Dirichlet series for finite combinatorial rank dynamics

TL;DR

This paper introduces a new class of group endomorphisms with slow orbit growth, uses Dirichlet series for exact orbit counting, and explores their analytic properties to understand asymptotic behavior.

Contribution

It defines finite combinatorial rank endomorphisms, derives an exact orbit counting formula via Dirichlet series, and analyzes their analytic properties for orbit-growth asymptotics.

Findings

Exact orbit counting formula derived

Dirichlet series has a closed rational form in the connected case

Orbit-growth asymptotics are polynomially bounded

Abstract

We introduce a class of group endomorphisms -- those of finite combinatorial rank -- exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to have a closed rational form. Analytic properties of the Dirichlet series are related to orbit-growth asymptotics: depending on the location of the abscissa of convergence and the degree of the pole there, various orbit-growth asymptotics are found, all of which are polynomially bounded.