Topological rigidity of linear cellular automaton shifts

TL;DR

This paper proves that linear cellular automaton shifts are topologically and algebraically isomorphic only when they are identical, and characterizes their automorphism groups as finitely generated abelian groups.

Contribution

It establishes a topological and algebraic equivalence criterion for linear cellular automaton shifts and describes their automorphism groups.

Findings

Topologically isomorphic linear cellular automaton shifts are algebraically isomorphic.

Distinct linear cellular automaton shifts are not isomorphic.

Automorphism group of such shifts is finitely generated abelian.

Abstract

We prove that topologically isomorphic linear cellular automaton shifts are algebraically isomorphic. Using this, we show that two distinct such shifts cannot be isomorphic. We conclude that the automorphism group of a linear cellular automaton shift is a finitely generated abelian group.