Dynamical Mahler Measure: A survey and some recent results

TL;DR

This paper surveys the dynamical Mahler measure of multivariate polynomials, presenting new formulas, analogues of classical results, and improvements in computational methods within dynamical systems and number theory.

Contribution

It provides a comprehensive survey of dynamical Mahler measure, introduces new formulas for computation, and extends classical results to the dynamical setting.

Findings

Formulas for computing dynamical Mahler measure in specific cases

Dynamical analogues of Kronecker's Lemma

Improved results for two-variable dynamical Mahler measure

Abstract

We study the dynamical Mahler measure of multivariate polynomials and present dynamical analogues of various results from the classical Mahler measure as well as examples of formulas allowing the computation of the dynamical Mahler measure in certain cases. We discuss multivariate analogues of dynamical Kronecker's Lemma and present some improvements on the result for two variables due to Carter, Lal\'in, Manes, Miller, and Mocz.