Two-variable polynomials with dynamical Mahler measure zero

TL;DR

This paper explores the dynamical Mahler measure of multivariate polynomials, establishing a weak Boyd--Lawton formula and characterizing polynomials with zero measure, assuming a dynamical Lehmer's conjecture.

Contribution

It introduces a dynamical version of the Boyd--Lawton formula and characterizes polynomials with zero dynamical Mahler measure in one and two variables.

Findings

Proves a weak dynamical Boyd--Lawton formula

Characterizes polynomials with zero dynamical Mahler measure

Assumes dynamical Lehmer's question for two-variable case

Abstract

We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of Boyd--Lawton formula and we characterize the polynomials with integer coefficients having dynamical Mahler measure zero both for the case of one variable (Kronecker's lemma) and for the case of two variables, under the assumption that the dynamical version of Lehmer's question is true.