Quenched decay of correlations for one dimensional random Lorenz maps
aa r X i v : . [ m a t h . D S ] F e b QUENCHED DECAY OF CORRELATIONS FOR ONE DIMENSIONALRANDOM LORENZ MAPS
ANDREW LARKIN
Abstract.
We study rates of mixing for small random perturbations of one dimensionalLorenz maps. Using a random tower construction, we prove that, for H¨older observables,the random system admits exponential rates of quenched correlation decay. Introduction
In 1963, Lorenz [16] introduced the following system of equations˙ x = − x + 10 y, ˙ y = 28 x − y − xz, ˙ z = − z + xy (1)as a simplified model for atmospheric convection. Numerical simulations performed byLorenz showed that the above system exhibits sensitive dependence on initial conditionsand has a non-periodic “strange” attractor. Since then, (1) became a basic example of achaotic deterministic system that is notoriously difficult to analyse. We refer to [3] for athorough account on this topic.It is now well known that a useful technique to analyse the statistical properties ofsuch a flow, and any nearby flow in the C -topology, is to study the dynamics of the flowrestricted to a Poincar´e section, which includes the equilibrium point, via a well definedPoincar´e map [4]. Such a Poincar´e map admits an invariant stable foliation; moreover,it is strictly uniform contracting along stable leaves [4]. Therefore, the dynamics of thePoincar´e map can be understood via quotienting along stable leaves; i.e., by studying thedynamics of its one dimensional quotient map along unstable leaves. The above techniquehave been employed to obtain statistical properties of Lorenz flows [14] and to prove thatsuch statistical properties of this family of flows is stable under deterministic perturbations[2, 7, 6]. This illustrates the importance of understanding the statistical properties of onedimensional Lorenz maps.In this paper, we study rates of mixing for small random perturbations of such one di-mensional maps. We use a random tower construction to prove that for H¨older observablesthe random system admits exponential rates of quenched correlation decay. The paper isorganised as follows. In section 2 we present the setup and introduce the random systemunder consideration. Section 2 also includes the main result of the paper, Theorem 2.1.Section 3 includes the proof of Theorem 2.1. Section 4 is an Appendix, which contains a Date : February 26, 2021.2020
Mathematics Subject Classification.
Primary 37A05, 37C10, 37E05.
Key words and phrases.
Lorenz attractors, random dynamical systems, quenched correlation decay. version of the abstract random tower result of [8, 13], which is used to deduce Theorem2.1.
Acknowledgment : I would like to thank Marks Ruziboev for our helpful discussionsthroughout this work. 2.
Setup
The unperturbed system.
We assume that the following conditions hold.(A1) T : I → I , I = [ − , ], is C on I \ { } with a singularity at 0 and one-sidedlimits T (0 + ) < T (0 − ) > T is piecewise monotone increasing on I \ { } .Furthermore, T is uniformly expanding, i.e. there are constants ˜ C > ℓ > | DT n ( x ) | > ˜ Ce nℓ for all n ≥ x / ∈ S n − j =0 T − j (0);(A2) There exists C > < λ < such that in a one sided neighborhood of 0 C − | x | λ − ≤ | DT ( x ) | ≤ C | x | λ − . (2)Moreover, 1 /DT is H¨older on [ − ,
0] and [0 , ] with exponent α ∈ (0 , T is transitive (for the construction, we use that pre-images of 0 is dense in I ).Notice that (A2) implies that DT is locally H¨older for any exponent 0 < ˜ α ≤ min { α, − λ } ,i.e. there exists a constant K > x, y ∈ I we have | DT ( x ) − DT ( y ) | < K | x − y | ˜ α | x | ˜ α | y | ˜ α . (3)2.2. The random system.
Let T : I → I be the map introduced in subsection 2.1.Also, let λ, ¯ λ ∈ (0 , ) satisfying λ < ¯ λ be fixed. We consider a family of Lorenz maps A ( ε ) = { T λ : I → I | λ ∈ [ λ, ¯ λ ] } such that (i) every T λ ∈ A ( ε ) is ε -close to a T in asuitable C α sense , (ii) every T λ ∈ A ( ε ) satisfies (A2) with λ ∈ [ λ, ¯ λ ] and thus thereexists α > T λ ∈ A ( ε ) is locally H¨older with exponent α . We fix such an α from here on.Let P be a Borel probability measure on A ( ε ), and let Ω = A ( ε ) Z . Additionally, let P = P Z , and let σ : Ω → Ω be the left shift map, i.e. ω ′ = σ ( ω ) if and only if ω ′ i = ω i +1 for i ∈ Z . Then σ is an invertible map that preserves P . Notice that ω denotes a bi-infinitesequence of maps from A ( ε ). To simplify the notation, we denote by T ω = T ω the zerothelement of ω and denote by λ ω the order of the singularity corresponding to T ω . We assumeuniform expansion on random orbits: for almost every ω ∈ Ω, | DT nω ( x ) | > ˜ Ce nℓ for all n ≥ , for some ˜ C > , ℓ > , (4)whenever x / ∈ S n − j =0 T − jω (0).We express the random dynamics of our system in terms of the skew product S : Ω × I → Ω × I, For example, T and T λ could be close in the usual C sense outside some neighbourhood of 0 as in[7]. UENCHED DECAY OF CORRELATIONS FOR ONE DIMENSIONAL RANDOM LORENZ MAPS 3 where S ( x, ω ) = ( σ ( ω ) , T ω ( x )) . Iterates of S are defined naturally as S n ( x, ω ) = ( σ n ( ω ) , T nω ( x )) , T nω ( x ) = T ω n − ◦ · · · ◦ T ω ( x ) . Here we are looking at the quenched statistical properties of S , i.e. we study statisticalproperties of the system generated by the compositions T nω on I for almost every ω ∈ Ω,which we refer to as T ω without confusion, since the underlying driving process (Ω , σ, P ) isfixed. We call a family of Borel probability measures { µ ω } ω ∈ Ω on I equivariant if ω µ ω is measurable and T ω ∗ µ ω = µ σω for P almost all ω ∈ Ω . The following theorem is the main result of the the paper.
Theorem 2.1.
The random system { T ω } admits a unique equivariant family of absolutelycontinuous measures { µ ω } for almost every ω ∈ Ω . Moreover, there exists a constant ρ ∈ (0 , such that for every ϕ ∈ C η ( I ) , η ∈ (0 , and ψ ∈ L ∞ ( I ) , we have (cid:12)(cid:12)(cid:12)(cid:12)Z ( ϕ ◦ T nω ) ψdµ ω − Z ϕdµ σ n ω Z ψdµ ω (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ϕ,ψ ρ n and (cid:12)(cid:12)(cid:12) Z ( ϕ ◦ T nω ) ψdµ σ − n ω − Z ϕdµ ω Z ψdµ σ − n ω (cid:12)(cid:12)(cid:12) ≤ C ϕ,ψ ρ n , for some constant C ϕ,ψ > . Proof of theorem 2.1
Strategy of the proof.
