Ergodic properties of systems with asymptotic average shadowing property

Abstract

In this paper, we explore a topological system f : M → M with asymptotic average shadowing property and extend Sigmund's results from Bowen's specification case. We show that every non-empty, compact and connected subset V ⊆ M inv (f ) coincides with some V f (y). Moreover, we show that the setAs consequences, we have several corollaries. One is that every invariant measure has generic points. Another is that the set consisting of those points for which the Birkhoff ergodic average does not exist (called irregular set) is either dense in Δ max (residual provided that Δ max = M ) or empty. In particular, we give an uncountable division of irregular set and obtain a refined characterization which can be as a substantial generalization of [18] with a different method. Remark that for systems with asymptotic average shadowing property, this article studies Birkhoff ergodic average from topological viewpoint but it is still unknown how about the theory from the perspective of dimension theory.

Generic points for dynamical systems with average shadowing

“…Since it is proved in [19] that the almost specification property implies the average shadowing property and in Section 4 we have shown that the same follows from the weak specification property, Theorem 21 immediately implies the following corollary, which can be also found in [10] (with a different proof).…”

“…As a corollary we get that if a system has the asymptotic average shadowing property (a notion introduced by Gu in [17], see below), then every invariant measure has a generic point, because in such a system every average asymptotic pseudo-orbit is followed by the orbit of some point. The same result is proved independently in [10] via a direct construction. We believe that using the Besicovitch pseudodistance provides a new perspective and leads to short proofs extending the theory (see the proof of Theorem 21 and Example 23).…”

“…It was proved in [27] that (10) implies that Y is a topologically mixing and proximal system with positive topological entropy. Let Y n = X (s1,t1) ∩ .…”

Entropy of irregular points for some dynamical systems

“…This result was generalized to maps satisfying the almost specification property ( [22], see also [16] for the context of flows) or the orbit gluing property (see [14] and further references therein). Moreover, by [6], assuming the asymptotic average shadowing property (AASP for short) and a certain condition on the measure center, the set X(f, ϕ) is either residual or empty. It is not known if AASP in general implies that X(f, ϕ), if nonempty, has full entropy.…”

Average Chain Transitivity and the Almost Average Shadowing Property

“…There are now various variants of this concept which exist in the literature, for instance, one can refer [2,3,4,5,10] and some equivalences are obtained for expansive homeomorphisms having the shadowing property on compact metric spaces [7,8,12].…”