Sharing a measure of maximal entropy in polynomial semigroups

TL;DR

This paper characterizes when all elements of a polynomial semigroup share a measure of maximal entropy, linking this property to the non-emptiness of the intersection of certain right ideals, under specific conditions on the generating polynomials.

Contribution

It provides a necessary and sufficient condition for the existence of a shared measure of maximal entropy in polynomial semigroups generated by non-monomial, non-Chebyshev polynomials.

Findings

All elements share a measure of maximal entropy iff the intersection of principal right ideals is non-empty.

The result applies to semigroups generated by polynomials not conjugate to monomials or Chebyshev polynomials.

The paper characterizes the structure of polynomial semigroups with a common measure of maximal entropy.

Abstract

Let $P_1,P_2,\dots, P_k$ be complex polynomials of degree at least two that are not simultaneously conjugate to monomials or to Chebyshev polynomials, and $S$ the semigroup under composition generated by $P_1,P_2,\dots, P_k$. We show that all elements of $S$ share a measure of maximal entropy if and only if the intersection of principal right ideals $SP_1\cap SP_2\cap \dots \cap SP_k$ is non-empty.