A version of Kac's lemma on first return times for suspension flows
aa r X i v : . [ m a t h . D S ] N ov A VERSION OF KAC’S LEMMA ON FIRST RETURNTIMES FOR SUSPENSION FLOWS
PAULO VARANDAS
Abstract.
In this article we study the mean return times to a given set forsuspension flows. In the discrete time setting, this corresponds to the classicalversion of Kac’s lemma [12] that the mean of the first return time to a set withrespect to the normalized probability measure is one. In the case of suspensionflows we provide formulas to compute the mean return time. Positive measuresets on cross sections are also considered.In particular, this varies linearly withcontinuous reparametrizatons of the flow and takes into account the meanescaping time from the original set. Relation with entropy and returns topositive measure sets on cross sections is also considered. Introduction
The construction of invariant measures and the study of their statistical proper-ties are the main goals of the ergodic theory in the attempt to describe the behaviorof a discrete and continuous time dynamical system. In the presence of invariantmeasures the celebrated Poincar´e’ s recurrence theorem guarantees that recurrenceoccurs almost everywhere and, from this qualitative statement, a natural questionis to ask for quantitative results on the recurrence. Given a measure preservingmap f : ( X, µ ) → ( X, µ ) and a positive measure set A ⊂ X consider the first returntime to the set A given by n A ( x ) = inf { k ≥ f k ( x ) ∈ A } . In 1947, Kac [12] established an expression to the expectation of the return timeproving that Z A n A ( x ) dµ A = 1 , where µ A ( · ) = µ ( · ∩ A ) /µ ( A ) denotes the normalized restriction of µ to A . Sucha quantitative estimate has been very useful tool in the study of other recurrenceproperties (see e.g. [7, 8, 9, 15, 16] and the references therein)In the time-continuous setting the situation is substantially different. It is clearthat if a flow ( X t ) t preserves a probability measure µ then it is an invariant proba-bility measure for discrete-time transformation obtained as the time- t map f = X t of the flow and Kac’ s lemma holds for ( f, µ ). However, this does not constitute atrue time-continuous quantitative recurrence estimate. In fact, given a continuousflow ( X t ) t , an open set A and a point x ∈ A , the first return time of x to the set A should be considered after the time e A ( x, t ) > x needs toexit A (c.f. Section 2 for the precise definition). Date : August 30, 2018.2010
Mathematics Subject Classification.
Primary: 37B20 Secondary: 28D10, 37C10 .
Key words and phrases.
Suspension flows, Poincar´e recurrence, hitting times, Kac’s lemma. ur purpose here is to prove a time-continuous quantitative Kac-like formulafor first return times for suspension flows for positive measure sets both in thecross-section and ambient spaces. Our first motivation follows from Ambrose andKakutani [3, 4] which proved that any aperiorid flow ( Y t ) t on a probability space( M, ν ) without singularities is metrically isomorphic to a suspension flow ( Z t ) t .More precisely, there exists a measurable transformation f : Σ → Σ, an f -invariantprobability measure µ Σ and a roof function τ : Σ → R +0 defining a suspensionflow ( Z t ) t on Σ τ that preserves µ = µ Σ × Leb/ R τ dµ Σ , and there exists a measurepreserving ψ : ( M, ν ) → (Σ τ , µ ) that is a bijection (modulo zero measure sets) andsatisfying Z t ◦ ψ = ψ ◦ Y t and Y t ◦ ψ − = ψ − ◦ Z t for all t ≥
0. Our secondmotivation is that, using finitely many tubular neighborhoods or the Hartman-Grobman theorem around singularities, smooth flows on compact manifolds thatexhibit some hyperbolicity often can be modelled by suspension flows. This is truefor hyperbolic flows (see e.g. [6, 14]) for geometric Lorenz and three-dimensionalsingular-hyperbolic flows (see e.g. [5]) and for smooth three-dimensional flows withpositive topological entropy (see [11]). A final motivation is that it is expected tohave a large utility and large amount of applications, including the study of largedeviations for return times in cylinders, fluctuations theorems and study of therecurrence properties for Axiom A flows. Our main results below assert that theexpectation of the first return time to open sets takes into account the mean returntime of the roof function. Moreover, the mean return time to subsets of the globalcross section can be understood as a quotient of the corresponding entropies of theflow and the first return time map on the base. The detailed statements will begiven in the next section.After this work was complete we were informed by J.-R. Chazottes of the un-noticed and very interesting work by G. Helberg [10] where the author addressesthe problem of return time estimates for positive measure sets for measurable andcontinuous flows. Since this is a rather unknown result, never quoted, we shalldescribe his main results in Section 2 for completeness.2.
