Shadowing for families of endomorphisms of generalized group shifts. (English) Zbl 1492.37023
Summary: 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.
MSC:
| 37B51 | Multidimensional shifts of finite type |
| 37B65 | Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems |
| 37B10 | Symbolic dynamics |
| 14L10 | Group varieties |
Keywords:
shadowing property; pseudo-orbit tracing property; group shift; subshift of finite type; infinite alphabet symbolic system; higher-dimensional Markov shift; Artinian group; Artinian moduleReferences:
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