Complexity of measure-induced maps

Abstract

A classical result by Glasner and Weiss states that a map $f$ has zero topological entropy if and only if its measure-induced map $f_{\ast }$, defined as the push-forward of a measure, does. In this work, we use generalized entropy to distinguish the complexity of these maps and prove that the measure-induced map is strictly more complex whenever the generalized entropy of the original map is nonzero and is not an abstract order of growth. Moreover, we show that the complexity of the measure-induced map is of at most polynomial type when the nonwandering set of $f$ is finite. For homeomorphisms of infinite spaces, this bound is attained, so measure-induced maps of Morse--Smale diffeomorphisms cannot be distinguished from one another.