Shadowing for families of endomorphisms of generalized group shifts

TL;DR

This paper proves a shadowing property for families of endomorphisms acting on certain group subshifts, extending the concept to generalized settings involving cellular automata and module structures.

Contribution

It introduces an intrinsic shadowing property for monoid actions on group subshifts and generalizes it to families of endomorphisms of admissible group subshifts.

Findings

Shadowing property holds for the valuation action of finitely generated monoids of cellular automata.

Results apply to group subshifts that are also subgroups or submodules of $A^G$.

Generalizations include families of endomorphisms of admissible group subshifts.

Abstract

Let $G$ be a countable monoid and let $A$ be an Artinian group (resp. an Artinian module). Let $\Sigma \subset A^G$ be a closed subshift which is also a subgroup (resp. a submodule) of $A^G$. Suppose that $\Gamma$ is a finitely generated monoid consisting of pairwise commuting cellular automata $\Sigma \to \Sigma$ that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the valuation action of $\Gamma$ on $\Sigma$ satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissible group subshifts.