Constant Slope Maps on the Extended Real Line

TL;DR

This paper investigates the conditions under which transitive countably piecewise monotone Markov interval maps can be conjugated to maps of constant slope, considering various properties like continuity, mixing, and domain type.

Contribution

It provides a comprehensive analysis of when conjugate maps of constant slope exist for different classes of Markov interval maps, extending understanding of their structural properties.

Findings

Existence of conjugate constant slope maps depends on continuity and mixing properties.

Results vary based on the domain being bounded, half-line, or entire real line.

Conditions for conjugacy are characterized for different map classes.

Abstract

For a transitive countably piecewise monotone Markov interval map we consider the question whether there exists a conjugate map of constant slope. The answer varies depending on whether the map is continuous or only piecewise continuous, whether it is mixing or not, what slope we consider, and whether the conjugate map is defined on a bounded interval, half-line or the whole real line (with the infinities included).