Topological Entropy, Entropy points and shadowing

TL;DR

This paper investigates topological entropy in non-compact, non-metrizable spaces, establishing that uniform shadowing implies positive entropy, extending known results from compact metric spaces.

Contribution

It extends the relationship between shadowing and entropy to more general spaces beyond compact metric spaces.

Findings

Shadowing property implies positive entropy in uniform spaces.

Extension of known compact space results to non-metrizable spaces.

Provides new insights into entropy behavior in non-compact settings.

Abstract

In this note we study some properties of topological entropy for non-compact non-metrizable spaces. We prove that if a uniformly continuous self-map $f$ of a uniform space has topological shadowing property then the map $f$ has positive uniform entropy, which extends the similar known result for homeomorphisms on compact metric spaces having shadowing property.