Orbits closeness for slowly mixing dynamical systems

Abstract Given a dynamical system, we prove that the shortest distance between two n-orbits scales like n to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author [On the shortest distance between orbits and the longest common substring problem. Adv. Math. 344 (2019), 311–339]. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit.


Introduction
The study of the statistical properties of dynamical systems is one of the main pillars of ergodic theory.In particular, one of the principal lines of investigation is to try and obtain quantitative information on the long-term behaviour of orbits (such as return and hitting times, dynamical extremal indices or logarithm laws).
In a metric space (X, d), the problem of the shortest distance between two orbits of a dynamical system T : X → X, with an ergodic measure μ, was introduced in [BLR].That is, for n ∈ N and x, y ∈ X, they studied M n (x, y) = M T ,n (x, y) := min 0≤i,j ≤n−1 d(T i (x), T j (y)) and showed that the decay of M n depends on the correlation dimension.

Orbits closeness for slowly mixing dynamical systems 1193
The lower correlation dimension of μ is defined by C μ := lim inf r→0 log μ(B(x, r)) dμ (x) log r , and the upper correlation dimension C μ is analogously defined via the limsup.If these are equal, then this is C μ , the correlation dimension of μ.This dimension plays an important role in the description of the fractal structure of invariant sets in dynamical systems and has been widely studied from different points of view: numerical estimates (e.g.[BB, BPTV, SR]), existence and relations with other fractal dimension (e.g.[BGT, P]), and relations with other dynamical quantities (e.g.[FV, M]).
It is worth mentioning that the problem of the shortest distance between orbits is a generalisation of the longest common substring problem for random sequences, a key feature in bioinformatics and computer science (see e.g.[W]).
In [BLR,Theorem 1], under the assumption C μ > 0, a general lower bound for M n was obtained, as follows.
To replace the inequality above with equality, in [BLR,Theorems 3 and 6], the authors assumed that C μ exists and proved lim inf , μ × μ-a.e.x, y, ( 1 . 1 ) using some exponential mixing conditions on the system.One could naturally wonder if this mixing condition could be relaxed or even dropped.In [BLR], a partial answer was given and it was proved that for irrational rotation (which are not mixing), the inequality in Theorem 1.1 could be strict.
In this paper, we extend the above results in equation (1.1) to discrete systems with no requirement on mixing conditions.The main tool in proving our positive results is inducing: the idea in the discrete case is to first take advantage of the fact that Theorem 1.1 holds in great generality (including, as we note later, to higher-dimensional hyperbolic cases) and then to show that if there is an induced version of the system satisfying equation (1.1), then this inequality passes to the original system.
Moreover, we also extend the results of [BLR] to flows.Thus, we first have to prove an analogue of Theorem 1.1 and observe that in the continuous setting, the correct scaling is C μ − 1.Then, using inducing via Poincaré sections, we also obtain an analogue of equation (1.1).
We will give examples for all of these results both in the discrete and continuous setting.We also give a class of examples in §5 where the conclusions of [BLR] fail to hold.This class is slowly mixing and also does not admit an induced version.
Finally, we emphasise that one of the obstacles to even wider application is proving that the correlation dimension C μ exists, see §3.1 for some discussion and results.For suspension flows, under some natural assumptions, we will show in §4.2 that if the correlation dimension of the base exists, then the correlation dimension of the invariant measure of the flow also exists.
2. Main results and proofs for orbits closeness in the discrete case 2.1.The main theorem in the non-uniformly expanding case.We will suppose that given (X, T , μ), there is a subset Y = i Y i ⊂ X and an inducing time τ : Y → N ∪ {∞}, constant on each Y i and denoted τ i , so that our induced map is F = T τ : Y → Y .We suppose that there is an F-invariant probability measure μ F with τ dμ F < ∞ which projects to μ by the following rule: (2.1) We call (Y , F , μ F ) an inducing scheme, or an induced system for (X, T , μ).For systems which admit an inducing scheme, we have our main theorem, as follows.
THEOREM 2.1.Assume that the inducing scheme (Y , F , μ F ) satisfies equation (1.1) and that C μ F = C μ .Then, In §5, we give an example of a class of mixing systems where the conclusion of this theorem fails.These systems do not have good inducing schemes, see Remark 5.2 below.
Remark 2.2.As can be seen from the proof of this theorem, as well as related results in this paper, in fact what we prove is that if there is an induced system satisfying lim inf with the analogous statements for flows in §4.
In §3, we will give examples of systems where {Y i } i is countable, μ and μ F are absolutely continuous with respect to Lebesgue, and Proof of Theorem 2.1.The main observation here is that it is sufficient to prove that lim k→∞ (log M T ,n k (x, y)/(− log n k )) ≥ 2/C μ along a subsequence (n k ) k which scales linearly with k.
For x ∈ Y , define τ n (x) Orbits closeness for slowly mixing dynamical systems 1195 These are nested sets and by Birkhoff's ergodic theorem, we have lim N→∞ μ F (U ε,N ) = 1.In particular, by equation (2.1), μ(U ε,N ) > 0 for N sufficiently large and hence [BLR,Theorem 3] implies that for any η > 0 and sufficiently large n, Putting together the facts that the n-orbit by F of T i (x) (respectively T j (y)) is a subset of the τ n (x)-(respectively τ n (y)-) orbit by T of T i (x) (respectively T j (y)) and that i, j , and thus Observing that lim n→∞ (log n τ + 2ε /log n) = 1 and taking limit in the previous equation, we deduce that lim n→∞ (log Since η can be choose arbitrary small, the theorem is proved.
2.2.The main theorem in the non-uniformly hyperbolic case.We next consider systems T : X → X with invariant measure μ which are non-uniformly hyperbolic in the sense of Young, see [Y1].Then there is some Y ⊂ X and an inducing time τ defining F = T τ : Y → Y , with measure μ F , which is uniformly expanding modulo uniformly contracting directions.We can quotient out these contracting directions to obtain a system F : Ȳ → Ȳ , which has an invariant measure μ F .
THEOREM 2.3.Assume that the induced system ( Ȳ , F , μ F ) satisfies equation ( 1.1) and that The proof is directly analogous to that of Theorem 2.1.

