Ergodic properties of systems with asymptotic average shadowing property

In this paper, we explore a topological system f:M→Mf: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⊆Minv(f)V⊆Minv(f) coincides with some Vf(y)Vf(y). Moreover, we show that the set MV={y∈M:Vf(y)=V}MV={y∈M:Vf(y)=V} is dense in ΔV=⋃ν∈Vsupp(ν)¯. In particular, if Δmax=⋃ν∈Minv(f)supp(ν)¯ coincides with M  , then Mmax={y:Vf(y)=Minv(f)}Mmax={y:Vf(y)=Minv(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 ΔmaxΔmax (residual provided that Δmax=MΔmax=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.