Projective metrics and mixing properties on towers

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AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 8, Pages 3371–3389 S 0002-9947(01)02786-6

Article electronically published on April 9, 2001

PROJECTIVE METRICS AND MIXING PROPERTIES ON TOWERS

V ´ERONIQUE MAUME-DESCHAMPS

Abstract. We study the decay of correlations for towers. Using Birkhoff’s projective metrics, we obtain a rate of mixing of the form:

cn(f, g)≤Ctα(n)kfk kgk¹

where α(n) goes to zero in a way related to the asymptotic mass of upper floors,kfkis some Lipschitz norm andkgk1 is someL¹ norm. The fact that the dependence ongis given by anL¹norm is useful to study asymptotic laws of successive entrance times.

Introduction

A powerful method to study nonuniformly hyperbolic systems or hyperbolic systems with singularities consists in conjugating the system to a tower (see the de- scription of towers below). Indeed, this strategy was initiated by F. Hofbauer ([H]) to study piecewise expanding maps of the interval. It was then used to study piecewise expanding maps in higher dimension ([Buz]), stochastic stability of unimodal maps (([Ba,V], [Ba,Y]), to obtain exponential decay of correlations for unimodal maps ([K,N],[Y1]), some billiards ([Y1], [Ch]), Henon maps ([B,Y]), partially hyperbolic diffeomorphisms ([D], [Ca]). In a recent paper, L.-S. Young ([Y2]) relates the decay of correlations for some mixing towers to the asymptotics of return times on the base of the tower. This gives, for example, polynomial decay of correlations for some interval maps with neutral fixed points. To this aim, she uses a coupling method.

We present a new approach to studying mixing properties on towers “`a la Young”.

Using Birkhoff cones and projective metrics instead of coupling, we relate explicitly the rate of mixing to the mass of upper floors of the tower (Theorem 1.5). More- over, our method gives strong mixing properties: the correlations depend of one observable solely through some L¹ norm. This is very useful to study asymptotic laws of return times and successive return times (see [C], [C,G,S], [G,S], [Sau], [P]).

Section 1 contains the setting and precise statement of the results on decay of correlations (Theorem 1.5).

In section 2, we briefly recall definitions and properties of Birkhoff’s cones and projective metrics.

Section 3 is devoted to the proof of the estimation of decay of correlations. The strategy is the following: we construct a sequence of cones of locally Lipschitz functionsCj such that for somek∈Nthe transfer operator L^k mapsCj intoCj+1

Received by the editors May 23, 1999 and, in revised form, January 13, 2000.

2000Mathematics Subject Classification. Primary 37A25, 37C30, 37C40.

Key words and phrases. Decay of correlations, tower, transfer operator, projective metrics.

2001 American Mathematical Societyc 3371

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with contractionγj <1. The rate of mixing is given by the product of the γj’s.

This product is easily related to the mass of the upper floors and may be sub- exponential. The rate of convergence of L^k to the spectral projection associated with the invariant density is obtained in the uniform norm on each floor.

In section 4, we discuss specific rates of mixing and lower bounds for uniform speed of convergence.

Acknowledgments. This article answers a question that Viviane Baladi posed to me. I thank her for this question and for many fruitful discussions on related topics. I also acknowledge support from the PRODYN program of the European Science Foundation.

1. Setting, statement of the results Let us describe the tower model. We follow Young’s setting.

A tower ∆ consists of floors Λ`,`∈N. The base Λ0of the tower is a probability space; letm0 be the probability measure on Λ0. The tower structure is given by a partition (Λ^j₀)j∈N (modm0) of Λ0 and a return time function defined on the base Λ0of the towerR: Λ0−→Nwhich is constant on each Λ^j₀. The floors Λ`are just copies of a part of Λ0:

Λ`={(x, `)/x∈Λ0, R(x)> `}. We will denote by Λ^j_` the copies of Λ^j₀:

Λ^j_` ={(x, `)/x∈Λ^j₀, R(x)> `} ⊂Λ`.

The dynamic on the tower is given byf0: Λ0 −→Λ0 such thatf0 : Λ^j₀−→Λ0 is bijective (modm0).

Remark 1.1. The assumption that f0 : Λ^j₀ −→ Λ0 is bijective may be replaced by the Markov assumption: f0(Λ^j₀) is a union of some Λ^k₀’s and the large image property: ∃η >0 such that inf

j∈Nm0(f0(Λ^j₀))≥η.

Let us considerF : ∆−→∆ defined by

F(x, `) = F(x, `+ 1) for (x, `)∈Λ` andR(x)> `+ 1

= f0(x,0) =:F^R(x,0) otherwise.

On each floor Λ` we put theσ-algebra of Λ0 pushed byF, so that ∆ is a measur- able space. We assume that the partition R ={Λ^j_`} generates in the sense that

_∞ i=0

F⁻ⁱR :=

(_∞ _

i=0

F⁻ⁱAi, Ai∈ R )

, is the partition into points. For k ∈ N, the elements of the partitionR^(k) =

k_−1 i=0

F⁻ⁱRare calledcylinder or k-cylinders. We denote byCk(x) the element ofR^(k)which containsx.

