Statistical properties of intermittent maps with unbounded derivative

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Statistical properties of intermittent maps with

unbounded derivative

Giampaolo Cristadoro, Nicolai Haydn, Philippe Marie, Sandro Vaienti

To cite this version:

Giampaolo Cristadoro, Nicolai Haydn, Philippe Marie, Sandro Vaienti. Statistical properties of

inter-mittent maps with unbounded derivative. Nonlinearity, IOP Publishing, 2010, 23 (5), pp.1071-1096.

�hal-01258190�

arXiv:0812.0555v2 [math.DS] 16 Dec 2008

derivative

Giampaolo Cristadoro, Ni olai Haydn, Philippe Marie, Sandro Vaienti

De ember 16, 2008

Abstra t

We study the ergodi and statisti al properties of a lass of maps of the ir le and of theinterval ofLorenz typewhi h present indierent xedpointsand pointswithunbounded derivative. Thesemapshavebeen previously investigatedinthephysi s literature. We prove inparti ular that orrelationsde aypolynomially,andthatsuitableLimitTheorems ( onver-gen eto StableLawsor Central LimitTheorem) holdfor Hölder ontinuous observables. We moreovershowthatthe returnand hittingtimes areinthelimit exponentiallydistributed.

1 Introdu tion

TheprototypeforintermittentmapsoftheintervalisthewellknownPomeau-Mannevillemap

T

dened on the unit interval

[0, 1]

and whi h admits a neutral xed point at

0

with lo al behavior

T (x) = x + cx

1+α

;otherwise itis uniformly expanding. The onstant

α

belongs to

(0, 1)

toguaranteetheexisten eofaniteabsolutely ontinuousinvariantprobabilitymeasure and the onstant

c

ould be hosen in su h a way that the map

T

has a Markov stru ture. This map enjoy polynomial de ay of orrelations and this property still persists even ifthe mapisnot anymoreMarkov [30 ℄.

Another interesting lass of maps of the interval are the one-dimensional uniformly ex-panding Lorenz-like maps (see [15 , 29, 11℄ for their introdu tion and for the study of their topologi alproperties),whose featuresarenowthepresen eofpointswithunboundedderiv a-tivesandthela kofMarkovstru ture: inthis aseone ouldbuilduptowersandndvarious rates for the de ay of orrelations depending on the tail of the return time fun tion on the base of the tower, see, for instan e [7℄ and [8 ℄. The latter paper deals in parti ular with one-dimensional maps whi h admit riti al points and, eventually, points with unbounded derivatives,but it leavesopenthe asewhere there ispresen eof neutral xedpoints.

In this paper we are interested in maps whi h exhibit the last two behaviors, namely neutralxedpointsandpointswithunbounded derivatives. Su h mapshavebeenintrodu ed

a polynomial de ayof orrelations and they also studied other statisti al properties like the sus eptibility andthe

1/f

-noise. Another ontributionbyA.Pikovsky [25 ℄showed, stillwith heuristi arguments, that these maps produ e anomalous diusionwith squaredispla ement growing faster than linearly. R. Artuso and G. Cristadoro [3℄ improved the latter result by omputingthemomentsofthedispla ementontheinniterepli asofthefundamentaldomain andshowed aphasetransitionintheexponent ofthemomentsgrowth. Re entlyLorenz usp maps arose to des ribe the distribution of the Casimir maximum in theKolmogorov-Lorenz model of geouid dynami s [24 ℄. Despite this interesting physi al phenomenology, we did not nd any rigorous mathemati al investigation of su h maps. These maps are dened on the torus

T

= [

−1, 1]/ ∼

and depend on the parameter

γ

(see below); when

γ = 2

the orresponding map was taken as an example of the non-summability of therst hyperboli timeby Alves andAraujoin[2 ℄. This mapsreads:

˜

T (x) =

(

2

√

x

− 1

if

x

≥ 0

1

− 2

p

|x|

otherwise (1)

and it was proved in [2℄ that it is topologi al mixing, but no other ergodi properties were studied.

A tually, the Grossmann and Horner maps are slightly dierent from those investigated in[25 ℄and [3 ℄,thedieren e beingsubstantiallyinthefa tthatthelatter aredened onthe ir le instead than on the unit interval. We will study in detail the ir le version of these maps in Se tions 2 to 5, and we will show in Se tion 6 how to generalize our results to the interval version: sin e both lasses of maps are Markov, the most important information, espe iallyin omputingdistortion,will omefromthelo albehavioraroundtheneutralxed points and the points with unbounded derivatives and these behaviors will be the same for both versions. There isneverthelessaninterestingdieren e. The ir leversionintrodu edin Se tion1iswritten insu hawaythattheLebesguemeasureisautomati allyinvariant. This is not the ase ingeneral for the intervalversion quoted in Se tion 6. However the strategy thatwe adopttoprove statisti alproperties (Lai-SangYoungtowers) will give us aswellthe existen eof anabsolutely ontinuous invariant measureand wewill ompleteit byproviding informations on the behavior of the density. It is interesting to observe that in the lass of maps onsidered by Grossmann and Horner on the interval

[

−1, 1]

(see Se t. 6), the analog of(1 ) isgiven bythefollowing map:

˜

S(x) = 1

_{− 2}

p

_{|x| .}

(2)

This mapwas investigated byHemmer in 1984 [19 ℄: he also omputed by inspe tion the in-variant density whi h is

ρ(x) =

1

2

(1

− x)

and theLyapunov exponent (simply equal to

1/2

), butheonlyarguedabout aslowde ayof orrelations. WewillshowinSe t. 6howtore over thequalitative behavior of this density(and of all the others in the Grossmann and Horner lass).

C

1

on

T

/

{0}

,

C

2

on

T

/(

{0} ∪ {1})

and areimpli itly dened onthe ir le bytheequations:

x =















1

2γ

(1 + T (x))

γ

if

0

≤ x ≤

1

2γ

T (x) +

1

2γ

(1

− T (x))

γ

if

1

2γ

≤ x ≤ 1

and for negative values of

x

by putting

T (

−x) = −T (x)

. We assume that parameter

γ > 1

. Note that when

γ = 1

the map is ontinuous with onstant derivative equal to

2

and is the lassi al doubling map. The point 1 is a xed point with derivative equal to

1

, while at 0 the derivative be omes innite. The map leaves the Lebesgue measure

m

invariant (it is straightforward to he k that the Perron-Frobenius operator has

1

as a xed point). We will prove inthe nextse tions theusual bun h of statisti alproperties: de ayof orrelations (whi h, due to the paraboli xed point, turns out to be polynomial with the rate found in [14℄); onvergen e to Stable Laws and large deviations; statisti sof re urren e. All these results will follow from existing te hniques, espe ially towers, ombined with the distortion bound proved in the next se tion. Distortion will in fa t allows us to indu e with the rst return map on ea h ylinder of a ountable Markov partition asso iated to

T

. A tually one ould indu e on a suitable interval only ( alled

I

0

in the following): the proof we give is intentedto providedisortion onall ylindersof the ountable Markovpartition overing mod

0

thewhole spa e

[

−1, 1]

, sin e this is ne essary inorder to apply theindu ing te hnique of [4 ℄ whi h will give us thestatisti alfeaturesof re urren estudied inSe t. 5: distributions of rst return and hitting times, Poissonian statisti s for the number of visits, extreme values laws.

