Homomorphisms from aperiodic subshifts to subshifts with the finite extension property
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909777352720384 |
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| author | Bland, Robert McGoff, Kevin |
| author_facet | Bland, Robert McGoff, Kevin |
| contents | Given a countable group $G$ and two subshifts $X$ and $Y$ over $G$, a continuous, shift-commuting map $ϕ: X \to Y$ is called a homomorphism. Our main result states that if every finitely generated subgroup of $G$ has polynomial growth, $X$ is aperiodic, and $Y$ has the finite extension property (FEP), then there exists a homomorphism $ϕ: X \to Y$. By combining this theorem with a previous result of Bland, we obtain that if the same conditions hold, and if additionally the topological entropy of $X$ is less than the topological entropy of $Y$ and $Y$ has no global period, then $X$ embeds into $Y$. We also establish some facts about subshifts with the FEP that may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18795 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homomorphisms from aperiodic subshifts to subshifts with the finite extension property Bland, Robert McGoff, Kevin Dynamical Systems Given a countable group $G$ and two subshifts $X$ and $Y$ over $G$, a continuous, shift-commuting map $ϕ: X \to Y$ is called a homomorphism. Our main result states that if every finitely generated subgroup of $G$ has polynomial growth, $X$ is aperiodic, and $Y$ has the finite extension property (FEP), then there exists a homomorphism $ϕ: X \to Y$. By combining this theorem with a previous result of Bland, we obtain that if the same conditions hold, and if additionally the topological entropy of $X$ is less than the topological entropy of $Y$ and $Y$ has no global period, then $X$ embeds into $Y$. We also establish some facts about subshifts with the FEP that may be of independent interest. |
| title | Homomorphisms from aperiodic subshifts to subshifts with the finite extension property |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2410.18795 |