On linear shifts of finite type and their endomorphisms
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arXiv
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| Format: | Preprint |
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2020
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| author | Ceccherini-Silberstein, Tullio Coornaert, Michel Phung, Xuan Kien |
| author_facet | Ceccherini-Silberstein, Tullio Coornaert, Michel Phung, Xuan Kien |
| contents | 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 $Σ\subset A^G$ and the dynamical behavior of linear cellular automata $τ\colon Σ\to Σ$. We say that $G$ is of $K$-linear Markov type if, for every finite-dimensional vector space $A$ over $K$, all linear subshifts $Σ\subset A^G$ 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 $τ$ 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 $Σ$ is topologically mixing, we show that $τ$ is nilpotent if and only if its limit set is finite-dimensional. A new characterization of the limit set of $τ$ in terms of pre-injectivity is also obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_14191 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On linear shifts of finite type and their endomorphisms Ceccherini-Silberstein, Tullio Coornaert, Michel Phung, Xuan Kien Dynamical Systems Group Theory Rings and Algebras 37B15, 37B20, 37B51, 20F65, 68Q80 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 $Σ\subset A^G$ and the dynamical behavior of linear cellular automata $τ\colon Σ\to Σ$. We say that $G$ is of $K$-linear Markov type if, for every finite-dimensional vector space $A$ over $K$, all linear subshifts $Σ\subset A^G$ 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 $τ$ 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 $Σ$ is topologically mixing, we show that $τ$ is nilpotent if and only if its limit set is finite-dimensional. A new characterization of the limit set of $τ$ in terms of pre-injectivity is also obtained. |
| title | On linear shifts of finite type and their endomorphisms |
| topic | Dynamical Systems Group Theory Rings and Algebras 37B15, 37B20, 37B51, 20F65, 68Q80 |
| url | https://arxiv.org/abs/2011.14191 |