Poincar\'e recurrence for observations

A high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities, called an observation. Following this idea and in the continuity of Boshernitzan's work, for a measure preserving system, we study Poincar\'e recurrence for the observation. The link between the return time for the observation and the Hausdorff dimension of the image of the invariant measure is considered. We prove that when the decay of correlations is super polynomial, the recurrence rates for the observations and the pointwise dimensions relatively to the push-forward are equal.


Introduction
The famous Zermelo paradox reveals that the classical Poincaré recurrence theorem has some implications out of physical sense. Indeed, if we start with all the particles in one side of a box, nobody will ever see all the particles coming back in one side of our box at the same time. Nevertheless, if we focus on a few number of these particles, this event will appear after a reasonable time. In the same way, when we study a high dimensional dynamical system we might not know all the aspects of the evolution but only a part or certain quantities of the system. This might be due to the difficulty to study a high dimensional system, but also to the lack of interest of an over-detailed description.
Recently, Ott and York tried to elaborate some Platonic formalism of dynamical systems [11]. The reality, the dynamical system (X, T, µ), is only known through a measurement or observation, that is a function defined on X taking values in (typically) a lower dimensional space. The following result by Boshernitzan [4] about Poincaré recurrence falls in this frame. If we have a measure preserving dynamical system (X, T, µ) and an observable f from X to a metric space (Y, d) then whenever the α-dimensional Hausdorff measure is σ-finite on Y we have lim inf n→∞ n 1/α d (f (x), f (T n x)) < ∞ for µ-almost every x.
The main aim of this paper is to prove a refinement of (1) and a generalization of [2,12] for recurrence rates for observations.
In Section 2, we give the precise definition of the recurrence rates for the observations and state an upper bound in term of dimension (Theorem 2 which is proved in Section 3), then under an additional assumption we state our main result (Theorem 5 which is proved in Section 4), and finally, we analyze in the case of the Lebesgue measure the existence of the pointwise dimension for its smooth image (Theorem 9 which is proved in Section 5).

2.1.
Definitions and general inequality. Let (X, A, µ, T ) be a measure preserving system (m.p.s.) i.e. A is a σ-algebra, µ is a measure on (X, A) with µ(X) = 1 and µ is invariant by T (i.e µ(T −1 A) = µ(A) for all A ∈ A) where T : X → X.
Let f : X → Y be a function, called observable (we will specify the space X and Y later). We introduce the return time for the observation and its associated recurrence rates.
Definition 1. Let f : X → Y be a measurable function, we define for x ∈ X the return time for the observation: where B(x, r) is the ball centered in x with radius r. We then define the lower and upper recurrence rate for the observation: We also define for p ∈ N the p-non-instantaneous return time for the observation: Then we define the non-instantaneous lower and upper recurrence rates for the observation: .
we denote by R f (x) the value of the limit.
The lower and upper pointwise or local dimension of a Borel probability measure ν on Y at a point y ∈ Y are defined by The pushforward measure f * µ(.) := µ(f −1 (.)) is a probability measure on Y and we define the lower and upper pointwise dimension for the observations with respect to µ at a point x ∈ X by ). If they are equal, we denote by d f µ (x) the common value.
Theorem 2. Let (X, A, µ, T ) be a m.p.s, let f : X → R N be a measurable function. Then This result is satisfactory in the sense that it holds for any dynamical system and observation. Moreover, under natural assumptions we will show that the equality is true. Still, these inequalities may be strict, the caricatural example is when T is the identity map.
Example 3. Let (Ω, F, P) be a probability space together with a P-preserving map θ and Y ⊂ R N a Borel set. The family (F ω ) ω∈Ω is called a random transformation, where for each ω, F ω is a map from Y to Y such that the map (w, y) → F ω (y) is F × B(R N )measurable. The map T : X = Ω × Y → X defined by T (ω, y) = (θω, F ω (y)) is called a skew product transformation. Let M P (X, T ) be the set of T -invariant probability measure having the marginal P on Ω. For any µ ∈ M P (X, T ), Theorem 2 applies with f the projection on Y , and gives an upper bound for the time needed by a typical random orbit F θ k ω • ... • F θω • F ω (y) to come back close to its starting point y.

