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- W4287586688 abstract "Let $G$ be a group and let $A$ be a finite-dimensional vector space over an arbitrary field $K$. We study finiteness properties of linear subshifts $Sigma subset A^G$ and the dynamical behavior of linear cellular automata $tau colon Sigma to Sigma$. We say that $G$ is of $K$-linear Markov type if, for every finite-dimensional vector space $A$ over $K$, all linear subshifts $Sigma subset A^G$ are of finite type. We show that $G$ is of $K$-linear Markov type if and only if the group algebra $K[G]$ is one-sided Noetherian. We prove that a linear cellular automaton $tau$ is nilpotent if and only if its limit set, i.e., the intersection of the images of its iterates, reduces to the zero configuration. If $G$ is infinite, finitely generated, and $Sigma$ is topologically mixing, we show that $tau$ is nilpotent if and only if its limit set is finite-dimensional. A new characterization of the limit set of $tau$ in terms of pre-injectivity is also obtained." @default.
- W4287586688 created "2022-07-25" @default.
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- W4287586688 date "2020-11-28" @default.
- W4287586688 modified "2023-09-27" @default.
- W4287586688 title "On linear shifts of finite type and their endomorphisms" @default.
- W4287586688 doi "https://doi.org/10.48550/arxiv.2011.14191" @default.
- W4287586688 hasPublicationYear "2020" @default.
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