p-adic Mahler measure and 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.