2.2.
Poincaré recurrence for observations. From now on, let assume that X is a metric space and A is its Borel σ-algebra. We can then introduce the decay of correlations: Definition 4. (X, T, µ) has a super-polynomial decay of correlations if, for all φ, ψ Lipschitz functions from X to R and for all n ∈ N * , we have: with lim n→∞ θ n n p = 0 for all p > 0 and where . is the Lipschitz norm.
The main result of our paper is: Taking the identity function for f , we recover the result of [2] and [12] under weaker assumptions. The main assumption of the theorem about decay of correlations is satisfied in a variety of systems with some hyperbolic behavior and studied in an abundant literature (e.g. [13,5,1]). Definition 6. We say that a probability measure ν is exact dimensional if there exists a constant d ν ∈ R such that It is well known that in this case many notion of dimension coincide (see Section 5.1 for details). In particular the Hausdorff dimension dim H ν satisfies Remark. We have the equivalence Proof of Corollary 8. If dim H f * µ = 0, then the conclusion follows from Theorem 2 and Proposition 7. In the general case, it is just a combination of Theorem 5 and Proposition 7.
This is a non trivial result because the image measure f * µ may be quite complicated and rather counter intuitive. Already in the one dimensional case, there exists f ∈ C ∞ (R, R) such that f ({f ′ = 0}) is an uncountable set of dimension 0 and f * (Leb| {f ′ =0} ) is a non null and non atomic measure. We emphasize that Theorem 9 applies to any C ∞ function, and not only for generic functions. This is essential in applications, where we are mostly interested in particular observables. Proof of Corollary 10. We apply Theorem 8 and Theorem 9 when d f µ > 0. When d f µ = 0, we use Theorem 2.

2.3.
On the necessity of the non-instantaneous recurrence rate. In this part, we give a simple example which illustrates the utility of non-instantaneous return times.
Let Ω := {0, 1} N and σ be the shift on Ω. Fix some 1-approximable α ∈ R (e.g. [6] for a nice perspective) i.e. δ(α) = 1 where Let ν be an invariant ergodic probability measure on Ω. Fix some measurable A ⊂ Ω such that 1 > ν(A) > 0 and set ϕ: Let T 1 denote the 1-dimensional torus and define on X := Ω × T 1 the map Let Leb be the Lebesgue measure on T 1 . We consider the T -invariant probability measure µ := ν ⊗ Leb. We examine below the recurrence rate of the system (X, T, µ) for the observable f given by the projection on the second variable i.e.
f : First, we need the following obvious result on the pushforward measure: since f * µ = Leb and the local dimension of the Lebesgue measure is one, the measure f * µ is exact dimensional and satisfies So, for all r > 0, τ f r (x) = 1 and then R f i (x) = 0.
We therefore need to introduce the non-instantaneous return time to avoid this kind of problem.
Proof. For k ∈ N and ω ∈ Ω, let q k (ω) : Since ν is ergodic, the Poincaré Recurrence Theorem gives, for ν-almost every ω ∈ Ω So, for ν-almost every ω ∈ Ω, we can choose p sufficiently large such that p ≥ k 0 and Thus for µ-almost every x = (ω, y) ∈ X, we have This is true for all ε > 0, thus R f (x) ≥ 1. The conclusion follows from Theorem 2 and equation (3).
Remark. We point out that indeed our example fulfills the conditions of Corollary 8 when, for example, ν is a Gibbs measure [7].

Majoration of the recurrence rate for measure preserving systems
The basic strategy of the proof of Theorem 2 follows [2]. We recall that any probability measure on R N is weakly diametrically regular [2]: Definition 13. A measure µ is weakly diametrically regular (wdr) on the set Z ⊂ X if for any η > 1, for µ-almost every x ∈ Z and every ε > 0, there exists δ > 0 such that if r < δ then µ (B (x, ηr)) ≤ µ (B (x, r)) r −ε .
Proof of Theorem 2. The measure f * µ is weakly diametrically regular on R N . We can remark that the function δ(f (·), ε) in the previous definition can be made measurable for every fixed ε. Let us fix ε > 0 and choose δ > 0 sufficiently small to have µ(G) > µ( For all r > 0, λ > 0, p ∈ N and x ∈ X we define the set 4r). Markov's inequality gives: Since τ f 4r,p (y, x) is bounded by the p th return time of y in the set f −1 B(f (x), 4r), by Kac's lemma we have: Using (7) and (8), we have: If d(f (x), f (y)) < 2r then Let C ⊂ f (G) a maximal 2r-separated set for f (G).
since f * µ is wdr and with η = 4 ≤ p r ε according to the definition of C.

