On $p$-adic entropy of some solenoid dynamical systems

TL;DR

This paper extends the concept of $p$-adic entropy to certain solenoid dynamical systems, providing explicit formulas and proving their existence, thus generalizing previous results relating entropy and Mahler measures.

Contribution

It introduces explicit formulas for $p$-adic entropy in new classes of dynamical systems, including those over number fields and solenoids, expanding the theoretical framework.

Findings

Proved the existence of $p$-adic entropy for these systems.

Derived explicit formulas for $p$-adic entropy.

Generalized previous entropy-Mahler measure relations.

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.