Suspension flows
Given a topological space Σ, a measurable map f : Σ → Σ and a roof function τ : Σ → R + that is bounded away from zero consider the quotient spaceΣ τ = { ( x, s ) ∈ Σ × R + : 0 ≤ s ≤ τ ( x ) } / ∼ obtained by the equivalence relation that ( x, τ ( x )) ∼ ( f ( x ) ,
0) for every x ∈ Σ. Thesuspension flow ( X t ) t on Σ τ associated to ( f, Σ , τ ) is defined by as the ’verticaldisplacement’ X t ( x, s ) = ( x, t + s ) whenever the expression is well defined. Moreprecisely, X s ( x, t ) = (cid:16) f k ( x ) , t + s − k − X j =0 τ ( f j ( x )) (cid:17) (2.1)where k = k ( x, t, s ) ≥ P k − j =0 τ ( f j ( x )) ≤ t + s < P kj =0 τ ( f j ( x )) . We shall refer to Σ as a cross-section to the flow. Since τ is bounded away fromzero there is a natural identification between the space M X of ( X t ) t -invariantprobability measures and the space M f ( τ ) of f -invariant probability measures µ o that τ ∈ L ( µ ). Namely, if m denotes the Lebesgue measure in R then the map L : M f ( τ ) → M X µ ¯ µ := ( µ × m ) | Σ τ R Σ τ dµ (2.2)is a bijection. For that reason in some situations one reduces some ergodic prop-erties from the suspension semiflow to the first return map to the global Poincar´esection Σ × { } . For instance, while the measure induced on the cross section candetect ergodicity of the flow it fails to detect the mixing properties.From now on, and if otherwise stated, we shall fix an f -invariant ergodic prob-ability measure µ and let ¯ µ denote the corresponding flow-invariant ergodic prob-ability measure. We start by recalling Kac’s qualitative recurrence estimates formeasurable maps. Theorem 2.1 (Kac’s lemma) . Let f : Σ → Σ be a measurable map preserving aprobability measure µ . For any measurable set A with µ ( A ) > the hitting time n A ( · ) : Σ → N defined by n A ( x ) = inf { k ≥ f k ( x ) ∈ A } satisfies Z A n A ( x ) dµ = 1 Equivalently, R A n A ( x ) dµ A = µ ( A ) is inversely proportional to the measure of A . In the end of the sixties, Helmberg [10] studied recurrence for arbitrary setssatisfying a ’boundary condition’ with respect to measurable and continuous flows.In general, if a set A is such that the boundary ∂A has positive ¯ µ -measure andadmits a fractal structure then it was not clear how to define properly returntimes to A . To overcome this difficulty Helmberg paper defined return times using(measurable) families of exit and entrance regions: for any s > exitregion A s by A s = { z ∈ M : ∃ ≤ ℓ < r ≤ s so that X ℓ ( z ) ∈ A and X r ( z ) / ∈ A } and analogousy the entrance region ˜ A s by ˜ A s = { z ∈ M : ∃ ≤ ℓ < r ≤ s so that X ℓ ( z ) / ∈ A and X r ( z ) ∈ A } . These are cleary decreasing families as s tends to zero. In this case, the return time is defined by r A ( z ) = inf { s > z ∈ ˜ A s } and it is proved to be almost everywhere well defined. More precisely, Theorem 2.2. [10, Theorem 5]
Let ( X t ) t ≥ be a semiflow of measurable transfor-mations on a probability space ( M, ¯ µ ) . If A ⊂ M is a positive ¯ µ -measure set and t > is such that lim n →∞ ¯ µ ( A tn ) = 0 then lim s → s Z A s r A ( z ) d ¯ µ = ¯ µ (cid:18) [ r ≥ X − r ( A ) (cid:19) − ¯ µ ( A ) . In particular, if ¯ µ is ergodic then the right hand side is equal to − ¯ µ ( A ) . The limit in the previous theorem follows by the strategy used by Helmberg ofapproximating return times for the flow by studying return times for individualtime- s maps T s for small s >