We prove Theorem 2.1 by showing that the random system { T ω } admits a Random Young Tower structure [8, 13] for every ω ∈ Ω with exponentialdecay for the tail of the return times. Thus, uniform (in ω ) rates leads to uniform (in ω )exponential decay of correlations. For convenience, we include the random tower theoremof [8, 13] in the Appendix, which we will refer to in the proof of Theorem 2.1. Similarto the work of [11, 12] in the deterministic setting, to construct a random tower, we firstconstruct an auxiliary stopping time called escaping time E : J → N on subintervals of I . Roughly speaking, E is a moment of time when the image of a small interval J ′ ⊂ J reaches a fixed interval length δ > T Eω ( J ′ ) does not intersect a fixedneighborhood ∆ of the singularity 0. We show that T Eω has good distortion bounds andthat Leb { E > n } , where Leb denotes Lebesgue measure, decays exponentially fast. Thenext step will be to construct a full return partition Q ω onto some small neighborhood∆ ∗ ⊂ ∆ of the singularity. Once this is obtained, we induce a random Gibbs-Markovmap F ω in the sense of Definition 1.2.1 of [13]. This means we will need to show that F ω : ∆ ∗ → ∆ ∗ , defined as F ω ( x ) = T τ ω ( x ) ω ( x ), where τ ω : ∆ ∗ → N a measurable return timefunction, has the following properties: • τ ω | Q is constant for every Q ∈ Q ω and every ω ∈ Ω. ANDREW LARKIN • F ω | Q : Q → ∆ ∗ is uniformly expanding diffeomorphism with bounded distortion forevery Q ∈ Q ω and every ω ∈ Ω. • |{ τ ω > n }| ≤ Be − bn for some constants B > b ∈ (0 , • there exists N ∈ N and two sequence { t i ∈ N , i = 1 , , . . . , N } , { ε i > , i =1 , . . . , N } such that g.c.d. { t i } = 1 and |{ x ∈ ∆ ∗ | τ ω ( x ) = t i }| > ε i for almost every ω ∈ Ω.3.2.
Escape time and partition.
Let r < r ∗ be sufficiently large numbers (we willchoose them later). Define δ ∗ = e − r ∗ and δ = e − r . Moreover, let ∆ = ( − δ, δ ) and∆ ∗ = ( − δ ∗ , δ ∗ ). Consider an exponential partition P = { I r } r ∈ Z of ∆ as in [9], where I r = ( e − r , e − ( r − ] , I − r = ( − e − ( r − , − e − r ] for r ≥ r and for | r | < r we set I r = ∅ . Furthermore, fix ϑ = (cid:2) α (cid:3) + 1, where α is the same as in2.2. We divide every I r into r ϑ equal parts. Let I r,m denote one of the small intervals, m = 1 , . . . , r ϑ . We use the usual notation x ω,i = T iω ( x ), J i,ω = T iω ( J ) for i ≥ J ⊂ I .Let J ⊂ I be an interval such that either J ∩ ∆ = ∅ and | J | ≥ δ/
5, or J = ∆ .We construct a stopping time E ω : J → N called escape time and a partition P ω ( J ) of J such that E ω | J ω is constant for every J ω ∈ P ω ( J ) and | T E ω ( J ω ) | ≥ δ (for simplicityof notation, from this point onwards we will denote J = J ω ).We call T E ω ( J ) an escapinginterval (or component).The construction is inductive and we construct E ω and P ω simultaneously. We have thefollowing proposition. Proposition 3.1.
Let δ > be a small constant. Consider δ ∗ ∈ (0 , δ ) , and considerintervals J ⊂ I such that either J = ∆ or J ∩ ∆ = ∅ and | J | ≥ δ/ . Then forevery ω ∈ Ω there exists an escape time E ω : J → N and a partition P ω ( J ) such that E ω is constant on the elements of P ω ( J ) . For every J ∈ P ω ( J ) and n ≤ E ω ( J ) such that T E ω ω ( J ) ∩ ∆ = ∅ , the image T E ω ω ( J ) does not intersect more than adjacent three intervalsof the form I m,r , r ≥ r , m = 1 , . . . , r ϑ . Moreover, there exist constants C δ > , γ > such that |{ E ω ( J ) ≥ n : J ∈ P ω ( J ) }| ≤ C δ e − γn | J | . We prove the proposition above in a series of lemmas. It is important to keep distortionunder control, which means that we have to keep track of visits of the orbits near thecritical point. Therefore, we use a chopping algorithm following [9].Let k ≥ J ∗ ∈ P ω ( J ) satisfying E ( J ∗ ) < n have alreadybeen constructed. Let J be an interval in the complement of { E ω ( J ∗ ) < n } . Fix ω ∈ Ωand consider the following cases depending on the position and length of J k,ω = T kω ( J ), k ∈ N . Non-chopping intervals.
Suppose that | J k,ω | < δ , ∆ ∩ J k,ω = ∅ and J k,ω does notintersect more than three adjacent intervals I r,m . Then k is called an inessential returntime for J . Define return depth as r ω = max {| r | : T kω J ∩ I r } . Notice that r ω depends onlyon ω , . . . , ω k − . UENCHED DECAY OF CORRELATIONS FOR ONE DIMENSIONAL RANDOM LORENZ MAPS 5
Chopping times.
Suppose that | J k,ω | < δ , ∆ ∩ J k,ω = ∅ and J k,ω intersects more thanthree adjacent intervals I r,m . Then k is called an essential return time for J . In this casewe chop J into pieces J r,j,ω ⊂ J in such that I r,j ⊂ T kω ( J r,j,ω ) ⊂ ˆ I r,j , where ˆ I r,j is the union of I r,j and its two neighbors. We say that J r,j,ω has the associatedreturn depth r ω .If J k,ω ∩ ∆ = ∅ , we call k free iterate. Definition 3.2 ( Escape times ) . Suppose that k is free iterate for J k,ω and | J k,ω | ≥ δ .We then set E ω ( J ) = k and add J to P ωm ( J ) for all m ≥ k . We call J k,ω escape interval.We let P ω ( J ) = W ∞ m =1 P ωm ( J ) . We notice that this partition depends on ω . The proof closely follows the approach of [11]. Suppose that J ∈ P ω ( J ), and supposethat J had s returns before escaping. We define the total return depth as R ω ( J ) = P si =1 r i,ω ( J ). Lemma 3.3.
There exists ˆ λ > ¯ λ depending on δ such that for any J ∈ P ω ( J ) | J | ≤ e − (1 − ˆ λ ) R ω ( J ) . Proof.
Suppose that R ω ( J ) = 0 and let d , d , ..., d s − be the free iterates between returns.Then E ( J ) = d + 1 + · · · + d s + 1. We have | T E ω ω ( J ) | = | T d +1+ ··· + d s +1 ω ( J ) | = | D ( T d +1+ ··· + d s +1 ω )( ξ ) | · | J | , for some ξ ∈ J . For every i = 1 , . . . . , s , x ∈ I and ω ∈ Ω, we have DT ν i ω ( x ) ≥ ˜ Ce ℓν i and DT ω ( T ν i ω ( x )) ≥ C − e (1 − ¯ λ ) r i,ω . Using the chain rule and the above two inequalities, we have | ( T d +1+ ··· + d s +1 ω ) ′ ( ξ ) | ≥ ˜ C s +1 e P si =0 d i C − s e (1 − ¯ λ ) P r i,ω > ˆ C s e (1 − ¯ λ ) R ω = e (1 − ¯ λ − sRω log ˆ C ) R ω . where ˆ C = min { C − , ˜ C ( s +1) /s } . Recall that r i,ω ≤ r , where e − r = δ . Thus R ω ≥ sr ,which implies 1 − ¯ λ − sR ω log ˆ C ≤ − ¯ λ − log ˆ C log δ − . We choose δ small enough such thatˆ λ = ¯ λ + log ˆ C log δ − < . Thus, we have1 ≥ | T E ω ω ( J ) | > e (1 − ˆ λ ) R ω | J | which finishes the proof. (cid:3) The above lemma allows us to prove the following exponential estimate.
