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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 set M V = { y ∈ M : V f ( y ) = V } is dense in Δ V = ⋃ ν ∈ V supp ( ν ) ¯ . In particular, if Δ max = ⋃ ν ∈ M inv ( f ) supp ( ν ) ¯ coincides with M , then M max = { y : V f ( y ) = M inv ( f ) } is residual in M . As 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. [ABSTRACT FROM AUTHOR]

We prove that the two-sided limit shadowing property is among the strongest known notions of pseudo-orbit tracing. It implies shadowing, average shadowing, asymptotic average shadowing and specification properties. We also introduce a weaker notion that is called two-sided limit shadowing with a gap and prove that it implies shadowing and transitivity. We show that those two properties allow to characterize topological transitivity and mixing in a class of expansive homeomorphisms and hence they characterize transitive (mixing) shifts of finite type. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to introduce the notion of ergodic shadowing for a continuous onto map which is equivalent to the map being topologically mixing and has the ordinary shadowing property. In particular, we deduce the chaotic behavior of a map with ergodic shadowing property. Moreover, we define some kind of specification property and investigate its relation to the ergodic shadowing property. [Copyright &y& Elsevier]