On averaged tracing of periodic average pseudo orbits

We propose a definition of average tracing of finite pseudo-orbits and show that in the case of this definition measure center has the same property as nonwandering set for the classical shadowing property. We also show that the average shadowing property trivializes in the case of mean equicontinuous systems, and that it implies distributional chaos when measure center is nondegenerate.

1. Introduction. A dynamical system is a pair (X, T ), where X is a compact metric space with a metric d and T : X −→ X is a continuous map. The notion of the average shadowing property was introduced in 1988 (see [2,3]) as a generalization of the shadowing property. The main motivation was a property suitable for dynamical systems obtained by random perturbations, where we cannot control error of pseudo-orbit in each iterate, but on average the error can be controlled. First examples of dynamical systems with the average shadowing property were obtained on manifolds in the class of Axiom A diffeomorphisms (see [2,14]) and their appropriately chosen random perturbations. Presently it is known that dynamical systems with the average shadowing property are much more common, since this property is a consequence of the specification property or its averaged variant called the almost specification property (e.g. see [6] and [18]). Since the average shadowing property in the system with a fully supported measure implies weak mixing (e.g. see [6,Theorem 4.3]), it is natural to expect that such systems will share some properties of systems with shadowing. One of standard properties in theory of shadowing (see [1]) is that shadowing of finite pseudo-orbits is equivalent to shadowing of infinite ones. It is also well known that when a dynamical system (X, T ) has the shadowing property, then its restriction to non-wandering set Ω(T ) has this property as well.
In the case of average pseudo-orbits a candidate to replace Ω(T ) is the measure center supp(X, T ), that is the smallest possible closed set with full measure for any invariant measure.
To deal with both problems, we define a potentially weaker version of average shadowing property which we call the average shadowing of periodic (or finite) average pseudo-orbits property (abbrev. FinASP). It is our attempt to define a notion which is similar to shadowing of finite pseudo-orbits, but involves averages. We show that this property behaves quite well, in the sense that (X, T ) has this property if and only so does (supp(X, T ), T ). We also show that a dynamical system with FinASP must have a trivial measure center, when it is mean equicontinuous and when it is not mean equicontinuous then it has in some sense rich dynamics, exhibiting distributional chaos.
The paper is organized as follows. Section 2 contains all basic definitions. In Section 3 we study main properties of FinASP, including its behavior on measure center and relations to the mean equicontinuity. In last Section 4 we deal with distributional chaos in dynamical systems with FinASP.

2.
Preliminaries. The set of real numbers, integers, natural numbers and nonnegative integers are denoted, respectively, by R, Z, N = Z∩(0, +∞) and N 0 = N∪{0}. If A is a set, then its complement is denoted A c and its closure A. The cardinality of a set A is denoted |A|.
Let M(X) denote the space of all Borel probability measures on X. A measure µ ∈ M(X) is invariant for T : X −→ X if µ(A) = µ(T −1 (A)) for any Borel set A ⊂ X. The classical Krylov-Bogolyubov theorem implies that every compact dynamical system (X, T ) has at least one such measure.
A subset A ⊂ X is measure saturated if for every open set U satisfying U ∩A = ∅, there exists an invariant measure µ such that µ(U ) > 0. The measure center of T which is denoted supp(X, T ) is the largest measure saturated subset. It is not difficult to verify that supp(X × X, T × T ) = supp(X, T ) × supp(X, T ).
When the map T is clear from the context, we simply write Λ . Similarly, we use the following simplified notation (for both Λ and Λ c ): Finally, we will denote the above sets restricted on a subset A of N 0 by (4) an asymptotic average pseudo-orbit (of T ) if A δ-average-pseudo-orbit is periodic if it is formed by a periodic sequence, i.e. x i = x i+s for some integer s > 0 and all i ∈ N 0 . We use the above notions of approximate trajectories to define three main shadowing properties of the paper. Definition 2.2. A dynamical system (X, T ) has (1) the average shadowing property (abbrev. ASP) if for any ε > 0 there exists δ > 0 such that every δ-average-pseudo-orbit {x i } ∞ i=0 is ε-shadowed on average by a point z ∈ X, i.e. lim sup (2) the average shadowing of periodic (or finite) pseudo-orbits property (abbrev. FinASP) if for any ε > 0 there exists δ > 0 such that every periodic δ-average pseudo-orbit is ε-shadowed on average by a point in X; (3) the asymptotic average shadowing property (abbrev. AASP) if every asymptotic average pseudo-orbit {x i } ∞ i=0 is asymptotically shadowed on average by a point z ∈ X, i.e.  The following simple fact shows that there is no much difference between finite and periodic average δ-pseudo orbits. The proof is simple and left to the reader.
and diam(X)/s < δ/2, then periodic sequence given by x js+i = w i for all j ∈ N 0 and 0 ≤ i < s is a δ-average pseudo-orbit.