Lemma 3.4.
The tail of the total return depth decays exponentially fast. Let J ⊂ I and R ω : J → N be the escape time for all ω ∈ Ω . Then we have |{ R ω ( J ) = k | J ∈ P ω ( J ) }| ≤ e − (1 − λ ) k . (5) and if J ≥ δ/ then |{ R ω ( J ) = k | J ∈ P ω ( J ) }| ≤ δ − e − (1 − λ ) | J | . (6) ANDREW LARKIN
Proof.
Recall that for every ω ∈ Ω and every sequence of return depths ( r , . . . , r s ) there areat most two escaping intervals. Thus number of escaping intervals N k,s having r + · · · + r s = k for sufficiently large r satisfies the following [10, Lemma 3.4] N k,s ≤ e ˆ λR ω . Combining these and Lemma 3.3 we have X J ∈P ω ( J ) R ω ( J )= k | J | ≤ e − (1 − λ ) k < δ − e − (1 − λ ) k | J | , where in the last inequality we have used | J | ≥ δ/ (cid:3) The following lemma relates the total return depth and the escape time.
Lemma 3.5.
Let J ∈ P ω ( J ) and let ( r , . . . , r s ) be its associated return depth. Also let R ω = P i | r i | . Then E ω ( J ) ≤ ℓℓ R ω ( J ) Proof.
Since intervals are not chopped at the inessential return times, we distinguish be-tween essential and inessential return times. Thus, let R ω ( J ) = R eω ( J ) + R ieω ( J ) be thecorresponding splitting into essential and inessential return times respectively. Also, define D e ( J ) = d e + . . . d eq , where d ei is the number of free iterates before essential return time.We have d ei ≥ d i , for all i = 0 , . . . q and D e ( J ) = d + · · · + d s . Thus, q ≤ s . Now, let J ( i − be an ancestor of J of order i −
1. Observe that | J ( i − | ≥ e − ( r i − − e − r i r ϑi ≥ e − ( r i − r ϑi . Therefore, at the next return time we have1 ≥ | T ν i +1 ω ( J ( i ) ) | ≥ | J ( i ) | · | DT ω ′ T d ei ( ξ ) | · | D ( T d ei ω )( ξ ) | ≥ | J ( i ) | e ℓd ei +(1 − ¯ λ ) r i ≥ e − r i r ϑi e ℓd ei +(1 − ¯ λ ) r i . Simplifying this inequality yields d ei ≤ r i ℓ for i = 1 , . . . s for sufficiently large r . A similarestimate also holds for d . Thus using the R ω ≥ s we obtain E ω ( J ) = d + q X i =0 d ei + s ≤ s X i =0 r i ℓ + R ω = 1 ℓ R ω + R ω = 1 + ℓℓ R ω . (cid:3) Finally, we are ready to prove Proposition 3.1.
Proof of Proposition 3.1.
By the above lemma, for every J in the sum, we have that n ≤ E ω ( J ) ≤ ℓℓ R ω . Thus, R ω ≥ nℓ ℓ . Combining this with lemma 3.4 we have that X J ∈P ω ( J ) E ω ( J ) ≥ n | J | ≤ X R ω ≥ nℓ ℓ | J | ≤ ∞ X j = nℓ ℓ δ − e − (1 − λ ) j | J | ≤ C δ e − (1 − λ ) nℓ ℓ | J | = C δ e γn | J | . UENCHED DECAY OF CORRELATIONS FOR ONE DIMENSIONAL RANDOM LORENZ MAPS 7
Notice that we need to fix δ >
0, since C δ = O ( δ − ) . Thus, P ω ( J ) given in definition 3.2 defines a partition of J and every element of it isassigned an escape time. Notice that the way we constructed the escape time immediatelyimplies that for every J ∈ P ω ( I ) satisfying n ≤ E ω ( J ) such that T E ω ω ( J ) ∩ ∆ = ∅ ,the image T E ω ω ( J ) does not intersect more than three adjacent intervals of the form I r,m , | r | ≥ r , m = 1 , . . . , r ϑ . (cid:3) Bounded Distortion.
In this subsection we prove that T nω has bounded distortion onevery interval J such that T kω ( J ) does not intersect more than three adjacent intervals I r,m for all 1 ≤ k ≤ n . Therefore, the escape map in Proposition 3.1 has bounded distortion.We prove the following lemma: Lemma 3.6.
For every ω ∈ Ω , let T ω ∈ A ( ǫ ) . Furthermore, let ∆ = ( − δ, δ ) be aneighborhood of 0 and P be the exponential partition of ∆ . For every interval J ⊂ I forwhich there exists n J ∈ N such that no ≤ k ≤ n J − is an essential return time, for a.e. ω we have max x,y ∈ J log | DT kω ( x ) || DT kω ( y ) | ≤ D , ≤ k ≤ n J (7) Proof.
