Binary constant-length substitutions and Mahler measures of Borwein polynomials

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.