0. While the result holds under big generality e.g. forall positive measure sets A so that ¯ µ ( ∂A ) = 0 it provides estimate on the limitingmean return time function on the exit components A s , that is, on points in theeminence of leaving the set A .In our time-continuous setting of suspension flows and geometrical objects likecylinders or balls whose boundary will have always always zero. We will computethe mean return time among points in the whole set A . Taking into account the ormalization given by equation (2.2) it is a natural question to understand inwhich sense the mean return time for suspension flows reflects its dependence onthe integral of the roof function. More precisely, given a set A if one makes areparametrization of the suspension flow leaving the set A invariant it is interestingto understand if the mean return times for both flows coincide or, if not, if theycan be related to each other. Moreover, we shall address some questions on positivemeasures sets on the cross section Σ, which have zero ¯ µ -measure on the flow whileare dynamically significant and contain non-trivial recurrence.Let us introduce some necessary notions. Given a set A ⊂ Σ τ define the escapetime e A ( · , · ) : Σ τ → R +0 to be given by e A ( x, t ) = inf { s > X s ( x, t ) A } for any ( x, t ) ∈ Σ τ . The escape time e A ( x, t ) is clearly zero for any point ( x, t )in the complement of A . We define also the hitting time to A as the function n A : Σ τ → R +0 given by n A ( x, t ) = inf { s > e A ( x, t ) : X s ( x, t ) ∈ A } . (2.3)The return time function to A consists of the restriction of n A ( · , · ) to the set A and,clearly, n A ( x, t ) ≥ e A ( x, t ) for every ( x, t ) ∈ A . Alternatively, we can define thereturn time ˜ n A ( x, t ) = inf { s > X s + e A ( x,t ) ( x, t ) ∈ A } , in which case the functionsverify n A = e A + ˜ n A and consequently to R A n A d ¯ µ and R A ˜ n A d ¯ µ just differ by theaverage escaping time R A e A d ¯ µ . For that reason we shall restrict to the study ofthe previously defined hitting time function. Our first result is a version of Kac’slemma for returns to cylinder sets. Theorem A.
Let f : Σ → Σ be a measurable invertible map on a topological space Σ preserving an ergodic probability measure µ and let τ : Σ → R +0 be a µ -integrableroof function. Denote by ( X t ) t the suspension semiflow over ( f, µ, τ ) and let ¯ µ be the corresponding ( X t ) t invariant probability measure given by (2.2) . Given apositive ¯ µ -measure cylinder set A = I × [ t , t ] ⊂ Σ × R + in Σ τ then Z A n A ( x, t ) d ¯ µ A = Z A e A ( x, t ) d ¯ µ A + (cid:0) − ¯ µ ( A ) (cid:1) µ ( I ) Z Σ τ dµ = m ([ t , t ])2 + 1 µ ( I ) Z Σ (cid:18) τ − ( t − t ) χ I (cid:19) dµ In particular, if the set A is kept unchanged and we change the roof function τ themean return time R A n A ( x, t ) dµ A varies lineary with respect to the mean of the rooffunction R Σ τ dµ .Proof. Let n I ( · ) be the first return time function to I by f . Given ( x, t ) ∈ Σ τ and s ≥ X s ( x, t ) = ( f k ( x ) , t + s − P k − j =0 τ ( f j ( x )))where k = k ( x, t, s ) ≥ P k − j =0 τ ( f j ( x )) ≤ t + s < P kj =0 τ ( f j ( x )) . Observe that if X s ( x, t ) ∈ A for s > e A ( x, t ) and k = k ( x, t, s ) is defined as abovethen necessarily f k ( x ) ∈ I . Reciprocally, if ( x, t ) ∈ Σ τ and f k ( x ) ∈ I then thereexists s < P kj =0 τ ( f j ( x )) − t such that X s ( x, t ) ∈ A . Now, since we are dealing ith a cylinder A = I × [ t , t ] then e A ( x, t ) = t − t and so n A ( x, t ) = (cid:2) n I ( x ) − X j =0 τ ( f j ( x )) − t (cid:3) + t = n I ( x ) − X j =0 τ ( f j ( x )) + e A ( x, t ) − m ([ t , t ]) (2.4)for any ( x, t ) ∈ A . Given n ≥ I n = { x ∈ I : n I ( x ) = n } and it is clear that I = ∪ I n modulo a zero measure set with respect to µ . For0 ≤ k ≤ n − I n,k = f k ( I n ). Since all points in I n,k have n − k as firsthitting time to A and f is invertible then f − k ( I n,k ) = I n and, consequently, by f -invariance of the measure it holds µ ( I n,k ) = µ ( I n ). Since µ -almost every x willeventually visit the set I ⊂ Σ, µ is ergodic and I n,k is family of disjoint sets then µ ( ∪ n ∪ ≤ k ≤ n − I n,k ) = 1. Hence, Z Σ τ dµ = X n ≥ n − X k =0 Z I n,k τ dµ = X n ≥ n − X k =0 Z I n τ ◦ f k dµ = X n ≥ Z I n n − X k =0 τ ◦ f k dµ = Z I n I ( x ) − X k =0 τ ◦ f k dµ. (2.5)In consequence using equations (2.4) and (2.5) together it follows that Z A n A ( x, t ) d ¯ µ = 1 R Σ τ dµ Z t t Z I h n I ( x ) − X j =0 τ ◦ f j i dµ dt + Z A e A ( x, t ) d ¯ µ − m ([ t , t ]) ¯ µ ( A )or, in other words, Z A n A ( x, t ) d ¯ µ = Z A e A ( x, t ) d ¯ µ + m ([ t , t ]) (cid:0) − ¯ µ ( A ) (cid:1) . (2.6)If ¯ µ A denotes as before the normalized probability measure ¯ µ | A ( · )¯ µ ( A ) the later becomes Z A n A ( x, t ) d ¯ µ A = Z A e A ( x, t ) d ¯ µ A + (cid:0) − ¯ µ ( A ) (cid:1) µ ( I ) Z Σ τ dµ (2.7)which proves the first equality in the theorem. A simple integral computation showsthat R t t ( t − t ) dt = m ([ t , t ]) / R A e A ( x, t ) d ¯ µ = ¯ µ ( A ) m ([ t ,t ])2 . So R A e A ( x, t ) d ¯ µ A = m ([ t ,t ])2 and replacing ¯ µ ( A ) = µ ( I ) R τ dµ m ([ t , t ]) in equation (2.7)we conclude Z A n A ( x, t ) d ¯ µ A = m ([ t , t ])2 + 1 µ ( I ) Z Σ (cid:18) τ − ( t − t ) χ I (cid:19) dµ where χ I denotes the indicator function of the set I ⊂ Σ. This proves the secondequality in the theorem. In particular, it follows from the previous expression thatif the set A is kept unchanged the mean return time varies linearly with R Σ τ dµ .This finishes the proof of the theorem. (cid:3) ome comments are in order. The first one concerns the ergodicity assumption.It is a simple consequence of Birkhoff’s ergodic theorem and the integrability ofthe roof function that ¯ µ is ergodic for the flow if and only if µ is ergodic for f . Ifergodicity fails it follows from the ergodic decomposition theorem that for µ almostevery x ∈ Σ there are ergodic probability measures µ x on Σ so that µ = R µ x dµ .In such case ¯ µ = R ¯ µ x d ¯ µ is the ergodic decompositon for ¯ µ , where ¯ µ x are thealmost everywhere defined ergodic measures ¯ µ x = ( µ x × m ) / R τ dµ x . Using this werecover analogous expression for the mean return time as in Theorem A withoutthe ergodicity assumption.A second remark is related with the heigh of the cylinders A . Indeed, the roleof the constants t and t was not important to guarantee recurrence to the set A since in our setting Σ is a global cross-section and recurrence is obtained simplyby assuming the projection I has positive µ -measure. Hence, as a consequenceof our strategy we deduce the following recurrence estimates for subsets of thecross-section. Corollary 1.
