Shadowing and the continuity of omega-limit sets

TL;DR

This paper explores how shadowing properties influence the continuity of omega-limit sets in dynamical systems, establishing conditions for semicontinuity and chain continuity with illustrative examples.

Contribution

It provides necessary and sufficient conditions linking shadowing phenomena to the semicontinuity and chain continuity of omega-limit sets in dynamical systems.

Findings

Shadowable points are characterized as upper or lower semicontinuity points of omega-limit sets.

Lower semicontinuity of omega-limit sets is equivalent to chain continuity under global shadowing.

Examples demonstrate the theoretical results in various dynamical contexts.

Abstract

This paper examines the relationship between shadowing phenomena and the continuity properties of $\omega$-limit sets in dynamical systems. We give a necessary and sufficient condition for a shadowable point to be an upper (resp. a lower) semicontinuity point of $\omega$-limit sets. Assuming global shadowing, we show that the lower semicontinuity of $\omega$-limit sets is equivalent to the chain continuity. We also show that the lower semicontinuity of $\omega$-limit sets is equivalent to the chain continuity in a general setting. Several examples are given to illustrate the results.