On finite spacer rank for words and subshifts
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arXiv
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| Autori principali: | , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866916300285018112 |
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| author | Gao, Su Jacoby, Liza Johnson, William Leng, James Li, Ruiwen Silva, Cesar E. Wu, Yuxin |
| author_facet | Gao, Su Jacoby, Liza Johnson, William Leng, James Li, Ruiwen Silva, Cesar E. Wu, Yuxin |
| contents | We define a notion of rank for words and subshifts that we call spacer rank, extending the notion of rank-one symbolic shifts of Gao and Hill. We construct infinite words of each finite spacer rank, of unbounded spacer rank, and show there exist words that do not have a spacer rank construction. We consider words that are fixed points of substitutions and give explicit conditions for the word to have an at most spacer rank two construction, and not to be rank one. We prove that finite spacer rank subshifts have topological entropy zero, and that there are zero entropy subshifts not defined by a word with a finite spacer rank construction. We also study shift systems associated with infinite words, including those associated to Sturmian sequences, which we show are spacer rank-two systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_05165 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On finite spacer rank for words and subshifts Gao, Su Jacoby, Liza Johnson, William Leng, James Li, Ruiwen Silva, Cesar E. Wu, Yuxin Dynamical Systems 37B02, 37B10, 37B40, 37A99, 37A50 We define a notion of rank for words and subshifts that we call spacer rank, extending the notion of rank-one symbolic shifts of Gao and Hill. We construct infinite words of each finite spacer rank, of unbounded spacer rank, and show there exist words that do not have a spacer rank construction. We consider words that are fixed points of substitutions and give explicit conditions for the word to have an at most spacer rank two construction, and not to be rank one. We prove that finite spacer rank subshifts have topological entropy zero, and that there are zero entropy subshifts not defined by a word with a finite spacer rank construction. We also study shift systems associated with infinite words, including those associated to Sturmian sequences, which we show are spacer rank-two systems. |
| title | On finite spacer rank for words and subshifts |
| topic | Dynamical Systems 37B02, 37B10, 37B40, 37A99, 37A50 |
| url | https://arxiv.org/abs/2010.05165 |