Orbital shadowing, $\omega$-limit sets and minimality

TL;DR

This paper investigates the behavior of pseudo-orbits in dynamical systems on compact Hausdorff spaces, demonstrating how they approximate omega-limit sets and characterizing minimal systems through shadowing properties.

Contribution

It generalizes existing shadowing results, showing that pseudo-orbits can trap omega-limit sets and characterizes minimal systems via pseudo-orbit properties.

Findings

Pseudo-orbits trap omega-limit sets within prescribed accuracy after uniform time.

Every system has the second weak shadowing property.

Minimal systems exhibit the strong orbital shadowing property.

Abstract

Let $X$ be a compact Hausdorff space, with uniformity $\mathscr{U}$, and let $f \colon X \to X$ be a continuous function. For $D \in \mathscr{U}$, a $D$-pseudo-orbit is a sequence $(x_i)$ for which $(f(x_i),x_{i+1}) \in D$ for all indices $i$. In this paper we show that pseudo-orbits trap $\omega$-limit sets in a neighbourhood of prescribed accuracy after a uniform time period. A consequence of this is a generalisation of a result of Pilyugin et al: every system has the second weak shadowing property. By way of further applications we give a characterisation of minimal systems in terms of pseudo-orbits and show that every minimal system exhibits the strong orbital shadowing property.