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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 withthe 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 compactabelian group. Let $d$ be a positive integer and let $A$ be a $d\timesd$-matrix with entries in $\mathbb{Z}_F$ and non-zero determinant. Themultiplication-by-$A$ map is a surjective endomorphism on $\mathbb{T}_F^d$.First, we compute the entropy of this endomorphism; the result and argumentsare analogous to those for the classical case$\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Second and most importantly, weresolve the algebraicity problem for the Artin-Mazur zeta function of all suchendomorphisms. As a consequence of our main result, we provide a completecharacterization and an explicit formula related to the entropy when the zetafunction is algebraic.

Endomorphisms of positive characteristic tori: entropy and zeta function