Let ω ∈ Ω. If we set x ω,i = T iω ( x ), then by using the chain rule and log(1 + x ) ≤ x for x ≥ | DT kω ( x ) || DT kω ( y ) | = log k − Y i =0 | DT ω ( x ω,i ) || DT ω ( y ω,i ) |≤ k − X i =0 | DT ω ( x ω,i ) − DT ω ( y ω,i ) || DT ω ( y ω,i ) | . (8)From (2) we can infer | DT ω ( y ω,i ) | ≥ C − | y ω,i | ˆ λ − > C − | y ω,i | − α , thus, combining this with(3) and letting κ = KC , we obtainlog | DT kω ( x ) || DT kω ( y ) | < k − X i =0 K | x ω,i − y ω,i | α | x ω,i | α | y ω,i | α · C − | y ω,i | − α < κ k − X i =0 | x ω,i − y ω,i | α | x ω,i | α . Let J = [ x, y ] ⊂ I , and let K = { ν < ν < ... < ν i < ... < ν p } be the ordered set of allinessential returns under the dynamics of T kω on J . The largest possible size for T ν i ω ( J ) isif it is contained in one interval of the form I r i ,m and two of the form I ( r i − ,m , thus | J ν i | ≤ | e − ( r i − − e − ( r i − | ( r i − ϑ + | e − ( r i − − e − ( r i ) | r ϑi ≤ r i − ϑ (cid:0) | e − r i ( e − e ) | + | e − r i ( e − | (cid:1) = c r i e − r i r ϑi (9) ANDREW LARKIN with c r i = r ϑi ( r i − ϑ (cid:0) e − e − (cid:1) .Let us examine time k . There are two possibilities: k ≤ ν p + 1 or ν p + 1 < k ≤ n J − ν s ( k ) denote the last return time before k , and let us split the sum in (8) accordingly: k − X i =0 | x ω,i − y ω,i | α | x ω,i | α = ν s ( k ) − X i =0 ,i/ ∈K | x ω,i − y ω,i | α | x ω,i | α + k − X i = ν s ( k ) +1 | x ω,i − y ω,i | α | x ω,i | α + s ( k ) X i =1 | x ω,ν i − y ω,ν i | α | x ω,ν i | α (10)where the first sum is all times up to time ν s ( k ) but excluding return times, the second sumis the times after ν s ( k ) , and the third sum is the return times.For the first sum, we use the fact that our expansion condition is equivalent to | J ν s ( k ) − i | ≤ ˜ C − e − il | J ν s ( k ) | and that | x ω,ν s ( k ) − i | ≥ δ for i = 1 , ..., ν s ( k ) − i / ∈ K to show that | x ω,ν s ( k ) − i − y ω,ν s ( k ) − i || x ω,ν s ( k ) − i | ≤ ˜ C − e − li | J ν s ( k ) | δ . (11)Thus, we have ν s ( k ) − X i =0 ,i/ ∈K | x ω,i − y ω,i | α | x ω,i | α ≤ ν s ( k ) − X i =0 ,i/ ∈K ˜ C − α e − l ( ν s ( k ) − i ) α | J ν s ( k ) | α δ α (cid:16) using (11) (cid:17) ≤ C ν s ( k ) − X i =0 ,i/ ∈K e − l ( ν s ( k ) − i ) α c − αr s ( k ) e − r s ( k ) α r ϑαs ( k ) δ α (cid:16) using (9) (cid:17) ≤ C C c αr s ( k ) r ϑαs ( k ) ν s ( k ) − X i =0 ,i/ ∈K e − l ( ν s ( k ) − i ) α (cid:16) using c r s ( k ) , r s ( k ) > e − l ( ν s ( k ) − i ) < (cid:17) ≤ C C c αr s ( k ) r ϑαs ( k ) ∞ X i =0 e − liα = C C c αr s ( k ) r ϑαs ( k ) · − e − lα = D . with C = ˜ C − α and C = (cid:16) e − rs ( k ) δ (cid:17) α . We note that D is independent of both k and δ as e − rs ( k ) δ < UENCHED DECAY OF CORRELATIONS FOR ONE DIMENSIONAL RANDOM LORENZ MAPS 9
For the second sum, we also have that | x ω,i | ≥ δ and | J i | ≤ ˜ C − e − l ( k − − i ) | J k − | for i = ν s ( k ) + 1 , .., k − . Thus k − X i = ν s ( k ) +1 | x ω,i − y ω,i | α | x ω,i | α ≤ k − X i = ν s ( k ) +1 ˜ C − α e − l ( k − − i ) α | J k − | α δ α ≤ C k − X i = ν s ( k ) +1 e − l ( k − − i ) α | J k − | α δ α ≤ C k − X i = ν s ( k ) +1 e − l ( k − − i ) α ≤ C ∞ X i =0 e − liα = C · − e − liα = D < ∞ , Finally, for the third sum, we define the set K r = { ν η < ν η < ... < ν η i < ... < ν η q } suchthat r η i = r for all ν η i ∈ K r . Clearly s ( k ) X i =1 | x ω,ν i − y ω,ν i | α | x ω,ν i | α = X r ≥ r δ X i ∈K r | x ω,i − y ω,i | α | x ω,i | α (12)Let M ( r ) denote the maximum value in K r . Then we have | J i | ≤ ˜ C − e − l ( M ( r ) − i ) | J M ( r ) | and | J M ( r ) | < c M ( r ) e − r r ϑ . Additionally, since i ∈ K r implies e − r ≤ | x ω,i | ≤ e − ( r − , we have | J i || x i | ≤ c M ( r ) ˜ C − e − l ( M ( r ) − i ) r ϑ (13)Thus, we have X i ∈K r | x ω,i − y ω,i | α | x ω,i | α ≤ X i ∈K r c αM ( r ) ˜ C − α e − l ( M ( r ) − i ) α r ϑα (cid:16) using (13) (cid:17) ≤ C c αM ( r ) r ϑα X i ∈K r e − l ( M ( r ) − i ) α ≤ C c αM ( r ) r ϑα ∞ X j =0 e − ljα = C c αM ( r ) r ϑα − e − lα ≤ C r ϑα with C = C c αM ( r ) 11 − e − lα . Thus, s ( k ) X i =1 | x ω,ν i − y ω,ν i | α | x ω,ν i | α = X r ≥ r δ X i ∈K r | x ω,i − y ω,i | α | x ω,i | α ≤ C X r ≥ r δ r θα = D < ∞ . This concludes the part of the proof for k ≤ ν p + 1.If we instead have ν p + 1 < k , then the above proof will not work. This is because,after ν p , the upper bound of | J i | < δ no longer applies so long as the i -th iterate does not intersect ∆ . Instead, we use the fact that, since J i does not intersect ∆ , we have | J i | < − δ to give us the following estimate: k − X i = ν p +1 | x ω,i − y ω,i | α | x ω,i | α ≤ k − X i = ν p +1 ˜ C − α e − l ( k − − i ) α | J k − | α δ α < C k − X i = ν s ( k ) +1 e − l ( k − − i ) α (cid:16) − δδ (cid:17) α ≤ C (cid:16) − δδ (cid:17) α k − X i = ν s ( k ) +1 e − l ( k − − i ) α (cid:16) since (cid:16) − δδ (cid:17) > (cid:17) = C (cid:16) − δδ (cid:17) α · − e − liα = D ( δ ) < ∞ , where D ( δ ) is a function dependent on δ . In order to make our upper bounded independentof δ , we restrict our subintervals to ˜ J ⊂ J such that | ˜ J | < δ . Thus, we set our distortionconstant D = κ ( D + D + D ). (cid:3) Full return partition.
Below we construct the full return partition of some neigh-borhood of 0. For this we use the following lemma:
Lemma 3.7.
