On the Shadowing Property and Shadowable Point of Set-valued Dynamical Systems

In this article, the authors introduce the concept of shadowable points for set-valued dynamical systems, the pointwise version of the shadowing property, and prove that a set-valued dynamical system has the shadowing property iff every point in the phase space is shadowable; every chain transitive set-valued dynamical system has either the shadowing property or no shadowable points; and for a set-valued dynamical system there exists a shadowable point iff there exists a minimal shadowable point. In the end, it is proved that a set-valued dynamical system with the shadowing property is totally transitive iff it is mixing and iff it has the specification property.     

Recurrence of Transitive Points in Dynamical Systems with the Specification Property

Let T:X → X be a continuous map of a compact metric space X. A point x ∈ X is called Banach recurrent point if for all neighborhood V of x, {n ∈ N:Tⁿ(x) ∈ V } has positive upper Banach density. Denote by Tr(T), W(T), QW(T) and BR(T) the sets of transitive points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points of (X, T). If (X, T) has the specification property, then we show that every transitive point is Banach recurrent and ∅≠W(T) ∩ Tr(T) _≠^⊂W^*(T) ∩ Tr(T) _≠^⊂ QW(T) ∩ Tr(T) _≠^⊂ BR(T) ∩ Tr(T), in which W^*(T) is a recurrent points set related to an open question posed by Zhou and Feng. Specifically the set Tr(T) ∩ W^*(T)\W(T) is residual in X. Moreover, we construct a point x ∈ BR\QW in symbol dynamical system, and demonstrate that the sets W(T), QW(T) and BR(T) of a dynamical system are all Borel sets.