A δ-chain from x to y is a finite δ-pseudo-orbit between these points, that is, a sequence x 1 , . . . , x n+1 such that d(T (x i ), x i+1 ) < δ for all i = 1, . . . , n, and x 1 = x, x n+1 = y. A map is chain transitive if for any δ > 0 and any two points x, y ∈ X there is a δ-chain from x to y. Chain transitivity is a natural generalization of transitivity. It is clear that if a map is chain transitive then it must be surjective as well. There is a surprising result [13] which shows that chains do not distinguish between total transitivity and mixing. Precisely speaking, if (X, T n ) is chain transitive for all n > 0 then it is chain mixing, that is, for any x, y ∈ X and δ > 0 there is N > 0 such that there is a δ-chain from x to y consisting of exactly n elements for every n > N . Recently, Wu et al. [19] proved that this also holds for iterated function systems. A point x ∈ X is chain recurrent if for every δ > 0 there exists a δ-chain from x to x. The set of all chain recurrent points is denoted CR(T ). By compactness, it is not difficult to check that CR(T ) is a closed set and T (CR(T )) = CR(T ).

2.2.
Furstenberg families and tracing. A (Furstenberg) family F is a collection of subsets of N 0 which is upwards hereditary, that is The dual family of F is Definition 2.3. A dynamical system (X, T ) has (ergodic) F -shadowing property if, for any ε > 0 there is δ > 0 such that every δ-ergodic pseudo-orbit ξ is F -εshadowed by some point z ∈ X, i.e.
In the special case of F =M 1 (resp., F = M 0 and M 1/2 ), we say that (X, f ) has the ergodic shadowing property (resp., d-shadowing property and d-shadowing property).
2.3. The (almost) specification property. The specification property was first introduced by Bowen [4]. It is one of the strongest mixing properties that can be expected from a dynamical system. Recently, Pfister and Sullivan introduced in [12] a property called the g-almost product property, which generalizes Bowen's specification in terms of average tracing. Inspired by [12], Thompson in [16] modified slightly this definition and proposed to call it the almost specification property, which in turn generalizes the notion of specification. In this paper, we adopt the concepts of [16]. First, we introduce some auxiliary notation.
Using the above notation, we are able to present the definition of the almost specification property.
Definition 2.4. A dynamical system (X, T ) has the almost specification property if there exists a mistake function g and a function k g : (0, ∞) −→ N such that for any m ≥ 1, any ε 1 , . . . , ε m > 0, any points x 1 , . . . , x m ∈ X, and any integers n 1 ≥ k g (ε 1 ), . . . , n m ≥ k g (ε m ) setting n 0 = 0 and one can find a point z ∈ X such that for every j = 1, . . . , m, In other words, the appropriate part of the orbit of z, ε j -traces with at most g(ε j , n j ), mistakes the orbit of x j , j = 1, . . . , m.

Distributional chaos. A very important generalization of the concept of Li-
Yorke chaos is distributional chaos which was introduced by Schweizer and Smítal in [15]. Let (X, T ) be a dynamical system. For any pair x, y ∈ X and any n ∈ N, Define the lower and upper distributional functions, R −→ [0, 1] generated by T , x and y, as x,y (t), respectively. Both functions Φ x,y and Φ * x,y are nondecreasing. For any pair x, y ∈ X, if Φ * x,y ≡ 1 and Φ x,y (η, T ) = 0 for some η > 0, then (x, y) is called a distributionally η-chaotic pair of T . A subset D ⊂ X containing at least two points is called a distributionally η-scrambled set for some η > 0 if any pair of its distinct points is distributionally η-chaotic. A dynamical system is said to be distributionally ηchaotic if there exists an uncountable distributionally η-scrambled set.
3. FinASP. In the case of standard definition of shadowing property, it is not much different situation if we know that every finite δ-pseudo orbit can be traced or we know that tracing is possible for infinite δ-pseudo orbits. Namely, using compactness it is easy to see that both approaches are equivalent. This argument does not work, however, in the case of averaged versions of shadowing. It is also not completely clear how finite pseudo-orbits and their tracing should be defined. A possible approach is introduced below. It is easy to see that this new notion is not stronger than ASP, however presently we still do not know how whether both definitions are equivalent.
