Dynamics of endomorphisms of algebraic groups

TL;DR

This paper generalizes fixed point formulas for endomorphisms of algebraic groups over finite fields, introducing a new framework for analyzing their zeta functions and orbit counts, with applications to dynamical systems.

Contribution

It extends Steinberg's fixed point formula to all smooth algebraic groups, introducing the concept of finite-adelically distorted sequences and analyzing their zeta functions.

Findings

Established a dichotomy for Artin-Mazur zeta functions.

Derived asymptotic formulas for counting periodic orbits.

Connected fixed point counts to p-adic properties and cohomological zeta functions.

Abstract

Let $\sigma$ denote an endomorphism of a smooth algebraic group $G$ over the algebraic closure of a finite field, and assume all iterates of $\sigma$ have finitely many fixed points. Steinberg gave a formula for the number of fixed points of $\sigma$ (and hence of all of its iterates $\sigma^n$) 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 $\sigma_n$ of $\sigma^n$ can depend on $p$-adic properties of $n$. We axiomatise the structure of the sequence $(\sigma_n)$ 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 $\leq N$. For an algebraic group $G$ we express the error term via the $\ell$-adic cohomological zeta function of $G$.