Let f : Σ → Σ be a measurable invertible map on a topological space Σ , µ be an f -invariant ergodic probability measure and let τ ∈ L ( µ ) be a rooffunction, and let ( X t ) t be the suspension semiflow over preserving ¯ µ . Then, forany positive µ -measure set A ⊂ Σ , Z A n A ( x, dµ A = 1 µ ( A ) Z Σ τ dµ where µ A of the normalization of the measure µ | A in the cross-section Σ . Inparticular, if f A : A → A denotes the first return time map then Z A n A ( x, dµ A = h µ A ( f A ) h ¯ µ ( X ) is the entropy of the first return time map quotiented by the entropy of the flow.Proof. For simplicity reasons we shall denote also by A the set A × { } ⊂ Σ τ . it isclear that e A ( · , · ) ≡
0. Hence It follows e.g. from equation (2.4) and (2.5) (taking I = A ) that n A ( x,
0) = P n A ( x ) − j =0 τ ( f j ( x )) and we deduce that R A n A ( x, dµ = R Σ τ dµ, from which expression the result immediately follows.In addition, on the one hand Abramov [1] proved that the entropy h µ A ( f A ) ofthe first return time map f A with respect to µ A satisfies h µ ( f ) = h µ A ( f A ) µ ( A ) . Onthe other hand, the formula established for the time- t map of the flow with relationwith the base map obtained by Abramov [2] h ¯ µ ( X t ) = | t | h µ ( f ) R Σ τ dµ and so Z A n A ( x, dµ A = 1 µ ( A ) Z Σ τ dµ = h µ A ( f A ) h µ ( f ) h µ ( f ) h ¯ µ ( X ) = h µ A ( f A ) h ¯ µ ( X ) . This finishes the proof of the corollary. (cid:3)
Remark . Let us mention that the assumption of Theorem 2.2 is satisfied in thecase of suspension flows and cylinder sets. In fact, if A = I × [ t , t ] ⊂ Σ τ and s > A s = I × [ t − s, t ] and ˜ A s = I × { t } . In particular ¯ µ ( A s ) → s ends to zero. Since A s ⊂ A then the return time r A defined in [10] coincides withthe definition given in equation 2.3 and it follows thatlim s → s Z A s r A ( z ) d ¯ µ = 1 − ¯ µ ( A ) . Now, let us observe that the case of flows cannot be obtained directly fromthe discrete time setting since it reflects the escaping times and the width of thecylinders as shown by the following immediate consequence of the theorem, in whichwe also use the mean return times to compute entropy of a measure in the crosssection.
Corollary 2.4.
Let (( X t ) t , ¯ µ ) be the suspension semiflow over ( f, µ, τ ) as in The-orem A and assume that the roof function τ is constant. If A = I × [0 , τ ] ⊂ Σ × R is a full cylinder set in Σ τ with positive ¯ µ -measure then Z A n A ( x, t ) d ¯ µ A = 1 µ ( I ) Z Σ (cid:18) τ − τ χ I (cid:19) dµ = τµ ( I ) (cid:18) − µ ( I )2 (cid:19) . Despite the fact that cylinder sets arise naturally for flows (e.g. from the tubularneighborhood theorem) sometimes we are interested in the return times to geometricobjects as balls which can be used to study dimension of measures among otherrelevant dynamical quantities. For that reason, we will now extend our result fora more general class of sets that include balls. Let A be the family of closed setswhose boundary are graphs of functions over Σ. More precisely, A ∈ A if and onlyif there are measurable functions h , h : π ( A ) → R +0 such that A = { ( x, t ) ∈ Σ τ : h ( x ) ≤ t ≤ h ( x ) } , where π : Σ τ → Σ denotes the natural projection on Σ. Forsimplicity we shall assume h ( x ) ≤ h ( x ) ≤ τ ( x ) for every x ∈ Σ ands denote suchsets by A h ,h . Just as a remark, for the purpose of recurrence properties to thesesets the return times for the flow coincide with the ones if one had considered setsof the form A = { ( x, t ) ∈ Σ τ : h ( x ) < t < h ( x ) } . We can now state our nextresult. Theorem B.
Let ( X t ) t be a suspension semiflow associated to ( f, Σ , τ ) and let ¯ µ be the ergodic ( X t ) t -invariant probability measure associated to the f -invariantergodic probability measure µ . Given A = A h ,h ∈ A with ¯ µ -positive measure then Z A n A ( x, t ) d ¯ µ A = Z A e A ( x, t ) d ¯ µ A + R I h ( x ) τ n I ( x ) dµ R I h ( x ) dµ + R I h ( x )[ h ( f n I ( x ) ( x )) − h ( x )] dµ R I h ( x ) dµ where h ( x ) = h ( x ) − h ( x ) , τ n I ( x ) := P n I ( x ) − k =0 τ ◦ f k ( x ) and also Z A e A ( x, t ) d ¯ µ A = R I h ( x ) (cid:2) h ( x ) − h ( x )2 (cid:3) dµ R I h ( x ) dµ . Proof.
