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arXiv · 1609.09439

A note on shadowing properties

Hahng-Yun Chu·Dae Hwan Goo·Se-Hyun Ku
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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 $\Lambda$.

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Hahng-Yun Chu, Dae Hwan Goo, Se-Hyun Ku. 2016-09-29. A note on shadowing properties. https://arxiv.org/abs/1609.09439

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