The measure m0 is pushed on each floor byF, we assume that R

Rdm0 <∞, so that we obtain a finite measure on ∆. This measure will also be denoted bym0

and we assume that it has been normalized (m0(∆) = 1).

The space ∆ may be endowed with a dynamical distance. Forxandyin Λ0, the separation times(x, y) is the greatest integern such that (F^R)^p(x) and (F^R)^p(y) belong to the same Λ^j₀ for all p≤ n. If x and y belong to Λ`, ` ≥1, then they

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have exactly one preimagex⁰andy⁰in Λ0, the separation time betweenxandyis s(x, y) =s(x⁰, y⁰), ifxandyare not in the same Λ`, thens(x, y) = 0. The distance on the tower is defined by : d(x, y) =β^s(x,y) for some 0< β <1. Endowed with this metric, the space ∆ is separable (the topology is generated by the cylinders).

We will always assume the following regularity condition onF:

F^R: Λ^j₀−→Λ0

and its inverse are nonsingular. Its JacobianJ F^R satisfies the following bounded distortion inequality: ∃C >0 such that∀j∈N∀x, y∈Λ^j₀

J F^R(x) J F^R(y)−1

≤C d(F^R(x), F^R(y)).

(BD)

The Ruelle-Perron-Frobenius operator or transfer operator associated to F is defined in the standard way:

L0f(x) = X

F(y)=x

J F(y)⁻¹f(y),

whereJ F is the Jacobian ofF. Remark that except on Λ0, this Jacobian is 1. The bounded distortion property implies that for anyx∈Λ^j_` withR(j) =`+ 1,

C⁻¹m0(Λ^j_`)≤J F(x)⁻¹≤Cm0(Λ^j_`).

(1.1)

More generally, for anyk∈N, for anyx∈∆ such thatF^k(x)∈Λ0, C⁻¹m0(Ck(x))≤J F^k(x)⁻¹≤Cm0(Ck(x)).

(1.2)

So L0 is well defined and acts continuously on the spaceCu(∆) of uniformly continuous and bounded functions on ∆, it also acts onL¹(m0).

The following facts on L0 directly follow from the definitions:

• Z

L0f·gdm0= Z

g◦F·f dm0for allf andginCu(∆) (i.e. m0is aconformal measure),

• a measureµ=h0m0,h0∈L¹(m) isF invariant if and only if L0h0=h0. We expect that the mixing properties of an invariant measure absolutely continuous with respect tom0are related to the spectral properties of L0. To be more precise, let us assume that there exists a fixed pointh0∈Cu(Σ) for L0 which is normalized (m0(h0) = 1) and letµ=h0m0. Forf ∈Cu(Σ) andg ∈L¹(m0), thecorrelations off andg measure the lack of independence betweenf andg◦Fⁿ with respect to the invariant measureµ: forn∈N,

cn(f, g) = Z

f(g◦Fⁿ)dµ− Z

f dµ Z

gdµ .

The measureµ is mixingif and only if the coefficients cn(f, g) go to zero for any f ∈Cu(Σ) andg∈L¹(m0). In this case, estimates on the speed of convergence to zero ofcn(f, g) or, equivalently, estimates on the decay of correlationsmay lead to the Central Limit Theorem (see [Li2]) and to the determination of asymptotic laws for entrance times (see [G,S] and [Sau]). The following trivial computation relates the decay of correlations to the asymptotic behavior of the iterates of L0:

cn(f, g) = Z

[ Lⁿ₀(f h0)−h0m0(f h0)]gdm0

(1.3)

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so that if Lⁿ₀f →h0m0(f) in some reasonable way, thenµis mixing and estimates on the speed of this convergence would lead to estimates on the decay of correlations.

We will denote byL the space of functions that areuniformly Lipschitz (u.L.) and bounded on ∆. That is, bounded functionsf satisfying the u.L. condition:

there existsL(f)>0, such that forxandy in the same Λ`,

|f(x)−f(y)| ≤L(f)d(x, y).

Letkfk= max(kfk∞, L(f)), this defines a norm onLwhich turnsLinto a Banach space.

1.1. Invariant measure and decay of correlations. Under some additional assumptions, we will prove existence and uniqueness on anF-invariant probability, absolutely continuous with respect to m0. We will also give some estimations on the decay of correlations when this invariant measure is mixing.

Following [A,D,U], we will say that the system (∆, F) is irreducible if for all i, j, `, `⁰, there exists an integernsuch thatF⁻ⁿΛ^j_`∩Λⁱ_`0 6=∅; we will say that the system (∆, F) isaperiodicif for alli, j, `, `⁰, there exists an integerNsuch that for all n≥N, F⁻ⁿΛ^j_`∩Λⁱ_`0 6=∅. Following Markov chains and positive matrices methods ([Se]), it may be proved that aperiodicity is equivalent to L.-S. Young’s condition: g.c.d. R= 1 ([Y2]).

To study mixing properties of L0, we will need to deal with more general poten- tials.

