Shadowing maps
Alfonso Artigue
TL;DR
The paper investigates shadowing maps for homeomorphisms on compact metric spaces by formalizing δ-pseudo-orbits and their shadowing points, and by developing a hierarchy of shadowing notions, including shift-invariant and L-shadowing variants. It adapts Bowen’s hyperbolic-bracket technique to construct self-tuning shadowing maps from hyperbolic brackets and applies these ideas to north–south dynamics, shift spaces, and cw-expansive settings, highlighting when shadowing maps exist and how they interact with dynamical invariants. The work connects local product structures, canonical coordinates, and bracket constructions to shadowing, expansivity, and topological stability, extending the classical Smale/Ruelle framework to broader contexts. Overall, it provides a coherent framework linking pseudo-orbits, shadowing maps, and bracket-induced dynamics to stability and recurrence in topologically hyperbolic and shift-like systems.
Abstract
This article is about the shadowing property of homeomorphisms on compact metric spaces and the map associating a point of the space to each pseudo-orbit, called 'shadowing map'. Based on some particular dynamical properties, as expansivity, we develop a brief theory and a hierarchy of such maps. We consider examples as odometers, shifts on infinite spaces, topologically hyperbolic homeomorphisms and north-shouth dynamics. We revisit a well-known technique for proving shadowing of expansive homeomorphisms with canonical coordinates due to R. Bowen, to obtain a shadowing map with the property we call 'self-tuning' from a hyperbolic bracket. This notion is introduced as part of the hierarchy of shadowing maps studied in this paper.