2022/08/31 by Jakub Byszewski, Gunther Cornelissen, Byszewski, Jakub +3
Computer Science · Mathematics · #11N45 #14F20 #20G40 #30B40 #37C25 #37C30 #37C35 Secondary 11B37 #37P55 #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary 14L10 #Topological and Geometric Data Analysis #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2209.00085
openalex publication_date 2022/08/31 · openalex created_date 2022/09/03 · openalex updated_date 2026/07/28
Let σ denote an endomorphism of a smooth algebraic group G over the algebraic closure of a finite field, and assume all iterates of σ have finitely many fixed points. Steinberg gave a formula for the number of fixed points of σ (and hence of all of its iterates σn) in the semisimple case, leading to a representation of its Artin-Mazur zeta function as a rational function. We generalise this to an arbitrary (smooth) algebraic group G, where the number of fixed points σn of σn can depend on p-adic properties of n. We axiomatise the structure of the sequence (σn) via the concept of a `finite-adelically distorted' (FAD-)sequence. Such sequences also occur in topological dynamics, and our subsequent results about zeta functions and asymptotic counting of orbits apply equally well in that situation; for example, to S-integer dynamical systems, additive cellular automata and other compact abelian groups. We prove dichotomies for the associated Artin-Mazur zeta function, and study the analogue of the Prime Number Theorem for the function counting periodic orbits of length ≤ N. For an algebraic group G we express the error term via the ℓ-adic cohomological zeta function of G.