# On pointwise periodicity in tilings, cellular automata and subshifts

## TL;DR

This paper investigates how properties like expansiveness and pointwise periodicity influence the structure of groups, automata, and tilings, establishing finiteness results for certain configurations and group actions.

## Contribution

It proves that finitely generated pointwise periodic groups of cellular automata are finite and that subshifts with finite orbits over finitely generated groups are finite, extending to Euclidean tilings.

## Key findings

- Finitely generated pointwise periodic groups of cellular automata are finite.
- Subshifts with finite orbits over finitely generated groups are finite.
- Results apply to tilings of Euclidean space.

## Abstract

We study implications of expansiveness and pointwise periodicity for certain groups and semigroups of transformations. Among other things we prove that every pointwise periodic finitely generated group of cellular automata is necessarily finite. We also prove that a subshift over any finitely generated group that consists of finite orbits is finite, and related results for tilings of Euclidean space.