2.3.
Requirements on the induced system.In [BLR], the main requirement for equation (1.1) to hold is that the system has some Banach space C of functions from X to R, θ ∈ (0, 1) and C 1 ≥ 0 such that for all ϕ, ψ ∈ C and n ∈ N, Some regularity conditions on the norms of characteristics on balls and the measures were also required, as well as a topological condition on our metric space (always satisfied for subset of R n with the Euclidean metric and subset of a Riemannian manifold of bounded curvature), but we leave the details to [BLR].We can also remark that for Lipschitz maps on a compact metric space with C = Lip, these regularity conditions can be dropped [GoRS].
In [BLR,Theorem 3], the main application was to systems where C = BV so, for example, we have a Rychlik interval map, and in [BLR,Theorem 6], the main application was to Hölder observables, so that the induced system is Gibbs-Markov, see for example [A, §3].

Examples in the discrete setting
Examples of our theory require an inducing scheme and, ideally, well-understood correlation dimensions.In [PW], correlation dimension is dealt with in the Gibbs-Markov setting in the case {Y i } i is a finite collection of sets, but under inducing, we usually expect this collection to be infinite (in which case much less is known), so this is not directly relevant here.
The simplest case in the context of our results is when the invariant probability measure μ for the system is d-dimensional Lebesgue, or is absolutely continuous with respect to Lebesgue (an acip) with a regular density, since in these cases, the correlation dimension for both μ and the corresponding measure for the system is d.

