Lyapunov exponents and rates of mixing for one-dimensional maps

LYAPUNOV EXPONENTS AND RATES OF MIXINGFOR ONE-DIMENSIONAL MAPSJOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIROAbstract. We show that one dimensional maps f with strictlypositive Lyapunov exponents almost everywhere admit an absolutelycontinuous invariant measure. If f is topologically transitivesome power of f is mixing and in particular the correlation ofHölder continuous observables decays to zero. The main objectiveof this paper is to show that the rate of decay of correlations is determined,in some situations, to the average rate at which typicalpoints start to exhibit exponential growth of the derivative.1. Introduction and statement of results1.1. Lyapunov exponents. The purpose of this paper is to study thestatistical properties of one-dimensional maps f : I → I with positiveLyapunov exponents, where I may be the circle S 1 or an interval. Suchmaps satisfy asymptotic exponential estimates for growth of the derivativebut do not necessarily exhibit exponential estimates for otherfeatures of the dynamics such as the decay of correlation. Our mainobjective here is to identify some criteria which distinguishes differentdegrees of expansivity and which is reflected in the rate of decay ofcorrelations of the system.Definition 1. We say that a map f : I → I has positive Lyapunovexponents almost everywhere if there exists some λ > 0 such thatlim infn→∞1n log |(f n ) ′ (x)| ≥ λ > 0for Lebesgue almost every point x ∈ I.(∗)Date: July 2002.1991 Mathematics Subject Classification. Primary 58F15, Secondary 58F11.Work carried out at the Federal University of Bahia, University of Porto andImperial College, London. Partially supported by CMUP, PRODYN, SAPIENSand UFBA.1

2 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIRO1.2. Decay of correlations. Positive Lyapunov exponents are knownto be a cause of sensitive dependence on initial conditions and otherdynamical features which give rise to a degree of chaoticity or stochasticityin the dynamics. One can formalize this idea through the notionof mixing with respect to some invariant measure:Definition 2. A probability measure µ defined on the Borel sets of Iis said to be f-invariant if µ(f −1 (A)) = µ(A) for every Borel set A ⊂ I.Definition 3. A map f is said to be mixing with respect to somef-invariant probability measure µ if|µ(f −n (A) ∩ B) − µ(A)µ(B)| → 0, when n → ∞,for any measurable sets A, B.One interpretation of this property is that the conditional probabilityof f −n (A) given B, i.e. the probability that the event B is aconsequence of the event A having occurred at some time in the past,is asymptotically the same as if the two events were completely independent.This is sometimes referred to as a property of loss of memory,and thus in some sense of stochasticity, of the system. A natural questionof interest both for application and for intrinsic reasons, therefore,is the speed at which such loss of memory occurs. Standard counterexamplesshow that in general there is no specific rate: it is alwayspossible to choose sets A and B for which mixing is arbitrarily slow.However this notion can be generalized in the following way.Definition 4. For a map f : I → I preserving a probability measureµ, and functions ϕ, ψ ∈ L 1 (µ), we define the correlation function∫∫ ∫C n = C n (ϕ, ψ) =∣ (ϕ ◦ f n )ψdµ − ϕdµ ψdµ∣ .Notice that choosing these observables to be characteristic functionsof Borel sets this gives the well-known definition of mixing when C n →0. By restricting the set of allowed observables, for example to theclass of Hölder continuous functions, it is sometimes possible to obtainspecific rates of decay of the correlation function C n which depend onlyon the map f (up to a multiplicative constant which is allowed todepend on the ϕ and ψ).Until recently, the only examples for which a specific rate of decayof correlations was known, were uniformly expanding maps in one dimensionand more generally, uniformly hyperbolic systems in higherdimensions [34, 16, 17, 15]. In these cases one always has exponentialdecay of correlation for Hölder continuous observables, an estimate

LYAPUNOV EXPONENTS AND RATES OF MIXING 3which can be expected due to the fact that essentially everything isexponential in these cases. Relaxing uniform expansion conditions hasproved extremely hard and until now estimates on decay for correlationsfor systems satisfying the asymptotic exponential expansion condition(∗) but strictly not uniformly expanding have been known only in somefairly specific classes of examples.The simplest example is that of Pomeau-Manneville one-dimensionalmaps which are expanding everywhere except at some neutral fixedpoint p for which f ′ (p) = 1. In certain cases (essentially depending onthe second derivative f ′′ (p)) there is an ergodic absolutely continuousmixing invariant measure µ. The neutral fixed point implies that themap cannot be uniformly expanding, but the existence of the absolutelycontinuous invariant measure and the fact that f ′ > 1 everywhereexcept at the fixed point imply that the Lyapunov exponent of typicalpoints is∫1limn→∞ n log |(f n ) ′ (x)| = log |f ′ (x)|dµ > 0.IThus f satisfies condition (∗). In these cases the correlation functiondecreases to zero but only at a polynomial [26, 23, 39] or even subpolynomial[22] rate depending again on the particular higher orderterms of f near the fixed point. Intuitively this subexponential decayrate is caused by the neutral fixed point slowing down the mixing process.Indeed nearby points are moving away, and thus “mixing”, ata slower rate than that at which they move away from other fixed orperiodic points.A more subtle slowing down effect occurs in smooth one-dimensionalmaps with critical points. Under some general conditions such mapsadmit an absolutely continuous invariant measure with positive Lyapunovexponents, and thus in particular satisfy condition (∗). Herehowever all periodic points are hyperbolic repelling and the rate ofmixing is essentially determined by the rate of growth of the derivativealong the critical orbits [11]. Indeed, points which come close to one ofthe critical points shadow its orbit, thus possibly slowing the mixingprocess, for a certain amount of time. The amount of time depends onthe strength with which they are repelled by the critical orbit, which inturn boils down to the rate of growth of the derivative along this orbit.If the critical orbit has an exponentially growing derivative, nearby orbitsare repelled exponentially fast and the rate of mixing continuesto be exponential as in the uniformly expanding case. If the rate ofgrowth of the derivative along the critical orbit is subexponential, thenit has an essentially neutral behaviour and the effect is very much like

4 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIROthat of a neutral fixed point and results in subexponential decay ofcorrelations.1.3. Degree of non-uniformity of the expansion. A natural questionsis what general characteristics of a system satisfying (∗) determinesthe rate of decay of correlations for Hölder continuous observables?The purpose of the present paper is to give a partial solution tothis question by relating the rate of decay of correlations to the degreeof non-uniformity of the expansivity, i.e. the time one has to wait fortypical points to start behaving as though the system were uniformlyexpanding.Definition 5. For 0 < λ ′ < λ, we define the expansion time functionE by{E(x) = min N : 1 }n log |(f n ) ′ (x)| ≥ λ ′ , ∀n ≥ N .Condition (∗) implies that E is defined and finite almost everywhere.We think of this as the waiting time before the exponential derivativegrowth kicks in. A map is uniformly expanding if E(x) is uniformlybounded for every x. In general however, E(x) is defined only almosteverywhere and unbounded, indicating that some points may exhibitno growth or even arbitrarily large loss of growth of the derivativealong their orbit for arbitrarily long time before starting to exhibit theexponential growth implied by condition (∗). Our results show that atleast in certain situations the properties of the function E(x) are closelyrelated to the rate of mixing of the map.1.4. Local diffeomorphisms. We begin with the simplest situationin order to highlight the main idea of the results.Theorem 1. Let f : I → I be a C 2 local diffeomorphism with somepoint having dense pre-orbit. Suppose that f satisfies condition (∗) andthat there exists γ > 1 such that for some 0 < λ ′ < λ we have∣∣{E(x) > n} ∣ ∣ ≤ O(n −γ ).Then there exists an absolutely continuous, f-invariant, probabilitymeasure µ on I. Some finite power of f is mixing with respect to µand the correlation function C n for Hölder continuous observables on IsatisfiesC n ≤ O(n −γ+1 ).We emphasize here that the asymptotic statements here do not dependon the choice of λ ′ . Thus the rate of decay of correlations depends

