On ball expanding maps

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.