American Institute of Mathematical Sciences

January  2022, 42(1): 285-299. doi: 10.3934/dcds.2021116

Shadowing for families of endomorphisms of generalized group shifts

 Département de mathématiques, Université du Québec à Montréal, Case postale 8888, Succursale Centre-ville, Montréal, QC, H3C 3P8, Canada

Received  April 2021 Revised  June 2021 Published  January 2022 Early access  August 2021

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 natural action of $\Gamma$ on $\Sigma$ satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissible group subshifts.

Citation: Xuan Kien Phung. Shadowing for families of endomorphisms of generalized group shifts. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 285-299. doi: 10.3934/dcds.2021116
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