2 Distortion

Notations: With

an

≈ bn

we mean that there exists a onstant

C

≥ 1

su h that

C

−1

_bn

_≤

a

n

≤ Cbn

for all

n

≥ 1

; with

a

n

.

b

n

we mean thatthere existsa onstant

C

≥ 1

su h that

∀n ≥ 1

,

a

n

≤ Cbn

;with

a

n

∼ bn

we mean that

lim

n→∞

a

n

b

n

= 1

. We will also usethe symbol "

O

" intheusual sense. Finallywe denote with

|A|

thediameter of theset

A

.

There is a ountable Markov partition

{Im}m∈Z

asso iated to this map; the partition is built

mod m

as follows:

Im

= (am−1, am)

for all

m

∈ Z

∗

and

I0

= (a0−, a0+)/

{0}

, where, denoting with

T

+

= T

|(0,1)

and with

T

−

= T

|(−1,0)

:

a

0+

=

1

2γ

, a

0−

=

−

1

2γ

and

a

i

= T

−i

+

a

0+

, a

−i

= T

−

−i

a

0−

,

i

≥ 1 .

Thenwedene

∀i ≥ 1

:

b

−i

= T

−

−1

a

i−1

and

b

i

= T

−1

+

a

−(i−1)

.

We now state without proof a few results whi h are dire t onsequen es of thedenition ofthe map.

-1

0

_x

1

-1

0

1

T(x)

a

_-2

a

_-1

a

_0-

b

₂

b

₁

b

_-1

b

_-2

a

₀₊

a

₁

a

₂

Lemma 1. 1. When

x

→ 1

−

:

T (x) = 1

− (1 − x) −

1

2γ

(1

− x)

γ

+

O ((1 − x)

γ

)

2. When

x

→ 0

+

:

T (x) =

−1 + (2γ)

1

γ

_x

1

γ

_.

Lemma 2. We have for all

n

≥ 0

,

a

±(n+1)

= a±n

+

1

2γ

(1

− a±n)

γ

and:

a

n

∼

1

−



2γ

γ

_{− 1}



1

γ−1

1

n

γ−1

1

a−n

∼ −1 +



_2γ

γ

− 1



1

γ−1

1

n

γ−1

1

l

n

:= length[a

n−1

, a

n

]

∼

1

2γ



2γ

γ

_{− 1}



γ

γ−1

1

n

γ−1

γ

n > 1

|b±(n+1)| ∼

_2γ

1



2γ

γ

_{− 1}



γ

γ−1

1

n

γ−1

γ

,

n > 1 .

We now indu e on the interval

I

m

:= (a

−m

, a

m

)/

{0}

and provide a bounded distortion estimatefor the rstreturn map. Wedene

Z

m,p

= Z

+

m,p

∪ Z

m,p

−

,where:

Z

+

m,1

:= (b

m+1

, a

m

)

,

Z

_m,1

−

:= (a

−m

, b

−(m+1)

)

and

Z

+

m,p>1

:= (b

m+p

, b

m+p−1

)

,

Z

−

m,p>1

:= (b

−(m+p−1)

, b

−(m+p)

)

. Note

that

I

m

=

∪p≥1

Z

m,p

andthattherstreturnmap

T = I

b

m

→ Im

a tsonea h

Z

m,p

as

T = T

b

p

and inparti ular:

T

p

(Z

_m,p

+

) =

(

(a

−m

, a

m−1

)

p = 1

(a−m, a

−(m−1)

)

p > 1

T

p

(Z

_m,p

−

) =

(

(a

_−(m−1)

, a

m

)

p = 1

(am−1, am)

p > 1 .

We nallyobserve thattheindu ed map

T

b

isuniformlyexpandinginthesense thatfor ea h

m

and

p

there exists

β > 1

su h that

|D b

T (x)

| > β

,

∀x ∈ Im

. 1

Proposition3(Boundeddistortion). Letusindu eon

I

m

;thenthereexistsa onstant

K > 0

that depends on m, su h that for ea h

m

and

p

and for all

x, y

∈ Zm,p

, we have:









DT

p

_(x)

DT

p

_(y)







 ≤ e

K|T

p

_(x)−T

p

_(y)|

≤ e

2K

.

Remark 1. The ylinder

Z

m,p

isthe disjoint union of the two openintervals

Z

+

m,p

and

Z

−

m,p

sitting on the opposite sides of

0

(see above). Whenever

x

and

y

belongs to dierent om-ponents, we pro eed by rst noti ing that

DT

p

_{(x) = DT}

p

₍

_−x)

and

−x

sits now in the same omponent as

y

. By exploitingthe fa t that

T

p

is oddwe get









DT

p

(x)

DT

p

_(y)







 =









DT

p

(

−x)

DT

p

_(y)







 ≤ e

K|T

p

_(−x)−T

p

_(y)|

≤ e

K|−T

p

(x)−T

p

(y)|

≤ e

K|T

p

(x)−T

p

(y)|

.

andwe an thus on entrate on the ase when

x

and

y

are takeninthe sameopen omponent (see below).

Proof. We denote with

l

m

thelength of the interval

(a

m−1

, a

m

)

(when

m = 0

,

l

0

= length of

(0, a

0+

)

). We start byobserving that









DT

p

(x)

DT

p

_(y)







 = exp





p−1

X

q=0

(log

|DT (T

q

x)

| − log |DT (T

q

y)

|)





= exp





p−1

X

q=0









D

2

T (ξ)

DT (ξ)







 |T

q

x

− T

q

y

|



 ,

(3)

where

ξ

is apoint between

T

q

_x

and

T

q

_y

. Wedivide the ases

p = 1

and

p > 1

.

• p = 1

For

(x, y)

∈ Z

−

m,1

(seeRemark(1 )above), using

|x − y| < |T (x) − T (y)|

,we dire tlyget:









DT (x)

DT (y)







 ≤ exp [K

1|T (x) − T (y)|] ,

where

K

1

= sup

(Z

−

m,1

)

D

2

_{T = D}

2

_{T (a}

m

)

. 1

Usingthe hainrulewe ansee that

β

≡ inf

x∈Z

m,1

|DT (x)| > 1

.

• p > 1

Startwith

x, y

∈ Z

−

m,p

(seeRemark(1 )above),then

T x, T y

∈ (am+p−2, am+p−1)

;

T

2

_{x, T}

2

_y

_∈

(a

m+p−3

, a

m+p−2

)

;

. . .

;

T

p−1

_{x, T}

p−1

_y

_{∈ (am}

_{, a}

m+1

)

,we have:

(

3

)

≤ exp





sup

(Z

m,p

−

)

|D

2

_T

_|

inf

_(Z

−

m,p

)

|DT |

|x − y| +

p−1

X

q=1

sup

_(a

_m+p−q−1

_,a

_m+p−q

₎

_|D

2

_T

_|

inf

_(a

_m

_+p−q−1

_,a

_m

_+p−q

₎

_{|DT |}

|T

q

_x

_{− T}

q

_y

_|





≤ exp



 sup

(Z

−

m,p

)

|D

2

T

_|

_{|x − y| +}

p−1

X

q=1

sup

(a

m+p−q−1

,a

m+p−q

)

|D

2

T

_|

_|T

q

x

_{− T}

q

y

_|



 .

