
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 oftheset
A ∗ = {a 0 a 1 · · · a n−1 | a i ∈ A, n ≥ 0}
and
A ∗
is alledthelanguageofA
. AlanguageL
onA
is asubsetofA ∗
.Let
A N
betheset ofallinnitesequen esofelementsinA
. Wedenethemap
σ : A N → A N
byσ : {a k } k∈N 7→ {a k+1 } k∈N
. A ylinderisasetoftheformk [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
issaidtobeasubshiftifitisinvariantunderσ
and losedinthetopologygeneratedbythe 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 integerm ≥ 0
su hthata k = i m+k
fork = 0, 1, . . . , n − 1
. Thesetofallowedwordsisalledthelanguageof
S
.InChapter2we onsiderthespe ial aseofthe Belykhmaps when
k = 0
and
γ
andλ
are arbitrary. In this ase the dynami s depends only on these 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)]
isthenthefollowing.
T (x) =
γx − (γ − 1)
ifx > 0, γx + (γ − 1)
ifx < 0.
Thegraph of
T
is in Figure 1.2. Bya hange of variablesT
anbewrittenin the form
x 7→ γx + α (mod 1)
wereα = 1 − γ/2
. This is similar to theβ
-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 termsoftheorbitof1
.Healsoprovedtheexisten eofanabsolutely ontinuousinvariantmeasureand
al ulatedthetopologi alentropy.
Let
[x]
denote the integer part of the numberx
and let{x}
denote thefra tionalpartof
x
. Letβ > 1
. Foranyx ∈ [0, 1]
weasso iatethe sequen e−(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)
,forβ = 3 4
.d(x, β) ∈ {0, 1, · · · , [β]} N
denedasfollows. Ifd(x, β) = {i k } ∞ k=0
thenforea hk ∈ N
wedenei 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 alledthe
β
-shift. Itisinvariantundertheleft-shiftσ : {i k } ∞ k=0 7→ {i k+1 } ∞ k=0
andthemap
d( ·, β)
satisesσ n (d(x, β)) = d(f β (x), β)
. IfweorderS β
with thelexi o-graphi alorderingthenthemap
d( ·, β)
isone-to-oneandmonotonein reasing.Parry [22℄ proved that the map
β 7→ d(1, β)
is monotone in reasing andinje tive. Forasequen e
{i k } ∞ k=0
thereisaβ > 0
su hthat{i k } ∞ k=0 = d(1, β)
if and only if
σ n ( {i k }) ≤ {i k }
for everyn ≥ 0
. The numberβ
is then theuniquepositivesolutionoftheequation
1 = X ∞ k=0
i k x −k .
The subshift
S β
is theset of sequen es{i k }
su h thatσ n ( {i k }) ≤ d(1, β)
forevery
n ≥ 0
. Ifx ∈ [0, 1]
thenx =
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 eisinS
ifandonly if there is apath in
G
whi h yields the same sequen e by readingofthelabelsof the verti es alongthe path.
Asubshift
S
issaidtobeso ifitisasso iatedwithanitedire tedgraphwithlabellededges.
A subshift
S
is said to be spe ied, or have the spe i ation property, if there exists a numberk
su h that ifa
andb
are words in the language ofS
thenthereisaword
c
oflengthk
su hthat thewordacb
isallowed.Notethatif
S
isof nitetypethenitisso . Therearesubshiftsthat arenotso and subshiftsthat arenotspe ied.
It is possible to hara terise the dierent types of subshifts
S β
in termsofthepropertiesofthesequen e
d(1, β)
andmake onne tionstothenumber- theoreti alpropertiesofβ
,seeforexample[7℄and[31℄fora olle tionofresultsonthesubje t. Amongotherresultsare
S β
is of nite type if andonly ifd(1, β)
either terminates with zeros orisperiodi . [22℄
S β
isso ifandonlyifd(1, β)
iseventually periodi , thatistheorbitof1
underf β
isnite. [6℄S β
isspe ied if andonly if thereis ann
su hthat there are non
on-se utive zerosin
d(1, β)
. [6℄If
β
isaPisot numberthenS β
isso . [22℄If
S β
isso thenβ
isaPerronnumber. [20℄, [9℄Themethodsfromthe
β
-expansion anbeappliedtothemapT
withsmallhanges. 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.
