p$ -adic Mahler measure and $\mathbb{Z}$ -covers of links

Let $p$ be a prime number. We develop a theory of $p$ -adic Mahler measure of polynomials and apply it to the study of $\mathbb{Z}$ -covers of rational homology 3-spheres branched over links. We obtain a $p$ -adic analogue of the asymptotic formula of the torsion homology growth and a balance formula among the leading coefficient of the Alexander polynomial, the $p$ -adic entropy and the Iwasawa $\unicode[STIX]{x1D707}_{p}$ -invariant. We also apply the purely $p$ -adic theory of Besser–Deninger to $\mathbb{Z}$ -covers of links. In addition, we study the entropies of profinite cyclic covers of links. We examine various examples throughout the paper.