# Proof of the theorem

I dokument The Family of Belykh Maps Persson, Tomas (sidor 49-57)

theset

### {y ∈ W s (x) | d(x, y) < δ}

isa onne ted urveoflength

thendene

### W δs (x) = {y ∈ W s (x) | d(x, y) < δ}

tobethestablemanifoldof

withlength

### 2δ

. Otherwisewesaythat

doesnotexist.

Sin e

### f

isnotinvertiblewe annotdene thelo alunstablemanifoldof

### x

intheusualway. Let

### D −δ = {x | d(f −n (x), S) > δγ −n ∀n ≥ 0}.

ThissethaspositiveLebesguemeasureif

### δ

istakensu ientlysmall,see[27℄.

Itiseventruethat

If

thenforany

with

and

we

have

forall

### n ≥ 0

. We anthusdenetheunstable

manifoldof

oflength

tobetheset

If

thenwesaythat

### W δu (x)

doesnotexist.

Lemma3.4.0.1. Let

bexedand

### I ⊂ (γ −1 +ε, 0.64)

. Thereisa onstant

su hthatfor any

andany

### {a i } ∞ i=0 ∈ {−1, 0, 1} N

thefollowingestimate

isvalid.

Let

,

bexedand

### I ⊆ (0.5, 0.64)

anyintervalsu hthat

,

wheretheset

### P

willbe hosenlater.

Let

andlet

### µ λ,s,xSBR (B r (t)) 2r

denotethelowerderivativeofthemeasure

### µ λ,s,xSBR

the onditionalmeasureof

### µ λSBR

withrespe ttothelo alstablemanifold

. Thepartitionof

### Q

into

lo al stable manifolds is learly measurable so the onditional measures exist

andwe anusethemasfollows. Wewanttoprovethatfora.e.

thereisaset

su hthat

and

(3.1)

holds for a.e.

### x

. This implies that the measure

### µ λ,s,xSBR

restri ted to the set

### Ω λ

is absolutely ontinuousfora.e.

### x

. Sin ethe onditionalmeasures on the unstablemanifoldsareabsolutely ontinuouswithrespe ttoLebesguemeasure,

this implies that

### µ λSBR | Ω λ

is absolutely ontinuous with respe t to Lebesgue

measure. Sin e

### µ λSBR (Ω λ ) > 0

, ergodi itythen impliesthatthis alsoholds for

themeasure

### µ λSBR

.

Fatou'slemmaimpliesthatin ordertoprove(3.1)itsu estoprovethat

### µ λ,s,xSBR (Ω λ ∩ B r (y))dµ λ,s,xSBR (y) < ∞.

Wemayrewritethisas

### χ {|y 1 −z 1 |<r} dµ λ,s,xSBR (z)dµ λ,s,xSBR (y) < ∞.

(3.2)

Wewill hoose a lassof fun tions

su h that

### x(λ) ∈ Ω λ

andprove

thatthefollowingestimateisvalid

(3.3)

### γ=1.858, λ=0.54, k=0

Figure3.3: Someinverseimagesof

### S v

.

This thenimplies that

for a.e.

usethat

### µ SBR = ˆ µ SBR ◦ π −1

andprovetheequivalent ondition

### χ {|y 1 −z 1 |<r} dˆ µ λ,s,ˆSBRx(λ) (ˆ z)dˆ µ λ,s,ˆSBRx(λ) (ˆ y)dλ < ∞.

(3.4)

Toprove(3.4)thesymboli oding

### Σ λ,γ,κ

willbeused. Sin ewehave

and

xedand vary

### λ

it isthedynami s taking pla ein the horizontaldire tion

thatis ru ial. Itishen ethedynami albehaviouroftheverti al omponent

ofthesingularity

### S v

thatisimportant. Below,wewill hooseaset

inwhi h

### S v

behavesasifitdoesnotexist.

One he ks withanumeri al al ulationthat for

,

and

### κ = 0

there are only nitely many points in the omplete ba kwardorbit of

theverti alpie e ofthesingularity,

### S v

. Numeri sareonlyused tosee thisin

aneasyway. Itdoesnotinuen eontherigorousityoftheproof. Inthis ase,

when

, wehave

### S v = {0}

. Figure3.3 and 3.4illustrate this byshowing

thetwopossiblepathsof

startingin

### Q 1

. Thepointsintheba kwardorbit

of

are marked with

### ×

in the gures. The dashed lines are the border of

thesets

and

### f (Q −1 )

. Thearrowsshowshowthepointsaremappedby

### f −1

. Ea hpathdrawnintheguresterminatesafternitelymanysteps,after

whi hthere arenomoreinverseimages.

### γ=1.858, λ=0.54, k=0

Figure 3.4: Yet someinverseimagesof

### S v

.

The numeri al evaluations of the points are in Table 3.1 and Table 3.2.

Comparingthese valueswith

### f (Q 1 ) = [1 − 2λ, 1] × [1 − γ, 1] = [ −0.08, 1] × [−0.858, 1], f (Q −1 ) = [ −1, 2λ − 1] × [−1, γ − 1] = [ −1, 0.08] × [−1, 0.858],

showsthattherearenomorepointsintheba kwardorbitof

### S v

andthatthese

pointsareboundedawayfromtheset

### f (S)

. Thisallowustodrawthefollowing

on lusion. Thereisanopenneighborhood

of

su hthat

for

### N

su ientlylarge.