There exists δ ∗ and t ∗ ∈ N depending on δ such that for every ω ∈ Ω andfor every interval J with | J | ≥ δ there exists ˜ J ⊂ J with the following properties: (i) There exists t < t ∗ such that T tω : ˜ J → ∆ ∗ = ( − δ ∗ , δ ∗ ) is a diffeomorphism; (ii) both components of J \ ˜ J have size greater than δ/ ; (iii) | ˜ J | ≥ β | J | , where β is a uniform constant.Proof. By assumption for ∪ n ∈ N T − n (0) is dense in I . Since all the maps in A ( ε ) are closeto T by taking ε sufficiently small, there exists t ¯ δ ∈ N such that ∪ n ≤ t ¯ δ T − nω (0) is ¯ δ -densein I and uniformly bounded away from 0 for all ω . Fix 0 < ¯ δ < δ/
5. Then the followingholds: (a) there is a preimage of 0 in the interval of length ¯ δ in the middle of J ; (b) thereexists δ ∗ sufficiently small such that | T − t ω (∆ ∗ ) | ≤ ¯ δ , where ∆ ∗ = ( − δ ∗ , δ ∗ ), and uniformlybounded away from 0 for all ω ∈ Ω and for t ≤ t ¯ δ ; (c) the distance from the boundary of J to J ∩ T − t ω (∆ ∗ ) is larger than δ/
5. We let ˜ J = J ∩ T − t ω (∆ ∗ ). Then item (i) is automatic.Item (ii) follows from (c). Since we consider only finitely many iterates T − t ω the distortionof it is bounded by a constant (that depends on t ¯ δ ). Thus T − t ω (∆ ∗ ) ≤ C for some C > β as in item (iii). (cid:3) The above lemma implies that for every interval of length δ or greater there exists δ ∗ > t ∗ such that every escape interval J has a subinterval which is mapped onto∆ ∗ = ( − δ ∗ , δ ∗ ) within a uniformly bounded number of iterates. We use this propertyrepeatedly in order to construct a full return partition Q ω (∆ ∗ ) of ∆ ∗ = ( − δ ∗ , δ ∗ ), i.e. weconstruct a countable partition Q ω (∆ ∗ ) such that for every J ∈ Q ω (∆ ∗ ) there exists anassociated return time τ ( J ) = τ ω ( J ) ∈ N such that T τ ( J ) ω : J → ∆ ∗ is a diffeomorphism UENCHED DECAY OF CORRELATIONS FOR ONE DIMENSIONAL RANDOM LORENZ MAPS 11 for every ω ∈ Ω . To this end, we start with an exponential partition of ∆ ∗ of the form P (∆ ∗ ) = { I ( r,m ) } | r |≥ r ∗ , where each I ( r,m ) is defined as in the beginning of subsection 3.2. Weconstruct an escape partition on each I r,m , which by extension naturally induces a partitionon ∆ ∗ , denoted by P E ,ω (∆ ∗ ). Thus, for every J ∈ P E ,ω (∆ ∗ ) we have J ⊂ I ( r,m ) for some r and m which has an associated escape time E ω, = E ω ( J ). Lemma 3.7 implies that each J ∈ P ωE (∆ ∗ ) contains a subinterval ˜ J that maps diffeomorphically to ∆ ∗ within a boundednumber of iterates. Thus, in order to construct the next partition P ωE , we divide J into ˜ J and the components of J \ ˜ J . To each of these we assign the first escape time E ω, = E ω ( J ),and we assign to the returning subinterval ˜ J the return time τ ( ˜ J ) = E ω, ( ˜ J ) + t ( ˜ J ), where t ( ˜ J ) is the number of iterates (bounded above by t ∗ ) such that T E ω ( J )+ t ( ˜ J ) ω ( ˜ J ) = ∆ ∗ . Weplace ˜ J in P ωE , and it will remain unchanged in all subsequent P ωE k ’s.Next, we apply the escape partition algorithm to the components of T E ω, ω ( J \ ˜ J ). Thisis possible since Lemma 3.7 guarantees that these components will be of size greater than δ/
5. Let J NR denote one of the non-returning components of J \ ˜ J , and let K ⊂ J NR be asubinterval that is obtained after applying the escape partition algorithm to T E ω, ω ( J NR ).We place K in P ωE , and again by Lemma 3.7 there exists ˜ K ⊂ K that also maps dif-feomorphically to ∆ ∗ after a bounded number of iterates. We divide K into ˜ K and thecomponents of K \ ˜ K , and to each of these we assign the second escape time E ω, = E ω ( K ),and to ˜ K we assign the return time τ ( ˜ K ) = E ω, ( ˜ K ) + t ( ˜ K ).We then repeat this process ad infinitum, by 1) taking each non-returning L ∈ P ωE k (forsome k ); 2) chopping L into its returning component and non-returning components; 3)assigning to each of these the k -th escape time E ω,k = E ω ( L ); 4) assigning to the returningcomponent ˜ L the return time τ ( ˜ L ) = E ω,k ( ˜ L ) + t ( ˜ L ) and placing ˜ L in P ωE k +1 ; and 4)performing the escape partition on the remaining non-returning components and placingthe resulting subintervals in P ωE k +1 .Note that the i -th escape time of a non-returning interval, i ≤ k , will be the same asthe escape time of its i -th ancestor.Finally, using the above we define the full partition as Q ω (∆ ∗ ) = ∨ ∞ k =0 P ωE k (14)i.e. the set of all possible intersections of all P ωE k ’s. Below we will show that Q ω defines amod 0 partition of ∆ ∗ and set τ ω ( J ) = E ω,k ( J ) + t ( J ) for every J ∈ Q ω .The following lemmas are useful for us. Notice that in principle we need to distinguish between different fibers and consider T ω : I × { ω } → I × { σω } . Then we obtain the inducing domain at fiber ω by ∆ ∗ ω = ∆ ∗ . Then the induced map is T τ ( ω,k ) ω : J k → ∆ ∗ σ τ ( ω,k ) ω . This extension does not cause any problem, since all the estimates are uniformin ω . Lemma 3.8.
Let ˜ J ⊂ J ∈ P ωE i be a non-returning subinterval of J which has had its i -thescape. Then there exists a constant C > such that X ˜ J ⊂ JE ω,i +1 ( ˜ J ) ≥ E ω,i ( J )+ n | ˜ J | ≤ C e − γn | J | . (15) Proof.
Since J ∈ P ωE i , we therefore know that T jω ( J ) is contained in at most three intervalsof the form I ( r,m ) for j ≤ E ω,k , which means we can apply our bounded distortion lemma3.6. We make use of the following property of bounded distortion: for a function f whichis of bounded distortion with distortion constant D , for any intervals we have | A || B | ≤ e D | T kω ( A ) || T kω ( B ) | , (16)where k ≤ min { n A , n B } . Indeed, since T ω is a differentiable function, we know from themean value theorem that there exist ξ ∈ A, ξ ∈ B such that DT kω ( ξ ) = | T kω ( A ) || A | , DT kω ( ξ ) = | T kω ( B ) || B | . Thus, | T kω ( B ) || B | . | T kω ( A ) || A | = | DT kω ( ξ ) || DT kω ( ξ ) | ≤ e D , and rearranging this we obtain inequality (16). Using this, we can write X ˜ J ⊂ JE ω,i +1 ( ˜ J ) ≥ E ω,i ( J )+ n | ˜ J | = X ˜ J ⊂ JE ω,i +1 ( ˜ J ) ≥ E ω,i ( J )+ n | ˜ J || J | · | J |≤ e D X ˜ J ⊂ JE ω,i +1 ( ˜ J ) ≥ E ω,i ( J )+ n | T E ω,i ( J ) ω ( ˜ J ) || T E ω,i ( J ) ω ( J ) | · | J | (cid:16) using (16) (cid:17) ≤ e D X ˜ J ⊂ JE ω,i +1 ( ˜ J ) ≥ E ω,i ( J )+ n | T E ω,i ( J ) ω ( ˜ J ) | (cid:16) using | J | < | T E ω,i ( J ) ω ( J ) | (cid:17) ≤ e D · C e − γn | J | (cid:16) using Proposition 3.1 (cid:17) = C e − γn | J | where C = e D · C . (cid:3) An important corollary of this is the following:
Corollary 3.9.
We denote by Q ( n ) ω ( E ω, , ..., E ω,i ) the set of all J ∈ Q ω such that J hasescape times { E ω, < E ω, < ... < E ω,i < n } and whose ( i + 1) -th escape time is after n . UENCHED DECAY OF CORRELATIONS FOR ONE DIMENSIONAL RANDOM LORENZ MAPS 13
Then we have X J ∈ Q ( n ) ω ( E ω, ,...,E ω,i ) | J | ≤ C i e − γn | ∆ ∗ | . (17) Proof.
