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arXiv · 2002.11199

Expansivity and unique shadowing

Chris Good·Sergio Macías·Jonathan Meddaugh·Joel Mitchell·Joe Thomas
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Abstract

Let $f\colon X\to X$ be a continuous function on a compact metric space. We show that shadowing is equivalent to backwards shadowing and two-sided shadowing when the map $f$ is onto. Using this we go on to show that, for expansive surjective maps the properties shadowing, two-sided shadowing, s-limit shadowing and two-sided s-limit shadowing are equivalent. We show that $f$ is positively expansive and has shadowing if and only if it has unique shadowing (i.e.\ each pseudo-orbit is shadowed by a unique point), extending a result implicit in Walter's proof that positively expansive maps with shadowing are topologically stable. We use the aforementioned result on two-sided shadowing to find an equivalent characterisation of shadowing and expansivity and extend these results to the notion of $n$-expansivity due to Morales.

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Chris Good, Sergio Macías, Jonathan Meddaugh, Joel Mitchell, Joe Thomas. 2020-02-25. Expansivity and unique shadowing. https://arxiv.org/abs/2002.11199

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