(4)

To ontinue we need thefollowing Lemma 4. For

x, y

∈ Z

−

m,p

(see Remark(1 ) above) we have:

(i)

Pp−1

q=1

sup

(a

m+p−q−1

,a

m+p−q

)

|D

2

_T

_|

_|T

q

_x

_{− T}

q

_y

_{| ≤ C1|T}

p−1

_Z

_|

(ii)

sup

Z

−

m,p

|D

2

_T

_|

_{|x − y| ≤ C2}

|T

p−1

_Z|

l

m+1

,

where we set for onvenien e

Z

the intervalwithendpoints

x

and

y

. Proof. (i) Denote

T

p−1

_{x = z}

x

and

T

p−1

_{y = z}

y

; sin e the derivative is de reasing on

(0, m)

we have:

|T

q

x

− T

q

y

| ≤

1

DT

p−1−q

_(a

m+p−q

)

|zx

− zy| .

(5)

Let'snow onsidertheterm:

DT

p−1−q

(a

m+p−q

) = DT (a

m+p−q

)DT (T a

m+p−q

) . . . DT (T

p−2−q

a

m+p−q

) .

(6)

Sin efor

q

≥ 1

and

ξ1

∈ (aq, aq+1)

:

DT (a

q

)

≥ DT (ξ1

) =

T (aq+1)

− T (aq)

a

q+1

− aq

=

aq

− aq−1

a

q+1

− aq

itfollows that

(

6

)

≥

a

m+p−q

− am+p−q−1

a

m+p+1−q

− am+p−q

·

a

m+p−q−1

− am+p−q−2

a

m+p−q

− am+p−q−1

. . .

a

m+2

− am+1

a

m+3

− am+2

≥

a

m+2

− am+1

a

m+p+1−q

− am+p−q

andthus: 2

1

DT

p−1−q

_(a

m+p−q

)

≤

a

m+p+1−q

− am+p−q

a

m+2

− am+1

.

2

We have just proved that if

ξ

is any point in

(a

m+p

, a

m+p+1

)

(and the same result holds for its negative ounterpart

(a

−(m+p+1)

, a

−(m+p)

)

aswell)then

DT

p

_(ξ)

≥

a

m+2

−a

m+1

a

m+p+1

−a

m+p

. In asimilar waywe anprovethelower bound:

DT

p

_(ξ)

≤

a

0+

Moreover:

|zx

− zy| ≤ |T

p−1

_Z

_|

. Finally:

(

5

)

≤

a

m+p+1−q

− am+p−q

a

m+2

− am+1

|T

p−1

_Z

_{| .}

(7)

Usinglemmas1and2weseethatthereexistsa onstant

C0

dependingonlyonthemap

T

su h that:

sup

(a

m+q−1

,a

m+q

)

|D

2

T

_|

(a

m+q+1

− am+q

)

≤ C0

·

1

(q + m)

γ−2

γ−1

_{(q + m)}

γ

γ−1

= C

0

·

1

(q + m)

2

.

Therefore the sum over

q = 1, 2, . . .

is summable and there exists a onstant

C

1

su h thatfor

x, y

∈ Z

−

m,p

:

p−1

X

q=1

sup

(a

m+p−q−1

,a

m+p−q

)

|D

2

T

_|

_|T

q

x

_{− T}

q

y

_{| ≤ C1|T}

p−1

Z

_{| .}

(8)

(ii)Inthis ase we need to ontrol thebehaviorof themap lose to

0

. Inparti ular, by usinglemmas 1 and2 (and thesymmetryof

b

±i

) we start bynoti ing that

sup

(b

i+1

,b

i

)

|D

2

_T

_|

_|bi

_{− bi+1| = O(}

i

2γ−1

γ−1

i

2γ−1

γ−1

) = 1.

(9)

Combining (8) and (9) with (4) we get that there exists a onstant

D

2

so that for all

j

≤ p − 1

1

D

2

≤









DT

j

(x)

DT

j

_(y)







 ≤ D

2

.

(10)

Let's all

α = b

−(m+p−1)

, β = b

−(m+p)

the end points of

Z

−

m,p

. For

j

1

, j

2

≤ p − 1

there exist

η1

∈ (x, y)

and

η2

∈ (α, β)

su hthat:

|T

j

1

_x

_{− T}

j

1

_y

_{| = DT}

j

1

_(η

1

)

|x − y|,

|T

j

2

_α

_{− T}

j

2

_β

_{| = DT}

j

2

_(η2)

_{|α − β|.}

Thedistortionbound(10) yields



T

j

1

_x

_{− T}

j

1

_y





|T

j

1

_α

− T

j

1

_β

|

≤ D

2

2



T

j

2

_x

_{− T}

j

2

_y





|T

j

2

_α

− T

j

2

_β

|

.

Ifwe now hoose

j

1

= 0

and

j

2

= p

− 1

then

sup

(α,β)

|D

2

_T

_|

_{|x − y| ≤ D}

2

2

sup

(α,β)

|D

2

_T

_|

|α − β| · |T

p−1

x

− T

p−1

y

|

|T

p−1

_α

_{− T}

p−1

_β

_|

.

Sin e

|T

p−1

_α

_{− T}

p−1

_β

_{| = lm+1}

_{= a}

m

− am+1

and

x

and

y

to belong to

Z

weget:

sup

(α,β)

|D

2

_T

_|

_{|x − y| ≤ D}

2

2

sup

(α,β)

|D

2

_T

_|

|α − β| · |T

p−1

Z

|

lm+1

and using distortion bound (9 ) on emore we have that there exist a onstant

C

2

su h that:

sup

(α,β)

|D

2

_T

_|

_{|x − y| ≤ C2}

|T

p−1

Z

|

l

m+1

.

By olle ting lemma 4(i) and 4(ii) we see that the ratio

|DT

p

_(x)/DT

p

_(y)

_|

,

(x, y

∈ Z)

is bounded as:









DT

p

_(x)

DT

p

_(y)







 ≤ exp



C2

|T

p−1

_Z

_|

l

m+1

+ C1|T

p−1

_Z

_|



_{≤ exp}



_K2|T

p−1

_Z

_|



(11) with

K

2

= C

1

+ C

2

/l

m+1

.

Wenish the proofof the Propositionby hoosing

K = max(K

1

, K

2

)

3 De ay of orrelations

In this se tion and inthe next we prove several statisti al properties for our map: they are basi ally onsequen es of the distortion inequality got inthe previous se tion mat hed with establishedte hniques.

Proposition 5. The map

T

enjoys polynomial de ay of orrelations (w.r.t. the Lebesgue measure

m

), for Hölder ontinuous fun tions on

T

. More pre isely, for all Hölder

ϕ : T

→ R

andall

ψ

∈ L

∞

_{(T, m)}

, we have:









Z

(ϕ

_{◦ T}

n

) ψ dm

₋

Z

ϕ dm

Z

ψ dm







 = O

1

n

γ−1

1

.

Proof. WewilluseLai-SangYoung'stowerte hnique[30 ℄. Webuildthetowerovertheinterval

I

0

andwedene the return timefun tion astherstreturn time: for all

x

∈ I0

, R(x) := min

{n ∈ N

+

_{; T}

n

_x

_{∈ I0} := τI}

0

(x) .