Set A = A h ,h and I = π ( A ) for simplicity. Given ( x, t ) ∈ A then clearlythe escape time is given by e A ( x, t ) = h ( x ) − t . Moreover, if there exists s ≥ X s ( x, t ) = (cid:0) f k ( x ) , t + s − P k − j =0 τ ( f j ( x )) (cid:1) ∈ A then f k ( x ) ∈ I and k = k ( x, t, s ) ≥ or ¯ µ -almost every ( x, t ) ∈ A we get n A ( x, t ) = n I ( x ) − X j =0 τ ( f j ( x )) + e A ( x, t ) − h ( x ) + h ( f n I ( x ) ( x )) , we write I = ∪ I n (mod µ ) and notice, by ergodicity, the family I n,k = f k ( I n ) pair-wise of disjoint sets verifies µ ( ∪ n ∪ ≤ k ≤ n − I n,k ) = 1. Hence, a simple computationshows that ¯ µ ( A ) = 1 R τ dµ Z I h ( x ) dµ and also Z A n A ( x, t ) d ¯ µ = 1 R τ dµ Z I h ( x ) τ n I ( x ) dµ + 1 R τ dµ Z I Z h ( x ) h ( x ) [ h ( x ) − t ] dt dµ + 1 R τ dµ Z I Z h ( x ) h ( x ) [ h ( f n I ( x ) ( x )) − h ( x )] dt dµ and the result follows by simple computations. (cid:3) Final remarks ’Almost’ cylinders.
The first remark is that computations are above clearly simplerin the case where A h ,h has ’parallel sides’ meaning h ( x ) = h ( x ) + c for some c >
0. In this case, using again the notation I = π ( A ) and noticing h ( x ) = c forevery x ∈ I it follows that ¯ µ ( A ) = c µ ( I ) / R τ dµ , that R A e A ( x, t ) d ¯ µ A = c and,using that the first return time map f n I : I → I preserves the (ergodic) normalizedprobability measure µ I Z A n A ( x, t ) d ¯ µ A = c µ ( I ) Z Σ τ dµ + 1 µ ( I ) Z I [ h ( f n I ( x ) ( x )) − h ( x ) − c ] dµ = c µ ( I ) Z Σ ( τ − cχ I ) dµ + Z I [ h ( f n I ( x ) ( x )) − h ( x )] dµ I = c µ ( I ) Z Σ ( τ − cχ I ) dµ Suspension semiflows.
We should mention that our results hold for suspension semi-flows ( X t ) t ≥ associated to the suspension for non-invertible maps f : Σ → Σ.Given an ergodic measure µ , for I ⊂ Σ we can decomposeΣ = [ n ≥ [ I n ∪ I ∗ n ] ( µ Σ mod 0)where I n = I ∩ { n I ( · ) = n } and I ∗ n = (Σ \ I ) ∩ { n I ( · ) = n } , and all elements inthe previous union are pairwise disjoint. Moreover, for every n ≥ µ ( I ∗ n ) = µ ( f − ( I ∗ n )) = µ ( I n +1 ) + µ ( I ∗ n +1 ) and consequently µ ( I ∗ n ) = P k ≥ n +1 µ ( I k ) and so1 = µ (Σ) = P k ≥ kµ ( I k ) = R n I ( · ) dµ . Taking I n,j = f j ( I n ) for 0 ≤ j ≤ n − ince I n,j ⊂ I ∗ n − j , it follows that µ (cid:0) [ n ≥ n − [ j =0 I n,j (cid:1) = µ (cid:0) [ n ≥ [ j ≥ I n + j,j (cid:1) = X n ≥ (cid:2) µ (cid:0) I n, (cid:1) + µ (cid:0) [ j ≥ I n + j,j (cid:1)(cid:3) = X n ≥ (cid:2) µ ( I n ) + µ ( I ∗ n ) (cid:3) = 1 . Thus, as in equation (2.5) and obtain R Σ τ dµ = R I P n I ( x ) − k =0 τ ◦ f k dµ. and thesame computations as in the proof of Theorems A and B follow straightforward. Acknowledgements.
This work was supported by a CNPq-Brazil postdoctoralfellowship at University of Porto. The author is grateful to M. Bessa, F. Rodriguesand J. Rousseau for some comments on the manuscript. The author is deeplygrateful to J.-R. Chazottes for his comments and for providing the reference [10].The author is also grateful to the organizers of the event “Probability in Dynamicsat UFRJ” where part of this work was developed.
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