Existence of the correlation dimension.
First of all, we will give a result which implies that the correlation dimension for regular acips exists.PROPOSITION 3.1.Let X ⊂ R d .If μ is a probability measure on X which is absolutely continuous with respect to the d-dimensional Lebesgue measure such that its density ρ is in L 2 , then Proof.The fact that C μ ≤ d follows, for example, from [FLR,Theorem 1.4].
To prove a lower bound, we start by defining the Hardy-Littlewood maximal function (see e.g.[SS,Ch. 2.4]) of ρ: Moreover, by Hardy-Littlewood maximal inequality, Mρ ∈ L 2 and there exists c 1 > 0 (depending only on d) such that Thus, using the Cauchy-Schwarz inequality, we have Orbits closeness for slowly mixing dynamical systems 1197 If the density of the acip is not sufficiently regular, the correlation dimension may differ from the correlation dimension of the Lebesgue measure, as in the following case.PROPOSITION 3.2.Let α ∈ (1/2, 1).Assume that μ is supported on [0, 1] and dμ = ρ dx with ρ(x) = x −α .Then, we have We estimate the first term from above by For the second term, we split the sum into However, since From now on, suppose that we are dealing with X = R d and μ, μ F being acips with m denoting normalized Lebesgue measure.We assume i Y i = Y .First notice that if F has bounded distortion (in the one-dimensional case, it is sufficient that F is C 1+α with uniform constants), dμ F /dm is uniformly bounded away from 0 and 1, so For C μ , we assume that dμ/dm = ρ.Moreover, we assume there is C > 0 with ρ(x) ≥ C for any x ∈ X and that ρ ∈ L 2 .Thus, C μ = d.
In the case α ∈ (1/2, 1), then the density is similar to that in Proposition 3.2 and a similar proof gives C μ = 2(1 − α) < 1 = C μ F , so our upper and lower bounds on the behaviour of M n do not coincide.
3.3.Multimodal and other interval maps.Our results apply to a wide range of interval maps with equilibrium states, for example, many of those considered in [DT], which guarantees the existence of inducing schemes under mild conditions.Here we will focus on C 3 interval maps f : I → I (where I = [0, 1]) with critical points with order in (1, 2), that is, for c with Df (c) = 0, there is a diffeomorphism ϕ : . Moreover, we assume that for each critical point c, |Df n (c)| → ∞ and that for any open set V ⊂ I , there exists n ∈ N such that f n (V ) = I .Then, as in the main theorem in [BRSS], the system has an acip and the density is L 2 , and hence Theorem 2.1 applies.

3.4.
Higher dimensional examples.We will not go into details here, but there is a large amount of literature on non-uniformly expanding systems in higher dimensions which have acips and which have inducing schemes with tails which decay faster than polynomially.A standard class of examples of this are the maps derived from expanding maps given in [ABV].

Orbits closeness for flows
In this section, we will extend our study to flows.First of all, as in Theorem 1.1, we will prove that an upper bound (related to the correlation dimension of the invariant measure) can be obtained in a general setting.Then, under some mixing assumptions, we will give an equivalent of Theorem 2.1 for flows.We will prove the abstract results before giving specific examples.
Let (X, t , ν) be a measure-preserving flow on a manifold.We will study the shortest distance between two orbits of the flow, defined by We assume that the flow has bounded speed: there exists K ≥ 0 such that for T > 0, d( t (x), t+T (x)) ≤ KT .
To obtain the lower bound, we will assume the existence of a Poincaré section Y transverse to the direction of the flow.We denote by τ (x) the first hitting time of x in Y, and obtain F = τ on Y, the Poincaré map and μ the measure induced on Y. THEOREM 4.2.Let (X, t , ν) be a measure-preserving Lipschitz flow with bounded speed.We assume that there exists a Poincaré section Y transverse to the direction of the flow such that the Poincaré map (Y , F , μ), or the relevant quotiented version ( Ȳ , F , μ), satisfies equation ( 1

.1). If C μ exists and satisfies
Proof.One can mimic the proof of Theorem 2.1 to prove that lim inf And the result is proved using Theorem 4.1.
We note that we are not aware of cases where C ν and C μ are well defined, but the condition C ν = C μ + 1 above fails.We give various examples in the remainder of this section of cases where these conditions hold.