Let Φ : ∆−→Rsatisfy the uniformly locally Lipschitz (u.l.L.) condition:

there exists a constantL >0 such that, for allxandy in the same Λ^j_`,

|Φ(x)−Φ(y)| ≤Ld(x, y).

The associated transfer operator LΦis well defined if

∀x∈∆, X

F y=x

e^Φ(y)<∞. It is then defined by

LΦf(x) = X

F y=x

e^Φ(y)f(y).

Under some general conditions, the operator LΦadmits a unique nonzero positive fixed point.

Theorem 1.2. Let Φ satisfy the u.l.L condition, have a conformal measure m⁰ (i.e. m⁰( LΦf) = m⁰(f) ∀f ∈ Cu(∆)), satisfy the “weak contribution of infinity”

condition:

∃n0 /∀n≥n0 sup

x∈Λ_n| LΦ1(x)| ≤1, (K)

andk LΦ1k∞<∞, then LΦ admits a nonzero positive fixed pointhΦ∈L.

If (∆, F) is irreducible, then hΦ >0, unique up to a multiplicative factor and the invariant measureµ=hΦm⁰ is ergodic.

If (∆, F) is aperiodic, then µ is mixing. Moreover Lⁿ_Φf converges to hΦm⁰(f) uniformly on each compact subset of ∆ and inL¹(m⁰)for any f ∈Cu(∆).

Remark 1.3. The function Φ is not necessarily bounded but the hypothesis k LΦ1k∞<∞implies that the functione^Φis bounded.

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The proof of Theorem 1.2 is now more or less standard. It follows arguments from [Sa] and [Ma1, Ma2] (see also [Y2] for a proof without transfer operator). We will just sketch the different steps. We begin with the following lemma.

Lemma 1.4. LetΦsatisfy the hypothesis of Theorem 1.2, then there existsM >0 such that ∀k∈N,k L^k_Φ1k_∞≤M.

Proof. First, let us note that the u.l.L condition implies the following bounded distortion property: ∃C >0 such that∀x, y in the same Λ`,

∀k∈N,

L^k_Φ1(x) L^k_Φ1(y)−1

≤Cd(x, y).

Let Φ satisfy the “weak contribution of infinity condition”. For anyk∈N,n∈N andx∈Λn, the bounded distortion property gives

L^k_Φ1(x) ≤ C 1 m⁰(Λn)

Z

Λn

L^k_Φ1dm⁰

≤ C 1 m⁰(Λn)

Z

∆

L^k_Φ1dm⁰

= C 1

m⁰(Λn) because m⁰ is a conformal probability.

So,M⁰>0 exists such that sup_x_∈_Λ_n L^k_Φ1(x)≤M⁰ for allk∈Nand for alln≤n0. It remains to control L^k_Φ1(x) for k ∈ N and x ∈ Λn, n ≥ n0. Let us denote by Mk = sup_x_∈_∆ L^k_Φ1(x) < ∞ because of k L^k1k < ∞ for all k. Since Φ satisfies the weak contribution of infinity assumption, if x belongs to Λn, n ≥ n0, then LΦ1(x)≤1 and

L^k+1_Φ 1(x)≤ LΦ1(x)·sup

∆

L^k_Φ1≤Mk.

So, Mk+1 ≤max(M⁰, Mk) which leads to Mk ≤ max(M⁰,1) which concludes the proof.

Sketch of Proof of Theorem 1.2. From Lemma 1.4, there exists M > 0 such that k L^kΦ1k∞ ≤ M for all k ∈ N. It is then easy to see that LΦ acts on L. Indeed, lettingf belong toLwe have the following inequality: ∀k∈N

| L^kΦf(x)− L^k_Φf(y)| ≤d(x, y) Ct [kfk_∞+L(f)]∀x, y in the same floor.

Ascoli’s theorem on separable spaces implies that there exists a sequence nk such that Qnk = _n¹

k

Pn_k−1

i=0 Lⁱ_Φ1 converges, uniformly on compact subsets of ∆ and in L¹(m⁰), to some limit hΦ ∈ L. Using the fact that m⁰ is conformal, we get that this limit is nonzero (indeed, m⁰(hΦ) = 1 by Lebesgue’s theorem) and satisfies LΦhΦ=hΦ. Letµ=hΦm⁰; this is aF-invariant probability.

If (∆, F) is irreducible, then [A,D,U] (Theorem 2.5) implies that µ = hm is ergodic andh >0.

Using [A,D,U] Theorem 3.2 (see also Young’s arguments [Y2]), aperiodicity of the system implies that (F, µ) is exact. If µ is mixing, then L^k_Φf converges to hΦm⁰(f) uniformly on each compact subset of ∆ (any accumulation point of L^k_Φf must behΦm⁰(f) because of mixing).

Recall that L0is the transfer operator associated to the potential−logJ F.

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Theorem 1.5. The operator L0 admits a nonzero positive fixed pointh0∈L.

If (∆, F)is irreducible, then h0>0 is unique up to a multiplicative factor and the invariant measureµ=h0m0 is ergodic.