LYAPUNOV EXPONENTS AND RATES OF MIXING 5essentially on the average time with which some given uniform exponentialrate of expansion is attained, and does not depend on whatturn out to be significantly more subtle characteristics of the systemsuch as the actual rate of convergence of the Lyapunov exponents tothe limit.Remark 1. The absolute continuity and ergodicity of µ follow from [3].Our argument gives an alternative proof of the absolute continuity of µand allows us to obtain the estimates on the rate of decay of correlationwhich are the main purpose of the paper.Remark 2. The statement about the rate of decay of the correlationsis of interest even if an absolutely continuous mixing f-invariant, probabilitymeasure µ is given to begin with. Then condition (∗) is justequivalent to the integrability condition∫log |f ′ |dµ > 0,since Birkhoff’s Ergodic Theorem then implies that the limit1λ = limn→∞ n log |(f n ) ′ 1 ∑n−1∫(x)| = lim log |f ′ (x)| = log |f ′ |dµ > 0n→∞ nexists for µ-almost every x ∈ I. In particular the expansion timefunction E(x) is also defined and finite almost everywhere and the conclusionsof the theorem hold under the given conditions on the rate ofdecay of |{E(x) > n}|.1.5. Multimodal maps. We can generalize our result to C 2 mapswith non flat critical points if we assume that almost all orbits haveslow approximation to the critical set C. Let dist δ (x, C) denote theδ-truncated distance from x to C defined as dist δ (x, C) = dist(x, C) ifdist(x, C) ≤ δ and dist δ (x, C) = 1 otherwise.Definition 6. We say that a map f : I → I, with a critical set C,satisfies the slow recurrence condition if given any ɛ > 0 there existsδ > 0 such that for Lebesgue almost every x ∈ Ilim supn→+∞1 ∑n−1nj=0i=0− log dist δ (f j (x), C) ≤ ɛ.(∗∗)Condition (∗∗) is an asymptotic statement, just like condition (∗),and we have no a-priori knowledge about how fast this limit is approachedor with what degree of uniformity for different points x. Thuswe introduce the analogue of the expansion time function as follows.

6 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIRODefinition 7. The recurrence time function is defined by{}R(x) = min N ≥ 1 : 1 ∑n−1− log dist δ (f j (x), C) ≤ 2ɛ, ∀n ≥ N .ni=0Condition (∗∗) implies that R is well-defined and finite almost everywherein I. For stating our results we also need the followingDefinition 8. For each n ≥ 1 define the tail of non-uniformity byΓ n = {x : E(x) > n or R(x) > n}This is the set of points which at time n have not yet achieved eitherthe uniform exponential growth or the uniform slow approximation toC given by (∗) and (∗∗).Theorem 2. Let f : I → I be a C 2 map with a set C of non-flat criticalpoints and with some point having dense pre-orbit in I \ C. Supposethat f satisfies the non-uniform expansivity condition (∗) and the slowapproximation condition (∗∗) to C and suppose that there exists γ > 1such that|Γ n | ≤ O(n −γ ).Then there exists an absolutely continuous, f-invariant, probabilitymeasure µ. Some finite power of f is mixing with respect to µ andthe correlation function C n for Hölder continuous observables on I satisfiesC n ≤ O(n −γ+1 ).Remark 3. As for condition (∗), also condition (∗∗) admits a formulationwhich may appear more natural if an ergodic f-invariant, absolutelycontinuous probability measure µ is given to begin with. Indeed,it then just reduces to an integrability condition on the log of the distancefunction to the critical set:∫|log dist(x, C)| dµ < ∞.This can be though of as saying that the invariant measure does notgive disproportionate weight to neighbourhoods of C.1.6. Maps with singular points. The way in which the control ofthe recurrence near critical points comes into play in our argument canbe used to deal similarly with other kinds of criticalities or singularities,such as discontinuities and/or points with infinite derivative. In fact, itis becoming increasingly clear (see for example references given aboveas well as [28]) that there is a kind of duality in one-dimensional maps

LYAPUNOV EXPONENTS AND RATES OF MIXING 7between critical points and discontinuities with infinite derivative. Regionsof in which the derivative is unbounded may appear at first tocontribute positively in the direction of proving some properties of themap which often follow from the expansivity. However it has beenknown for some time that discontinuities and unbounded derivativescreate non-trivial obstructions to the geometric structures and distortionestimates which usually form the basis for the study of dynamicalproperties. Therefore these regions need to be avoided, or at least therecurrence in these regions needs to be controlled, in a very similar wayto the way in which the recurrence near critical points is controlled. Infact it is possible to formulate a general set of conditions in which wecan talk about a singular set S without specifying the nature of thesingularities which may be critical points, discontinuities, or points atwhich the norm of the derivative goes to infinity. In this context weneed to generalize the notion of non-flatness for critical points.Definition 9. We say that the singular set S is non-degenerate if(s 1 ) f behaves like a power of the distance to S: there are constantsB > 1 and β > 0 such that for every c ∈ S and x ∈ I \ S1B |x − c|β ≤ |f ′ (x)| ≤ B|x − c| −β ;(s 2 ) log |f ′ | is locally Lipschitz at points x ∈ I \ S with Lipschitzconstant depending on the distance to S: for every c ∈ S andx, y ∈ I \ S with |x − y| < |x − c|/2 we have|log |f ′ (x)| − log |f ′ (y)| | ≤B|x − y|.|x − c|βWe can define the slow recurrence condition (∗∗) with respect to ageneral non-degenerate singular set exactly in the same way as for thesimpler case of non-flat smooth critical points. Similarly we extend thedefinition of the recurrence function R and of the tail of non-uniformityΓ n . With these definitions we can now formulate our results in themost general setting. Theorems 1 and 2 are then special cases of thefollowingTheorem 3. Let f : I → I be a C 2 local diffeomorphism outside anon-degenerate critical/singular set S with some point having densepre-orbit in I \ S. Suppose that f satisfies the non-uniform expansivitycondition (∗) and the slow approximation condition (∗∗) to the criticalset and suppose that there exists γ > 1 such that|Γ n | ≤ O(n −γ ).