3.1. FinASP and measure center. It is known that if (X, T ) has the shadowing property, then also its restriction to the nonwandering set (Ω(T ), T ) has the shadowing property. A natural candidate to replace Ω(T ) in the case of average shadowing is the measure center. We prove that this intuition is true in the case of FinASP. Unfortunately our argument is not sufficient to work for ASP. Proof. (⇐=). Without loss of generality, assume that diam(X) = 1. For any ε > 0, let 0 < δ < ε/4 be such that every periodic δ-average pseudo-orbit contained in supp(X, T ) is ε/4-traced on average by a point in supp(X, T ). Applying [6,Lemma 5.4] implies that there is 0 < δ 1 < δ/4 such that for every δ 1 -average-pseudo-orbit For any periodic δ 1 -average pseudo-orbit {x i } ∞ i=0 ⊂ X, take a δ/4-average pseudoorbit {y i } ⊂ supp(X, T ) satisfying condition (2) and let p be a period of {x i } such that at least p > 12/δ. There is an integer K > p such that for any k ≥ K, Take a sequence ξ = ξ 0 ξ 1 ξ 2 · · · = y 0 y 1 · · · y Kp−1 y 0 y 1 · · · y Kp−1 · · · .
For any n ≥ Kp and any j ∈ N 0 , applying (3), it can be verified that This shows that ξ is a periodic δ-average pseudo-orbit contained in supp(X, T ). Then there is z ∈ supp(X, T ) such that Combining this with (3) and (4), it follows that for any n ≥ Kp, and therefore lim sup (=⇒). Suppose that there is ε > 0 such that for every δ > 0 there is a periodic δaverage pseudo-orbit contained in supp(X, T ) which cannot be ε-traced in average by any point in supp(X, T ). Let δ > 0 be provided to ε/2 > 0 by FinASP and let {x i } ∞ i=0 ⊂ supp(X, T ) be a periodic δ-average pseudo-orbit such that for every x ∈ supp(X, T ) we have lim sup n→∞ By our assumption, z ∈ supp(X, T ). Take γ < ε/8 and let p be the least period of {x i } ∞ i=0 . There is an integer N > 0 such that p diam(X)/N < γ. Since lim sup n→∞ Let M = max i=1,...,s n i and let η > 0 be such that if d(x, y) < η then d(T j (x), T j (y)) < γ for j = 0, . . . , M . Note that T | supp(X,T ) is onto, hence the set U i ) is a neighborhood of supp(X, T ). We assume that η is sufficiently small, so that where B η (A) is the η-neighborhood of a subset A of X, i.e., B η (A) = x∈A {y ∈ X : d(x, y) < η}. Let ξ < η/4 be such that 2ξ is a Lebesgue number for each of the covers T −k ({U 1 , . . . , U s }), k = 0, . . . , p of the set B η (supp(X, T )). Observe that A = X \ B ξ (supp(X, T )) is a universally null set such that dist(A, supp(X, T )) > 0, hence there is a setK of density d(K) = 1 such that T j (z) ∈ B ξ (supp(X, T )) for every j ∈K.
Define a sequence {t j } ∞ j=0 of integers in the following way. Put t 0 = 0 and next, if T tj (z) ∈ A then we put t j+1 = t j + 1. If T tj (z) / ∈ A then there is y j ∈ supp(X, T ) such that d(y j , T tj (z)) < ξ. By the definition of Lebesgue number and condition 2ξ < η, there is 1 where 0 ≤ k < p is a number such that x tj +i+k = x i for every i ≥ 0. Then we put t j+1 = t j + n i . Since sequence {t j } ∞ j=0 is syndetic, there is a set K ⊂ N such that d(K) = 1 and {t j : j ∈ K} ⊂K. Note that there exists j ∈ K such that which is a contradiction. Let j ∈ K be an integer such that There is a unique 0 ≤ k < p such that {x i } ∞ i=0 = {x i } ∞ i=tj +k . Since T tj (z) ∈ A, by the definition of t j there are y = y j ∈ supp(X, T ) and 1 ≤ m ≤ s such that Then for r = 0, . . . , n m − 1 we have d(T tj +r (z), T r (y)) < γ and hence which is a contradiction. The proof is completed.
By the fact that (supp(X, T ), T ) is a surjection and statement of Theorem 3.1, a slight change in the proof of [6, Lemma 3.1] leads to the following result. We leave details to the reader.

