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