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.

Genetics of iterative roots for PM functions

“…So we need to identify which iterative roots are genetic or not. This problem was considered in [11] for a special class of PM functions of height 1, each of which has a characteristic interval bounded by an endpoint of the domain and is homeomorphic onto the characteristic interval but does not reach an endpoint of the characteristic interval outside. Theorem 2 of [12] indicates that all its iterative roots are in the simplest mode of 1-extension.…”