8 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIROThen there exists an absolutely continuous, f-invariant, probabilitymeasure µ. Some finite power of f is mixing with respect to µ andthe correlation function C n for Hölder continuous observables on I satisfiesC n ≤ O(n −γ+1 ).We expect this relation between the rate of decay of the tails of theexpansivity and recurrence functions and the rate of decay of correlationsto hold also for faster rates such as exponential and stretchedexponential. For the moment however, the kind of statistical argumentswe use do not allow us to include these cases in our statements.1.7. The Central Limit Theorem. As a by product of our resultswe do however get some further interesting statistical properties of themaps under consideration. In particular we can apply some general results,e.g. [39, 27], which show that under certain conditions includingsufficiently fast decay of correlations one can prove the Central LimitTheorem which states that the probability of a given deviation of theaverage values of an observable along an orbit from the asymptoticaverage is essentially given by a Normal Distribution: given a Höldercontinuous function φ which is not a coboundary (φ ≠ ψ ◦ f − ψ forany ψ) there exists σ > 0 such that for every interval J ⊂ R,µ{x ∈ M :1 ∑n−1√ nj=0In our context we have(φ(f j (x)) −∫) }φdµ ∈ J→ 1σ √ 2π∫Je −t2 /2σ 2 dt.Theorem 4. Let f : I → I be a C 2 local diffeomorphism outside anon-degenerate critical/singular set S with some point having densepre-orbit in I \ S. Suppose that f satisfies the non-uniform expansivitycondition (∗) and the slow approximation condition (∗∗) to S, andsuppose that|Γ n | ≤ O(n −γ ), for some γ > 2.Then there exists an absolutely continuous, f-invariant, probabilitymeasure µ and the Central Limit Theorem holds.Notice that we have strengthened our condition on the rate of decayof the tail of |Γ n | to γ > 2 rather than just γ > 1.Remark 4. The hypotheses of f having a point whose pre-orbit is denseand disjoint from the singular set is not needed in all its strength inall our theorems; cf. Remark 5. Actually one can prove that this is aconsequence of the transitivity of the map.

LYAPUNOV EXPONENTS AND RATES OF MIXING 91.8. Strategy and overview of the paper. There exists at least twobasic methodologies for studying the statistical properties of dynamicalsystems, and in particular the decay of correlations. One of these is thefunctional-analytic one on which the problem is formulated in terms ofspectral properties of the Ruelle-Perron-Frobenius operator. This approachhas been extremely successful in a variety of examples and generalsituations. Another one is more geometric/probabilistic and usesthe idea of coupling measures in a somewhat more direct approach takingadvantage of the idea of decay of correlations as being related tothe convergence to equilibrium of arbitrary densities. Both methodsnecessarily rely to some extent on the geometric structure of the systemunder consideration; for example the pioneering work of Sinai-Ruelle-Bowen for uniformly hyperbolic systems combined the information onthe existence of finite Markov partitions with the functional-analyticapproach. More recently, Young [38, 39], has made the profound observationthat the rate of decay of correlations can be deduced via acoupling argument from the existence of a generalized Markov structureand some analytic information about this structure. The existence ofsuch a structure and the necessary analytic estimates are highly nontrivialin general and of independent interest. This is the approachwhich we take in this paper.We start by choosing essentially arbitrarily some small interval ∆ 0centered at a point p with “sufficiently dense pre-images”. This willbe the domain of definition of our induced map. We then attempt(successfully !) to implement the naive strategy of iterating ∆ 0 untilwe find some good return iterate n 0 such that f n 0(∆ 0 ) completely covers∆ 0 and some bounded distortion property is satisfied. There exists thensome interval J ⊂ ∆ 0 such that f n 0(J) = ∆ 0 . This interval is then bydefinition an element of the final partition of ∆ 0 for the induced Markovmap and has an associated return time n 0 . We then continue iteratingthe complement ∆ 0 \ J until more good returns occur. Some subtletyis required in the construction to keep track of the various pieces, theirsizes, and the number of iterates associated to each piece. However thebasic strategy essentially works. Most importantly, the constructionalso yields some information about the statistics of the return timefunction. More specifically we shall show that there exists an interval∆ ⊂ I \ S, a countable partition P (mod 0) of ∆ into intervals J, anda return time function R : ∆ → N piecewise constant on elements of Psatisfying the following properties:(1) Markov: for each J ∈ P and R = R(J), f R : J → ∆ 0 isa C 2 diffeomorphism (and in particular a bijection). Thus an

10 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIROinduced mapF : ∆ 0 → ∆ 0given by F (x) = f R(x) (x)is defined almost everywhere and satisfies the classical Markovproperty.(2) Uniform expansivity: there exists λ 0 > 1 such that for almostall x ∈ ∆ 0 we have|F ′ (x)| ≥ λ 0 .In particular the separation time s(x, y) given by the maximuminteger such that F i (x) and F i (y) belong to the same elementof the partition P for all i ≤ s(x, y), is defined and finite foralmost every pair of points x, y ∈ ∆ 0 .(3) Bounded distortion: There exist a constant K > 0 such that forany pair of points x, y ∈ ∆ 0 with ∞ > s(x, y) ≥ 1 we have∣ F ′ (x) ∣∣∣∣F ′ (y) − 1 ≤ Kˆλ −s(F (x),F (y)) .(4) Polynomial decay of tail of return times:|{R > n}| ≤ O(n −γ ).The argument is divided into two natural sections. In section 3we give the basic algorithm for choosing the base interval ∆ 0 and forconstructing the indued map of ∆ 0 to itself. This includes variousestimates concerning the time it takes for small regions to grow tosome fixed size and for these to return to ∆ 0 . In this section we alsoprove that the partition elements constructed by this method satisfy therequired expansivity and distortion estimates. In section 4 we addressthe more analytical/statistical issues in the argument. We analyze therelative sizes of various combinatorially distinct regions and concludethat almost every point effectively does belong to some region of ∆ 0which eventually has a good return, i.e. the algorithm used for theconstruction does give a mod 0 partition of ∆ 0 . We also give thestatistical argument which allows us to relate more explicitly the tailof the return time function for the induced map to the tail of nonuniformityof the system.2. An induced Markov map2.1. Attaining large scale: Hyperbolic Times. The key idea underlyingour strategy is that of hyperbolic times first introduced in [1].Let B > 1 and β > 0 be as in the hypotheses (s 1 )-(s 2 ). In what followsb is any fixed constant satisfying 0 < b < min{1/2, 1/(4β)}.

LYAPUNOV EXPONENTS AND RATES OF MIXING 11Definition 10. Given σ < 1 and δ > 0, we say that n is a (σ, δ)-hyperbolic time for a point x ∈ I if for all c ∈ S and 1 ≤ k ≤ n,|(f k ) ′ (f n−k (x))| ≥ σ −k and |f n−k (x) − c| ≥ σ bk .For each n ≥ 1 we defineH n = H n (σ, δ) = {x ∈ I : n is a (σ, δ)-hyperbolic time for x}.Lemma 1. Given σ < 1 and δ > 0, there exist δ 1 , κ, D 1 > 0, dependingonly on σ, δ and f, such that for any x ∈ I and n ≥ 1 a (σ, δ)-hyperbolictime for x, there exists an open interval V n (x) containing x with thefollowing properties:(1) f n maps V n (x) diffeomorphically onto (f n (x) − δ 1 , f n (x) + δ 1 );(2) for 1 ≤ k < n and y, z ∈ V n (x), |f n−k (y)−f n−k (z)| ≤ σ k/2 |f n (y)−f n (z)|;(3) V n (x) is contained in B(x, κ −n );(4) f n |V n (x) has distortion bounded by D 1 : for every y, z ∈ V n (x),1≤ |(f n ) ′ (y)|D 1 |(f n ) ′ (z)| ≤ D 1 .Proof. See [3, Lemma 5.2] and [3, Corollary 5.3].We shall often refer to the sets V n (x) as hyperbolic pre-intervals andto their images f n (V n (x)) as hyperbolic intervals. Notice that the latterare intervals of length 2δ 1 . The existence of hyperbolic times for pointssatisfying conditions (∗) and (∗∗) is assured by the next lemma.Lemma 2. There exists θ > 0, δ > 0 and 0 < σ < 1, depending only onf, λ and λ ′ , such that for every x ∈ I \ Γ n there exist (σ, δ)-hyperbolictimes 1 ≤ n 1 < · · · < n l ≤ n for x with l ≥ θn.Proof. See [3, Lemma 5.4].3. An induced Markov structure3.1. The partition. We start by outlining the order in which the mainconstants used in the construction are chosen. We take the three constantsθ > 0, δ > 0 and 0 < σ < 1 given by Lemma 2, which dependonly on the map f and the constants λ and λ ′ which we consider to begiven. We then fix a small constant δ 1 = δ 1 (σ, δ) > 0 given by Lemma1. We finally take small constants δ 0 = δ 0 (δ 1 , θ) > 0 and ε = ε(δ 0 ) > 0,and a large positive integer R 0 = R 0 (θ).⋃Let us now fix once and for all p ∈ I and N 0 ∈ N for whichN0j=0 f −j ({p}) is δ 1 /3 dense in I and disjoint from the critical set S.□□