Bythe ontinuousdependen eontheparametersofthenitelymanypoints

in the ba kward orbit of

### S v

, there exists an open ball

around

### (γ, λ, κ) = (1.858, 0.54, 0)

su h that theba kwardorbit of

behavesin the

same way for any

### (γ, λ, κ) ∈ P

in the followingsense. Forany

### (γ, λ, κ) ∈ P

the ba kward orbit of

### S v

ontains nitely many pie es, ea h bounded away

from

### f (S)

, and there areuniform numbers

and

su h that

satises

### f −N (U ) = ∅

. This implies that

forany

### (γ, λ, κ) ∈ P

.

Re allthat wehavetheparameters

and

xedand

isanintervalsu h

that

. Partition

### I

into sub intervals

, su h that

### |I t | < 15 1 α

. This anbedonesothat

. Forea h

### t = 1, . . . , p

xa

iterate oordinates

numeri al

values ontainedin

0

0

### −0.9746 Q \ (f(Q 1 ) ∪ f(Q −1 ))

Table3.1: ThepointsinFigure 3.3.

iterate oordinates

numeri al

values ontainedin

0

0

### −0.8773 Q \ (f(Q 1 ) ∪ f(Q −1 ))

Table3.2: ThepointsinFigure3.4.

### λ t ∈ I t

. Thefollowinglemmawillprovidesu ient ontrolwhen hangingthe

parameter

### λ

.

Lemma3.4.0.2. Forany

andany

### λ, λ ′ ∈ I t

the symboli spa es

and

oin ide.

Proof. Apoint

liesin

### Λ ˆ λ t

ifandonlyifthereisasequen e

su hthat

and

forall

. Let

. Weshowthat for

any

thereisapoint

### x ˆ ′ = (x ′1 , x ′2 , x ′3 ) ∈ ˆ Λ λ ′

su hthatthe orresponding sequen e

satises

for all

### n ∈ Z

. Inthiswaywedeneamap

by

### Ξ λ t ,λ ′ : ˆ x 7→ ˆx ′

.

Itsu estoshowthatthepoint

dened by

satises

forall

### n ∈ Z

. Thenthisimpliesthat

. A hange

of

### λ

hasonlyinuen eonthese ond oordinateof

### f (ˆ ˆ x)

. Sin ethelo alstable

manifoldsare parallel and oriented in the dire tion of the se ond oordinate

it su esto he k that when hanging

### λ

, these ond oordinate nevermove

overtheverti aldis ontinuity

### S v

. Wedothisbyanestimateofthederivative

### n=0 i −n λ n 

. Asimple al ulationgivesthat

if

### 2 < λ < 34

. Thisimpliesthat

Thismeansthat

### ˆ x ′

doesnot rosstheverti alpie eofthesingularityandhen e stayson thesame side ofthe singularityas

### x ˆ

. Similarlyoneshowsthat any

iterate

### f ˆ λn′ (ˆ x ′ )

ofstaysonthesamesideofthesingularityas

### f ˆ λnt (ˆ x)

.

Remark. The partition of

### I

intosubintervalsisarbitrary soinfa t the sym-boli spa es oin ide for any

### λ, λ ′ ∈ I

.

We have shown that the symboli spa e

### Σ λ,γ,κ

does not hange when

### λ

varies. Wealsoneedtoestimatehowthemeasure

### µ ˆ λSBR

hanges.

Lemma 3.4.0.3. There is a onstant

### c 1 > 0

su h that for any

and any

### c −11 µ ˆ λSBRt (C λ t ) ≤ ˆµ λ SBR (C λ ) ≤ c 1 µ ˆ λSBRt (C λ t ),

(3.5)

forany ylinder setoftheform

,

### k < n

.

Proof. Sin ethereareonlynitelymanyinverseimagesoftheset

### S v

,allother

inverseimagesofthesingularitywill onsistofahorizontalline. As

variesover

### I

thesehorizontallinesarenot hanged,onlytheinverseimagesof

### S v

hanges.

Thereisthusa onstant

,independentof

,su hthatforany

### λ ∈ I t

andany

ylindersetoftheform

,

we

have

Espe ially

for any

andso

### c −11 µ ˆ λSBRt (C λ t ) ≤ ˆµ λ SBR (C λ ) ≤ c 1 µ ˆ λSBRt (C λ t ),

for any ylinder set of the form

with

### k < n

.

Remark. Lemma 3.4.0.3 is also valid for the onditional measures on the

stable manifold.

Sin e the entropy of the onditional measures are a.s. equal that of the

measure, the Shannon-M Millan-Breiman theorem implies that given

### ε > 0

,

thereexists anumber

su hthat

### > 1 − ε.

Lusin'sTheoremimpliesthatthereisanumber

andaset

su hthat

when

and

. Weput

and

Then

.

Put

for

### λ ∈ I t

. ByLemma3.4.0.3itfollowsthat if

then

and

Takea

and

. Dene

andfor

dene

sothat

. Then

### x ˆ

is ontinuous. ByLemma3.4.0.1

Sin e

### R ˆ ki1 (λ) × ˆ R i−1k (λ)

we anpro eedasfollows

Hen e

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

(3.6)

Thisisindependentofthe hoi e of

### ˆ x : I → Q

so thisprovesTheorem3.3.0.2.

I dokument The Family of Belykh Maps Persson, Tomas (sidor 49-57)