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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.

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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