Positive Entropy Through Pointwise Dynamics

TL;DR

This paper introduces pointwise properties in topological dynamics and provides conditions under which systems exhibit positive topological entropy, especially when approximated by maps with shadowing.

Contribution

It offers new pointwise criteria for positive entropy and links approximation by shadowing maps to entropy properties.

Findings

Pointwise properties can determine positive topological entropy.

Sufficient conditions are established for systems approximated by shadowing maps.

The results connect local dynamics to global entropy characteristics.

Abstract

We define some pointwise properties of topological dynamical systems and give pointwise conditions for such a system possesses positive topological entropy. We give sufficient conditions to obtain positive topological entropy for maps which are approximated by maps with the shadowing property in a uniform way.