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

TL;DR

This paper investigates the gluing-orbit property in dynamical systems, showing its implications for stable/unstable sets, induced hyperspace dynamics, and conditions for topological mixing, with key results on space structure and orbit properties.

Contribution

It establishes new links between the gluing-orbit property and the structure of stable/unstable sets, as well as the behavior of induced hyperspace dynamics, highlighting differences with shadowing and specification properties.

Findings

Local stable/unstable sets contain perfect subsets under gluing-orbit property.

Positively countably expansive homeomorphisms with this property are single periodic orbits.

Induced hyperspace maps may lack gluing-orbit, contrasting with shadowing and specification cases.

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.