A note on shadowing properties

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 $\Lambda$.

9 pages. arXiv admin note: text overlap with arXiv:1305.2977 by other authors