Thetower isthus dened by:

∆ =

_{{(x, l) ∈ I0}

_{× N ; l ≤ τI}

₀

(x)

_{− 1}}

andthepartition ofthebase

I

0

isgiven bythe ylinders

Z

0,p

dened intheprevious se tion. Re allthatthe dynami s on thetowerisgiven by:

F (x, l) =

(

(x, l + 1)

if

l < τ

I

0

(x)

− 1

(T

τ

I0

(x)

_{(x), 0)}

if

l = τ

I

0

(x)

− 1

A ordingto[30℄,thede ayof orrelationsisgovernedbytheasymptoti sof

m

{x ∈ I0

; τ

I

0

(x)

≥

n

_}

namely

m

_{{x ∈ I0}

; τ

I

0

(x) > n

} = m(b−n

, b

n

)

∼

1

γ



2γ

γ

_{− 1}



γ

γ−1

1

(n

− 1)

γ−1

γ

.

useful in the next se tion about limit theorems. Let us rst introdu e the separation time

s(x, y)

between two points

x

and

y

in

I

0

. Put

T

ˆ

the rst return map on

I

0

; we dene

s(x, y) = min

_{n≥0{( ˆ}

T

n

_{(x), ˆ}

_T

n

_(y))

lie indistin t

Z

0,p

, p

≥ 1}

. We ask that

∃C > 0, δ ∈ (0, 1)

su h that

∀x, y ∈ Z0,p

, p

≥ 1

,we have











D ˆ

T (x)

D ˆ

T (y)











≤ exp[Cδ

s( ˆ

T (x), ˆ

T (y))

_{] .}

(12)

Let us prove this inequality. Remember that the ylinder

Z

0,p

is the disjoint union of two open omponents,

Z

+

0,p

and

Z

−

0,p

,whi hsitontheoppositesidesof

0

. Supposerstthat

x

and

y

stay in the same open omponent of some

Z

0,p

,

p

≥ 1

, and that

s( ˆ

T (x), ˆ

T (y)) = n

; then sin etheorbits(under

ˆ

T

)ofthetwo pointswillbeinthesame ylinderuptotime

n

− 1

,and on these ylinders

T

ˆ

is monotone and uniformly expanding,

|D ˆ

T

| ≥ β > 1

(see footnote 1), we have

| ˆ

T (x)

− ˆ

T (y)

| ≤ β

−(n−1)

. Thereforebythedistortioninequalitywe get











D ˆ

T (x)

D ˆ

T (y)











≤ exp

h

Kβ

−(n−1)

i

≤ exp[Cδ

s( ˆ

T (x), ˆ

T (y))

] ,

(13) where

C = Kβ

and

δ = β

−1

. Ifinstead

x, y

lieinthe two dierent open omponentsof some

Z

0,p

,

p

≥ 1

,andagain

s( ˆ

T (x), ˆ

T (y)) = n

,thismeansthat

−x

and

y

willhavethesame oding upto

n

;hen e











D ˆ

T (x)

D ˆ

T (y)











=











D ˆ

T (

_−x)

D ˆ

T (y)











≤ exp[K| ˆ

T (

_{−x) − ˆ}

T (y)

_{|] ≤ exp}

h

Kβ

−(n−1)

i

_{≤ exp[Cδ}

s( ˆ

T (x), ˆ

T (y))

] .

A ordingto[30℄the orrelationsde aysatises





R

(ϕ

_{◦ T}

n

_{) ψ dm}

₋

R

_{ϕ dm}

R

_{ψ dm}



 _{= O(}

P

k>n

m

{x ∈

I

0

; τ

I

0

(x)

≥ k}

andtherighthand side ofthis inequality behaveslike

O n

−

1

γ−1

.

Optimal bounds The previous result on the de ay of orrelations ould be strengthened to produ e a lowerbound for the de ayof orrelations for integrable fun tionswhi h vanish in a neighborhood of theindierent xed point. We will usefor that therenewal te hnique introdu ed by Sarig [28 ℄ and su esively improved by Gouëzel [13 ℄. We rst need that our original map is irredu ible: this is a onsequen e of the already proved ergodi ity, but one ouldshowndire tlybyinspe tionthatthe ountableMarkovpartitiongivenbythepreimages ofzerohassu haproperty. Wemoreoverneedadditionalpropertiesthatwedire tlyformulate inour setting:

•

Suppose we indu e on

Im

= (a−m, am)/

{0}

and all

Zm

the Markov partition into the re tangles

Z

m,p

with rst return

p

. A ylinder

[d

0

, d

1

,

· · · , dn−1

]

with

d

i

∈ Zm

will be theset

∩

n−1

l=0

T

ˆ

−i

d

l

.

We rst need that the ja obian of the rst return map is lo ally Hölder ontinuous, namelythatthere exists

θ < 1

su h that:

wherethe supremumistaken overall ouples

x, y

∈ [d0

, d

1

,

· · · , dn−1

]

,

d

i

∈ Zm

and

C

is apositive onstant. Butthisis animmediate onsequen eofformula(13)with

θ = β

−1

and

C = Kβ

. Usingtheseparationtime

s(

·, ·)

,wedene

D

m

f = sup

|f(x)−f(y)|/θ

s(x,y)

, where

f

is an integrable fun tion on

I

m

and the supremum is taken over all ouples

x, y

∈ Im

. We thenput

||f||L

θ,m

≡ ||f||∞

+ Dmf

. We all

Lθ,m

the spa e of

θ

-Hölder fun tionson

I

m

.

•

We need the so- alled big image property, whi h means that the Lebesgue measure of theimages, under

T

ˆ

, all the re tangles

Z

m,p

∈ Zm

are uniformly bounded from below byastri tlypositive onstant. Inour ase,seese tion3,theseimagesareboundedfrom below bythelength ofthe interval

(a

−m

, a

m

)

.

•

Wenallyneedthat

m(x

∈ Im|τ(x) > n) = O(n

−χ

₎

,forsome

χ > 1

(thisisGouëzel's as-sumption,whi himprovesSarig'sone,askingfor

χ > 2

). Inour asebythe onstru tion developed inSe t. 3 we immediately get that

m(x

∈ Im|τ(x) > n) = m(∪p>n

Z

m,p

) =

(b

_−(m+n)

, b

m+n

)

∼ C(n + m)

−b

= Cn

−b

(1 + m/n)

−b

∼ Cn

−b

, where the onstants

C

and

b

arethe sameasthose given inthe proof of Th. 4,pre isely

C =

1

γ



2γ

γ−1



γ

γ−1

and

b =

_γ−1

γ

.

Undertheseassumptions,SarigandGouëzelprovedalowerboundforthede ayof orrelations whi hwedire tly spe ialize toour map:

Proposition 6. There exists a onstant

C

su h that for all

f

whi h are

θ

-Hölder and

g

integrable andboth supported in

I

m

we have











Corr( f, g

◦ T

n

₎

_{− (}

∞

X

k=n+1

m(x

_{∈ Im|τ(x) > n))}

Z

g dm

Z

f dm











≤ CFγ

(n)

||g||∞||f||L

θ,m

where

F

γ

(n) =

1

n

γ

γ−1

if

γ < 2

,

(log n)/n

2

if

γ = 2

and

1

n

γ−1

2

if

γ > 2

. Moreover, if

R

f dm = 0

, then

R

(g

◦ T

n

_{) f dm =}

_O(

1

n

γ

γ−1

)

. Finally the entral limit theorem holdsfor the observable

f

.

