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

Published online by Cambridge University Press:  29 June 2018

JUN UEKI
JUN UEKI*
Affiliation:
Department of Mathematics, School of System Design and Technology, Tokyo Denki University, 5 Senju Asahi-cho, Adachi-ku, Tokyo, 120-8551, Japan email uekijun46@gmail.com
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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.

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Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 1 , January 2020 , pp. 272 - 288
Copyright
© Cambridge University Press, 2018 

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