De ay of orrelations

Sin e

Λ ˆ λ × ˆ Λ λ = [ ∞ k=0

[

i

R ˆ ^k _i1 (λ) × ˆ R _i−1 ^k (λ)

we anpro eedasfollows

Z

I

Z

Ω ˆ λ

Z

Ω ˆ λ

χ {|y 1 −z 1 |<r} dˆ µ ^λ,s,ˆ _SBR ^x(λ) (z)dˆ µ ^λ,s,ˆ _SBR ^x(λ) (y)dλ

= X p t=1

X ∞ k=0

X

i

T i,t,k

≤ X p t=1

X ∞ k=0

X

i

c 3 r(γ ⁻¹ + ε) ^−k (γ − ε) ^−k µ ˆ ^λ _SBR ^{t ,s,ˆ x(λ t )} ( ˆ R ^k _i1 (λ t ))

≤ X p t=1

X ∞ k=0

c 3 r(γ ⁻¹ + ε) ^−k (γ − ε) ^−k ≤ X p t=1

c 4 r = c 5 r.

Hen e

lim inf

r→0

1 r

Z

I

Z

Ω ˆ λ

Z

Ω ˆ λ

χ {|y 1 −z 1 |<r} dˆ µ ^λ,s,ˆ _SBR ^x(λ) (ˆ z)dˆ µ ^λ,s,ˆ _SBR ^x(λ) (ˆ y)dλ ≤ c ⁵ .

^(3.6)

Thisisindependentofthe hoi e of

ˆ x : I → Q

^{so thisprovesTheorem3.3.0.2.}

andahyperboli produ tstru ture;

Θ = [

ω ^s ∈Γ ^s

ω ^s

!

\ [

ω ^u ∈Γ ^u

ω ^u

! ,

where

Γ ^s

^and

Γ ^u

^are olle tionsofstableandunstable urvesrespe tively,and areturntime

R : Θ → N

^{su h that}

f ^R(·) ( ·) : Θ → Θ

^.

For

ω ∈ Γ ^u

^let

ν ω

^denotethe onditionalmeasureoftheLebesguemeasure

ν

^{withrespe ttothe urve}

ω

^{. Ifthe onditions}

ν ω (ω ∩ Θ) > 0,

^{forea h}

ω ∈ Γ ^u

ν {x ∈ Θ | R(x) > n} < Cθ ⁿ ,

^forsome

C

^and

θ < 1, (f ⁿ , Q)

^{isergodi forea h}

n > 0,

andsomeotherregularity onditionsaresatisedthenthisissu ientto

on- ludeexponentialde ayof orrelationsforHölder ontinuousfun tions.

Amongotherexamplesin[38℄,Youngshowshowtondtheset

Θ

^andthe

returntime

R

^{for a lass ofpie ewise}

C ²

^{hyperboli mapsin two}dimensions.

This lassofmaps isdierentfromour lassbutstillthemethod anbeused

to onstru t

Θ

^and

R

^{. Wegivethe} onstru tionof

Θ

^and

R

^below.

The fa t that

(λ, γ, κ) ∈ P

^{makes the} onstru tion of

Θ

^{easier. This is}

be ausethereisauniform estimateof thelengthofthelo alstable manifolds

forthese parameters. From Se tion3.4 we on ludethat

W _α ^s (x)

^{existsfor all}

x ∈ Λ

^{. Theorem 3.5.0.3istruealsoif}

(λ, γ, κ) 6∈ P

^{but thenwehaveto work}

withtheset

D _δ ⁺ = {x | d(f ⁿ (x), S) > δγ ⁻ⁿ ∀n ≥ 0},

whi hwouldhavemadethe onstru tionof

Θ

^{somewhatmorete hni al.}

Wewill nowpro eedwith the onstru tionof

Θ

^and

R

^{. Itisimportantto}

gatherenough expansionin the unstabledire tion. Forthis purpose wetake

an

N ∈ N

^{su hthat}

γ ^N > 2e ^{1 e}

^.

Choose

δ > 0

^{sothat forany urve}

ω

^{in theunstabledire tion withlength}

notgreaterthan

δ

^theset

f ^N (ω)

^{onsistsofatmosttwo onne ted} omponents.

This an be done sin ewe an hoose

δ

^{to be the smallestdistan e between}

thelinesintheset

∪ ^N n=0 f ⁻ⁿ (S h )

^.

Take

0 < δ 0 < ¹ ₆ δ

^{tobesosmallthat theset}

A δ 0 = {x ∈ Θ | W 2δ ^u 0 (x)

^exists

}

has positive Lebesgue measure. For any

x ∈ A ^{δ 0}

^{we dene}

Ω(x) = W _δ ^u ₀ (x)

^.

