Binary constant-length substitutions and Mahler measures of Borwein polynomials

TL;DR

This paper links the Mahler measure of Borwein polynomials to the maximal Lyapunov exponent of a matrix cocycle from binary constant-length substitutions, connecting number theory and dynamical systems.

Contribution

It establishes a novel equivalence between Mahler measures of Borwein polynomials and spectral properties of substitution systems, expanding the understanding of Mahler measures in dynamics.

Findings

Mahler measure expressed as a Lyapunov exponent

Connection between Lehmer's problem and spectral theory

Extension of Mahler measures' role in dynamics

Abstract

We show that the Mahler measure of every Borwein polynomial -- a polynomial with coefficients in $ \{-1,0,1 \}$ having non-zero constant term -- can be expressed as a maximal Lyapunov exponent of a matrix cocycle that arises in the spectral theory of binary constant-length substitutions. In this way, Lehmer's problem for height-one polynomials having minimal Mahler measure becomes equivalent to a natural question from the spectral theory of binary constant-length substitutions. This supports another connection between Mahler measures and dynamics, beyond the well-known appearance of Mahler measures as entropies in algebraic dynamics.