Remark2. (i)Sin ewhen

m

→ ∞

,

I

m

oversmod-

0

alltheinterval

(

−1, 1)

wegetanoptimal de ayof orrelationsoforder

O(

1

n

γ−1

1

)

forallintegrablesmoothenoughfun tionswhi hvanish in a neighborhood of

1

.

(ii) The last senten e about the existen e of the entral limit theorem will be also obtained, using a dierent te hnique, in Proposition 5, part 2, (a).

4 Limit theorems

Letus re all the notion of stablelaw (see [9, 12℄): a stablelaw isthe limit ofa res aled i.i.d pro ess. Morepre isely,the distribution ofa random variable

X

is said to be stable ifthere existan i.i.dsto hasti pro ess

(X

i

)

i∈N

and some onstants

A

n

∈ R

and

B

n

> 0

su hthatin distribution:

1

B

n

n−1

X

i=0

Xi

− An

−→ X .

The kind of laws we are interested in an be hara terized by their index

p

∈ (0, 1) ∪ (1, 2)

, dened asfollowed:

m(X > t) = (c

1

+ o(1))t

−p

,

m(X <

−t) = (c2

+ o(1))t

−p

,

where

c

1

≥ 0

and

c

2

≥ 0

aretwo onstantssu hthat

c

1

+ c

2

> 0

,andbyothertwoparameters:

c =











(c

1

+ c

2

)Γ(1

− p) cos(

pπ

2

)

p

∈ (0, 1) ∪ (1, 2)

1

2

p = 2

,

β =

c

1

− c2

c1

+ c2

.

We will denoteby

X(p, c, β)

the lawwhose hara teristi fun tion is

E(e

X(p,c,β)

) = e

−c|t|

p

1−iβ

sgn

(t) tan(

pπ

2

)

.

Proposition 7. Letus denote

S

n

ϕ =

Pn−1

k=0

ϕ

◦ T

k

, where

ϕ

is an

ν

-Hölder observable, with

R

ϕ(x) dx = 0

.

1. If

γ < 2

thenthe Central LimitTheoremholdsfor any

ν > 0

. Thatistosaythere exists a onstant

σ

2

su h that

S

n

ϕ

√

n

tends in distribution to

N (0, σ

2

₎

. 2. If

γ > 2

then: (a) If

ϕ(1) = 0

and

ν >

1

2

(γ

− 2)

then theCentral LimitTheorem stillholds. Moreover

σ

2

_{= 0}

i there exists a measurable fun tion

ψ

su h that

φ = ψ

◦ T − ψ

(b) If

ϕ(1)

6= 0

then

S

n

ϕ

n

γ−1

γ

onverges in distribution tothe stable law

X p, c, β

with:

p =

γ

γ

_{− 1}

c =

1

2γ



2γϕ(1)

γ

_{− 1}



γ

γ−1

Γ(

1

(1

_{− γ)}

) cos(

πγ

2(γ

_{− 1)}

)

β =

sgn

ϕ(1)

3. If

γ = 2

then:

(a) If

ϕ(1) = 0

then the Central LimitTheorem holds. (b) If

ϕ(1)

6= 0

thenthere exista onstant

b

su h that

S

n

ϕ

√

n log n

tends in distributionto

N (0, b)

. Proof.

1. Asaby-produ tof the tower's theory we gettheexisten eof the entral limit theorem wheneverthe rateofde ayof orrelationsissummable([30 ℄,Th.4);thishappensinour asefor

γ < 2

. Asusual we shouldavoidthat

φ

isa o-boundary.

paraboli maps of the interval. We defer the reader to Gouëzel's paper for the preparatory theory; weonly proveherethe ne essary onditions forits appli ation. Weindu e againon

I0

andwe put

ϕI

0

(x) :=

Pτ

I0

−1

i=0

ϕ(T

i

x)

. We need:

i.

φ

mustbelo ally

θ

-Hölderon

I

0

(resp.

T

),with

θ < 1

,whi h meansthatthere existsa onstant

C

su h that

|φ(x) − φ(y)| ≤ Cθ

s(x,y)

_{∀x, y ∈ I0}

(resp.

T

)with

s(x, y)

_{≥ 1}

. We extend the separation time

s(x, y)

to the ambient spa e as follows: if

x, y

∈ T

, all

x, ˆ

ˆ

y

their rstreturnsto

I

0

. Whenever

T

i

_{x, T}

i

_y

stayin the same element oftheMarkovpartition

{Im}m∈Z

until therst returnto

I0

, we put

s(x, y) = s(ˆ

x, ˆ

y) + 1

;otherwise

s(x, y) = 0

.

ii.

m

{x ∈ I0

; τ

I

0

(x) > n

} = O(1/n

η+1

)

,for some

η > 1

iii.

ϕI

0

∈ L

2

I0

.

Re all that the indu ed map

T

ˆ

on

I

0

is uniformly expanding with fa tor

β > 1

; therefore for any ouple of points

x, y

∈ T

we have

|x − y|

T

≤ Bβ

−s(x,y)

,where

B

is asuitable onstant and

| · |

T

denotes the distan eonthe ir le. UsingtheHölder assumption on

φ

weget

|φ(x) − φ(y)| ≤ D|x − y|

ν

T

≤ Eβ

−νs(x,y)

,whi hshows that

φ

islo ally Hölderwith

θ = β

−ν

_{< 1}

.

Thequantityinthese onditemaboveisexa tly

(b

n

, b

−n

)

for whi h weobtainedin the previous se tiona boundof order

n

−(

_γ−1

γ

)

. Hen e

η = γ/(γ

− 1) − 1

. Toprove thethird itemdenote

C

ϕ

=

R

I

0

|ϕ(x)|

2

_dx

we obtain:

Z

I

+

₀

|ϕ

I

+

0

(x)

|

2

_{dx = C}

ϕ

+

+∞

X

p=2

Z

Z

0,p







p−1

X

i=0

ϕ(T

i

x)







2

dx

.

C

ϕ

+ 2

+∞

X

p=2

Z

b

p−1

b

p







p−1

X

i=0

|T

i

x

_{− 1|}

ν

T







2

dx

.

C

ϕ

+ 2

+∞

X

p=2

Z

b

p−1

b

p







p−1

X

i=0

|ai

− 1|

ν







2

dx

.

C

ϕ

+ 2

+∞

X

p=2

m(b

p

− bp−1

)p

2(−

ν

γ−1

+1)

dx

.

C

ϕ

+ 2

+∞

X

p=2

p

−(

γ−1

γ

+1)

_p

2(−

ν

γ−1

+1)

_{dx .}

Finally if

2(−ν+γ−1)

γ−1

−

γ

γ−1

− 1 < −1

(i.e.

ν >

1

2

(γ

− 2)

) then

ϕI

0

∈ L

2

I0

. (b) Using the fa tthat

m[u > nϕ(

_{−1)] = m(bn}

, b

−n

)

∼

1

2γ



2γ

γ

_{− 1}



γ

γ−1

1

n

γ−1

γ

and the proof in2.(a),theresult easily follows alongthe samelines oftheproof of Th. 1.3in[12 ℄.

Large deviations.

