2021/09/16 by Andrew N. W. Hone, Hone, Andrew N. W.
Biochemistry, Genetics and Molecular Biology · Chemistry · Mathematics · #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Molecular spectroscopy and chirality #Number Theory (math.NT) #Protein Structure and Dynamics
paper · doi:10.48550/arxiv.2109.08217
openalex publication_date 2021/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 A1) we prove a precise formula for the leading order asymptotics of the logarithmic Mahler measure, which grows linearly.