Shadowing for families of endomorphisms of generalized group shifts

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ G $\end{document}</tex-math></inline-formula> be a countable monoid and let <inline-formula><tex-math id="M2">\begin{document}$ A $\end{document}</tex-math></inline-formula> be an Artinian group (resp. an Artinian module). Let <inline-formula><tex-math id="M3">\begin{document}$ \Sigma \subset A^G $\end{document}</tex-math></inline-formula> be a closed subshift which is also a subgroup (resp. a submodule) of <inline-formula><tex-math id="M4">\begin{document}$ A^G $\end{document}</tex-math></inline-formula>. Suppose that <inline-formula><tex-math id="M5">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> is a finitely generated monoid consisting of pairwise commuting cellular automata <inline-formula><tex-math id="M6">\begin{document}$ \Sigma \to \Sigma $\end{document}</tex-math></inline-formula> that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the natural action of <inline-formula><tex-math id="M7">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M8">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissible group subshifts.</p>