Restri tion to one dimension
2.1 Denition of the system
Put
Q = [ −1, 1] 2
andS = ([ −1, 0] × {−k}) ∪ ({0} × [−|k|, |k|]) ∪ ([0, 1] × {k})
.Let
Q 1
andQ −1
betheupperrespe tivelythelower onne ted omponentof thesetQ \ S
.Considerthe lassofmaps
f : Q \ S → Q
denedbyf (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
and1 < γ ≤ 1+|k| 2
. ThesearetheBelykhmaps.
2.2 The one-dimensional ase
Here we onsider the ase
k = 0
. In this ase the dynami s in the se ondoordinate 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 mapT : I γ → I γ
, whereI γ = [ −(γ − 1); (γ − 1)]
bedened byT (x) =
γx − (γ − 1)
ifx > 0, γx + (γ − 1)
ifx ≤ 0.
Wehavedened
T
tobe(γ − 1)
at0
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)
andI 1 = [0; (γ − 1)]
. Foranyx ∈ I γ
we asso iateasequen e
i = {i k } ∞ k=0 ∈ {−1, 1} N
dened byT k (x) ∈ I i k
for anyk ∈ N
. Thenx
andi
satisfyx = γ − 1 γ
X ∞ k=0
i k
γ k .
We let
Σ γ
denote the losure of the set of all su h sequen es and dene themap
π γ : Σ γ → I γ
byπ γ (i) = γ − 1 γ
X ∞ k=0
i k
γ k .
Theleft-shift
σ
isdenedbyσ( {i k } ∞ k=0 ) = {i k+1 } ∞ k=0
. Itiseasytoseethatπ γ (σ n (i)) = T n (π γ (i))
. ThesetΣ γ
isinvariantunderσ
andishen easubshift.Weendowthesubshift
Σ γ
withthelexi ographi alordering,denotedby≤
.Be ause
T
ispie ewisemonotonein reasingthismakesthemapπ γ
monotonein reasing.
Wedenote by
γ = {γ k } ∞ k=0 ∈ Σ γ
thesequen ethatsatisesγ − 1 = π γ (γ)
.If we let
−γ = {i k }
denotethe sequen e su h thati k = −γ k
for ea hk
then−(γ − 1) = π γ ( −γ)
andΣ γ
isthesetΣ 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 isnon
su hthatσ n (γ) = −γ
.It ishoweverpossiblethat
σ n (γ) = γ
. Ifwehad denedT (0)
to be−(γ − 1)
thenwewouldhavetheopposite ase.
Welet
Ξ : (1, 2) → {−1, 1} N
denote themap that mapsγ ∈ (1, 2)
to theupperkneadingsequen eof
Σ γ
. ThemapΞ
satisesπ γ (Ξ(γ)) = γ − 1
.Let
K
denotethesetofkneadingsequen es. DeneP n = {γ ∈ K | γ = (γ 1 γ 2 · · · γ n ) ∞
forsomeγ 1 γ 2 · · · γ n }
andlet
P = S ∞
n=1 P n
. Foranyγ ∈ K
deneL(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
andk, l ∈ N
wedenotebyk [A] k+l
the ylindersetdened 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
.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 existsanumberk
su hthatifa
andb
arewordsinthelanguageofS
thenthereisaword
c
of lengthk
su hthatthe wordacb
isallowed.Inthis se tion wewill provethe followingtheorem, that hara terisesthe
threetypesofsubshiftsinDenition2.2.2.1intermsofthekneadingsequen e.