For each J ∈ Q ( n ) ω ( E ω, , ..., E ω,i ) there exists a sequence of ancestors J ⊂ J ( i ) ⊂ J ( i − ⊂ ... ⊂ J (2) ⊂ J (1) ⊂ I ( r,m ) . Note that we can write E ω,i +1 ( J ) ≥ E ω,i ( J ( i ) ) + ( E ω,i +1 ( J ) − E ω,i ( J ( i ) )) E ω,i ( J ( i ) ) ≥ E ω,i − ( J ( i − ) + ( E ω,i ( J ( i ) ) − E ω,i − ( J ( i − ))... E ω, ( J (1) ) ≥ E ω, ( J (0) ) + ( E ω, ( J (1) ) − E ω, ( J (0) )= E ω, ( J (1) )Let us define M j = { J ⊂ J ( j ) : E ω,j +1 ( J ) ≥ E ω,j ( J ( j ) ) + ( E ω,j +1 ( J ) − E ω,j ( J ( j ) )) } , and thenapply (17) recursively: we have X J ⊂ J ( i ) J ∈ M i | J | ≤ C e − γ ( E ω,i +1 ( J ) − E ω,i ( J ( i ) )) | J ( i ) | and thus by recursion we have X J ∈ Q ( n ) ω ( E ω, ,...,E ω,i ) | J | ≤ C i e − γE ω,i +1 | I ( r,m ) | < C i e − γn | ∆ ∗ | . (cid:3) Furthermore, we make use of the following lemma:
Lemma 3.10.
Let η ∈ (0 , and R n,q = { ( n , n , ..., n q ) : n i ≥ ∀ i = 1 , ..., q : P qi =0 t i = k } . Then for q ≤ ηk there exists a positive function ˆ η ( η ) such that ˆ η → as η → , and R n,q ≤ e ˆ ηR . (18)3.5. The tail of the return times.
To establish mixing rates, we need to obtain decayrates for the tails of the return times. By construction, the tail of the return times dependon the number of escape times that occurred before returning. We have the followinglemma:
Lemma 3.11.
Fix ω ∈ Ω and let ( n , n , . . . , n i ) be the sequence of escape times of aninterval J ∈ Q ω before time n ≥ such that P ij =0 n i = n. Then there exist constants
B, b > such that |{ J ∈ Q ω : τ ( J ) > n }| ≤ Be − bn . (19) Proof.
Let us define the following: Q ( n ) = { J ω ∈ Q ω : τ ( J ) > n } (20) Q ( n ) i = { J ∈ Q ( n ) : E ω,i − ( J ) ≤ n < E ω,i ( J ) } (21) Q ( n ) i ( n , . . . , n i ) = { J ∈ Q ( n ) i : k X j =1 n j = E ω,k ( J ) , ≤ k ≤ i − } (22)We decompose Q ( n ) into the following sums: | Q ( n ) | = X i ≤ n | Q ( n ) i | = X i ≤ ζn | Q ( n ) i | + X ζn ζ n ), which the two sums above represent respectively. For the manyescapes, we have | Q ( n ) i | ≤ (1 − β ) i | ∆ ∗ | = 2 δ (1 − β ) i , (23)and for big enough i we have X ζn n }| ≤ C e − γ β n + C e (ˆ η − γ ) n ≤ Be − bn for positive constants B and b , as long as we have chose small enough ζ . (cid:3) Thus, we have that τ ( x ) is finite for a.e. x ∈ ∆ ∗ . UENCHED DECAY OF CORRELATIONS FOR ONE DIMENSIONAL RANDOM LORENZ MAPS 15
Gibbs-Markov.
Let ∆ ∗ , Q ω and τ ω be as in subsection 3.4. By definition, T τ ω ω :∆ ∗ → ∆ ∗ has fiber-wise Markov property: every J ∈ Q ω is mapped diffeomorphically onto∆ ∗ × σ τ ( J ) ω . Notice that if τ ω ( J ) = k then τ ω depends only the first k − ω . This implies that if τ ω ( x ) = k and ω ′ i = ω i for 0 ≤ i ≤ k − τ ω ( x ) = τ ω ′ ( x ), i.e. τ ω ( x ) is a stopping time. The uniform expansion is immediate in our case. The tail of thereturn times are obtained in subsection 3.5. We still need to show bounded distortion andaperiodicity, which we address below. Set F ω ( x ) = T τ ω ( x ) for ( ω, x ) ∈ Ω × ∆ ∗ . As usualwe introduce a separation time for x, y ∈ ∆ ∗ ω by setting s ω ( x, y ) = min { n | F nω ( x ) ∈ J, F nω ( y ) ∈ J ′ , J = J ∈ Q ω } . Lemma 3.12.
There exists constants ˜ D and β ∈ (0 , such that for all J ∈ Q ω and forall x, y ∈ J log | DF ω ( x ) || DF ω ( y ) | ≤ ˜ Dβ s ω ( F x,F y ) . Proof.
Let Q ωn = ∨ n − j =0 F − nω Q ω . Then for every x, y ∈ J ωn ∈ Q ωn , F iω ( x ) and F iω ( y ) stayin the same element of Q ω for i = 1 , , . . . , n − F nω ( J n ) = ∆ ∗ . Definediam( Q ω ) = sup ω ∈ Ω sup {| J | : J ω ∈ Q ω } . We show that there exists κ ∈ (0 ,
1) such thatfor all ω ∈ Ω and J n ∈ Q ωn holds | J n | ≤ κ n diam( Q ω ) . (24)We prove this inequality via induction. For n = 1 notice that Q ω = Q ω and proceed asfollows: fix ω ∈ Ω and let J ∈ Q ω be such that J ⊂ J ∈ Q ω . Since F ω J = ∆ ∗ and F ω J ∈ Q ω . we have | J \ J || J | ≥ e −D | F ω ( J \ J ) || F ω ( J ) | ≥ e −D δ ∗ − diam( Q ω )2 δ ∗ = ¯ κ ∈ (0 , . Thus, we obtain | J | ≤ (1 − ¯ κ ) | J | . Iterating the the process we obtain (24).Consider x, y ∈ J with n = s ω ( x, y ) and F ω ( x ) , F ω ( y ) ∈ J n ∈ Q ωn and suppose that J n ⊂ J ∈ Q ω . By equation (16) and Lemma 3.6 we have | T kω ( J n ) || T kω ( J ) | ≤ e D | F ω ( J n ) || F ω ( J ) | for all k ≤ n and ω ∈ Ω. Thus proceeding as in the proof of Lemma 3.6 we havelog | DF ω ( x ) || DF ω ( y ) | = τ ω ( J ) X i =0 | T iω ( J n ) | α | T iω ( y ) | α ≤ e α D τ ω ( J ) X i =0 | T iω ( J ) | α | T iω ( y ) | α | F ω ( J n ) | α | F ω ( J ) | α ≤ e α D δ ∗ α | F ω ( J n ) | α τ ω ( J ) X i =0 | T iω ( J ) | α | T iω ( y ) | α . The sum on the right hand side can be bounded by a constant D as in the proof of Lemma3.6. Therefore, using (24) and diam( Q ω ) ≤ δ ∗ we havelog | DF ω ( x ) || DF ω ( y ) | ≤ e α D D κ αn = ˜ Dβ s ( x,y ) . for suitable constant ˜ D and β . (cid:3) Aperiodicity.
