Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property

TL;DR

This paper investigates the relationship between Mahler measure growth and algebraic entropy in discrete dynamical systems with the Laurent property, providing evidence for a conjectured equivalence through theoretical and numerical analysis.

Contribution

It formulates and supports the conjecture that Mahler measure growth rate equals algebraic entropy in systems with the Laurent property, with explicit calculations and proofs.

Findings

Mahler measure growth matches algebraic entropy in studied systems

Exact formula derived for Laurent polynomials from the Kronecker quiver

Numerical evidence supports the conjecture across multiple examples

Abstract

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