Theorem2.2.2.1. Thesubshift
Σ γ
isofnitetypeifandonlyifγ
isperiodi .The subshift
Σ γ
isso ifandonly ifγ
iseventually periodi .If
γ > √
2
thenthe subshiftΣ γ
isspe iedif andonly ifγ ∈ K n
,for someinteger
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)
isniteIf
γ > √
2
thenthesubshiftΣ γ
isspe iedifandonlyifthe orbitof(γ − 1)
isboundedaway from
−(γ − 1)
.Considerthe ase
γ = √ 2
. LetA = 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
andT −1 (B) = A
. Hen eΣ γ
is notspe iedeventhoughγ = 11( −11) ∞ ∈ K 1
.If
γ < √
2
thenf (A) ⊂ B
andf (B) ⊂ A
andΣ γ
an not be spe ied.Howeverthereisno
γ ∈ (1, √
2)
su hthatγ
isperiodi oreventuallyperiodi .Let
A
be an alphabet andL ⊆ A ∗ = {a 1 a 2 · · · a n | a k ∈ A, n ∈ N}
alanguage. For
x, y ∈ L
wedenetherelation∼
byx ∼ y ⇐⇒ (axb ∈ L
ifandonlyifayb ∈ L, ∀a, b ∈ A ∗ ).
Denition2.2.2.2. Thelanguage
L
issaidtoberationalifthequotientgroupwithrespe t totherelation
∼
isnite.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 thenσ n (γ)
isperiodi forsomen
.Assume that
Σ γ
is so andσ n (γ)
is not periodi for anyn ∈ N
. Thenthereexists aninnitesequen e
i 1 < i 2 < · · ·
su hthatthesequen esγ 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 hthatγ i k = γ i l , . . . , γ i k +j = γ i l +j
andγ i k +j+1 > γ i l +j+1 .
Thesequen e
γ i k γ i k +1 γ i k +2 · · ·
prolongthewordγ 0 γ 1 · · · γ i k −1
butitdoesnotprolongtheword
γ 0 γ 1 · · · γ i l −1
. Hen e thequotientgroupisnotniteandΣ γ
isnotso . Thisprovesthat
σ n (γ)
isperiodi forsomen
ifΣ γ
isso .Wenowprovethatif
γ
isperiodi thenΣ γ
isofnitetype.Thesubshift
Σ γ
isofnite typeifand onlyifit isasso iatedwithanitegraphwithlabelledverti es. Assumethat
γ
isperiodi . We onstru tagraphin ordertoprovethatthesubshift isofnitetype.
Assumethat
γ
isn
-periodi . LetV = {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 ea = a 0 a 1 a 2 · · ·
is in
Σ γ
if and only if thewordsa 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 wordsa k+nm · · · a k+n(m+1)−1
anda k+nm · · · a k+n(m+2)−1
,k = 0, 1, . . .
,m = 0, 1, 2, . . .
are allowedandthereforethereis a orresponding pathinG
. Anexampleofthe onstru tionisin Example2.2.2.1below.Itisnowtimetoprovethatif
σ n (γ)
isperiodi forsomen
thenΣ γ
isso .Assumethat
σ n (γ)
isperiodi forsomen
.If
γ
isperiodi thenΣ γ
isofnite typeandhen eso . Assumethatγ
isnotperiodi . Write
γ = α 0 α 1 · · · α m−1 (β 0 β 1 · · · β n−1 ) ∞ = γ 0 γ 1 · · ·
. Wemayassumethat
α 0 · · · α m−1 > β 0 · · · β n−1
.Foranyniteword
a 0 · · · a N −1
wedenethestate(k, l)
;Putk ′ = 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 ′
ifk ′ ≤ m + n,
l = l ′
ifl ′ ≤ m + n,
k = m + n + r
ifk ′ = m + pn + r, p ≥ 1
andl = m + n + r
ifl ′ = m + pn + r, p ≥ 1.
Then
0 ≤ k, l ≤ m + 2n − 1
. LetS
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
byE =
(k 1 , l 1 ) → (k L 2 , l 2 ) | L ∈ {−1, 1}, a 0 · · · a s−1 L
isanallowedwordwithS(a 0 · · · a s−1 ) = (k 1 , l 1 )
andS(a 0 · · · a s−1 L) = (k 2 , l 2 ),
where
a 0 · · · a s−1 = γ 0 · · · γ k 1 −1
ifk 1 > l 1
anda 0 · · · a s−1 = ( −γ 0 ) · · · (−γ l 1 −1 )
ifl 1 > k 1
.
