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

Andrew N. W. Hone

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.

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

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 we prove a precise formula for the leading order asymptotics of the logarithmic Mahler measure, which grows linearly.

Paper Structure

This paper contains 5 sections, 3 theorems, 63 equations, 4 figures.

Table of Contents

  1. Introduction, definitions and main conjecture
  2. Families of Laurent polynomials in rank 2
  3. Recurrence for Markoff numbers
  4. Somos-4 recurrence
  5. Concluding remarks

Key Result

Proposition 1.9

Suppose that $\varphi$ is a cluster map of recurrence type. Let $\boldsymbol{1} = (1,\ldots,1)$ denote the $N$-tuple consisting of $N$ 1s, and let ${\cal E}_H(\boldsymbol{1})$ denote the Diophantine entropy of the orbit of $\varphi$ with initial data $\boldsymbol{1}$. Then (assuming it exists) the M Suppose further that $\deg x_n=C{\lambda}^n(1+o(1))$, $m(x_n)=C'{\lambda}^n(1+o(1))$ for some real

Figures (4)

  • Figure 1: The case $r=2$: numerical approximation to $m(x_n)$ versus $n$.
  • Figure 2: Numerical approximation to $\log m(x_n)$ versus $n$ for $r=3,4,5$.
  • Figure 3: Markoff recurrence: numerical approximation to $\log m(x_n)$ versus $n$ .
  • Figure 4: Numerical approximation for the Somos-4 case.

Theorems & Definitions (12)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Conjecture 1.7
  • Remark 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 2 more