Endomorphisms of positive characteristic tori: entropy and zeta function

TL;DR

This paper studies endomorphisms of positive characteristic tori over finite fields, calculating their entropy and resolving the algebraicity problem for their Artin-Mazur zeta functions, providing explicit formulas and characterizations.

Contribution

It introduces the first computation of entropy for these endomorphisms and solves the algebraicity problem for their zeta functions, extending classical results to positive characteristic settings.

Findings

Entropy of endomorphisms is computed and analogous to classical cases.

The algebraicity problem for the Artin-Mazur zeta function is resolved.

Explicit formulas and characterizations are provided for cases with algebraic zeta functions.

Abstract

Let $F$ be a finite field of order $q$ and characteristic $p$. Let $\mathbb{Z}_F=F[t]$, $\mathbb{Q}_F=F(t)$, $\mathbb{R}_F=F((1/t))$ equipped with the discrete valuation for which $1/t$ is a uniformizer, and let $\mathbb{T}_F=\mathbb{R}_F/\mathbb{Z}_F$ which has the structure of a compact abelian group. Let $d$ be a positive integer and let $A$ be a $d\times d$-matrix with entries in $\mathbb{Z}_F$ and non-zero determinant. The multiplication-by-$A$ map is a surjective endomorphism on $\mathbb{T}_F^d$. First, we compute the entropy of this endomorphism; the result and arguments are analogous to those for the classical case $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Second and most importantly, we resolve the algebraicity problem for the Artin-Mazur zeta function of all such endomorphisms. As a consequence of our main result, we provide a complete characterization and an explicit formula related to the entropy when the zeta function is algebraic.