Conjugacy between piecewise monotonic functions and their iterative roots

Abstract

It was proved that all continuous functions are topologically conjugate to their continuous iterative roots in monotonic cases. An interesting problem reads: Does the same conclusion hold in non-monotonic cases? We give a negative answer to the problem by presenting a necessary condition for the topological conjugacy, which helps us construct counter examples. We also give a sufficient condition as well as a method of constructing the topological conjugacy.