12 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIRORemark 5. Incidentally this is the only point where we use the existencea point with dense pre-orbit. Actually, δ 1 /3 dense is enough for ourpurposes.Then we introduce the intervals ∆ 0 0 = ∆ 0 = (p − δ 0 , p + δ 0 ) and∆ 1 0 = (p−2δ 0 , p+2δ 0 ), ∆ 2 0 = (p− √ δ 0 , p+ √ δ 0 ), ∆ 3 0 = (p−2 √ δ 0 , p+2 √ δ 0 ).LetI k = { x ∈ ∆ 1 0 : δ 0 (1 + σ k/2 ) < |x − p| < δ 0 (1 + σ (k−1)/2 ) } , k ≥ 1,be a partition (mod 0) into countably many rings of ∆ 1 0 \ ∆ 0 . The nextlemma shows that there is some “scale” after which we can guaranteea good return to ∆ 0 within a fixed number of iterates.Lemma 3. There exist constants C 0 > 1 and D 0 > 0 depending onlyon f, σ, δ 1 and the point p, such that for any interval U ⊂ I of length2δ 1 there exist an open interval V ⊂ U and an integer 0 ≤ m ≤ N 0such that(1) f m maps V diffeomorphically onto ∆ 3 0;(2) f m |V has distortion bounded by D 0 ;(3) the m-preimages of ∆ 3 0 are uniformly bounded away from S forall 1 ≤ m ≤ N 0 and for x belonging to any such m-preimagewe have1C 0≤ |(f m ) ′ (x)| ≤ C 0 .Proof. Since ⋃ N 0j=0 f −j ({p}) is δ 1 /3 dense in I and disjoint from S,choosing δ 0 > 0 sufficiently small we have that each connected componentof the preimages of ∆ 3 0 up to time N 0 are bounded away from thecritical set S and are contained in an interval of length 2δ 1 /3. Thisimmediately implies that any interval U ⊂ I of length 2δ 1 contains apreimage V of ∆ 3 0 which is mapped diffeomorphically onto ∆ 3 0 in atmost N 0 iterates. Moreover, since the number of iterations and thedistance to the critical region are uniformly bounded, the distortion isuniformly bounded.Observe that δ 0 and N 0 have been chosen in such a way that all theconnected component of the preimages of ∆ 3 0 up to time N 0 satisfy theconclusions of the lemma. In particular, they are uniformly boundedaway from the critical set S, and so there is some constant C 0 > 1depending only on f and δ 1 such that1C 0≤ |(f m ) ′ (x)| ≤ C 0 .for all 1 ≤ m ≤ N 0 and x belonging to an m-preimage of ∆ 3 0.□

LYAPUNOV EXPONENTS AND RATES OF MIXING 13Now we describe the inductive construction of the partition P of ∆ 0 .For the sake of clearness let us explain the meaning of the objects thatwe will introduce below. At each step n we take ∆ n = ∆ 0 \ {R ≤ n}and write ∆ n as the disjoint union of sets A n and B n . A n may beunderstood as the set of points that is ready for returning at timen + 1, and t n : ∆ n → N gives the waiting time for points in B n . Wealso introduce a neighborhood A ε n of A n whose interest will becomeclear in Section 4.We ignore any dynamics occurring up to time R 0 . We assume thatsets ∆ i , A i , A ε i , B i , {R = i} and functions t i : ∆ i → N are definedfor all i ≤ n − 1. For i ≤ R 0 we just let A i = A ε i = ∆ i = ∆ 0 ,B i = {R = i} = ∅ and t i ≡ 0. Now let (Jn,j) 3 j be the connectedcomponents of f −n (∆ 0 ) ∩ A ε n−1 contained in hyperbolic pre-intervalsV m , with n − N 0 ≤ m ≤ n, which are mapped onto ∆ 3 0 by f n . TakeJn,j i = Jn,j 3 ∩ f −n ∆ i 0 for i = 0, 1, 2, and set R(x) = n for x ∈ Jn,j. 0 Takealso ∆ n = ∆ n−1 \ {R = n}. The function t n : ∆ n → N is defined asfollows:⎧⎪⎨ s if x ∈ Jn,j 1 \ Jn,j 0 and f n (x) ∈ I s for some j,t n (x) = 0 if x ∈ A n−1 \ ⋃ j⎪⎩J n,j,1t n−1 (x) − 1 if x ∈ B n−1 \ ⋃ j J n,j.1Finally letA n = {x ∈ ∆ n : t n (x) = 0}, B n = {x ∈ ∆ n : t n (x) > 0}andA ε n = {x ∈ ∆ n : dist(f n+1 (x), f n+1 (A n )) < ε}.At this point we have described the inductive construction of the setsA n , A ε n, B n and {R = n}. The proof that this gives indeed a partition(mod 0) of ∆ 0 is left to Section 4.Associated to each component Jn−k 0 of {R = n − k}, for some k > 0,we have the two intervals of Jn−k 1 \ J n−k 0 on both sides of it; knowingthat the new components of {R = n} “do not intersect too much”Jn−k 1 \ J n−k 0 is important for preventing overlaps on the new intervalsconstructed. This is indeed the case as long as ε > 0 is taken smallenough.Lemma 4. If ε > 0 is sufficiently small, then J 1 n ∩ {t n−1 > 1} = ∅ foreach J 1 n.Proof. Take some k > 0 and let Jn−k 0 be a component of {R = n − k}.Let Q k be the part of Jn−k 1 that is mapped by f n−k onto I k and assumethat Q k intersects Jn. 1 Recall that, by construction, Q k is precisely the

