On ball expanding maps

TL;DR

This paper investigates the properties of ball expanding maps on compact metric spaces, revealing their dynamical complexity, entropy characteristics, and chain recurrence behavior, with implications for mixing and structure of the system.

Contribution

It provides new insights into the structure and entropy of ball expanding maps, including conditions for density of periodic points and finiteness of chain components.

Findings

Periodic points are dense in the chain recurrent set.

Zero entropy implies a finite chain recurrent set.

Connected spaces lead to mixing behavior.

Abstract

We study the dynamical properties of ball expanding maps, a class of continuous self-maps defined on compact metric spaces. For a ball expanding map, we show that: (1) the set of periodic points is dense in the chain recurrent set; (2) if the topological entropy of the map is zero, then the chain recurrent set is finite; (3) the map has only finitely many chain components; (4) if the space is perfect, then the topological entropy of the map is positive; and (5) if the space is connected, then the map is locally eventually onto and hence mixing. Several examples are also provided.