Oriented and standard shadowing properties on closed surfaces

TL;DR

This paper demonstrates the equivalence of oriented and standard shadowing properties for certain flows on closed surfaces and classifies isolated singularities with these properties.

Contribution

It establishes the equivalence of shadowing properties on closed surfaces and characterizes singularities with these properties.

Findings

Oriented and standard shadowing are equivalent on closed surfaces with finite critical elements.

Isolated singularities with shadowing are either stable or have hyperbolic sectors.

The results apply to topological flows with specific nonwandering set structures.

Abstract

We prove that oriented and standard shadowing properties are equivalent for topological flows on closed surfaces with the nonwandering set consisting of the finite number of critical elements (i.e., singularities or closed orbits). Moreover, we prove that each isolated singularity of a topological flow on a closed surface with the oriented shadowing property is either asymptotically stable, backward asymptotically stable, or admits a neighborhood which splits into two or four hyperbolic sectors.