14 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIROpart of Jn−k 1 on which t n−1 takes the value 1. Letting q 1 and q 2 bethe two endpoints of one (of the two) connected components of Q k , wehave by Lemma 1 and Lemma 3We also have|f n−k (q 1 ) − f n−k (q 2 )| ≤ C 0 σ (k−N 0)/2 |f n (q 1 ) − f n (q 2 )|. (1)|f n−k (q 1 ) − f n−k (q 2 )| ≥ δ 0 (1 + σ (k−1)/2 ) − δ 0 (1 + σ k/2 )which combined with (1) gives= δ 0 σ k/2 (σ −1/2 − 1),|f n (q 1 ) − f n (q 2 )| ≥ C −10 σ N 0/2 δ 0 (σ −1/2 − 1).On the other hand, since J 1 n ⊂ A ε n−1 by construction of J 1 n, takingwe have J 1 n ∩ {t n−1 > 1} = ∅.ε < C −10 σ N 0/2 δ 0 (σ −1/2 − 1)□3.2. Expansivity. Recall that by construction, the return time R(J)for J an element of the partition P of ∆ 0 , is formed by a certain numbern of iterations given by the hyperbolic time of a hyperbolic pre-intervalV n ⊃ J, and a certain number m ≤ N 0 of additional iterates which isthe time it takes to go from f n (V n ) which could be anywhere in Ito f n+m (V n ) which covers ∆ 0 completely. By choosing R 0 sufficientlylarge it then follows from Lemma 3 that there exists a constant ˆλ > 1and a time n 0 such that for any hyperbolic time n ≥ n 0 and any pointx ∈ V n and 1 ≤ m ≤ N 0 , we have|(f n+m ) ′ (x)| ≥ ˆλ > 1We immediately have the required uniform expansivity property|F ′ (x)| = | ( f R(x)) ′(x)| ≥ ˆλ > 1.In particular, this implies that for any x, y ∈ ∆ 0 which have the samecombinatorics, i.e. which remain in the same elements of the partitionP for some number s(x, y) of iterates of the induced map F , we have|x − y| ≤ ˆλ −s(x,y) . (2)3.3. Bounded distortion. The distortion estimate required followsimmediately from (2) above and the following more classical formulationof the bounded distortion property:

LYAPUNOV EXPONENTS AND RATES OF MIXING 15Lemma 5. There exists a constant D > 0 such that for any x, y belongingto the same element J ∈ P with return time R, we have∣ logF ′ (x)∣∣∣∣ ( ) fR ′(x)∣F ′ (y) ∣ = log (f R ) ′ (y) ∣ ≤ D|f R (x) − f R (y)|.Proof. Recall that by construction, the return time R(J) for J an elementof the partition P of ∆ 0 , is formed by a certain number n of iterationsgiven by the hyperbolic time of a hyperbolic pre-interval V n ⊃ J,and a certain number m ≤ N 0 of additional iterates which is the timeit takes to go from f n (V n ) which could be anywhere in I, to f n+m (V n )which covers ∆ completely. Some standard formal manipulation basedon the chain rule gives( ) ∣ fR ′(x)∣∣∣∣( ) fR−n ′(f nlog∣ (f R ) ′ (y) ∣ = log (x))∣ ∣∣∣(f R−n ) ′ (f n (y)) ∣ + log (f n ) ′ (x)(f n ) ′ (y) ∣Since f i (x) and f i (y) are uniformly bounded away from S for n ≤ i ≤ R(recall Lemma 3), we may write( ) fR−n ′(f n (x))log∣ (f R−n ) ′ (f n (y)) ∣ ≤ B 1|f R (x) − f R (y)|where B 1 is some constant not depending on x, y or R. On the otherhand, by construction of V n (see the proof of Lemma 5.2 in [3]), theremust be some z ∈ V n for which n is a hyperbolic time and such that, for0 ≤ j < n, the distance from f j (z) to either f j (x) or f j (y) is smallerthan |f n (x) − f n (y)|σ (n−j)/2 , which is much smaller than σ b(n−j) ≤dist(f j (z), S). Thus, by (s 2 ) we havelog(f n ) ′ n−1(x)∣(f n ) ′ (y) ∣ ≤ ∑logf ′ (f j (x))∣f ′ (f j (y)) ∣ ≤ |f ∑n−1n (x) − f n (y)|j=0Since bβ < 1/2, there must be some B 2 > 0 such thatlog(f n ) ′ (x)∣(f n ) ′ (y) ∣ ≤ B 2|f n (x) − f n (y)|.j=02B σ(n−j)/2σ bβ(n−j) .Using again that f i (y) and f i (y) are uniformly bounded away from S(for n ≤ i ≤ R (cf. Lemma 3) it follows that|f n (x) − f n (y)| ≤ B 2 |f R (x) − f R (y)|,where B 2 is some constant not depending on x, y or R. This completesthe proof of the lemma.□

16 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIRO4. The statistical argumentWe now come to the main analytic estimates, i.e. the relation betweenthe decay of |∆ n | and that of |Γ n |.4.1. Proportion of points in A n−1 ∩H n returning at time of ordern. In the first part of the argument we use the strategy introduced in[38] for uniformly expanding maps, together with the properties ofhyperbolic times, and give a lower bound for the flow of mass fromA n−1 ∩ H n to the new intervals of the partition. We start with anauxiliary lemma.Lemma 6. For each ε > 0 there exists N ε > 0 such that any intervalB ⊂ I with |B| ≥ 2ε contains a hyperbolic pre-interval V n with n ≤ N ε .Proof. Given ε > 0 and an interval (z − ε, z + ε), choose N ε′ largeenough so that any hyperbolic pre-interval V n associated to a hyperbolictime n ≥ N ε ′ will be contained in an interval of length ε/5 (N ε ′ ∼κ −1 log(10ε −1 )). Now notice that each point has an infinite number ofhyperbolic times and therefore we have that∣∣I \ ⋃ n∣ → 0 as n → ∞.j=N ′ ε H jTherefore it is possible to choose{N ε = min n ≥ N ε ′ ∣: ∣I \ ⋃ nj=N ′ ε H j}∣ ≤ ε/10.This ensures that there is a point ˆx ∈ (z−ε/2, z+ε/2) with a hyperbolictime n ≤ N ε and associated hyperbolic pre-ball V n (x) ⊂ (z − ε, z +ε).□Proposition 1. There exist c 0 > 0 and N = N(ε) such that for everyn ≥ 1 ∣ ∣∣ ⋃ N{ } ∣ i=0 R = n + i ∣ ≥ c0 |A n−1 ∩ H n )|.Proof. Take r = 5δ 0 C N 00 , where N 0 and C 0 are given by Lemma 3. Let{z j } be a maximal set in f n (A n−1 ∩ H n ) with the property that theintervals (z j − r, z j + r) are pairwise disjoint. By maximality we have⋃(z j − 2r, z j + 2r) ⊃ f n (A n−1 ∩ H n ).jLet x j be a point in H n such that f n (x j ) = z j , and consider the hyperbolicpre-interval V n (x j ) associated to x j . Observe that f n sendsV n (x j ) diffeomorphically onto an interval of length 2δ 1 centered at z j ,as in Lemma 1. In what follows, given B ⊂ (z j − δ 1 , z j + δ 1 ), we willsimply denote (f n |V n (x j )) −1 (B) by f −n (B).