Finally we address the problem of aperiodicity; i.e. there exists N ∈ N and two sequence { t i ∈ N , i = 1 , , . . . , N } , { ε i > , i = 1 , . . . , N } such that g.c.d. { t i } =1 and |{ x ∈ ∆ ∗ | τ ω ( x ) = t i }| > ε i for almost every ω ∈ Ω. To show this, we recall thatthe original Lorenz system and all sufficiently close systems are mixing [17]. Thus, wecan proceed as in [1, Remark 3.14]. Since the unperturbed map admits a unique invariantprobability measure, it can be lifted to the induced map over ∆ ∗ constructed followingthe algorithm in the previous 2 subsections. Moreover, the lifted measure is invariant andmixing for the tower map. Therefore, there exists partition Q of ∆ ∗ and a return time τ : ∆ ∗ → N such that τ i = τ ( Q i ), Q i ∈ Q such that g.c.d. { τ i } N i =1 = 1 for some N > ε if necessary, we can ensure that the first N elements of the partition Q ω satisfy | Q ωi ∩ Q i | ≥ | Q i | / τ ω ( Q ωi ) = τ i for i = 1 , . . . N and for all ω ∈ Ω. Thus,we may take ε i = | Q i | /
2. Notice that we define only finitely many domains in this way.Therefore, the tails estimates, distortion, etc. are not affected, and we can stop at some ε >
0, thereby proving aperiodicity. Furthermore, by applying the results of [13], weobtain the proof of Theorem 2.1.
Proof. (Proof Theorem 2.1) In the above we have shown that conditions (C2)-(C6) of theAppendix hold. Condition (C1) is true by construction. Using the this and Theorem 4.1 inthe Appendix we are now ready to prove Theorem 2.1. We begin the proof by defining the tower projection π ω : ∆ ω → I for almost every ω ∈ Ω as π ω ( x, ℓ ) = T ℓσ − ℓ ω ( x ). One shouldnote that π σω ◦ ˆ F ω = T ω ◦ π ω . Then µ ω = ( π ω ) ∗ ν ω provides an equivariant family of measuresfor { T ω } and the absolute continuity follows from the fact that T ω are non-singular. Now,‘lift’ the observables ϕ ∈ L ∞ ( X ) and ψ ∈ C η ( X ) to the tower. Let ¯ ϕ ω = ϕ ◦ π ω and¯ ψ ω = ψ ◦ π ω respectively. Now, notice that Z ( ϕ ◦ T nω ) ψdµ ω = Z ( ϕ ◦ ( T nω ◦ π ω ) ◦ π − ω ) ψdµ ω = Z ( ϕ ◦ ( π σ n ω ◦ ˆ F n ) ◦ π − ω ) ψdµ ω = Z ( ¯ ϕ σ n ω ◦ ˆ F n ◦ π − ω ) ψd ( ν ω ◦ π − ω )= Z ( ¯ ϕ σ n ω ◦ ˆ F n ) ¯ ψ ω h ω dm. where dν ω = h ω · dm . Equally, we have Z ϕdµ σ n ω = Z ϕ ◦ π σ n ω ◦ π − σ n ω dµ σ n ω = Z ¯ ϕ σ n ω ◦ π − σ n ω d ( ν σ n ω ◦ π − σ n ω ) = Z ¯ ϕ σ n ω dν σ n ω (25)and Z ψdµ ω = Z ψ ◦ π ω ◦ π − ω dµ = Z ¯ ψ ω ◦ π − ω d ( ν ω ◦ π − ω ) = Z ¯ ψ ω h ω dm. (26) UENCHED DECAY OF CORRELATIONS FOR ONE DIMENSIONAL RANDOM LORENZ MAPS 17
Thus, we have (cid:12)(cid:12)(cid:12) Z ( ϕ ◦ T nω ) ψdµ ω − Z ϕdµ σ n ω Z ψdµ ω (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z ( ¯ ϕ σ n ω ◦ ˆ F nω ) ¯ ψh ω dm − Z ¯ ϕ σ n ω dν σ n ω Z ¯ ψ ω h ω dm (cid:12)(cid:12)(cid:12) . This means that if we can show that ¯ ϕ ω ∈ L K ω ∞ and ¯ ψ ω h ω ∈ F K ω γ , then we can apply theorem(4.1) and thereby obtain exponential decay of correlations on the original dynamics. Thefirst condition is trivial since ϕ ∈ L ∞ ( X ). To show the second condition, since F ω isuniformly expanding; i.e., | ( F ω ) ′ | ≥ κ >
1, we have | x − y | ≤ ( κ ) s ( x,y ) . Hence, for any( x, ℓ ) , ( y, ℓ ) ∈ ∆ ω we have the inequality | ¯ ψ ( x, ℓ ) − ¯ ψ ( y, ℓ ) | ≤ k ψ k η ( 1 κ ) η · s ( x,y ) ≤ k ψ k η ( 1 κ η γ ) s ( x,y ) γ s ( x,y ) . (27)Now since s (( x, ℓ ) , ( y, ℓ )) = s ( x, y ) the inequality (27) implies | ( ¯ ψh ω )( x, ℓ ) − ( ¯ ψh ω )( y, ℓ ) | ≤ k ψ k ∞ k h ω k F Kωγ γ s (( x,ℓ ) , ( y,ℓ )) + k h ω k L ∞ k ψ k η γ s (( x,ℓ ) , ( y,ℓ )) . This completes the proof. (cid:3) Appendix
Abstract random towers with exponential tails.
