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Dynamically affine maps in positive characteristic. (English) Zbl 1444.14082

Moree, Pieter (ed.) et al., Dynamics: topology and numbers. Conference, Max Planck Institute for Mathematics, Bonn, Germany, July 2–6, 2018. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 744, 125-156 (2020).
Let \(V\) be an algebraic variety over an algebraically closed field of characteristic \(p\). A morphism \(f \colon V \rightarrow V\) is said to be confined if the number of fixed points, \(f_n\), of its \(n\)th iterate is finite for all \(n\).
Given such a confined morphism \(f \colon V \rightarrow V\), the tame and Artin-Mazur dynamical zeta functions are respectively defined as \[ \zeta^*_f(z) := \exp\left( \sum_{p \nmid n} f_n z^n / n \right) \] and \[ \zeta_f(z) := \exp \left( \sum_{n \geq 1} f_n z^n / n\right) \text{.} \]
In the article under review, the authors study such zeta functions for the case of dynamically affine morphisms, in the sense of J. Silverman [Graduate Texts in Mathematics 241. New York, NY: Springer (2007; Zbl 1130.37001)]. In particular, dynamically affine morphisms are confined.
The authors’ first result treats the case of dynamically affine maps \(f \colon \mathbb{P}^1 \rightarrow \mathbb{P}^1\). They prove that the tame zeta function \(\zeta^*_f(z)\) is algebraic and satisfies a first order differential equation over \(\mathbb{C}(z)\). Further, they show that if \(f\) is coseparable, then the Artin-Mazur dynamical zeta function \(\zeta_f(z)\) is a rational function.
The authors’ second result applies to dynamically affine maps \(f \colon V \rightarrow V\), for \(V\) the Kummer variety of an abelian variety \(A\). It is required that \(f\) is induced by the multiplication map \([m]_A \colon A \rightarrow A\) for some integer \(m \geq 2\). Within this context, the authors establish that the tame zeta function \(\zeta^*_f(z)\) is algebraic and satisfies a first order differential equation over \(\mathbb{C}(z)\). Further, if \(m\) is coprime to \(p\), then the Artin-Mazur dynamical zeta function \(\zeta_f(z)\) is rational.
The results of the present article complement those of J. Byszewski and G. Cornelissen [Algebra Number Theory 12, No. 9, 2185–2235 (2018; Zbl 1419.37092)].
For the entire collection see [Zbl 1448.37001].

MSC:

14K15 Arithmetic ground fields for abelian varieties
37P55 Arithmetic dynamics on general algebraic varieties
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.

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References:

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