On injective endomorphisms of symbolic schemes
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arXiv
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| Format: | Preprint |
| Published: |
2017
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| _version_ | 1866910397430235136 |
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| author | Ceccherini-Silberstein, Tullio Coornaert, Michel Phung, Xuan Kien |
| author_facet | Ceccherini-Silberstein, Tullio Coornaert, Michel Phung, Xuan Kien |
| contents | Building on the seminal work of Gromov on endomorphisms of symbolic algebraic varieties [10], we introduce a notion of cellular automata over schemes which generalize affine algebraic cellular automata in [7]. We extend known results to this more general setting. We also establish several new ones regarding the closed image property, surjunctivity, reversibility, and invertibility for cellular automata over algebraic varieties with coefficients in an algebraically closed field. As a byproduct, we obtain a negative answer to a question raised in [7] on the existence of a bijective complex affine algebraic cellular automaton $τ\colon A^{\mathbb Z} \to A^{\mathbb Z}$ whose inverse is not algebraic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1712_05716 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On injective endomorphisms of symbolic schemes Ceccherini-Silberstein, Tullio Coornaert, Michel Phung, Xuan Kien Algebraic Geometry Dynamical Systems Group Theory 37B15, 14A10, 14A15, 37B10, 68Q80 Building on the seminal work of Gromov on endomorphisms of symbolic algebraic varieties [10], we introduce a notion of cellular automata over schemes which generalize affine algebraic cellular automata in [7]. We extend known results to this more general setting. We also establish several new ones regarding the closed image property, surjunctivity, reversibility, and invertibility for cellular automata over algebraic varieties with coefficients in an algebraically closed field. As a byproduct, we obtain a negative answer to a question raised in [7] on the existence of a bijective complex affine algebraic cellular automaton $τ\colon A^{\mathbb Z} \to A^{\mathbb Z}$ whose inverse is not algebraic. |
| title | On injective endomorphisms of symbolic schemes |
| topic | Algebraic Geometry Dynamical Systems Group Theory 37B15, 14A10, 14A15, 37B10, 68Q80 |
| url | https://arxiv.org/abs/1712.05716 |