Shadowing for families of endomorphisms of generalized group shifts
Xuan Kien Phung · Discrete and Continuous Dynamical Systems · 2021
Let \begin{document}$ G $\end{document} be a countable monoid and let \begin{document}$ A $\end{document} be an Artinian group (resp. an Artinian module). Let \begin{document}$ \Sigma \subset A^G $\end{document} be a closed subshift which is also a subgroup (resp. a submodule) of \begin{document}$ A^G $\end{document} . Suppose that \begin{document}$ \Gamma $\end{document} is a finitely generated monoid consisting of pairwise commuting cellular automata \begin{document}$ \Sigma \to \Sigma $\end{document} that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the natural action of \begin{document}$ \Gamma $\end{document} on \begin{document}$ \Sigma $\end{document} satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissible group subshifts.