Semiconjugacy to a map of a constant slope

TL;DR

This paper extends the classical semiconjugacy result from continuous to piecewise continuous and graph maps, establishing conditions under which maps are conjugate to constant slope maps and analyzing the properties of the associated operator.

Contribution

It generalizes semiconjugacy to broader classes of maps and introduces an operator linking transitive maps to constant slope maps, analyzing its continuity.

Findings

Semiconjugacy extends to piecewise continuous maps.

The conjugacy operator is not continuous.

Results apply to graph maps with positive entropy.

Abstract

It is well known that a continuous piecewise monotone interval map with positive topological entropy is semiconjugate to a map of a constant slope and the same entropy, and if it is additionally transitive then this semiconjugacy is actually a conjugacy. We generalize this result to piecewise continuous piecewise monotone interval maps, and as a consequence, get it also for piecewise monotone graph maps. We show that assigning to a continuous transitive piecewise monotone map of positive entropy a map of constant slope conjugate to it defines an operator, and show that this operator is not continuous.