Shadowing and the continuity of omega-limit sets

TL;DR

This work investigates how shadowing interacts with the continuity of $ω$-limit sets for continuous maps on compact metric spaces. It introduces shadowable points and precise (upper and lower) semicontinuity criteria for $ω$-limit sets, proving that for shadowable $x$, $x\in USC(ω_f)$ iff $ω_f(x)=Ω_f(x)$ and $x\in LSC(ω_f)$ iff $ω_f(x)\subset ω_f(y)$ for all $y\in Ω_f(x)$. Under global shadowing, the authors show that lower semicontinuity of $ω$-limit sets is equivalent to chain continuity, and they provide general extensions with several corollaries and illustrative examples. The results sharpen prior global-shadowing conclusions, clarify the recurrence/minimal structure along chain components, and illuminate how shadowing constrains the stability of $ω$-limit sets in dynamical systems.

Abstract

This paper examines the relationship between shadowing phenomena and the continuity properties of $ω$-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 $ω$-limit sets. Assuming global shadowing, we show that the lower semicontinuity of $ω$-limit sets is equivalent to the chain continuity. We also show that the lower semicontinuity of $ω$-limit sets is equivalent to the chain continuity in a general setting. Several examples are given to illustrate the results.

Theorems & Definitions (54)

• Remark 1.1
• Remark 1.2
• Theorem 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.1
• Theorem 3.1
• Lemma 3.1