If (∆, F) is aperiodic, then µ is mixing. Moreover Lⁿ₀f converges to h0m0(f) uniformly on each compact subset of ∆ and inL¹(m0) for anyf ∈Cu(∆).

For any increasing sequence (vn)n∈N such that P

`∈Nm0(Λ`)·v` <∞, for any f ∈L andg such thatgv∈L¹(m0)(gv is defined bygv(x) =g(x)v` ifxbelong to Λ`), there exists0< γ <1 such that

cn(f, g)≤Ct max[(vn)⁻¹, γⁿ]kfk kgvk1.

We also estimate the speed of convergence of Lⁿ0f toh0m0(f)in the uniform norm on each floor.

In particular,

• if there exist constants0< e1 and0< θ <1 such that for all `∈N,m(Λ`)≤ e1θ^`, then there exists0< γ <1such that for anyθ⁰< θ, forgsuch that ^g_θ ∈L¹(m) (where _θ^g₀(x) = ^g(x)_θ_0` for x∈Λ`),

cn(f, g)≤ Ct (θ⁰)γⁿ kfk g θ⁰

1,

• if there exist constants 0 < e1, 0 < θ < 1 and 0 < β < 1 such that for all

`∈N,m(Λ`)≤e1θ^`^β, then for any β⁰< β, let v`=θ⁻^`^β⁰, cn(f, g)≤Ct (β⁰)θⁿ^β⁰ kfk kgvk1,

• if there exist constants0< e1, β >1 such that for all `∈N,m(Λ`)≤e1`⁻^β, then for any γ >1, let v`=

h(ln`)^γ

`^β−1

i₋1

,

cn(f, g)≤Ct (γ)(lnn)^γ

n^β⁻¹ kfk kgvk1.

Remark 1.6. On the one hand, L-S. Young obtained a decay of correlations in O(n⁻^(β⁻¹⁾) in the polynomial case above. Our result is a little bit slower in this case. Except for this, we rediscover her results. On the other hand, we get that the decay of correlations is of the formcn(f, g)≤(vn)⁻¹C(f, g) where the dependence ofC(f, g) ong is given by kgvk1. This kind of property is very useful in studying asymptotic laws of successive return times (see [C,G,S], [G,S], [Sau], [P]) and was not obtained using coupling methods as in [Y2].

Existence and mixing properties of an invariant measure absolutely continuous with respect to m0 follow from Theorem 1.2. Remark that (1.1) and (BD) imply that the hypothesis of Lemma 1.4 is satisfied for Φ = −logJ F (obviously from definitions,m0is a conformal probability).

Our estimation of the decay of correlations is based on Birkhoff’s cones and projective metrics. Let us briefly recall basic definitions and facts on this theory.

2. Birkhoff’s cones and projective metrics

The theory of cones and projective metrics of G. Birkhoff [Bi1] is a powerful tool to study linear operators. P. Ferrero and B. Schmitt [F,S1] applied it to estimate the correlation decay for random dynamical systems. This strategy had been used by many authors. Let us mention C. Liverani [Li1] and M. Viana [V] for Anosov and Axiom A diffeomorphisms. They used Birkhoff cones to obtain exponential

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decay of correlations. In [K,M,S] the Birkhoff cones techniques were used in a different way to obtain sub-exponential decay of correlations. The way we use cone techniques to prove Theorem 1.5 follows some ideas of P. Ferrero and B. Schmitt ([F,S2]) and [Ma1, Ma2].

Let us recall definitions and properties of cones and projective metrics (see [Li1]

for a more complete presentation).

LetB be a vector space andC⊂B a cone with the following properties.

• C is convex,

• C∩ −C={0},

• ifαn is a sequence of real numbers such thatαn →αand x−αny ∈C ∀n thenx−αy∈C. This property is called “integral closure”.

For such a cone, the pseudo-metric θC on C is defined in the following way. Let x, y∈C,

µ(x, y) = inf{β >0 such thatβx−y∈C}, l(x, y) = sup{α >0 such thaty−αx∈C},

with the convention: µ(x, y) = ∞ and l(x, y) = 0 if the corresponding sets are empty. LetθC(x, y) = log^µ_l. θCis called pseudo-metric because it is not necessarily finite. Moreover, it is a projective pseudo-metric: ifxandx1are proportional, then for anyy∈C,θC(x, y) =θC(x1, y).

The following two results reveal the usefulness of projective metrics. LetC and C⁰ be two cones,P a linear operatorP :C →C⁰. Let ∆ denote the diameter of P C in C⁰:

∆ = sup

f,g∈C

θC0(P f, P g).

Theorem 2.1 ([Bi1]). For any f, gin C, we have θC⁰(P f, P g)≤tanh

∆ 4

θC(f, g).

This theorem implies thatP :C→C⁰is always a contraction (in the wide sense) for the projective metrics. If ∆<∞, then it is a strict contraction.

The following result relies the metricθC to certain norms onB. A normk kon B is anorm adapted toC if forf andg inB such that iff +g belongs toC and f −g belongs toC, then kgk ≤ kfk. ρ is a homogeneous form adapted to C ifρ maps C to R⁺, for anyl >0 and f ∈ C, ρ(lf) =lρ(f) and if f−g ∈C implies ρ(g)≤ρ(f).