LYAPUNOV EXPONENTS AND RATES OF MIXING 17Our aim now is to prove that f −n ((z j − r, z j + r)) contains somecomponent of {R = n + k j } with 0 ≤ k j ≤ N ε + N 0 . We start byshowing thatt n+kj |f −n ((z j − ε, z j + ε)) > 0 for some 0 ≤ k j ≤ N ε + N 0 . (3)Assume by contradiction that t n+kj |f −n ((z j − ε, z j + ε)) = 0 for all0 ≤ k j ≤ N ε + N 0 . This implies that f −n ((z j − ε, z j + ε)) ⊂ A ε n+k jfor all 0 ≤ k j ≤ N ε + N 0 . Using Lemma 6 we may find a hyperbolicpre-interval V m ⊂ (z j −ε, z j +ε) with m ≤ N ε . Now, since B = f m (V m )is an interval of length 2δ 1 , it follows from Lemma 3 that there is someV ⊂ B and m ′ ≤ N 0 with f m′ (V ) = ∆ 0 . Thus, taking k j = m + m ′we have that 0 ≤ k j ≤ N ε + N 0 and f −n (V m ) is an element of {R =n + k j } inside f −n ((z j − ε, z j + ε)). This contradicts the fact thatt n+kj |f −n ((z j − ε, z j + ε)) = 0 for all 0 ≤ k j ≤ N ε + N 0 , and so (3)holds.Let k j be the smallest integer 0 ≤ k j ≤ N ε +N 0 for which t n+kj |f −n (B(z j , ε)) >0. Sincef −n ((z j − ε, z j + ε)) ⊂ A ε n−1 ⊂ {t n−1 ≤ 1},there must be some element Jn+k 0 j(j) of {R = n + k j } for whichf −n ((z j − ε, z j + ε)) ∩ J 1 n+k j(j) ≠ ∅.Recall that by definition f n+k jsends Jn+k 1 j(j) diffeomorphically onto∆ 1 0, the ball of radius (1+s)δ 0 around p. From time n to n+k j we mayhave some final “bad” period of length at most N 0 where the derivativeof f may contract, however being bounded from below by 1/C 0 in eachstep. Thus, the diameter of f n (Jn+k 1 j(j)) is at most 4δ 0 C N 00 . Since(z j − ε, z j + ε) intersects f n (Jn+k 1 j(j)) and ε < δ 0 < δ 0 C N 00 , we have bydefinition of rf −n ((z j − r, z j + r)) ⊃ Jn+k 0 j(j).Thus we have shown that f −n ((z j − r, z j + r)) contains some componentof {R = n + k j } with 0 ≤ k j ≤ N ε + N 0 . Moreover, since n isa hyperbolic time for x j , we have by the distortion control given byLemma 1and|f −n ((z j − 2r, z j + 2r))||f −n ((z j − r, z j + r))||f −n ((z j − r, z j + r))||J 0 n+k j(j)|≤ D 14r2r = 2D 1 (4)2r≤ D 1|f n (Jn+k 0 j(j))| . (5)Here we are implicitly assuming that 2r < δ 1 . This can be done just bytaking δ 0 small enough. Note that the estimates on N 0 and C 0 improvewhen we diminish δ 0 .

18 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIROFrom time n to time n + k j we have at most k j = m 1 + m 2 iterateswith m 1 ≤ N ε , m 2 ≤ N 0 and f n (Jn+k 0 j(j))) containing some pointw j ∈ H m1 . By the definition of (σ, δ)-hyperbolic time we have thatdist δ (f i (x), S) ≥ σ bNε for every 0 ≤ i ≤ m 1 , which by the uniformdistortion control implies that there is some constant D = D(ε) > 0such that |(f i ) ′ (x)| ≤ D for 0 ≤ i ≤ m 1 and x ∈ f n (Jn+k 0 j(j)). On theother hand, since the first N 0 preimages of ∆ 0 are uniformly boundedaway from S we also have some D ′ > 0 such that |(f i ) ′ (x)| ≤ D ′ forevery 0 ≤ i ≤ m 2 and x belonging to any i preimage of ∆ 0 . Hence,which combined with (5) gives|f n (J 0 n+k j(j))| ≥ 1DD ′ |∆ 0|,|f −n ((z j − r, z j + r))| ≤ C|J 0 n+k j(j)|,with C only depending on D 1 , D, D ′ and δ 0 . We also deduce from (4)that|f −n ((z j − 2r, z j + 2r))| ≤ C ′ |f −n ((z j − r, z j + r))|with C ′ only depending on D 1 .Finally let us compare the measure of the sets ⋃ N{ }i=0 R = n + i andA n−1 ∩ H n . We have|A n−1 ∩H n | ≤ ∑ j|f −n ((z j −2r, z j +2r))| ≤ C ′ ∑ j|f −n ((z j −r, z j +r))|.On the other hand, by the disjointness of the intervals (z j − r, z j + r)we have∑|f −n ((z j − r, z j + r))| ≤ C ∑ |J 0 n+k j(j)| ≤ C ∣ ⋃ N}i=0{ ∣ R = n + i ∣ .jjWe just have to take c −10 = CC ′ . □4.2. Relative proportion of A n and B n in ∆ n . We now prove acouple of useful lemmas. The first one gives a lower bound for the flowof mass from B n−1 to A n , and second one gives a lower bound for theflow of mass from A n−1 to B n and {R = n}.Lemma 7. There exists a 1 > 0 such that for every n ≥ 1|B n−1 ∩ A n | ≥ a 1 |B n−1 |.Moreover, a 1 is bounded away from 0 independently from δ 0 .

LYAPUNOV EXPONENTS AND RATES OF MIXING 19Proof. It is enough to see this for each component of B n−1 at a time.Let C be a component of B n−1 and Q be its outer interval correspondingto t n−1 = 1. Observe that by Lemma 4 we have Q = C ∩A n . Moreover,there must be some k < n and a component Jk 0 of {R = k} such thatf k maps C diffeomorphically onto ⋃ ∞i=k I i and Q onto I k , both withuniform bounded distortion (not depending on δ 0 or n). Thus, it issufficient to compare the lengths of ⋃ ∞i=k I i and I k . We have|I k || ⋃ ∞i=k I i| = δ 0(1 + σ (k−1)/2 ) − δ 0 (1 + σ k/2 )≈ 1 − σ 1/2 .δ 0 (1 + σ (k−1)/2 ) − δ 0Clearly this proportion does not depend on δ 0 .□The second item of the lemma below is apparently counterintuitive,since our main goal is to make the points in ∆ 0 have small return times.However, this is needed for keeping |A n | uniformly much bigger than|B n |. This will help us in the statistical estimates of the last section.Lemma 8. There exist b 1 = b 1 (δ 0 ) > 0 and c 1 = c 1 (δ 0 ) > 0 withb 1 + c 1 < 1 such that for every n ≥ 1(1) |A n−1 ∩ B n | ≤ b 1 |A n−1 |;(2) |A n−1 ∩ {R = n}| ≤ c 1 |A n−1 |.Moreover, b 1 → 0 and c 1 → 0 as δ 0 → 0.Proof. It is enough to prove this for each neighborhood of a componentJn 0 of {R = n}. Observe that by construction we have Jn 3 ⊂ A ε n−1, whichmeans that Jn 2 ⊂ A n−1 , because ε < δ 0 < √ δ 0 . Using the uniformbounded distortion of f n on Jn 3 given by Lemma 6 and Lemma 3 weobtain|Jn 1 \ Jn|0|Jn 2 \ Jn| ≈ |∆1 0 \ ∆ 0 0|1 |∆ 2 0 \ ∆ 1 0| ≈ δ 0√ ≪ 1,δ0which gives the first estimate. Moreover,and this gives the second one.|Jn|0|Jn 2 \ Jn| ≈ |∆0 0|1 |∆ 2 0 \ ∆ 1 0| ≈ δ 0√ ≪ 1,δ0The next result is a consequence of the estimates we obtained in thelast two lemmas. The proof is essentially the same of the uniformlyhyperbolic case; see [38]. Here we need to be more careful on theestimates.Lemma 9. There exists a 0 = a 0 (δ 0 ) > 0 such that for every n ≥ 1Moreover, a 0 → 0 as δ 0 → 0.|B n | ≤ a 0 |A n |.□

