Oriented and standard shadowing properties for topological flows

TL;DR

This paper establishes the equivalence of oriented and standard shadowing properties for certain topological flows and explores conditions under which the product of flows inherits the shadowing property.

Contribution

It proves the equivalence of shadowing properties for flows with finite singularities under stability conditions and characterizes when product flows have the shadowing property.

Findings

Oriented and standard shadowing are equivalent for flows with finite singularities under stability conditions.

Product flows inherit the shadowing property under specified conditions.

The results apply to flows with finite singularities that are Lyapunov stable or unstable.

Abstract

We prove that oriented and standard shadowing properties are equivalent for topological flows with finite singularites that are Lyapunov stable or Lyapunov unstable. Moreover, we prove that the direct product $\phi_1 \times \phi_2$ of two topological flows has the oriented shdowing property if $\phi_1$ with finite singuralities has the oriented shadowing property, while $\phi_2$ has the limit set consisting of finite singularities that are Lyapunov stable or Lyapunov unstable.