This appendix is based on thework and results of [8, 13]. See also the related work of [5, 15] on random towers. Using thenotation and definitions in section 3, we can introduce a random tower for almost every ω as follows:∆ ω = { ( x, ℓ ) ∈ ∆ ∗ × N | x ∈ Q σ − ℓ ω (∆ ∗ ) , j, ℓ ∈ N , ≤ ℓ ≤ τ σ − ℓ ω ( x ) − } . We can also define the induced map ˆ F ω : ∆ ω → ∆ σω asˆ F ω ( x, ℓ ) = ( ( x, ℓ + 1) , ℓ + 1 < τ σ − ℓ ω ( x )( T ℓ +1 ω ( x ) , , ℓ + 1 = τ σ − ℓ ω ( x ) . Notice that this allows us to construct a partition on the random tower as Z ω = { ˆ F ℓσ − ℓ ( J σ − ℓ ω ) | J σ − ℓ ω ∈ Q σ − ℓ ω (∆ ∗ ) , τ ω | J σ − ℓ ω ≥ ℓ + 1 , ℓ ∈ Z + } . Assume:(C1)
Return and separation time: the return time function τ ω can be extended tothe whole tower as τ ω : ∆ ω → Z + with τ ω constant on each J ∈ Z ω , and thereexists a positive integer p such that τ ω ≥ p . Furthermore, if ( x, ℓ ) and ( y, ℓ ) areboth in the same partition element J ∈ Z ω , then s ω (( x, , ( y, ≥ ℓ , and for every( x, , ( y, ∈ J ∈ Z ω we have s ω (( x, , ( y, τ ω ( x,
0) + s σ τω ω ( ˆ F τ ω ( x, ( x, , ˆ F τ ω ( y, ( y, (C2) Markov property: for each J ∈ Q ω (∆ ∗ ) the map ˆ F τ ω ω | J : J → ∆ ∗ is bijective.(C3) Bounded distortion:
There exist constants 0 < γ < D > J ∈ Z ω and all ( x, ℓ ) , ( y, ℓ ) = x, y ∈ J (cid:12)(cid:12)(cid:12) D ˆ F τ ω ω ( x ) D ˆ F τ ω ω ( y ) − (cid:12)(cid:12)(cid:12) ≤ D γ s ( ˆ F τω ( x, , ˆ F τω ( y, . (C4) Weak forwards expansion: the diameters of the partitions W nj =0 ˆ F − jω Z ω tend tozero as n → ∞ .(C5) Uniform return time asymptotics: there exist constants
B, b, u, v >
0, a full-measure subset Ω ⊂ Ω such that for every ω ∈ Ω we have m ( { x ∈ ∆ ∗ | τ ω > n } ) ≤ Be − bn We also have for almost every ω ∈ Ω m (∆ ω ) = X ℓ ∈ Z + m ( { τ σ − ℓ ω > ℓ } ) < ∞ which gives us the existence of a family of finite invariant sample stationary measures.We also have for almost every ω ∈ Ωlim ℓ →∞ X ℓ ≥ ℓ m (∆ σ ℓ ω,ℓ ) = 0 , which implies the previously mentioned family of measures are mixing.(C6) Aperiodicity: there exists N ≥
1, a full-measure subset Ω ⊂ Ω and a set { t i ∈ Z + : i = 1 , , ..., N } such that g.c.d. { t i } = 1 and there exist ǫ i > ω ∈ Ω and every i ∈ { , ..., N } we have m ( { x ∈ Λ | τ ( x ) = t i } ) > ǫ i .Let us also define the following function spaces: F + γ = { ϕ ω : ∆ ω → R | ∃ C ϕ > ∀ J ∈ Z ω , either ϕ ω | J ≡ ϕ ω | J > (cid:12)(cid:12)(cid:12) log ϕ ω ( x ) ϕ ω ( y ) (cid:12)(cid:12)(cid:12) ≤ C ϕ ω γ s ω ( x,y ) , ∀ x, y ∈ J } . For almost every ω let K ω : Ω → R + be a random variable which satisfies inf Ω K ω > P ( { ω | K ω > n } ) ≤ e − un v , where u and v are the same as in (P4). We then define the spaces L K ω ∞ = { ϕ ω : ∆ ω → R | ∃ C ′ ϕ ω > , sup x ∈ ∆ ω | ϕ ω ( x ) | ≤ C ′ ϕ ω K ω } (29) F K ω γ = { ϕ ω ∈ L K ω ∞ | ∃ C ϕ ω > , | ϕ ω ( x ) − ϕ ω ( y ) | ≤ C ϕ ω K ω γ s ω ( x,y ) , ∀ x, y ∈ ∆ ω } (30) Theorem 4.1.
Let ˆ F ω satisfy (C1)-(C6), and let K ω satisfy the above condition. Thenfor almost every ω ∈ Ω there exists an absolutely continuous ˆ F ω − equivariant probabilitymeasure ν ω = h ω m on ∆ ω , satisfying ( ˆ F ω ) ∗ ν ω = ν σω , with h ω ∈ F + γ . Furthermore, there UENCHED DECAY OF CORRELATIONS FOR ONE DIMENSIONAL RANDOM LORENZ MAPS 19 exist constants
C > , v > and a full-measure subset Ω ⊂ Ω such that for every ω ∈ Ω , ϕ ω ∈ L K ω ∞ and ψ ω ∈ F K ω γ there exists a constant C ϕ,ψ such that for all n we have (cid:12)(cid:12)(cid:12) Z ( ϕ σ n ω ◦ ˆ F nω ) ψ ω dm − Z ϕ σ n ω dν σ n ω Z ψ ω dm (cid:12)(cid:12)(cid:12) ≤ C ϕ,ψ e − bn (31) and (cid:12)(cid:12)(cid:12) Z ( ϕ ω ◦ ˆ F nσ − n ω ) ψ σ − n ω dm − Z ϕ ω dν ω Z ψ σ − n ω dm (cid:12)(cid:12)(cid:12) ≤ C ϕ,ψ e − bn . (32) References [1] J.F. Alves, W. Bahsoun, M. Ruziboev.
Almost sure rates of mixing for partially hyperbolic attractors ,arXiv:1904.12844.[2] J.F. Alves, M. Soufi, M.
Statistical stability of geometric Lorenz attractors . Fund. Math. 224(3), 219–231(2014).[3] V. Ara´ujo, M. J. Pac´ıfico.
Three-dimensional flows.
Ergebnisse der Mathematik und ihrer Grenzgebiete.3. Folge. A Series of Modern Surveys in Mathematics. Results in Mathematics and Related Areas. 3rdSeries, vol. 53. Springer, Berlin, Heidelberg (2010).[4] V. Ara´ujo, M. J. Pac´ıfico, E. R. Pujals, M. Viana.
Singular-hyperbolic attractors are chaotic . Trans.Amer. Math. Soc.361 (5) (2009), 2431–2485.[5] W. Bahsoun, C. Bose, M. Ruziboev.
Quenched decay of correlations for slowly mixing systems . Trans.Amer. Math. Soc. 372 (2019), no. 9, 6547–6587.[6] W. Bahsoun, I. Melbourne and M. Ruziboev.
Variance continuity for Lorenz flows . Annales HenriPoincar´e 21 (2020), 1873–1892.[7] W. Bahsoun, M. Ruziboev.
On the statistical stability of Lorenz attractors with a C α stable foliation .Ergodic Theory Dynam. Systems 39 (2019), no. 12, 3169–3184.[8] V. Baladi, M. Benedicks, V. Maume-Deschamps. Almost sure rates of mixing for i.i.d. unimodal maps .Ann. Sci. ´Ecole Norm. Sup. (4) 35 (2002), no. 1, 77–126.[9] M. Benedicks, L. Carleson. The dynamics of the H´enon map, Ann. of Math. (2), 133, (1991), no. 1,73–169.[10] H. Bruin, S. Luzzatto, S. van Strien.
Decay of correlations in one-dimensional dynamics . Ann. Sci.´Ecole Norm. Sup. (4) 36 (2003), no. 4, 621–646.[11] K. D´ıaz-Ordaz.
Decay of correlations for non-H¨older observables for one-dimensional expandingLorenz-like maps.
Discrete Contin. Dyn. Syst. Series A, Vol. 15, (2006), no. 1, 159–176.[12] K. D´ıaz-Ordaz, M. Holland, S. Luzzatto.
Statistical properties of one-dimensional maps with criticalpoints and singularities.
Stoch. Dyn vol. 6, (2006), no. 4, 423–458.[13] Z. Du. On mixing rates for random perturbations, Ph.D. Thesis, National University of Singapore,2015.[14] M. Holland, I. Melbourne.
Central limit theorems and invariance principles for Lorenz attractors . J.Lond. Math. Soc. (2) 76 (2007), no. 2, 345–364.[15] X. Li, H. Vilarinho. Almost sure mixing rates for non-uniformly expanding maps.
Stoch. Dyn.
Deterministic nonperiodic flow . J. Atmosph. Sci. 20 (1963), 130–141.[17] S. Luzzatto. I. Melbourne, F. Paccaut,
The Lorenz attractor is mixing . Comm. Math. Phys. 260 (2005),no. 2, 393–401.
Department of Mathematical Sciences, Loughborough University, Loughborough, Le-icestershire, LE11 3TU, UK
Email address ::