Two-dimensional polynomials with dynamical Mahler measure 0

Abstract: The Mahler measure of an integer polynomial in one or more variables is an important quantity that shows up many places in number theory and mathematics more generally. The integer polynomials with Mahler measure 0 have been classified -- in the one-variable case, these are products of cyclotomic polynomials. We introduce a dynamical generalization of multivariable Mahler measure, and investigate the polynomials for which this dynamical Mahler measure vanishes. In the two-variable case, we characterize polynomials with dynamical Mahler measure 0 assuming a dynamical analogue of Lehmer's conjecture. This is joint work with Annie Carter, Matilde Lalín, Michelle Manes, and Lucia Mocz.

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