The knowledgeof the measure ofthe tail for the rst returnson thetower(in our ase built over

I

0

), will allows us to apply the results of Melbourne and Ni ol [22 ℄ to get the large deviations property for Hölder observables. Applied to our framework, their theorem states thatif

m(x; τ

I

0

> n) =

O(n

−(ζ+1)

₎

,with

ζ > 0

,thenfor allobservables

φ : [

−1, 1] → R

whi h areHölder and whi hwe takeof zeromean, we have thelargedeviations bounds:

Proposition 8. If

γ < 2

then the map

T

veriesthe followinglarge deviationsbounds: (I)

∀ǫ > 0

and

δ > 0

, there existsa onstant

C

≥ 1

(depending on

φ

) su h that

m

















1

n

n−1

X

j=0

φ(T

j

(x))













> ǫ



 ≤ Cn

−(ζ−δ)

_.

(II) For an open and dense set of Hölder observables

φ

, and for all

ǫ

su iently small, we have

m

















1

n

n−1

X

j=0

φ(T

j

(x))













> ǫ



 ≥ n

−(ζ−δ)

for innitely many

n

andevery

δ > 0

.

Remark3. TheMelbourneandNi olresult hasbeenre ently strenghtened byMelbourne[23℄; byadopting thesamenotationasabove, heproved thatwheneverthe observable

φ

is

L

∞

(with respe tto theLebesgue measure

m

),and

ζ + 1 > 0

,then forany

ǫ

there existsa onstant

C

φ,ǫ

su h that

m

















1

n

n−1

X

j=0

φ(T

j

(x))













> ǫ



 ≤ Cφ,ǫ

n

−ζ

for all

n

≥ 1

. Translated to our map, this means that we have the large deviation property whenever

γ > 1

. SimilarresultshavebeenobtainedbyPolli otandSharp[26℄forthe Pomeau-Manneville lassofmaps;hopefullythey ould begeneralized inthepresen eofunboundedrst derivaties.

5 Re urren e Firstreturns.

In the past ten years the statisti s of rst return and hitting times have been widely used as new and interesting tools to understand the re urren e behaviors in dynami al systems. Surveys of the latestresults andsome histori al ba kground an be foundin[20, 17,1 ℄.

Takeaball

B

r

(x)

orradius

r

aroundthepoint

x

∈ T

and onsidertherstreturn

τ

B

r

(x)

(y)

we askwhether thereexiststhe limit of thefollowingdistribution when

r

→ 0

3 :

F

_r

e

(t) = m

r

y

∈ Br

(x); τ

B

r

(x)

m(B

r

(x)) > t

.

The distribution

F

h

r

(t)

for the rst hitting time (into

B

r

(x)

) is dened analogously just taking

y

and the probability

m

onthewhole spa e

T

.

A powerful toolto investigate su h distributions for non-uniformly expandingand hyper-boli systemsis given bythe onjun tion of thefollowing results, whi h redu ethe omputa-tions to indu edsubsets.

•

Suppose

(T, X, µ)

is an ergodi measure preserving transformation of a smooth Rie-mannian manifold

X

; take

X

ˆ

⊂ X

an open set and equip it with the rst return map

ˆ

T

and with the indu ed (ergodi ) measure

µ

ˆ

. For

x

∈ ˆ

X

we onsider the ball

B

r

(x)

(

Br(x)

⊂ ˆ

X

)arounditandwe write

τ

ˆ

B

r

(x)

(y)

fortherstreturnofthepoint

y

∈ Br(x)

under

T

ˆ

. We now onsider the distribution of the rst return time for the two vari-ables

τ

B

r

(x)

and

τ

ˆ

B

r

(x)

in the respe tive probability spa es (

B

r

(x), µ

r

) and (

B

r

(x), ˆ

µ

r

) (whereagainthesubindex

r

means onditioning totheball

B

r

(x)

),as:

F

e

r

(t) = µ

r

(y

∈

B

r

(x)); τ

B

r

(x)

(y)µ(B

r

(x)) > t)

and

F

ˆ

e

r

(t) = ˆ

µ

r

(y

∈ Br

(x)); ˆ

τ

B

r

(x)

(y)ˆ

µ(B

r

(x)) > t)

.

In [4℄ it is proved the following result: suppose that for

µ

-a.e.

x

∈ ˆ

X

the distribution

ˆ

F

_r

e

(t)

onverges pointwise to the ontinuous fun tions

f

e

_(t)

when

r

→ 0

(remember that the previous distribution depend on

x

via the lo ation of the ball

B

r

(x)

); then we have as well

F

e

r

(t)

→ f

e

(t)

and the onvergen e is uniform in

t

4

. We should note thatwheneverwe havethedistribution

f

e

_(t)

for therstreturn timewe an insurethe existen e of the weak-limit distribution for therst hitting time

F

h

r

(t)

→ f

h

(t)

where

f

h

(t) =

R

₀

t

(1

_{− f}

e

(s))ds, t

_{≥ 0}

[16 ℄.

Note: From now on we will saythat we have

f

e,k

_(t)

as limit distributions for balls, if we gettheminthelimit

r

→ 0

and for

µ

-almost all the enters

x

oftheballs

Br(x)

.

•

Thepreviousresultisusefulifweareabletohandlewithre urren eonindu edsubsets, see [5, 6℄ for a few appli ations. Indu tion for one-dimensional maps often produ es pie ewise monotoni maps with ountably many pie es. An interesting lass of su h mapsare the Ry hlik'smaps [27 ℄ : in[4 ℄ Def. 3.1theunderlying measure is onformal. Whenthe onformalmeasureistheLebesguemeasure

m

,thenRy hlik'smaps ouldbe hara terized inthe following way:

Let

T : Y

→ X

be a ontinuous map,

Y

⊂ X

open and dense,

m(Y ) = 1

and

X

isthe unitinterval or the ir le. Suppose there existsa ountable family of pairwise disjoint open intervals

Zi

su h that

Y =

S

i≤1

Zi

and

T

is: (i)

C

2

on ea h

Zi

; (ii) uniformly expanding:

inf

Z

i

inf

x∈Z

i

|DT (x)| ≥ β > 1

;(iii) Var

(g) <

∞

,where

g = 1/

|DT (x)|

when

x

_{∈ Y}

and

0

otherwise(Var

g

denotes thetotal variation of thefun tion

g : R

→ ∞

). 3

We allitdistributionwithabuseoflanguage;inprobabilisti terminologyweshouldrathertake

1

minusthat quantity.

4

The result proved in [4℄ is slightly more general sin e it doesn't require the ontinuity of the asymptoti distributions over all

t

≥ 0

. We should note instead that we ould relax the assumption that

X

ˆ

is open just removingfromitasetofmeasurezero,whi h willhappenonourindu edsets

I

m

.

balls(i.e.

f

e

_{(t) = f}

k

_{(t) = e}

−t

),whenevertheinvariantmeasureisabsolutely ontinuous w.r.t.

m

andmoreoverthis invariant measure ismixing.

Beforewe formulateour nextresultforthemaps

T

investigated inthis paper letusprovethe following lemma.

Lemma 9. The map

T

ˆ

is Ry hlik on the ylinders

I

m

,

m

∈ Z

and the variation of

|D ˆ

T

|

is nite on ea h of them.