Put

Γ ^s (x) = {W ^s (y) | y ∈ Ω(x)},

Γ ^u (x) = {W ^u (z) | z ∈ A ^{δ 0} , W ^u (z) ∩ W ^s 6= ∅, ∀W ^s ∈ Γ ^s (x) }.

Welet

Θ(x)

^{bethehyperboli setwiththeprodu tstru turedenedby}

Γ ^s (x)

and

Γ ^u (x)

^.

For any

x ∈ A ^{δ 0}

^{we let}

Q(x)

^{be the smallest open re tangle ontaining}

Θ(x)

^{. Theopensets}

{Q(x)} ^x∈A δ0

^{form anopen overingof}

A δ 0

^{. Sin e}

A δ 0

^is

ompa twe ansele tanite sub over

{Q(x i ) } ^r i=1

^{. Thesets}

{Θ(x i ) } ^r i=1

^will

then over

A δ 0

^{. Wewillwrite}

Θ i

^for

Θ(x i )

^.

We dene thereturn time

R

^to

∪Θ ⁱ

^{ona subsetof the sets}

Θ i

^{. Let}

i

^be

xed. We iteratethe map

f ^N

^{and onsider the onne ted omponentsof the}

set

(f ^N ) ⁿ (Ω(x i ))

^.

We onstru tforea h

n ∈ N

^apartition

P n ⁱ

^of

Ω(x i ) \{R ≤ n}

^{into onne ted}

urveswiththepropertythatif

ω ∈ P n ⁱ

^then

(f ^N ) ⁿ (ω)

^{isa onne ted urveof}

length

< 6δ 0

^.

Let

P 0 ⁱ

^{bethetrivialpartition,}

P 0 ⁱ = {Ω(x ⁱ ) }

^{. Assumethat}

P n−1 ⁱ

^isdened.

Let

ω ∈ P n−1 ⁱ

^{. Sin e}

(f ^N ) ⁿ⁻¹ (ω)

^{isa onne ted urveoflength}

< 6δ 0

^theset

(f ^N ) ⁿ (ω)

^{onsistsofatmosttwo onne ted} omponents. Let

{ω ^′ j } ^j=1,2

^bethe

orresponding omponents of

ω

^{. If}

(f ^N ) ⁿ (ω j )

^haslength

< 6δ 0

^{then weput}

ω j

ⁱⁿ

P n ⁱ

^{. Ifhowever}

(f ^N ) ⁿ (ω j )

^haslength

≥ 6δ ⁰

^{then there isa}

k

^{su hthat}

(f ^N ) ⁿ (ω j )

^rosses

Q(x k )

^{with segments of length}

≥ δ ⁰

^{sti king out on ea h}

sideof

Q(x k )

^{. Sin e}

|(f ^N ) ⁿ (ω j ) | > 6δ ⁰

^wehave

(f ^N ) ⁿ (ω j ) ∈ Γ ^u (x k )

^{and this}

impliesthat

(f ^N ) ⁿ (ω j ) ∩Q(x ^k ) ⊂ Θ ^k

^{. Wedene}

R = n

^on

ω j ∩(f ^N ) ⁻ⁿ (Θ k )

^{. In}

thiswaywegettwopie esoflength

> δ 0

^{onea hsideof}

(f ^N ) ⁿ (ω j \ {R = n})

^.

Wepartition

ω \ {R = n}

^into

{ω j,k } ^m k=1

^{su hthat}

δ 0 < (f ^N ) ⁿ (ω j,k ) < 6δ 0

^and

put

{ω j,k }

ⁱⁿ

P n ⁱ

^.

Finally,if

ω ∈ P ⁿ⁻¹

^thenwedene

R = n

^ontheset

S ω = [

ω ^s ∈Γ ^s (ω)

ω ^s

!

\ [

ω ^u ∈Γ ^u

ω ^u

!

⊂ Θ ⁱ ,

where

Γ ^s (ω)

^{isthesetoflo alstablemanifoldsin}

Γ ^s (x i )

^{whi hhasnon-empty}

interse tion with

ω

^{. The uniform estimate on the length of}

ω s ∈ Γ ^s

^implies

that

(f ^N ) ⁿ (S ω ) ⊂ Θ ^k

^.

Lemma3.5.0.4. Let

Ω = Ω(x i )

^{for some}

x i

^{. There exists}

C > 0

^and

θ 1 < 1

su hthat

ν Ω {R > n} ≤ Cθ ⁿ 1 .