20 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIROProof. We have by Lemma 8Letting η = 1 − b 1 − c 1 we define|A n | ≥ (1 − b 1 − c 1 )|A n−1 |. (6)â = b 1 + c 1and a 0 = (1 + a 1)b 1 + c 1.a 1 a 1 ηThe fact that a 0 → 0 when δ 0 → 0 is a consequence of b 1 → 0 andc 1 → 0 when δ 0 → 0 and a 1 being bounded away from 0. Observe that0 < η < 1 and â < a 0 . Now the proof of the proposition follows byinduction. The result obviously holds for n up to R 0 . Assuming that itholds for n−1 ≥ R 0 we will show that it also holds for n, by consideringseparately the cases |B n−1 | > â|A n−1 | and |B n−1 | ≤ â|A n−1 |.Assume first that |B n−1 | > â|A n−1 |. We may writewhich by Lemma 7 givesSince we also have|B n−1 | = |B n−1 ∩ A n | + |B n−1 ∩ B n |,|B n−1 ∩ B n | ≤ (1 − a 1 )|B n−1 |. (7)|B n | = |B n ∩ B n−1 | + |B n ∩ A n−1 |,it follows from (7) and Lemma 8 that|B n | ≤ (1 − a 1 )|B n−1 | + b 1 |A n−1 |,which according to the case we are considering leads to|B n | ≤ (1 − a 1 )|B n−1 | + b 1a 1b 1 + c 1|B n−1 | < |B n−1 |.On the other hand, we have by (6) that |A n | < |A n−1 |, which togetherwith the last inequality and the inductive hypothesis yields|B n ||A n | < |B n−1||A n−1 | ≤ a 0,which gives the result in the first case.Assume now that |B n−1 | ≤ â|A n−1 |. Since we haveit follows from Lemma 8 that|B n | = |B n ∩ B n−1 | + |B n ∩ A n−1 |,|B n | ≤ |B n−1 | + b 1 |A n−1 |.Hence|B n ||A n | < |B n−1| + b 1 |A n−1 |η|A n−1 |which gives the result also in this case.≤ â + b 1η= a 0 ,□

LYAPUNOV EXPONENTS AND RATES OF MIXING 21Recalling that ∆ n = A n ∪ B n , it easily follows from Proposition 1and Lemma 9 that there is some constant b 0 > 0 not depending on nsuch that∣ ⋃ Ni=0{ } ∣ |A n−1 ∩ H n |R = n + i ∣ ≥ b0 |∆ n−1 |.|A n−1 |This immediately implies that()|A n−1 ∩ H n ||∆ n+N | ≤ 1 − b 0 |∆ n−1 |.|A n−1 |It is not hard to deduce from this last formula that()|∆ n+N | ≤ exp − b n∑0 |A j−1 ∩ H j ||∆ 0 |. (8)N + 1 |A j−1 |j=R 0On the other hand, it follows from Lemma 8 and Lemma 9 that thereis some constant c 2 > 0 not depending on n such that for every n ≥ 1|∆ n | ≤ c 2 |∆ n+1 |. (9)4.3. The tail estimates. Recall that θ > 0 was obtained in Lemma 2and gives a lower bound for the frequency of hyperbolic times; it onlydepends on the non-uniform expansion coefficient λ and the map f.Let us now derive a consequence of Lemma 2.Corollary 1. For every n ≥ 1 and every A ⊂ I \ Γ n with |A| > 0 wehave1n∑ |A ∩ H j |≥ θ.n |A|j=1Proof. Take n ≥ 1 and A ⊂ I \ Γ n with positive Lebesgue measure.Let ξ n be the measure in {1, . . . , n} defined by ξ n (J) = #J/n, for eachsubset J. Then, using Fubini’s theorem1nn∑|A ∩ H j | =j=1=∫ (∫∫A(∫A)χ(x, i) dx dξ n (i))χ(x, i) dξ n (i) dx,where χ(x, i) = 1 if x ∈ H i and χ(x, i) = 0 otherwise. Now, Lemma 2means that the integral with respect to dξ n is larger than θ > 0. So,the last expression above is bounded from below by θ|A|. □

22 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIROLet γ > 0 be some positive number (to be determined later) andtake 0 < α < (θ/12) γ+1 . Then we choose δ 0 > 0 small so that a 0 =a(δ 0 ) < 2α. We define, for each n ≥ 1,andE n =F ={j ≤ n: |A j−1 ∩ H j ||A j−1 |{n ∈ N: #E nn}< α ,> 1 − θ }.12In the next proposition we establish the relation between the statisticsof hyperbolic times and the geometrical structure of sets arisingfrom the construction of the partition described above. In the polynomialcase this establishes an essentially optimal link between the rateof decay of the expansion/recurrence function and the rate of decay ofcorrelations. The nature of the argument does not immediately extendto the exponential case.Proposition 2. Take any n ∈ F with n ≥ R 0 > 12/θ. If |A n | ≥ 2|Γ n |,then there is some 0 < k = k(n) < n for which |A n |/|A k | < (k/n) γ .Proof. We have for j ≤ n|A n ∩ H j ||A n |≥ |A n \ Γ n ||A n |· |(A n \ Γ n ) ∩ H j ||A n \ Γ n |≥ 1 2 · |(A n \ Γ n ) ∩ H j |,|A n \ Γ n |which together with the conclusion of Corollary 1 for the set A n \ Γ ngives1n∑ |A n ∩ H j |≥ θ n |A n | 2 . (10)LetSince n ∈ F , we have1nn∑j=1G n =|A n ∩ H j ||A n |j=1{j ∈ E n : |A j−1||A n |≤≤θ12 + 1 nθ12 + 1 n> θ }.12α∑ |A n ∩ H j ||A n |j∈E n∑|A n ∩ H j ||A n |j∈E n\G n+ #G nn .

Now, for j ∈ E n \ G n ,|A n ∩ H j ||A n |LYAPUNOV EXPONENTS AND RATES OF MIXING 23= |A n ∩ H j |· |A j−1||A j−1 | |A n |( |An ∩ A j−1 ∩ H j |≤|A j−1 |(|Aj−1 ∩ H j |≤|A j−1 |+ |(A )n \ A j−1 ) ∩ H j | |Aj−1 ||A j−1 | |A n |+ a 0) θ12α .For this last inequality we used the fact that (A n \ A j−1 ) ⊂ B j−1 andj /∈ G n . Hence1n∑ |A n ∩ H j |≤θn |Aj=1 n | 12 + 1 ∑ |A j−1 ∩ H j | θn |A j−1 | 12α + a θ012α + #G nnj∈E n\G n< θ12 + α θ12α + a θ012α + #G nn .By the choice of a 0 we have that the third term in the last sum aboveis smaller than θ/6. So, using (10) we obtainNow, definingwe have#G nn > θ 6 . (11)k = max(G n ) − 1,|A n | < 12αθ |A k|.It follows from (11) that k + 1 > θn/6, and so k/n > θ/12, becausen ≥ R 0 > 12/θ. Since we have chosen α < (θ/12) γ+1 , it follows that( kn) γ> 12θ( θ12This completes the proof of the result.) γ+1> 12αθ .By Lemma 9 we have |∆ n | ≤ const|A n |, and so it is enough to derivethe tail estimate for |A n | in the place of |{R > n}| = |∆ n |. Given anylarge integer n, we consider the following situations:(1) If n ∈ N \ F , then by (8) and (9) we have(|∆ n | ≤ c N 2 exp −b )0θα12(N + 1) (n − R 0) |∆ 0 |.(2) If n ∈ F , then we distinguish the following two cases:(a) If |A n | < 2|Γ n |, then nothing has to be done.□