Theorem 2.2 ([Bi1], [Li1]). Let C be a cone, let k kand ρbe adapted to C. For any f andg inC such that ρ(f) =ρ(g)6= 0 we have

kf−gk ≤(e^θ(f,g)−1) min(kfk,kgk).

3. Proof of Theorem 1.5

We will now prove Theorem 1.5. Let us begin with the following remark.

Remark 3.1. Let v = (v`)`∈N be an increasing sequence of positive numbers such

that X

`∈N

m0(Λ`)·v`<∞.

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We will consider such avas a function on ∆: v(x) =v`ifxbelongs to Λ`. Let L be the transfer operator associated to the change of potentialv: Lf =v⁻¹ L0(f v). The measure m=vm0 is conformal for L and it follows easily from (1.1) and Lemma 1.4 that sup_k_∈Nk L^k1k_∞<∞. Moreover,h= ^h_v⁰ is an eigenvector for L. Theorem 1.2 implies that for anyf ∈Cu(∆),

L^kf −→hm(f) uniformly on each compact set.

Throughout this section, we fix a strictly increasing sequencev = (v`)`∈N such thatm(∆)<∞and the sequence (_v^v^`

`+1)`∈Nis also increasing. We will assume that mandhare normalized (m(∆) = 1 andm(h) = 1). We will work with the transfer operator L.

As usual in these topics, the main tool is a “Lasota-Yorke” type inequality.

Before proving it, let us study the distance on ∆ a bit.

3.1. Some properties of the distance on ∆. Let us consider the following function on ∆: ρ(x) =β ifxbelongs to Λ0andρ(x) = 1 otherwise. Fork ∈N, we will denote byρ^(k)(x) =

Yk i=0

ρ(Fⁱx). The relation between ρ^(k)and the distance on

∆ is the following:

ifxandy are in Λ0, their preimages byF^k are paired. Ifx⁰ is a preimage ofx, we will denote byy⁰ the preimage ofy which belongs toCk(x⁰). We have

d(x⁰, y⁰) =ρ^(k)(x⁰)·d(x, y).

The following lemma proves thatρ^(k)(x) is almost exponential.

Lemma 3.2. There exists 0< γ <1 such that for all η >0, there exists Ωη ⊂∆ of m measure greater than 1−η and k0(η) such that for all k≥ k0 and x∈Ωη, ρ^(k)(x)≤γ^k.

Proof. This is an easy consequence of Birkhoff’s ergodic theorem and Egoroff’s theorem applied to logρ.

Remark 3.3. A finer analysis onρ^(k)should allow one to avoid the use of Birkhoff’s and Egoroff’s theorems and so give a constructive bound fork0.

3.2. A Lasota-Yorke type inequality. We are now in position to prove a Lasota- Yorke type inequality.

Lemma 3.4. For any ε >0, there exists C >0,N ∈Nand k0 ∈Nsuch that for allk > k0, for allf ∈L, we have

1. for `≥k,xandy in Λ`

L^kf(x)− L^kf(y)≤v`−k

v`

d(x, y)L(f), 2. for N ≤` < k,xandy inΛ`

L^kf(x)− L^kf(y)≤ε d(x, y) [L(f) + sup|f|], 3. for ` < N,xand y inΛ`

L^kf(x)− L^kf(y)≤d(x, y) [εL(f) +Csup|f|].

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Proof. Let f belong to L. Item 1 directly follows from the definition of L and d(x, y).

Letxandybe in Λ`,` < k, and let us denote byx⁰∈Λ0(resp. y⁰) the preimage ofx(resp. y) byF^k⁻^`. We have

L^kf(x)− L^kf(y)=v0

v`

L^k⁻^`f(x⁰)− L^k⁻^`f(y⁰). (3.4)

Let us considerxandy in Λ0; their paired preimages will be denoted byx⁰ andy⁰.

| L^kf(x)− L^kf(y)|= 1 v0| X

F^kx0=x

v(x⁰)

J F^k(x⁰)f(x⁰)− v(y⁰) J F^k(y⁰)f(y⁰)|

≤ 1 v0

X

F^kx⁰=x

v(x⁰)

J F^k(x⁰)|f(x⁰)−f(y⁰)|

+ 1 v0

X

F^kx⁰=x

v(x⁰)

J F^k(x⁰)|f(y⁰)|

J F^k(x⁰) J F^k(y⁰)−1

≤L(f)d(x, y) X

F^kx⁰=x

v(x⁰)J F^k(x⁰)⁻¹ρ^(k)(x⁰)

| {z }

(1)

+C d(x, y) sup|f| X

F^kx⁰=x

v(x⁰)J F^k(x⁰)⁻¹

| {z }

(2)

.

The term (2) is bounded above byk L^k1k∞≤M.

Let us turn our attention to term (1). Letη >0 to be fixed later, let Ωηbe given by Lemma 3.2 and assumek ≥k0(η). Recall (1.2): for z such that F^k(z)∈ Λ0, J F^k(z)⁻¹ ≤ Cm0(Ck(z)) and remark that ρ^(k) is constant on each k-cylinder.

