Invariant measures for $\mathscr{B}$-free systems revisited
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917600075710464 |
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| author | Dymek, Aurelia Kułaga-Przymus, Joanna Sell, Daniel |
| author_facet | Dymek, Aurelia Kułaga-Przymus, Joanna Sell, Daniel |
| contents | For $ \mathscr{B} \subseteq \mathbb{N} $, the $ \mathscr{B} $-free subshift $ X_η $ is the orbit closure of the characteristic function of the set of $ \mathscr{B} $-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_η $, have their analogues for $ X_η $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures ([Keller, G. Generalized heredity in $\mathcal B$-free systems. Stoch. Dyn. 21, 3 (2021), Paper No. 2140008]). A central assumption in our work is that $ η^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_η $) is regular. From this we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_η $ from above and below. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02134 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Invariant measures for $\mathscr{B}$-free systems revisited Dymek, Aurelia Kułaga-Przymus, Joanna Sell, Daniel Dynamical Systems 28C10, 37A05, 37A44, 37B40, 37B10 For $ \mathscr{B} \subseteq \mathbb{N} $, the $ \mathscr{B} $-free subshift $ X_η $ is the orbit closure of the characteristic function of the set of $ \mathscr{B} $-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_η $, have their analogues for $ X_η $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures ([Keller, G. Generalized heredity in $\mathcal B$-free systems. Stoch. Dyn. 21, 3 (2021), Paper No. 2140008]). A central assumption in our work is that $ η^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_η $) is regular. From this we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_η $ from above and below. |
| title | Invariant measures for $\mathscr{B}$-free systems revisited |
| topic | Dynamical Systems 28C10, 37A05, 37A44, 37B40, 37B10 |
| url | https://arxiv.org/abs/2307.02134 |