On rational functions sharing the measure of maximal entropy

TL;DR

This paper investigates rational functions sharing the measure of maximal entropy by linking it to solutions of a specific functional equation, providing new insights into their structure and solutions.

Contribution

It reduces the problem of describing such rational functions to solving a particular functional equation, offering new results on these solutions.

Findings

Reduction of the problem to a functional equation

New results on solutions of the functional equation

Insights into the structure of rational functions sharing maximal entropy

Abstract

We show that describing rational functions $f_1,$ $f_2,$ $\dots,f_n$ sharing the measure of maximal entropy reduces to describing solutions of the functional equation $A\circ X_1=A\circ X_2=\dots=A\circ X_n$ in rational functions. We also provide some results about solutions of this equation.