Weprovethatthegraph
G = (V, E)
determinesthesubshiftΣ γ
. Observethattheword
A(β 0 · · · β n−1 )B
isallowedifandonlyifthewordsA(β 0 · · · β n−1 ) i B
,i = 1, 2, 3, . . .
are allowed. This implies that for anya ∈ Σ γ
and anyj
thestate
S(a 0 · · · a j−1 )
is dened and from the vertexS(a 0 · · · a j−1 )
there is auniqueedgelabelledwith
a j
goingtothevertexS(a 0 · · · a j )
. Hen ethesubshiftdeterminedby
G
ontainsΣ γ
.Conversely,let
a
be asequen edetermined byapathinG
. Then foranyi = 1, 2, 3, . . .
the worda i−s+1 · · · a i = ±(γ 0 · · · γ s−1 )
wheres = max {k, l}
,(k, l) = S(a 0 · · · a i )
. Clearlya 0
isanallowedword. Assume thata 0 · · · a i
isanallowedword. Theword
a i−s+1 · · · a i
isallowedandbythe onstru tionofG
itis learthattheword
a i−s+1 · · · a i a i+1
isallowed. Hen ea 0 · · · a i+1
isallowedandbyindu tion
a ∈ Σ γ
.Thegraph
G
isobviouslynite soΣ γ
is indeedso . SeeExample2.2.2.2belowforanexampleofthe onstru tion.
We annowprovethat if
Σ γ
isofnitetypethenγ
isperiodi .If
Σ γ
is of nite type then it is so andσ n (γ)
is periodi for somen
.Assume
γ
isnotperiodi . Thenwe anwriteγ
asγ = α 0 · · · α m−1 (β 0 · · · β n−1 ) ∞ ,
where
β 0 · · · β n−1 < α 0 · · · α m−1
. Thenthereisak < n
su hthatβ 0 · · · β k−1 1 >
β 0 · · · β n−1
and(β 0 · · · β n−1 ) N β 1 · · · β k−1 1
isallowedforanyN
. ForanyN
theword
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
isofnitetypethenγ
isperiodi .Ifthereisno
n
su hthatγ ∈ K n
thenforanyn
thereisanm
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 wordC
of lengthm
there arek, l
withk + l < n
su hthateitherCγ 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 ewehaveeitherf m+k+1 (π γ ([C])) ⊃ h 0, γ − 1
γ + 1
i ,
orf m+k+1 (π γ ([C])) ⊃ h
− γ − 1 γ + 1 , 0 i
.
For
γ > √
2
there isanN
,dependingonγ
,su hthatf 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. Ifwelet1 = 11 −1
,2 = 1 −11
,3 = 1 −1−1
,4 = −1−11
,5 = −11−1
and6 = −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 subshiftdeterminedby
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℄. Weestimatethenumberofallowedwordsoflength
n
inthesubshiftΣ γ
andusethisestimatetoestimate the Lebesgue measure of pre-imagesof any ylinder. In this way
(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
pointof thesequen e
1 n
n−1 X
k=0
ν ◦ T −n ,
where
ν
denotes the Lebesgue measure. Thelimit measure isthe measure ofmaximalentropy.
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 satisesd γ (a, b) =
|π γ (a) − π γ (b) |
.Theorem2.2.3.1. Let
a, b ∈ Σ γ
. Then2d Σ γ (a, b) ≥ d γ (a, b)
.Proof. Assume that
d Σ γ (a, b) = γ −n
. Thend 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 lengthm
wehaved 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]
oflengthm
wehaved Σ γ ([C]) ≥ γ −n−m−1
.Proof. Let
j > 0
besu h that there isnoa, b ∈ [C]
withd Σ γ (a, b) ≥ γ −m−j
.Therearenumbers
k
andl
su hthatthelastk
lettersofC
areγ 0 · · · γ k−1
andthe last
l
letters ofC
are( −γ 0 ) · · · (−γ l−1 )
. Sin eγ ∈ K n
anya ∈ [C]
hasthe property that one of these hains of
γ 0 · · · γ k−1
and( −γ 0 ) · · · (−γ l−1 )
ina
ends after at leastn
letters. This impliesthat we annda, b ∈ [C]
withd(a, b) ≥ γ −m−n−1
.Theorem2.2.3.2. Assumethat
γ ∈ K n
. Then forany ylinder[C]
su hthatd Σ γ ([C]) > 0
we haveγ −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
withk ≤ n − 1
su hthat eithera 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]
. Ifm
is the lengthofC
then a dire t al ulation givesd γ ([C]) ≥ γ −m−n γ−1 γ+1
andhen ed γ ([C])
d Σ γ ([C]) ≥ γ −n+1 γ − 1 γ + 1 .
