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Dynamics on abelian varieties in positive characteristic. (English) Zbl 1419.37092

Summary: We study periodic points and orbit length distribution for endomorphisms of abelian varieties in characteristic \(p>0\). We study rationality, algebraicity and the natural boundary property for the dynamical zeta function (the latter using a general result on power series proven by Royals and Ward in the appendix), as well as analogues of the prime number theorem, also for tame dynamics, ignoring orbits whose order is divisible by \(p\). The behavior is governed by whether or not the action on the local \(p\)-torsion group scheme is nilpotent.

MSC:

37P55 Arithmetic dynamics on general algebraic varieties
11N45 Asymptotic results on counting functions for algebraic and topological structures
14G17 Positive characteristic ground fields in algebraic geometry
14K02 Isogeny
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
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