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June 2021 On $p$-adic entropy of some solenoid dynamical systems
Yu Katagiri
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Kodai Math. J. 44(2): 323-333 (June 2021). DOI: 10.2996/kmj44207

Abstract

To a dynamical system is attached a non-negative real number called entropy. In 1990, Lind, Schmidt and Ward proved that the entropy for the dynamical system induced by the Laurent polynomial algebra over the ring of the rational integers is described by the Mahler measure. In 2009, Deninger introduced the $p$-adic entropy and obtained a $p$-adic analogue of Lind-Schmidt-Ward's theorem by using the $p$-adic Mahler measures. In this paper, we prove the existence and the explicit formula about $p$-adic entropies for two dynamical systems; one is induced by the Laurent polynomial algebra over the ring of the integers of a number field $K$, and the other is defined by the solenoid.

Acknowledgment

The author would like to thank my supervisor Professor Takao Yamazaki so much for his advice and helpful comments. This paper is based on the author's master thesis. This work was supported in part by the WISE Program for AI Electronics, Tohoku University.

Citation

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Yu Katagiri. "On $p$-adic entropy of some solenoid dynamical systems." Kodai Math. J. 44 (2) 323 - 333, June 2021. https://doi.org/10.2996/kmj44207

Information

Received: 22 April 2020; Revised: 20 December 2020; Published: June 2021
First available in Project Euclid: 29 June 2021

Digital Object Identifier: 10.2996/kmj44207

Subjects:
Primary: 37P35
Secondary: 11R06, 11S82, 37A35

Rights: Copyright © 2021 Tokyo Institute of Technology, Department of Mathematics

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Vol.44 • No. 2 • June 2021
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