Theorem2.2.3.3. Let
N (k)
denote the numberof allowedwordsof lengthk
.Then
2
γ γ k ≤ N(k) ≤ 4 γ − 1 γ k .
Proof. Thereare
N (k + 1) − N(k)
allowedwordsW
oflengthk
su hthatbothW 1
andW −1
are allowed. For ea h su h wordW −1γ, W 1−γ ∈ Σ γ
. Sin eπ γ (W −1γ) = π γ (W 1 −γ)
wemaythink ofthewordW
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 utivewordsW 1 , W 2 , W 3
. It anhappenthatW 1
andW 2
orW 2
andW 3
arevery lose. However
d γ (W 1 , W 3 ) ≥ γ−1 γ k
. Hen eN (k + 1) − N(k) 2
γ − 1
γ k ≤ 2(γ − 1)
and
N (k) ≤ 4
γ − 1 γ k .
Considertheallowedword
i 0 i 1 · · · i k−1
oflengthk
. The ylinder[i 0 · · · i k−1 ]
hasthepropertythat
d γ ([i 0 · · · i k−1 ]) ≤ γ(γ−1) γ k
. Hen eN (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
islog γ
.Corollary 2.2.3.2. The map
Ξ : (1, 2) → {−1, 1} N
is monotone in reasingandinje tive.
Proof. If
γ < δ
thenΣ γ ⊂ Σ δ
by(2.1). Thisimpliesthath top (Σ γ ) ≤ h top (Σ δ )
andsowemusthave
γ ≤ δ
.Theorem2.2.3.4. Let
I ⊆ I γ
beanyinterval. Then for anym 1
m
m−1 X
k=0
ν(T −k (I)) ≤ 4 γ − 1 ν(I),
where
ν
denotes theLebesguemeasure. Ifγ ∈ K n
andγ > √
2
thenthereexistsa onstant
c > 0
,dependingonγ
,su hthat1
m
m−1 X
k=0
ν(T −k (I)) ≥ cν(I).
Proof. Theset
T −k (I)
onsistsofatmostN (k)
disjointintervalea hofmeasurelessorequalto
ν(I)
γ k
. Thisimpliesthatν(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
andγ > √
2
. ThenΣ γ
isspe i and thereexists anN
su hthat foranyallowedwordi 0 i 1 · · · i n−1
thereexistsawordW
oflengthN
su h thati 0 i 1 · · · i n−1 W C
is anallowedword. SoT −k (π γ ([C])
onsists ofatleast
N (k − N)
intervalsandbyLemma2.2.3.2and Theorem2.2.3.2thereexists a onstant
c 0 > 0
su h that ea h of these intervalshas lengthnot lessthan
c 0 ν(π γ ([C]))γ −k
. Hen eν(T −k (π γ ([C])) ≥ N(k − N)c 0 ν(π γ ([C]))γ −k ≥ cν(π γ ([C])),
where
c
dependsonγ
butnoton[C]
. ThisimpliesthestatementoftheTheo-rem.
Corollary2.2.3.3. Thereexistsaninvariantmeasure
µ
,absolutely ontinuouswith respe ttothe Lebesguemeasure
ν
su hthatµ(A) = lim
n→∞
1 n
n−1 X
k=0
ν(T −k (A)) ≤ 4
γ − 1 ν(A).
