Shadowing for families of endomorphisms of generalized group shifts

Xuan Kien Phung

doi:10.3934/dcds.2021116

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2022, Volume 42, Issue 1: 285-299. Doi: 10.3934/dcds.2021116

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Shadowing for families of endomorphisms of generalized group shifts

Xuan Kien Phung

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

Received: March 31, 2021

Revised: May 31, 2021

Early access: August 2021

Published: January 2022

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

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Mathematics Subject Classification: Primary: 37B51, 37B10, 37B15; Secondary: 14L10, 68Q80.

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