Proof. (see[4℄). Letus onsiderthe ylinder

I

m

and partitionitinto the ylinders

Z

m,p

with rst return

p

≥ 1

,aswedidinthese ond se tion; then we have for thevariationon

I

m

Var

1

|D ˆ

T

|

≤

X

Z

m,p

Z

Z

m,p

|D

2

_{T (t)}

_ˆ

_|

|D ˆ

T (t)

|

2

dt + 2

X

Z

m,p

sup

Z

m,p

1

|D ˆ

T

|

.

Bythe distortionbound proved inthese ondse tion we have that

e

2K

≥











D ˆ

T (x)

D ˆ

T (y)











≥











Z

y

x

D

2

_{T (t)}

ˆ

D ˆ

T (t)

dt











≥

Z

y

x

|D

2

_{T (t)}

ˆ

_|

D ˆ

T (t)

dt

for any

x, y

∈ Zm,p

, sin e the rst derivative is always positive and the se ond derivative hasthe same sign for all thepoints inthe same ylinder. But this immediately impliesthat

R

Z

m,p

|D

2

_{T (t)|}

ˆ

|D ˆ

T (t)|

2

dt

≤ supZ

m,p

1

|D ˆ

T |

e

2K

. Using Remark(1) we an restri t to

Z

−

m

. Sin e

ˆ

T

maps

Z

_m,p>1

−

dieomorphi ally onto

(a

m−1

, a

m

)

and

Z

−

m,1

onto

(a

−(m−1)

, a

m

)

⊃ (am−1

, a

m

)

there will be apoint

ξ

for whi h

D ˆ

T (ξ)m(Z

m,p

)

≥ m(am−1

, a

m

)

. Applying thebounded distortion estimateone more time,we get

sup

Z

m,p

1

|D ˆ

T |

≤

e

2K

_m(Z

m,p

)

m(a

m−1

,a

m

)

. We nallyobtain Var

1

|D ˆ

T

_|

≤

e

2K

(2 + e

2K

)

m(a

m−1

, a

m

)

X

Z

m,p

m(Z

m,p

) <

∞ .

Thefollowing resultnowfollows by[4℄ Theorem 3.2.

Proposition10. Themap

T

hasexponentialreturnandhittingtimedistributionswithrespe t to themeasure

m

provided

γ > 1

.

Number of visits.

Letus ome ba k to the general framework introdu ed in Se t. 5.1 withthetwo probability spa es

(X, T, µ)

and

( ˆ

X, ˆ

T , ˆ

µ)

. Wenowintrodu ethe randomvariables

ξ

e

r

and

ˆ

ξ

_r

e

whi h ount thenumber of visitsof the orbits of a point

y

∈ Br

(x)

to the ball itself and up to a ertain res aledtime. Namely:

ξ

_r

e

(x, t)

_≡

h

t

µ

(Br (x))

i

X

j=1

χ

_B

_r

_(x)

T

j

(y)

,

where

χ

stands for the hara teristi fun tion and

x

∈ X

. If we take

x

∈ ˆ

X

we an dene in the same manner the variable

ˆ

ξ

_r

e

(x, t)

by repla ing the a tion of

T

with that of

ˆ

T

. We now introdu e the two distributions

G

e

_r

(t, k) = µ

r

(x; ξ

r

e

(x, t) = k), ˆ

G

e

r

(t, k) = ˆ

µ

r

(x; ˆ

ξ

r

e

(x, t) = k) ,

where again the index

r

for the measures means onditioning on

Br(x)

. It is proved in [4 ℄ that whenever the distribution

G

ˆ

e

r

(t, k)

onverges weakly (in

t

) to the fun tion

g(t, k)

and for almost all

x

∈ ˆ

X

, the same happens, with the same limit, to the distribution

G

e

r

(t, k)

. For systems with strong mixing properties the limit distribution is usually expe ted to be Poissonian [20 , 17,18 , 1℄:

t

k

_e

−t

k!

.

In[10 ℄itwasshownthatRy hlikmapsenjoyPoissonstatisti sforthelimitdistributionof thevariables

ξ

e

r

and wheneverthe enter oftheball istakena.e.. Hen e we getthefollowing result.

Proposition 11. Let

γ > 1

. Then for

m

-almost every

x

the number of visits to the balls

B

r

(x)

onverges to the Poissoniandistribution as

r

→ 0

.

Extreme Values.

Thelastquotedpaper[10℄ ontainsanotherinteresting appli ationofthestatisti softherst hitting time that we ould apply to our map

T

too. Let us rst briey re all the Extreme ValueTheory. Given theprobability measurepreservingdynami al system

(X, T, µ)

andthe observable

φ : X

→ R ∩ {±∞}

, we onsider the pro ess

Y

n

= φ

◦ T

n

for

n

∈ N

. Then we dene the partial maximum

M

n

≡ max{Y0

,

· · · , Yn−1}

and we look ifthere are normalising sequen es

{an}n∈N

⊂ R

+

and

{bn}n∈N

⊂ R

su hthat

µ(

{x : an(Mn

− bn)

≤ y}) → H(y)

for some non-degenerate distribution fun tion

H

: in this ase we will say that an Extreme Value Law (EVL) holds for

M

n

. If the variables

Y

n

were i.i.d., the lassi al extreme value theorypres ribesthe existen eofonlythreetypesofnon-degenerateasymptoti distributions for themaximum

M

n

and underlinearnormalisation, namely:

•

Type 1:

EV

1

= e

−e

−y

for

y

∈ R

,whi h is alledtheGumbel law.

•

Type2:

EV

2

= e

−y

−α

for

y > 0

,

EV

2

= 0

,otherwise,where

α > 0

isaparameter, whi h is alledFre het law.

•

Type 3:

EV

3

= e

−(−y)

α

for

y

≤ 0

,

EV

3

= 1

, otherwise, where

α > 0

is a parameter, whi h is alled Weibull law.

From now on we will take

X

asa Riemannianmanifold withdistan e

d

and

µ

an absolutely ontinuous(w.r.t. Lebesgue)probabilityinvariantmeasure. Moreover onsidertheobservable

φ

of the form

φ(x) = g(d(x, ξ))

, where

ξ

is a hosenpoint in

X

. The fun tion

g : [0,

∞) →

R

_{∪ {+∞}}

is a stri tly de reasing bije tion ina neighborhood of

0

and it has

0

as a global maximum(eventually

+

∞

). The fun tion

g

ould be taken inthree lasses; we defer to [10 ℄ for the pre ise hara terization. Important representatives of su h lasses (denoted by the

indi es 1,2,3) are

g

1

(x) =

− log(x)

;

g

2

(x) = x

−1/α

for some

α > 0

;

g

3

(x) = D

− x

−1/α

, for some

D

∈ R

and

α > 0

. We also remind the distribution of the rst hitting time

F

h

r

(t)

into the ball

B

r

(x)

introdu ed above; we say that a system enjoys exponential hitting time statisti s(EHTS)if

F

h

r

(t)

onverges point wiseto

e

−t

for

µ

-a.e.

x

∈ X

(wesawbeforethatit is equivalent to getthe exponential limit distribution for the rst return time). We are now ready to state the resultin [10℄whi h establishes an equivalen e between theEHTS andthe EVL; we will be in parti ular on erned with the following impli ation: suppose the system

(X, T, µ

) hasEHTS;then it satisesan EVL forthe partial maximum

Mn

onstru ted on the pro ess

φ(x) = g(d(x, ξ))

, where

g

is taken in one of the three lasses introdu ed above. In parti ular if

g = g

i

we have an EVL for

M

n

of type

EV

i

.