3.2.
A remark on the ergodic shadowing property. In [5, Theorem A], Fakhari and Gane proved that for a surjective dynamical system, ergodic shadowing is equivalent to pseudo-orbital specification. Now, we shall show that ergodic shadowing implies almost specification. Before we are able to present a simple proof, we need the following facts from [18].   Proof. Clearly, the ergodic shadowing property implies the thick shadowing property, i.e., F t -shadowing property. According to the proof of [10,Theorem 4.5], it follows that every point in CR(T ) can be presented as a limit of minimal points in X. This, together with [10, Theorem 4.5 (1)], implies that the measure center of (X, T ) is CR(T ), i.e., supp(X, T ) = CR(T ) and that (CR(T ), T ) has the shadowing property. It follows from Theorem 3.3 and Corollary 3.2 that (CR(T ), T ) is chain mixing. Applying [5,Theorem A] yields that (CR(T ), T ) has the specification property as (CR(T ), T ) is surjective. Hence the result follows by Theorem 3.4.
As a direct consequence of the proof of Theorem 3.5, we have Corollary 3.6. If a dynamical system (X, T ) has the ergodic shadowing property, then (supp(X, T ), T ) has the specification property. Note that if (X, T ) is chain transitive on CR(T ) and x has well defined backward extension, i.e. there exists a sequence {x −i } ∞ i=0 such that x = x 0 and T (x i−1 ) = x i for every i ≤ 0 then x ∈ CR(T ). Then the only situation when (X, T ) with the ergodic shadowing property has the almost specification property but does not have the specification property occurs when there are starting points in X, that is points

3.3.
Mean equicontinuity and FinASP. In [7], the authors introduced the notion of mean equicontinuity which is equivalent to mean-L-stability and proved that every ergodic invariant measure of a mean equicontinuous system has discrete spectrum. According to them, a dynamical system (X, T ) is called mean equicontinuous if for every ε > 0, there exists δ > 0 such that whenever x, y ∈ X with d(x, y) < δ, A pair (x, y) ∈ X×X is proximal if lim inf n→∞ d(T n (x), T n (y)) = 0. A dynamical system (X, T ) is called proximal if any pair of two points in X is proximal. If (x, y) ∈ X × X is not proximal, then it is said to be distal. Clearly each equicontinuous system is mean equicontinuous system. For example it can be periodic orbit or an odometer (systems having classical shadowing property) or irrational rotation of the circle (which does not have shadowing property). Another example of mean equicontinuous map (which is not equicontinuous) can be the circle map induced from [0, 1] by x → x 2 with both endpoints 0, 1 identified. Clearly this circle map has the unique invariant measure concentrated on the fixed point 0, hence its measure center is trivial.
Here, we shall show that for a mean equicontinuous dynamical system, almost specification, AASP, ASP, FinASP, d-shadowing and d-shadowing are all equivalent. Strictly speaking, we prove that if any of the above properties is present in mean equicontinuous dynamical system, then it is trivial from measure theoretic point of view.
Theorem 3.8. If a dynamical system (X, T ) is mean equicontinuous and has Fi-nASP, d-shadowing or d-shadowing, then (X, T ) has trivial measure center. In particular, (X, T ) is proximal.
Proof. As a preliminary step we prove that (X, T ) is proximal. First, we consider the case of FinASP. Assume on the contrary that (X, T ) is not proximal. Let x, y be a distal pair and put ε = 1 3 inf{d(T n (x), T n (y)) : n ∈ N 0 } > 0. Let δ > 0 be such that if d(p, q) < δ then lim sup Let η > 0 be such that every periodic η-average pseudo-orbit is δ/3-traced on average by some point. Take N ∈ N sufficiently large, so that 3 diam(X)/N < η.
