Orbit-counting for nilpotent group shifts

TL;DR

This paper investigates the asymptotic behavior of orbit-counting functions for actions of finitely-generated torsion-free nilpotent groups, revealing complex growth patterns and providing bounds and numerical evidence for non-cyclic groups.

Contribution

It introduces new asymptotic formulas for orbit counts in nilpotent group shifts and explores the complexity of their growth, extending previous understanding of dynamical systems.

Findings

Derived a single asymptotic form for the sum over finite orbits

Established bounds for the orbit-counting function

Provided numerical evidence for complex growth in non-cyclic group actions

Abstract

We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens' theorem for the full $G$-shift for a finitely-generated torsion-free nilpotent group $G$. Using bounds for the M{\"o}bius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape \[ \sum_{|\tau|\le N}\frac{1}{e^{h|\tau|}}\sim CN^{\alpha}(\log N)^{\beta} \] where $|\tau|$ is the cardinality of the finite orbit $\tau$. For the usual orbit-counting function we find upper and lower bounds together with numerical evidence to suggest that for actions of non-cyclic groups there is no single asymptotic in terms of elementary functions.