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arXiv · 2405.17574

Gluing-orbit property, local stable/unstable sets, and induced dynamics on hyperspace

Mayara Antunes·Bernardo Carvalho·Welington Cordeiro·José Cueto
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Abstract

We prove that local stable/unstable sets of homeomorphisms of an infinite compact metric space satisfying the gluing-orbit property always contain compact and perfect subsets of the space. As a consequence, we prove that if a positively countably expansive homeomorphism satisfies the gluing-orbit property, then the space is a single periodic orbit. We also prove that there are homeomorphisms with gluing-orbit such that its induced homeomorphism on the hyperspace of compact subsets does not have gluing-orbit, contrasting with the case of the shadowing and specification properties, proving that if the induced map has gluing-orbit, then the base map has gluing-orbit and is topologically mixing.

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Mayara Antunes, Bernardo Carvalho, Welington Cordeiro, José Cueto. 2024-05-27. Gluing-orbit property, local stable/unstable sets, and induced dynamics on hyperspace. https://arxiv.org/abs/2405.17574

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