Then the periodic sequence is a periodic η-average pseudo-orbit because if we fix any n ≥ 2N and k ∈ N 0 , then there is j such that jN ≤ n < (j + 1)N and then 1 n Let z ∈ X be a point which δ/3-traces ξ on average. Suppose that for every k ∈ N 0 and we either have d(T kN +j (z), T j (x)) ≥ δ for every j = 0, . . . , N − 1 or d(T kN +j+N (z), T j+N (y)) ≥ δ for every j = 0, . . . , N − 1. Then for each s > 2 and each 2sN ≤ n < 2(s + 1)N we have which is impossible. Therefore there exist K ≥ 0 and 0 ≤ i, j < N such that This, together with (5), implies that lim sup and therefore 2ε ≥ lim sup which is a contradiction. Indeed (X, T ) is proximal. Next assume that supp(X, T ) is nontrivial. By Theorem 3.1 without loss of generality we may assume that X = supp(X, T ). Let p be the unique fixed point in X (as (X, T ) is proximal) and let U be a nonempty open set such that dist(p, U ) := inf{d(p, y) : y ∈ U } = η > 0. By ergodic decomposition theorem there is an ergodic measure µ such that µ(U ) = τ > 0. Let x be a generic point for µ. Then there exists N > 0 such that for every n ≥ N , we have 1 n {0 ≤ i < n : T i (x) ∈ U } > τ /2. Take any ε > 0 such that ε < τ η/6.
Increasing N if necessary and repeating previous arguments, we can find a point z, k ≥ 0 and 0 ≤ i, j < N such that d(T kN +i (z), T i (x)) < ε and d(T kN +N +j (z), T N +j (p)) < ε.
Similarly as before we obtain that which is a contradiction. The proof of the case of d-shadowing or d-shadowing is analogous, with the main difference in definition of pseudo-orbit ξ. Let m 0 = m 0 = 0, m 1 = 2, m 1 = 1, m n = 2 m1+···+mn−1 and m n = m n−1 +n for n ≥ 2 and take We leave verification of the details to the reader. Corollary 3.9. Let (X, T ) be a mean equicontinuous dynamical system. Then, the following statements are equivalent: (1) (X, T ) has the almost specification property; (2) (X, T ) has AASP; (3) (X, T ) has ASP; (4) (X, T ) has FinASP; (5) (X, T ) has the d-shadowing property; (6) (X, T ) has the d-shadowing property. (7) (X, T ) has trivial measure center.
4. FinASP and distributional chaos. In [11], Oprocha andŠtefánková proved that a dynamical system with the specification property and with a pair of distal points is distributionally chaotic. Very recently, Wang et al. [17] proved that a dynamical system with AASP having a distal pair is distributionally chaotic. Here, we shall show that this also holds for FinASP which is weaker that AASP. By [6,Lemma 8.2], it follows that there exists a proximal dynamical system with the average shadowing property, hence we need an additional assumption of distal pair to have a chance to prove distributional chaos.
Proof. By [8, Corollary 69], we obtain (1) ⇐⇒ (3). It is proved in [9] that proximal system does not have distributionally chaotic pairs. Therefore, it suffices to check that (2) =⇒ (1). Let (p, q) be a distal pair. Then ω(T × T, (p, q)) ∩ ∆ = ∅, therefore there is a minimal set M for T × T such that M ∩ ∆ = ∅. Taking any (u, v) ∈ M we see that (u, v) is a distal pair, and since (u, v) is uniformly recurrent for T × T , both u, v are uniformly recurrent. But each minimal set is contained in measure center, hence u, v ∈ supp(X, T ), that is supp(X, T ) contains a distal pair.
Fix a distal pair p , q ∈ supp(X, T ) and let η = 1 3 inf{d(f n (p ), f n (q )) : n ∈ N 0 }. For M = 1, 2, . . ., we define sets P M , S M ⊂ supp(X, T )×supp(X, T )in the following way: (x, y) ∈ S M (resp. (x, y) ∈ P M ) if there exist n > M and a > 0 such that Φ (n) . It is not difficult to verify that each set P M and S M is open in supp(X, T ) × supp(X, T ). We are going to show that each set S M and P M is also dense in supp(X, T ) × supp(X, T ).
Let µ be an invariant measure such that supp(µ) = supp(X, T ) and let µ 2 = µ×µ be the product measure on X × X.
On the other hand we have it is easy to see that (T k (z 1 ), T k (z 2 )) ∈ P M which implies (U × V ∩ (supp(X, T ) × supp(X, T ))) ∩ P M = ∅.
Next, replacing definition of ξ in (6) by ξ i = (T × T ) i (u, v), i ∈ [0, n), (T × T ) i (p , q ), i ∈ [n, (3M + 1)n), and repeating previous argument, we easily obtain that there is a positive integer k such that (T k (z 1 ), T k (z 2 )) ∈ S M , i.e., (U ×V ∩(supp(X, T )×supp(X, T )))∩S M = ∅. Since all the sets P M , S M are open, we obtain that the set is residual in supp(X, T ) × supp(X, T ). Note that every pair contained in R is distributionally η-chaotic, hence the result follows by application of Mycielski theorem.