Proof. Let

T 1 (x)

^{bethesmallest}

n ≥ 1

^{su hthatif}

ω ∈ P ⁿ⁻¹

^{isthe omponent}

ontaining

x

^then

|(f ^N ) ⁿ (ω) | ≥ 6δ ⁰

^{. If there is no su h}

n

^{then wesay that}

T 1 (x)

^{isnotdened. Notethat}

T 1 ≤ R

Suppose that

T k (x)

^{has been dened. Then we dene}

T k+1 (x)

^{to be the}

smallest

n > T k (x)

^{su hthat if}

ω ∈ P ⁿ⁻¹

^{is the omponent ontaining}

x

^then

|(f ^N ) ⁿ (ω) | ≥ 6δ ⁰

^{. Let}

T ^k = {T ^k

^isdened

}

^.

Ea htimeasegmentisstret hedtoalengthover

6δ 0

^{apie eoflength}

2δ 0

returnstooneofthesets

Θ i

^{. Thisimpliesthatif}

ω ∈ P ⁿ⁻¹

^and

T 1 | ^ω = n

^then

|(f ^N ) ⁿ (ω) | < γ ^N 6δ 0

^andso

ν Ω (ω ∩ {R = T ¹ })

ν Ω (ω) > 2δ 0

6δ 0 γ ^N = 1 3γ ^N .

Hen e

ν Ω (ω ∩ {R > T ¹ })

ν Ω (ω) < 1 − 1 3γ ^N = θ 2 .

Thismeansthat

ν Ω (T 2 )

ν Ω (T 1 ) < θ 2

^{. Similarlyweget}

ν Ω ( T ^k+1 ) ν Ω ( T ^k ) < θ 2 .

Thisimpliesthat

ν Ω ( T ^k ) < θ ^k ₂

^and

ν Ω {R > T ^k } < θ ^k 2

^.

Forany

k

{R > n} ⊂ {T ^k ≥ n} ∪ {T ^k < n < R }.

^(3.7)

There is a number

M

^{su h that if}

ω ∈ P ⁿ⁻¹

^then

ω \ {R = n}

^{an be}

overedbylessthan

M

^elementsfrom

P ⁿ

^.

Let

K p = {k ⁱ } ^p i=1

^where

k 1 < k 2 < · · · < k ^p

^with

k p ≥ n

^and

k p−1 < n

^.

Considerthe set

A K p = {T ⁱ = k i , i = 1, 2, . . . , p }

^{. It anbe overedby less}

than

2 ^{k p} M ^p

^elementsfrom

P ^k p

^{. Sin ethelengthof}

Ω

^is

2δ 0

^wehave

ν Ω (A K p ) ≤ 2 ^{k p} M ^p 6δ 0 (γ ^N ) ^{−k p} 2δ 0

= 3M ^p  2 γ ^N

 k p

.

If

p ≪ n

^{thenbyStirling'sformula}

n p



≈ n ^{n+ 1 2} e ⁻ⁿ

p!(n − p) ^{n−p+ 1 2} e ^−n+p = e ^−p n ^p p!

 1 − p n

 p−n

andsoif

ε

^{issmallenough}

X [εn]

p=0

n p



≤ C ¹ X [εn]

p=0

e ^−p n ^p p!

 1 − p n

 p−n

< C 1 (1 − ε) ⁻ⁿ X [εn]

p=0 n

e (1 − ε)
p

p! < C 1 (1 − ε) ⁻ⁿ e ^{n e (1−ε)} .

Choose

ε

^{sosmall that}

θ 3 = ^e

1−ε e M ^ε 2

(1−ε)γ ^N < 1

^{. This anbedone be auseof the}

hoi eof

N

^{. Then}

ν Ω {T ^[εn] > n } ≤

X [εn]

p=0

X

K p

ν Ω (A K p ) ≤ X [εn]

p=1

n p

 3M ^p

X ∞ k p =n

 2 γ ^N

 k p

≤ C ² e ^{1−ε e} M ^ε 2 (1 − ε)γ ^N

! n

= C 2 θ ⁿ ₃ .

Weuse(3.7)toapproximate

ν Ω {R > n}

^{. We hoose}

k = εn

^andget

ν Ω {R > n} ≤ C ² θ ⁿ ₃ + θ ^εn ₂ ≤ Cθ ⁿ 1 .

Wehaveproved that thereturn time

R

^{de ays} exponentially. Thisis not quitewhatwewant.

R

^{isthetimeneededforapie eof}

Θ i

^{toreturntosome}

Θ j

^{, but we would need}

R

^{to be the return time from}

Θ i

^to

Θ i

^{. Arguingas}

in [38℄, we an hoose

Θ ∗ = Θ i

^{for some}

i

^{su h that the return time}

R ∗

^of

Θ ∗

^satises

ν ω {R > n} < Cθ ⁿ

^{for some}

θ < 1

^{. This is su ientto on lude}

Theorem3.5.0.3. Notethatthe

SBR

^-measure onstru tedwiththemethodin [38℄isthesamemeasureasthe

SBR

^-measure onstru tedinSe tion 3.2.

