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We call that a flow has the orbital shadowing property if for any ε > 0 there is d > 0 such that, for any d -pseudo orbit g ( t ) there exists an orbit Orb ( x ) satisfying d i s t H ( g ( t ) ‾ , Orb ( x ) ‾ ) < ε . In this paper, we show that the C 1 -interior of the set of 3-dimensional flows having the orbital shadowing property is contained in the set of Ω-stable 3-flows. [ABSTRACT FROM AUTHOR]

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]