Asymptotic average shadowing property, almost specification property and distributional chaos

Abstract

It is proved that a nontrivial compact dynamical system with asymptotic average shadowing property (AASP) displays uniformly distributional chaos or distributional chaos in a sequence. Moreover, distributional chaos in a system with AASP can be uniform and dense in the measure center, that is, there is an uncountable uniformly distributionally scrambled set consisting of such points that the orbit closure of every point contains the measure center. As a corollary, the similar results hold for the system with almost specification property.

On averaged tracing of periodic average pseudo orbits

“…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.…”

On distributional chaos in non-autonomous discrete systems

“…Remark 4.7. The methods used in the proofs of Theorems 4.4 and 4.6 are motivated by those used in the proofs of Theorem 3.2 in [37] and Theorem 1 in [35], respectively.…”

“…The asymptotic average shadowing property was introduced in [10]. Recently, Wang et al proved that the asymptotic average shadowing property implies distributional chaos for a continuous map with two almost period points in a compact metric space [37]. Note that Xiong-chaos, introduced in [40], is induced by a topologically mixing map.…”

“…For ADSs, many elegant results about distributional chaos have been obtained [1,14,16,17,20,21,[34][35][36][37]42]. Most of them are about relations between distributional chaos and other concepts of chaos.…”