A Note on a Theorem of Parry

TL;DR

This paper generalizes Parry's theorem, showing that essentially transitive mappings with positive entropy are conjugate to uniformly piecewise linear maps, extending known results for mappings with one turning point.

Contribution

It extends Parry's theorem to essentially transitive mappings, broadening the class of systems where conjugacy to linear maps applies.

Findings

Generalization of Parry's theorem to essentially transitive mappings

Converse holds for mappings with one turning point and slope > 1

Establishes a broader class of conjugate systems

Abstract

A well-known result of Bill Parry shows that a topologically transitive continuous piecewise monotone mapping with positive topological entropy is conjugate to a uniformly piecewise linear mapping with slope determined by the entropy. In this note we generalise Parry's result somewhat to what we call the class of essentially transitive mappings. This generalisation is of some interest in as much as for mappings with one turning point the converse also holds, i.e., a uniformly piecewise linear mapping with one turning point and with slope greater than 1 is essentially transitive.