Growth of Mahler Measure and Algebraic Entropy of Dynamics with the Laurent Property

We consider the growth rate of the Mahler measure in discrete dynamical\nsystems with the Laurent property, and in cluster algebras, and compare this\nwith other measures of growth. In particular, we formulate the conjecture that\nthe growth rate of the logarithmic Mahler measure coincides with the algebraic\nentropy, which is defined in terms of degree growth. Evidence for this\nconjecture is provided by exact and numerical calculations of the Mahler\nmeasure for a family of Laurent polynomials generated by rank 2 cluster\nalgebras, for a recurrence of third order related to the Markoff numbers, and\nfor the Somos-4 recurrence. Also, for the sequence of Laurent polynomials\nassociated with the Kronecker quiver (the cluster algebra of affine type\n$\\tilde{A}_1)$ we prove a precise formula for the leading order asymptotics of\nthe logarithmic Mahler measure, which grows linearly.\n