A note on shadowing properties

TL;DR

This paper proves that vector fields with the average or limit shadowing property in an isolated set have specific dynamical characteristics, such as no proper attractors or topological transitivity, extending previous results with direct proofs.

Contribution

It provides a direct proof that shadowing properties imply certain dynamical behaviors in isolated sets for $C^{1}$-vector fields, refining earlier results by Gu and Ribeiro.

Findings

Vector fields with average shadowing have no proper attractors in the isolated set.

Vector fields with limit shadowing are topologically transitive.

Shadowing property holds in the isolated set for these vector fields.

Abstract

Let $\mathfrak{X}^{1}(M)$ be the space of $C^{1}$-vector fields on $M$ endowed with the $C^{1}$-topology and let $\Lambda$ be an isolated set for a $X\in\mathfrak{X}^{1}(M)$. In this paper, we directly prove that every $X\in\mathfrak{X}^{1}(M)$ having the (asymptotic) average shadowing property in $\Lambda$ has no proper attractor in $\Lambda$. Our proof is a direct version of the results by Gu and Ribeiro. We also show that every $X\in\mathfrak{X}^{1}(M)$ having the (two-sided) limit shadowing property with a gap in $\Lambda$ is topologically transitive and has the shadowing property in $\Lambda$.