On dynamical finiteness properties of algebraic group shifts

TL;DR

This paper introduces algebraic group subshifts over groups and demonstrates their properties, extending classical results to broader non-compact group alphabets, especially for polycyclic-by-finite groups.

Contribution

It defines algebraic group subshifts and proves their equivalence with sofic and finite type subshifts over certain groups, generalizing existing dynamical systems results.

Findings

Algebraic group subshifts generalize existing classes of subshifts.

The descending chain condition holds for $V^G$ when $G$ is polycyclic-by-finite.

Equivalence of algebraic, sofic, and finite type subshifts in this setting.

Abstract

Let $G$ be a group and let $V$ be an algebraic group over an algebraically closed field. We introduce algebraic group subshifts $\Sigma \subset V^G$ which generalize both the class of algebraic sofic subshifts of $V^G$ and the class of closed group subshifts over finite group alphabets. When $G$ is a polycyclic-by-finite group, we show that $V^G$ satisfies the descending chain condition and that the notion of algebraic group subshifts, the notion of algebraic group sofic subshifts, and that of algebraic group subshifts of finite type are all equivalent. Thus, we obtain extensions of well-known results of Kitchens and Schmidt to cover the case of many non-compact group alphabets.