[1℄ J. C.Alexander,J. A. Yorke,Fatbaker's transformations, Ergod. Th. &

Dynam.Sys.4(1984),123

[2℄ J. Buzzi, Absolutely ontinuous invariant measures for generi

multi-dimensional pie ewise ane expanding maps, Internat. J. Bifur. Chaos

Appl.S i.Engrg.9(1999),17431750

[3℄ J. Buzzi, Absolutely ontinuous invariant probability measures for

arbi-traryexpanding pie ewise

R

^{-analyti mappingsoftheplane,Ergod.Th.&}

Dynam.Sys.20(2000),697708

[4℄ J.Buzzi,GKeller,Zetafun tionsandtransferoperatorsfor

multidimen-sional pie ewiseane andexpanding maps,Ergodi TheoryDynam.

Sys-tems21(2001),689716

[5℄ V.P.Belykh,Modelsofdis retesystemsofphasesyn hronization,Systems

ofPhaseSyn hronization.Eds. V. V.Shakhildyanand L.N. Belynshina.

RadioiSvyaz,Mos ow,1982,61176

[6℄ A. Bertrand-Mathis, Développement en base

θ

^; répartition modulo unde la suite

(xθ ⁿ ) n≥0

^{; langages odéset}

θ

^{-shift, Bull. So .Math. Fran e114}

(1986),271323

[7℄ F.Blan hard,

β

-expansionsandsymboli dynami s,Theor.Comp.S i.65 (1989),131141

[8℄ R. Bowen, Bernoulli maps of the interval, Israel J. of Math. 28 (1977),

161168

[9℄ M. Denker, C. Grillenberger, K. Sigmund, Ergodi Theory on Compa t

spa es,Le tureNotesin Math.,527,Springer,Berlin,1976

[10℄ S.Eilenberg,Automata,languagesandma hines,vol.A,A ademi press,

London, 1974

[11℄ P.Erdös,Onafamilyofsymmetri Bernoulli onvolutions,Amer.J.Math.

61(1939),974976

[12℄ F. Hofbauer,

β

^{-Shifts Have Unique Maximal Measure, Mh. Math. 85}

(1978),189198

[13℄ F. Hofbauer, On intrinsi ergodi ity of pie ewise monotoni

transforma-tionswithpositive entropy, IsraelJ.ofMath.34(1979)213237

[14℄ F. Hofbauer, On intrinsi ergodi ity of pie ewise monotoni

transforma-tionswithpositive entropy II,IsraelJ.ofMath.38(1981),107115

[15℄ F. Hofbauer, The stru ture of Pie ewise monotoni transformations,

Er-god. Th.&Dynam.Sys.1(1981),159178

[16℄ F. Hofbauer, The maximal measure for linear mod one transformations,

J.LondonMath.So .2(1981),92112

[17℄ F. Hofbauer, Pie ewise Invertible Dynami al Systems, Probab. Th. Rel.

Fields72(1986),359386

[18℄ G.Keller,Equilibriumstatesinergodi theory, LondonMathemati al

So- iety Student Texts 42, Cambridge University Press, Cambridge, 1998,

ISBN:0-521-59420-0

[19℄ T.Li,J.Yorke,Ergodi transformationsfromanintervalintoifself,Trans

Amer.Math.So .235(1978),183192

[20℄ D. A.Lind,The entropiesof topologi al Markov shifts andarelated lass

ofalgebrai integers,Ergod. Th.&Dynam.Sys.4(1984),283300

[21℄ J. C. Oxtoby Measure and ategory, Graduate Texts in Mathemati s 2,

Springer-Verlag,NewYork-Berlin,1971

[22℄ W. Parry, On the

β

^{-expansion of real numbers, A ta Math. A ad. S i.}

Hung.11(1960),401416

[23℄ W.Parry,Representationsforrealnumbers,A taMath.A ad.S i.Hung.

(1964),95105

[24℄ W. Parry, Symboli dynami s and transformations of the unit interval,

Trans.Amer.Math. So .122(1966),368278

[25℄ W. Parry, The Lorenz attra tor and a related population model, Le ture

NotesinMath.,729,Springer,Berlin,1979

[26℄ Y. Peres, B. Solomyak, Absolute ontinuity of Bernoulli onvolutions, a

simpleproof, Math.Resear hLetters3(1996),231236.

In document The Family of Belykh Maps Persson, Tomas (Page 57-65)