If
γ ∈ K n
andγ > √
2
thenµ
isequivalenttoν
.Parry[25℄ hasshownthatthemeasure
µ
hasthedensityh(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,andifγ > √
2
thenh(x) > D γ(γ−1) γ 2 −2
.Theorem 2.2.3.5. The measure
µ
is the measureof maximal entropy. Thatis
h µ (T ) = h top (T )
.Proof. It su es to show that
h µ (T ) ≥ log γ
sin eh µ (T ) ≤ h top (T )
for anymeasureand
h top (T ) = log γ
. Theorem2.2.3.4andLemma2.2.3.1impliesthatforany ylinder
[C]
oflengthn
wehaveµ([C]) ≤ 4γ 1−n
. IfC n
isthepartitionof
I γ
into ylindersoflengthn
wehave1
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 showsthath µ (T ) ≥ log γ
.Example2.2.3.1. If
Σ γ
isspe iedthenthereexistsasequen ea ∈ Σ γ
su hthat
{σ n (a) } n∈N
isdenseinΣ γ
withrespe ttothemetrid γ
.Let
{a n } n∈N = {a n,1 a n,2 · · · } n∈N
bedenseinΣ γ
. DeneA n = a n,1 a n,2 · · · a n,n .
The ylindersets
[A n ]
are allnon-emptysin ea n ∈ [A n ]
. Sin eΣ γ
has nitememorythere existswords
A T,n
oflengthT n ≤ T
su hthatb = 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
{σ k n (b) } n∈N
is denseinΣ γ
withrespe ttod 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
. Ifγ ∈ K N
thenT
ismixing.Proof. Bowen[8℄hasshownthatanypie ewise
C 2
fun tiononanintervalwithonedis ontinuityandderivativelargerthan
√ 2
isweaklymixing.IfT
isweaklymixingthen
T
ismixingifandonlyifthere existsa onstantK
su hthatlim sup
n→∞ ν(A ∩ T −n B) ≤ Kν(A)ν(B),
foranymeasurablesets
A
,B
.Takeanytwo ylinders
[A]
and[B]
. Weprovethatd γ ([A] ∩ T −n [B]) ≤ Kd γ ([A])d γ ([B]),
(2.2)for
n
large enough. Letn A
be the length of the wordA
. Assume thatd Σ γ ([A]) = γ −N A
andd Σ γ ([B]) = γ −N B
.Let
n > n A
. Then[A] ∩ T −n [B] = {a ∈ Σ γ | a = A · · · | {z }
n
lettersB · · · } = [
k
[C k ],
where
[C k ]
are disjoint ylinders of the form[Ai n A · · · i j−1 B]
. There are atmost
N (n − n A )
su hnon-empty ylinders.ByLemma 2.2.3.2wehave
d Σ γ ([A]) ≥ γ −n A −N −1
. Wegetd 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]).
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
thenT
istopologi ally mixing.Proof. Weshowthatforanynon-empty ylinders
[A]
thereexists anumbern
su hthat
σ n ([A]) = Σ γ
. ItthenfollowsthatT
istopologi allymixing.Let
n A
bethelengthofthewordA
. Letn γ
bethelargestnumbersu hthat( −γ 0 ) · · · (−γ n γ −1 )
is awordinγ 0 · · · γ n A −1
. Clearly,n γ < n A
. There existsk, l < n γ
su hthateitherAγ 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 ]
orf n A +k (π γ ([A])) ⊃ [− γ−1 γ+1 , 0]
. For anyγ > √
2
thereisanumberN
su hthatf 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 measureµ
is the unique measure ofmaximal entropy.
Proof. In[13℄, Hofbauershowed that provided
T
is transitivethen there is auniquemeasureofmaximalentropy.
Theorem2.2.4.3.
T
isergodi withrespe ttothe measureµ
.Proof. In [19℄ it is shown that that any transformation of a ertain lass of
transformations with
n
dis ontinuities has at mostn
ergodi measures ea habsolutely ontinuous. Thus
µ
isergodi .Theorem2.2.4.4. If
γ > √
2
thenµ
isBernoulli.Proof. Bowen [8℄ has shown that provided
T
is weakly mixing, the measureµ
is Bernoulli. As mentioned in the proof of Theorem 2.2.4.1,T
is weaklymixing.