Using Lemma 3.2 we get

(1) ≤ X

F^kx⁰=x ρ^(k)(x⁰)≤γ^k

v(x⁰)J F^k(x⁰)⁻¹ρ^(k)(x⁰) + X

F^kx⁰=x x⁰∈(Ωη)^c

v(x⁰)J F^k(x⁰)⁻¹ρ^(k)(x⁰)

≤ γ^kM +Cm(Ω^cη)≤C(γ^k+η).

We have proved that forxandyin Λ0, for anyη >0, providedkis large enough,

| L^kf(x)− L^kf(y)| ≤Cd(x, y)

(γ^k+η)L(f) + sup|f| . Remark that we have proved also the following weak inequality for allk:

| L^kf(x)− L^kf(y)| ≤Cd(x, y) [L(f) + sup|f|]

Letεbe fixed and choose :

• η such thatCη < ε/2, letk1=k0(η) be given by Lemma 3.2,

• N >0 such thatC_v^v⁰

N < ε,

• k0> k1+N and fork > k0−N,Cγ^k< ε/2.

Letk > k0andN ≤` < k, the choices above and (3.4) give L^kf(x)− L^kf(y)≤ v0

v`

Cd(x, y) [L(f) + sup|f|]≤εd(x, y) [L(f) + sup|f|].

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Letting`≤N, we have

L^kf(x)− L^kf(y) ≤ v0

v`

Cd(x, y)

(γ^k⁻^`+η)L(f) + sup|f|

≤ d(x, y)[εL(f) +Ckfk∞].

Remark 3.5. This is clear from the proof of Lemma 3.4 that trying to get a Lasota- Yorke inequality for the transfer operator L0instead of L would lead to some “weak Lasota-Yorke inequality”, i.e. the coefficient in front ofL(f) would not be strictly smaller than 1.

3.3. Construction of cones. As stated in the introduction, to prove Theorem 1.5, we are going to use cone techniques. Let us explain a bit how to construct the cones and how sub-exponential decay of correlations may be obtained.

We start by recalling the classical way of using cones (see [F,S1] and [Li1] for details). To get exponential decay of correlations, it is sufficient to find a coneC and an integerksuch that L^k mapsC into itself and the diameterd0 of L^kC into C is finite. If the fixed point hof L belongs to C then Theorem 2.1 gives, for any integerj,

θC( L^kjf , h)≤γ^j⁻¹d0 whereγ= tanhd0

4 <1.

Then Theorem 2.2 gives forf ∈C, thatk L^pf − hm(f)kgoes to zero exponentially fast for k k an adapted norm, provided f 7→ m(f) is adapted. Then one has to extend this result from the cone to a Banach space.

The starting point of the construction of cones is usually a Lasota-Yorke inequality. Looking at Lemma 3.4, one sees that in our context the Lasota-Yorke inequality depends on the floors of the tower, it is not uniform unless if v` = ^Ct_θ`

for some 0 < θ <1, this is equivalent to the fact that m0(Λ`)≤Ctθ⁰^` for some 0 < θ⁰ < θ. In general for fixed k, lim sup_`_→∞^v^`_v⁻^k

` = 1. This is the reason why we have to deal with a sequence of cones instead of a single one and why we get sub-exponential decay of correlations.

Roughly speaking, for some fixed integerkand any integerj, we will consider a coneCj of functionsf for which we control the Lipschitz constant on the first kj floors (see condition2 below). Thanks to the definition of L and to Lemma 3.4, iff belongs toCjthen we will control the Lipschitz constant of L^kf on the firstk(j+ 1) floors and the control is improved by a factor _v^v^kj

k(j+1) <1. This should imply that L^k maps Cj intoCj+1 with finite diameterdj such that tanh^d₄^j ≤ _v_k(j+1)^v^kj . Then, using Theorems 2.1 and 2.2, we get that for f in C0, L^kjf goes to m(f)hat rate

v0

v_kj in any adapted norm, provided that h belongs to all the cones Cj and that f 7→m(f) is adapted. This is the philosophy of the construction.

Unfortunately, things are not so simple. Firstly, to control L^kf on Λ0, . . . ,Λk, we need to controlf on all the floors (not only the firstkj’s) because of the definition of L.

Secondly, to ensure that a cone C satisfies properties of section 2 and more specifically the conditionC∩−C={0}, some positivity for the functions in the cone is needed. Also, ifC⊂ {f ≥0}=:C+, then for anyf,g inC,θC(f, g)≥θC+(f, g)

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(use Theorem 2.1 withP =Id) and

θC+(f, g) = supf inff ·supg

infg.

