Entropy and preimage sets

Abstract

We study the relation between topological entropy and the dispersion of preimages. Symbolic dynamics plays a crucial role in our investigation. For forward expansive maps, we show that the two pointwise preimage entropy invariants defined by Hurley agree with each other and with topological entropy, and are reflected in the growth rate of the number of preimages of a single point, called a preimage growth point for the map. We extend this notion to that of an entropy point for a system, in which the dispersion of preimages of an $\varepsilon$-stable set measures topological entropy. We show that for maps satisfying a weak form of the specification property, every point is an entropy point and that every asymptotically h-expansive homeomorphism (in particular, every smooth diffeomorphism of a compact manifold) has entropy points. Examples are given of maps in which Hurley's invariants differ and of homeomorphisms with no entropy points.