Finally:
n,e −n <δ Then, thanks to the Borel-Cantelli lemma, for µ-almost every x ∈ G for any n sufficiently large. Then Observing that for all a > 0 we have: and since ε can be chosen arbitrarily small , we have the result if we take the limit inferior or the limit superior and then the limit over p in (11).

Recurrence rate and dimension for mixing systems
Despite some similarities with [12], we emphasize that the proof of Theorem 5 is relatively different. In particular we make no assumption on the entropy of the system.
Since we can choose ε arbitrarily small, the lemma is proved.

5.
Dimensions of the smooth image of Lebesgue measure 5.1. Hausdorff and packing dimensions. In this section, we recall the notion of Hausdorff dimension, packing dimension and pointwise dimension and the link between each other (see [9] for more details).
Let (X, d) be a metric space. Let U be a non-empty set, its diameter is Let E be a subset of X and s ≥ 0, for δ > 0, we define: We then define the Hausdorff s-dimensional outer measure of E as There exists a unique t such that H s (E) = ∞ if s < t and H s (E) = 0 if s > t which is called the Hausdorff dimension of E i.e.
If µ is a probability measure on X, we define the Hausdorff dimension of µ Remark. We warn the reader that this definition of the Hausdorff dimension of a measure differs from the one given by Falconer [9] but it is the most used in Ergodic Theory.
Given ε > 0, the collection {B(x i , r i )} i∈I is called a ε-packing of E if I is a finite or countable set, for all i ∈ I we have x i ∈ E, r i ≤ ε and the balls are disjoints. For s ≥ 0, we write P s ε (E) = sup i∈I (r i ) s : {B(x i , r i )} i∈I is a ε-packing of E and P s 0 (E) = lim ε→0 P s ε (E).
We then introduce the s-dimensional packing outer measure and the packing dimension of E is defined as Hausdorff dimension For a probability measure µ, we also have a packing dimension of µ There is a link between Hausdorff dimension, packing dimension and pointwise dimension: Proposition 17. Assume that X ⊂ R N for some N , and There is a decomposition {A i } i∈N of C κ such that for each i ∈ N there exist two subspaces This decomposition will be instrumental to prove an analogue result but for the packing dimension.
Lemma 19. If f ∈ C ∞ (R M , R N ) then the packing dimension of the critical set satisfies Let j ∈ J and l ∈ J, in (24), we take S := {x j , x l }, then . If ε is sufficiently small (depending only on k), we have 1 2C (r j + r l ) ≤ 1 − 1 2 This inequality holds for the packing measure computed with the metric d i and thus this is also true (possibly with another constant) for the packing measure computed with the euclidean metric d since they are equivalent. Therefore Finally, taking a sequence of compacts K n such that R M = ∪ n∈N K n , we obtain: On the other hand, by definition of d ν , ∀x ∈ Z, ∃J x ⊂ R + with 0 ∈ J x , such that ∀r ∈ J x , ν (B(x, r)) ≥ r ρ . We notice that {B(x, r) , x ∈ Z , r ∈ J x ∩ [0, 1]} cover Z so, by Besicovitch covering Theorem, there exists a subcovering {B(x i , r i )} i∈I with I countable and m 0 a constant depending only on N such that Z ⊂ ∪ i∈I B(x i , r i ) and the multiplicicty of the subcovering is bounded by m 0 . Hence i∈I r ρ i ≤ i∈I ν (B(x i , r i )) ≤ m 0 ν(K) which implies dim H Z ≤ ρ. But this is in contradiction with the fact that dim H Z ≥ κ.