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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2010.01967 |
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| _version_ | 1866912124427567104 |
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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 $V$ be an algebraic variety over an algebraically closed field $K$. Let $A$ denote the set of $K$-points of $V$. We introduce algebraic sofic subshifts $Σ\subset A^G$ and study endomorphisms $τ\colon Σ\to Σ$. We generalize several results for dynamical invariant sets and nilpotency of $τ$ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $τ$ is nilpotent if and only if its limit set, i.e., the intersection of the images of its iterates, is a singleton. If moreover $G$ is infinite, finitely generated and $Σ$ is topologically mixing, we show that $τ$ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_01967 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts Ceccherini-Silberstein, Tullio Coornaert, Michel Phung, Xuan Kien Dynamical Systems Algebraic Geometry Let $G$ be a group and let $V$ be an algebraic variety over an algebraically closed field $K$. Let $A$ denote the set of $K$-points of $V$. We introduce algebraic sofic subshifts $Σ\subset A^G$ and study endomorphisms $τ\colon Σ\to Σ$. We generalize several results for dynamical invariant sets and nilpotency of $τ$ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $τ$ is nilpotent if and only if its limit set, i.e., the intersection of the images of its iterates, is a singleton. If moreover $G$ is infinite, finitely generated and $Σ$ is topologically mixing, we show that $τ$ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values. |
| title | Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts |
| topic | Dynamical Systems Algebraic Geometry |
| url | https://arxiv.org/abs/2010.01967 |