24 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIRO(b) If |A n | ≥ 2|Γ n |, then we apply Proposition 2 and get somek 1 < n for which( ) γ k1|A n | < |A k1 |.nThe only situation we are left to consider is 2(b). In such case, eitherk 1 is in situation 1 or 2(a), or by Proposition 2 we can find k 2 < k 1 forwhich( ) γ k2|A k1 | < |A k2 |.k 1Arguing inductively we show that there is a sequence of integers 0 0 is arbitrary. So, the only casewe are left to consider is (II).Assume that |Γ n | ≤ O(n −γ ) for some γ > 0. In this case there mustbe some C > 0 such that k γ |Γ k | ≤ C for all k ∈ N, which applied to k sin case (II) leads to |A n | ≤ O(n −γ ).References[1] J. F. Alves, SRB measures for non-hyperbolic systems with multidimensionalexpansion, Ann. Scient. Éc. Norm. Sup., 4e série, 33 (2000), 1-32.[2] J. F. Alves, V. Araújo, Random perturbations of nonuniformly expanding maps,to appear in Astérisque.[3] J. F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systemswhose central direction is mostly expanding, Invent. Math. 140 (2000),351-398.[4] J. F. Alves, S. Luzzatto, V. Pinheiro, Markov structures and decay of correlationsfor non-uniformly expanding dynamical systems, preprint 2002.[5] J. F. Alves, M. Viana, Statistical stability for robust classes of maps with nonuniformexpansion, Ergod. Th. & Dynam. Sys. 22 (2002), 1-32.[6] R. Adler, B. Weiss, Proc. Entropy, a complete metric invariant for automorphismsof the torus, Nat. Acad. Sci. USA, 57 (1967), 1573-1576.[7] R. Adler, B. Weiss, Similarity of automorphisms of the torus, Mem. Amer.Math. Soc. 98 (1970), Amer. Math. Soc. Providence, RI, USA.[8] V. Baladi, M. Viana, Strong stochastic stability and rate of mixing for unimodalmaps, Ann. Scient. Éc. Norm. Sup., 4e série, 29 (1996), 483-517.

LYAPUNOV EXPONENTS AND RATES OF MIXING 25[9] M. Benedicks, L. Carleson, On iterations of 1 − ax 2 on (−1, 1), Ann. Math.122 (1985), 1-25.[10] M. Benedicks, L. Carleson, The dynamics of the Hénon map, Ann. Math. 133(1991), 73-169.[11] H. Bruin, S. Luzzatto, S. van Strien, Decay of correlations in one-dimensionaldynamics, to appear in Ann. Scient. Éc. Norm. Sup.[12] M. Benedicks, L.-S. Young, SRB-measures for certain Hénon maps, Invent.Math. 112 (1993), 541-576.[13] M. Benedicks, L.-S. Young, Markov extensions and decay of correlations forcertain Hénon maps, Astérisque 261 (2000), 13-56.[14] C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems withmostly contracting central direction, Israel J. Math. 115 (2000), 157-193.[15] R. Bowen, D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29(1975), 181-202.[16] R. Bowen, Markov partitions for Axiom diffeomorphisms, Amer. J. Math. 92(1970), 725-747[17] R. Bowen, Equilibrium states and the ergodic theory of Axiom A diffeomorphisms,Lecture Notes in Mathematics, 480 (1975), Springer.[18] X. Bressaud, Subshifts on an infinite alphabet, Ergodic Theory & Dynam.Systems 19 (1999), no. 5, 1175–1200.[19] L. A. Bunimovich, Ya. G. Sinaĭ and N. I. Chernov, Statistical properties oftwo-dimensional hyperbolic billiards, Russian Math. Surveys 46 (1991), no. 4,47–106.[20] J. Buzzi, O. Sester, M. Tsujii, Weakly expanding skew-products of quadraticmaps, preprint 2001.[21] N. Chernov, L.-S. Young, Decay of correlations for Lorentz gases and hardballs, Enciclopaedia Math. Sci. 101, Springer, Berlin, 2000.[22] M. Holland, Slowly mixing systems and intermittency maps, preprint 2002.[23] H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixedpoints, preprint.[24] M. Jakobson, Absolutely continuous invariant measures for one-parameterfamilies of one-dimensional maps, Comm. Math. Phys. 81 (1981), 39-88.[25] T. Krüger, S. Troubetzkoy, Markov partitions and shadowing for nonuniformlyhyperbolic systems with singularities, Ergodic Theory Dynam. Systems12 (1992), no. 3, 487–508.[26] C. Liverani, B. Saussol, S. Vaienti, A probabilistic approach to intermittency,Ergodic Th. Dynam. & Syst. 19 (1999), 671–685.[27] C. Liverani. Central limit theorem for deterministic systems, In F. Ledrappier,J. Lewowicz, and S. Newhouse, editors, Procs. Intern. Conf. on DynamicalSystems (Montevideo 1995) – A tribute to Ricardo Mañé, pages 56–75, Boston,1996. Pitman.[28] S. Luzzatto, W. Tucker, Non-uniformly expanding dynamics in maps with singularitiesand criticalities, Publ. Math. IHES 89 (1999), 180–t226.[29] R. Mañé, Ergodic theory and differentiable dynamics, Springer-Verlag, 1987.[30] V. Maume-Deschamps, Correlation decay for Markov maps on a countablestate space Ergodic Theory and Dynamical Systems 21 (2001), 165–196.[31] T. Nowicki, D. Sands. Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math. 132 (1998), no. 3, 633–680.

26 JOSÉ F. ALVES, STEFANO LUZZATTO, AND VILTON PINHEIRO[32] D. Ruelle, A measure associated with Axiom A attractors, Amer. Jour. Math.98 (1976), 619-654.[33] O. Sarig. Subexponential decay of correlations, preprint 2002.[34] Y. Sinai, Markov partitions and U-diffeomorphisms, Funkcional Anal. iPrilozen 2, 32 (1968), 70-80.[35] Y. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surv. 27, n. 4, (1972),21-69.[36] M. Viana, Multidimensional non-hyperbolic attractors, Publ. Math. IHES 85(1997), 63-96.[37] L.-S. Young, Decay of correlations for certain quadratic maps, Comm. Math.Phys. 146 (1992), 123-138.[38] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity,Ann. Math. 147 (1998), 585-650.[39] L.-S. Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999),153-188.Departamento de Matemática Pura, Faculdade de Ciências do Porto,Rua do Campo Alegre 687, 4169-007 Porto, PortugalE-mail address: jfalves@fc.up.ptURL: http://www.fc.up.pt/cmup/home/jfalvesMathematics Department, Imperial College, 180 Queen’s Gate, LondonSW7, UKE-mail address: stefano.luzzatto@ic.ac.ukURL: http://www.ma.ic.ac.uk/∼luzzattoDepartamento de Matemática, Universidade Federal da Bahia, Av.Ademar de Barros s/n, 40170-110 Salvador, Brazil.E-mail address: viltonj@ufba.br