As already explained, we want the invariant densityhto belong to the cones. We know that for x ∈ Λ`, h(x) = ^h⁰_v^(x)

` where h0 is the fixed point of L0 which is bounded by Theorem 1.5. Thus, there is no hope of controlling the quantity ^sup_inf_h^h. These two problems are solved by considering a finite partition of ∆ which is de- composed into finitely manys-cylinders (the “compact” part) and the complement of the union of these s-cylinders (the “noncompact” part). Then we require the positivity of some kind of conditional expectation of f with respect to this finite partition (see condition 1 below). This together with the control of the Lipschitz constant on the low floors leads to a good control off on the atoms of the compact part. Then, we require another kind of control on the noncompact part (see condition 3 and 4 below).

Let us introduce the finite partition of ∆. Letη andtbe given, letsbe greater than k0(η) given by Lemma 3.2. These three parameters are used to control the noncompact part of ∆. LetP be the finite partitionP=P1∪ P2 defined by:

• P2=



 [

R(j)≥t

[

`∈N

Λ^j_` ∪ {x / ρ^(s)(x)> γ^s}



=:{P2},

• P1 is the partition ofP₂^c intos-cylinders.

Because of Lemma 3.2, the mmeasure ofP2 is as small as we want, providedt is large enough andη is small enough:

m(P2)≤m( [

R(j)≥t

[

`∈N

Λ^j_`) +η.

We will denote byD1 the diameter ofP1 (which is smaller thanγ^s by definition) and byD2 themmeasure ofP2.

Theorem 1.2 implies the following mixing property: forα <1< α⁰ there exists q0such that for any integerkgreater thanq0, forP andP⁰ elements ofP,

α < m(P∩F⁻^kP⁰) m(P)µ(P⁰) < α⁰. (3.5)

Let a, b, c, d be positive numbers and k an integer to be chosen later. For any j∈N,C(a, b, c, d, j) is the cone of functions inL such that:

1. 0≤ 1 µ(P)

Z

P

f dm≤a Z

f dm∀P ∈ P, 2. for`≤jk, for any elements xandy of Λl,

|f(x)−f(y)| ≤d(x, y)b Z

f dm, 3. for`≤jk, for any element xof Λl∩P2,|f(x)| ≤c

Z f dm, 4. for` > jk,|f(x)| ≤ ^v_v^kj_` d

Z

f dmand

|f(x)−f(y)| ≤ ^v_v^kj_` d(x, y)d Z

f dm, for allx, y∈Λ`.

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Note that the cone also depends ons,t,k,η. We will choosea,b,c,d,k,η,sand t in such a way that L^k is a contraction fromC(a, b, c, j) to C(a, b, c, d, j+ 1). Let us denote byδj= _v^v^kj

k(j+1) <1.

Remark 3.6. It will be clear from the proofs that if there exists 0 < β < 1 such that 0< δj< βthen condition 4 in the definition of the cone is useless. This means that it is possible to work with the coneC(a, b, c) of functionsf such that:

1. 0≤ 1 µ(P)

Z

P

f dm≤a Z

f dm∀P ∈ P, 2. L(f)≤b

Z f dm,

3. forx∈Λ`∩P2 |f(x)| ≤c Z

f dm.

Proposition 3.7. For any 0 < σ < δj, for any positive numbers a0, b0, c0, d0, there exist an integer k0, positive numbers a ≥ a0, b ≥ b0, c ≥ c0, d ≥d0, and integers sandt such that for all k > k0,

L^kC(a, b, c, d, j)⊂ C(σa, δjb, δjc, δjd, j+ 1).

Moreover, forf ∈ L^kC(a, b, c, d, j), we have 1

µ(P) Z

P

f dm≥Afor all P∈ P,

whereA can be chosen arbitrarily provided it is strictly smaller than1/2.

Remark 3.8. In fact, it is clear from the proof that by modifying slightly the ar- gument the constant A above may be chosen arbitrarily close to 1. This may be important to optimize the rate of convergence in the exponential case.

Before proving Proposition 3.7, let us explain where the contraction of each of the four conditions comes,

• the contraction of condition 1 will follow from the mixing and the contraction rateσmay be obtained as close to 0 as we wish, providedkis large enough,

• the contraction of condition 2 will follow from the Lasota-Yorke inequality,

• the contraction of condition 3 will follow mainly from the definition of L: for

`≥k, L^kf(x) =^v^`_v⁻^k

` f(x⁰) forx∈Λ` andF(x⁰) =x,x⁰ ∈Λ`−k,

• the contraction of condition 4 follows directly from the definition of L.

We begin the proof of Proposition 3.7 with the following lemma which gives a control on the sup. of functions in C(a, b, c, d, j). When there is no ambiguity, we will forget about the dependence on a, b, c, d and j and write C instead of C(a, b, c, d, j) orC(j) if we wish to emphasize the dependence onj.

Lemma 3.9. For any f ∈ C, for`∈NandP ∈ P1, for xinΛ`∩P we have

|f(x)| ≤ 1 m(P)

Z

P

f dm+ max[b, d]|P| Z

f dm, (3.6)

|f(x)| ≤[Da+D1max[b, d]]

Z f dm.

(3.7)

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(whereD= sup_P_∈P_m(P)^µ(P)). For anyf ∈ C, for x∈P2∩Λ`,`∈N,

f(x) ≥ 1

m(P2) Z

P₂

f dm−2 max[c, d]

Z f dm, f(x) ≤ max[c, d]

Z f dm.