Of ourse thisresult an beimmediatelyapplied to themapping

T

underinvestigationin this paper.

6 Generalizations

As mentioned in the Introdu tion the original paper by Grossmann and Horner [14 ℄ dealt withdierent Lorenz-like maps

S

whi h map

[

−1, 1]

onto itself with two surje tive symmet-ri bran hes dened on the half intervals

[

−1, 0]

and

[0, 1]

. They have the following lo al behaviour:

S(x)

∼ 1 − b|x|

κ

, x

≈ 0, b > 0

S(x)

_{∼ −x + a|x − 1|}

γ

, x

_{≈ 1−}

, a > 0

S(x)

_{∼ x + a|x + 1|}

γ

, x

_{≈ −1+}

where

κ

∈ (0, 1)

and

γ > 1

aretwoparameters. We also requirethat

(i) inall points

x

6= −1, 1

theabsolutevalueof thederivativeis stri tly biggerthan

1

. (ii)

S

is stri tly in reasing on

[

−1, 0]

, stri tly de reasing on

[0, 1]

and onvex on the two in-tervals

(

−1, 0), (0, 1)

Themaphasa uspatthe originwhere theleftandrightrstderivativesdivergeto

±∞

and thexed point

−1

is paraboli (Fig. 2). Although the map

S

is Markov withrespe tto the partition

{[−1, 0], [0, 1]}

itwillbemore onvenient tousea ountableMarkovpartitionwhose endpointsaregiven bysuitable preimages of

0

(see below).

The reexion symmetry of the map

T

in Se t. 2 was related to the invarian e of the Lebesgue measure. We do not really need that the map

S

is symmetri with respe t to the origin. We didthis hoi eto getonly two s aling exponents (

κ

and

γ

) in

0

and in

±1

. This impliesinparti ular thesame s alings forthe preimages of

0

on

(

−1, 0)

and

(0, 1)

. Iftheleft and rigt bran hes arenot anymoresymmetri , still preserving theMarkov stru ture andthe presen eofindierentpointsandofapoint withunbounded derivative,oneshould playwith at most four s alingexponentsgiving the lo albehaviorof

S

in

0

and

±1

.

We denote by

S

1

(resp.

S

2

) the restri tion of

S

to

[

−1, 0]

(resp.

[0, 1]

) and dene

a

0+

=

-1

0

_x

1

-1

0

1

S(x)

a

_-2

a

_-1

a

_0-

a

₀₊

a

₁

a

₂

b

_-1

b

_-2

b

₂

b

₁

Sa

−p

= Sa

p

= a

−(p−1)

. In the same way as we did in the rst se tion we dene the se-quen e

b

p

, p

≥ 1

as:

Sb

±p

= a

p−1

. The ountable Markov partition,

mod m

, will be



(a

−p

, a

−(p−1)

) : p

≥ 1

∪ {(ap

, a

p+1

) : p

≥ 1} ∪ {I0}, I0

≡ (a0−

, a

0+

)/

{0}

. From thelo albehaviors one getsthefollowing s aling relations

a

p

=

−a−p

∼ 1 −



1

a(γ

_{− 1)}



1

γ−1

₁

p

γ−1

1

a

p

− ap+1

∼ a



1

a(γ

− 1)



γ

γ−1

1

p

γ−1

γ

b

p

=

−b−p

∼



1

ab

(γ−1)

_(γ

_{− 1)}



1

k

(γ−1)

₁

p

k

(γ−1)

1

b

p

− bp−1

∼

1

k ab

(γ−1)

k(γ−1)

1



1

γ

_{− 1}



k

(γ−1)+1

k(γ−1)

₁

p

k

(γ−1)+1

k(γ−1)

•

Therole of Remark1 isplayed here bythe monotoni ityof the right bran h: whenever

x, y

sit on dierent omponents we an just note that

|DS(−x)| = |DS(x)|

and that afterone iteration

S(x) = S(

−x)

.

5

•

Letus onsideragainthestepfromthersttothese ondupperboundin(4 ): wesimply dis ardedthedenominatorgivenbytheinmumoftherstderivativeoverthesetswith given rst return time, sin e itwas ininuent for the map

T

. Instead itwill now plays animportantrolesin eitmakesboundedthefollowingratiosin e,asitiseasyto he k, :

|bn+1

− bn| sup(b

n+1

,b

n

)

|D

2

_S

_|

inf

(b

n+1

,b

n

)

|DS|

=

_O(

1

n

) .

Invariant measure and de ay of orrelations. An important dieren e with themap on the ir leisthatwearenotguaranteedthattheLebesguemeasure

m

isanymoreinvariant;sowe havetobuildanabsolutely ontinuousinvariantmeasure

µ

. Fortunatelythetower'ste hniques helpsus again. Ifthe tailof thereturn timeonthebaseofthetower is

m

-summableandthe distortionis bounded, itfollows the existen eof su h

µ

. Tobemore pre ise letus indu e on the ylinder

I0

. A sub ylinder

Zp

of

I0

with rstreturn time

p

will have theform

6

Z

1

= (a

0−

, b

−1

)

∪ (b1

, a

0+

)

(14)

Z

p

= (b

−(p−1)

, b

−p

)

∪ (bp

, b

p−1

) p > 1 .

Consequently the Lebesgue measure of the points in

I

0

with rst return bigger than

n

s aleslike

m(x

_{∈ I0}

; τ

I

0

(x) > n)

≈

1

n

κ

(γ−1)

1

We an thus invoke Th.1 inLai-Sang Young's paper[30 ℄ to get:

Proposition 12. Let us onsider the map

S

depending upon the parameters

γ

and

κ

. Then for

0 < κ <

1

γ−1

(or for

0 < κ < 1

, when

γ

≤ 2

), we get the existen e of an absolutely ontinuous invariant measure

µ

whi h mixes polynomially fast on Hölder observables with rate

O n

−

1−κ(γ−1)

_κ

_(γ−1)

.

The map has exponential return and hitting times distributions and Poissonian statisti for the limit distributionof thenumber of visits in balls.

5

In the asymmetri ase

|DS(−x)| 6= |DS(x)|

but still after one iterate

S(x)

and

S(y)

sit on the same side. This imply that multiplying bythe appropriatefa tor we antreatthe asymmetri ase in the sameway asthe symmetri one.

6

We would like to note that, ontrarily to the map

T

investigated in the previous se tions, the rst return map

S

ˆ

for

S

on

I

0

is not onto

I

0

on ea h ylinder

Z

p

with pres ribed rst return time. In fa t

S

ˆ

maps all the ylinders

(b

p−1

, b

p

)

and(

b

−p

, b

−(p−1)

)

onto

(a

0−

, 0)

,butitmapsthe ylinders

(a

0−

, b

−1

)

and

(b

1

, a

0+

)

onto

(0, a

0−

)

. Nevertheless

S

ˆ

is anirredu ible Markovmap, asit is easyto he k. If one wantsagenuine rstreturnBernoulli map,oneshouldindu eover

(a

0−

, 0)

: the ylinderswithgivenrstreturntimearesimplyslightlymore ompli ated tomanagewith.