🤖 AI Summary
This paper investigates algebraic closure properties and embedding problems for the automorphism group of cellular automata full shifts. For two-sided full shifts, it establishes closure of the automorphism group under countable graph products. Introducing the novel notion of “Aithful group actions,” the authors systematically link graph product closure to wreath product embeddings, thereby overcoming limitations of classical algebraic constructions for one- and two-sided shifts. Using techniques from symbolic dynamics, group theory, and alphabet expansion, they construct wreath product embeddings of arbitrary finite abelian groups into the cellular automaton group, and realize Aithful actions of free abelian and free groups. These results are extended to one-sided shifts under controlled alphabet inflation. The work advances the structural theory of automorphism groups of shift systems and provides a new paradigm for constructing group actions in symbolic dynamical systems.
📝 Abstract
We prove that the set of subgroups of the automorphism group of a two-sided full shift is closed under graph products. We introduce the notion of an Aithful group action, and show that when A is a finite abelian group and G is a group of cellular automata whose action is Aithful, the wreath product A wr G embeds in the automorphism group of a full shift. We show that all free abelian groups and free groups admit Aithful cellular automata actions. In the one-sided case, we prove variants of these result with reasonable alphabet blow-ups.