(3.8)

Proof. Let f belong toC. For x∈ P ∈ P1, equations (3.6) and (3.7) for` ≤ kj directly follow from definitions. Just write that forxandy inP ∈ P,

f(y)−L(f)d(x, y)≤f(x)≤f(y) +L(f)d(x, y)

and integrate these inequalities on P with respect tom. Letx, y∈P2∩Λ` with

`≤kj,

f(y)−2 sup

Λ_`∩P₂|f| ≤f(x)≤ sup

Λ_`∩P₂|f|;

this leads to (3.8) for`≤kj. Inequalities for` > kjfollow from 4 in the definition of the cone.

Let us begin with the proof of Proposition 3.7.

Proof of Proposition 3.7. We are going to check how the four conditions in the definition of C evolve under the action of L^k. First of all, condition 4 of the cone is very easily checked: if x ∈ Λ`, ` > k(j + 1), then L^kf(x) = ^v^`−k_v

` f(x⁰), with x⁰ ∈Λ`−k so| L^kf(x)| and| L^kf(x)− L^kf(y)| are less than ^v^k(j+1)_v

` δjdR

f dm, forx and y in Λ`, ` > kj. It just remains to check the first three conditions. Assume thatdis fixed, bandcwill be chosen to haved≤min[b, c].

We takef ∈ C. We fix 0< σ < δj andε < σ,α <1< α⁰ andq0∈Nsuch that (3.5) is satisfied fork > q0. In what follows,kis greater thanq0.

The following remark will be used many times in the proof: iff ∈ C, then since we choosed≤max(b, c) we have that sup|f| ≤cR

f dmandL(f)≤bR

f dm(use also that if` > kj, then ^v_v^kj

` ≤1).

Condition 1. LetP ∈ P, 1

µ(P) Z

P

L^kf dm = 1 µ(P)

Z

F^−kP

f dm= X

P⁰∈P

1 µ(P)

Z

P⁰∩F^−kP

f dm

= X

P0∈P1

1 µ(P)

Z

P⁰∩F^−kP

f dm+ 1 µ(P)

Z

P₂∩F^−kP

f dm.

Using Lemma 3.9, we get forP⁰∈ P1: m(F⁻^kP∩P⁰)

µ(P)m(P⁰)



Z

P⁰

f dµ−m(P⁰)D1b Z

f dm



 ≤ 1 µ(P)

Z

P⁰∩F^−kP

f dm

and 1 µ(P)

Z

P⁰∩F^−kP

f dm ≤ m(F⁻^kP∩P⁰) µ(P)m(P⁰)



Z

P⁰

f dµ+m(P⁰)D1b Z

f dm



,

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using (3.5), this leads to α



Z

P⁰

f dm−α⁰bD1m(P⁰) Z

f dm



 ≤ 1 µ(P)

Z

P⁰∩F^−kP

f dm

and 1 µ(P)

Z

P0∩F−kP

f dm ≤ α⁰



Z

P0

f dm+bD1m(P⁰) Z

f dm



.

ForP2, we get:

α Z

P₂

f dm−2cD2α⁰ Z

f dm≤ 1 µ(P)

Z

P₂∩F^−kP

f dm≤α⁰cD2

Z f dm.

Summing these inequalities over P ∈ P1 and P2, we get (one has to notice that m(1P₂·f)≥0):

[α−α⁰D1b−2α⁰cD2] Z

f dm≤ 1 µ(P)

Z

P

L^kf dm

and 1 µ(P)

Z

P

L^kf ≤[1 +D1b+cD2]α⁰ Z

f dm (3.9)

Condition 2. Fixε >0.

Letxandybelong to Λ`,`≤k(j+ 1), using the fact that ifx∈Λ`,`≤k(j+ 1), thenF^kx⁰ =xwithx⁰ ∈Λ`−k and Lemma 3.4, we findk0andN∈Nsuch that for k > k0,

| L^kf(x)− L^kf(y)| ≤ v`−k

v`

d(x, y)b Z

f dmif`≥k

≤ εd(x, y)[b Z

f dm+ sup|f|] ifN ≤` < k

≤ d(x, y)[εb Z

f dm+Csup|f|] otherwise.

Lemma 3.9 gives sup|f| ≤max[c, bD1+aD]

Z

f dmso we get

| L^kf(x)− L^kf(y)| ≤ v`−k

v`

d(x, y)b Z

f dmif`≥k (3.10)

≤εd(x, y) Z

f dm[b+ max(c, bD1+aD)] ifN ≤` < k (3.11)

≤d(x, y) Z

f dm[εb+Cmax(c, bD1+aD)] otherwise.

(3.12)

Condition 3. Letx∈Λ`∩P2 with`≤k(j+ 1), if`≥k, we have

| L^kf(x)| ≤ v`−k

v`

c Z

f dm.

Let` > k, we denote byx⁰ the preimage ofxbyF^`,x⁰∈Λ0, L^kf(x) =v0

v`

L^k⁻^`f(x⁰).

(3.13)