$p$-adic Mahler measure and $\mathbb{Z}$-covers of links. (January 2020)

• Record Type: Journal Article
• Authors: UEKI, JUN
• Is Part Of: Ergodic theory and dynamical systems. Volume 40:Number 1(2020)
• Page Start: 272
• Page End: 288
• Publication Date: 2020-01
• Subjects: Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2018.35
• Languages: English
• ISSNs: 0143-3857

Abstract

Abstract : 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.