Examples of flows.
Examples where C ν exists and there is a Poincaré section, as in Theorem 4.2, with a measure μ such that C μ exists include Teichmüller flows [AGY] and a large class of geodesic flows with negative curvature, see [BMMW], a classic example being the geodesic flow on the modular surface.In these cases, the relevant measure for (the tangent bundle on) the flow is Lebesgue, and the measure on the Poincaré section is an acip.
In the case of conformal Axiom A flows, the conditions of Theorem 4.2 hold for equilibrium states of Hölder potentials, see the proof of [PS,Theorem 5.2].4.2.Suspension flows.For Theorem 4.2, we assume that C ν = C μ + 1. Obtaining this equality in a general setting is an open and challenging problem.In this section, we will prove that, under some natural assumptions, for suspension flows, this equality holds.
Let T : X → X be a bi-Lipschitz transformation on the separable metric space (X, d).
Let ϕ : X → (0, +∞) be a Lipschitz function.We define the space Orbits closeness for slowly mixing dynamical systems 1201 where (u, ϕ(u)) and (T u, 0) are identified for all u ∈ X.The suspension flow or the special flow over T with height function ϕ is the flow which acts on Y by the following transformation: The metric on Y is the Bowen-Walters distance, see [BW].First, we recall the definition of the Bowen-Walters distance d 1 on Y when ϕ(x) = 1 for every x ∈ X.Let x, y ∈ X and t ∈ [0, 1], so the length of the horizontal segment [(x, t), (y, t)] is defined by Let (x, t), (y, s) ∈ Y be on the same orbit, so the length of the vertical segment [(x, t), (y, s)] is defined by α v ((x, t), (y, s)) = inf{|r| : r (x, t) = (y, s) and r ∈ R}.
Let (x, t), (y, s) ∈ Y , so the distance d 1 ((x, t), (y, s)) is defined as the infimum of the lengths of paths between (x, t) and (y, s) composed by a finite number of horizontal and vertical segments.When ϕ is arbitrary, the Bowen-Walters distance on Y is given by d Y ((x, t), (y, s)) , y, s ϕ (y) .
For more details on the Bowen-Walters distance, one can see [BS, Appendix A].
Let μ be a T-invariant Borel probability measure in X.We recall that the measure ν on Y is invariant for the flow where Y gdν = X ϕ(x) 0 g(x, s) ds dμ(x) X ϕ dμ for every continuous function g : Y → R.Moreover, any -invariant measure is of this form.For an account of equilibrium states for suspension flows, see for example [IJT].THEOREM 4.3.Let X be a compact space and T : X → X a bi-Lipschitz transformation.We assume that for the invariant measure μ, the correlation dimension exists.If is a suspension flow over T as above, then with respect to the Bowen-Walters distance.
Remark 4.4.Under the same assumptions, one can observe that if C μ does not exist, then we have Before proving the theorem, we will recall some properties of the Bowen-Walters distance.First of all, for (x, s) and (y, t) ∈ Y , we define d π ((x, s) ((x, s), (y, t)) ≤ d Y ((x, s), (y, t)) ≤ cd π ((x, s), (y, t)).
In this system, the second coordinate is translated by α if the first coordinate belongs to I.
We make the standing assumption on our choice of ϕ that: • (NA) for any u ∈ [−π , π ], the equation f e iuϕ = λf • T , where f is Hölder (on the subshift) and λ ∈ S 1 , has only the trivial solutions λ = 1 and f constant.The simple case, where the I which defines ϕ is a non-empty union of size 1 cylinders such that both I and I c contain a fixed point, fulfils this assumption.
Definition 5.1.Given an irrational number α, we define the irrationality exponent of α as the following (possibly infinite) number: γ (α) = inf β : lim inf q→∞ q β qα > 0 , where • indicates the distance to the nearest integer number in R.
Remark 5.2.By [GRS,Theorem 19], if γ (α) > d μ + 1, then the hitting time statistics is typically degenerate.This is an indirect way of seeing that there cannot be an inducing scheme satisfying equation (1.1), otherwise [BSTV,Theorem 2.1] would be violated; it also suggests that the conclusions of Theorem 2.1 will not hold here, which we show below is indeed the case.The theorem is proved taking arbitrary small.