Intermediate $β$-shifts as greedy $β$-shifts with a hole
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908760470978560 |
|---|---|
| author | Langeveld, Niels Samuel, Tony |
| author_facet | Langeveld, Niels Samuel, Tony |
| contents | We show that every intermediate $β$-transformation is topologically conjugate to a greedy $β$-transformation with a hole at zero, and provide a counterexample illustrating that the correspondence is not one-to-one. This characterisation is employed to (1) build a Krieger embedding theorem for intermediate $β$-transformation, complementing the result of Li, Sahlsten, Samuel and Steiner [2019], and (2) obtain new metric and topological results on survivor sets of intermediate $β$-transformations with a hole at zero, extending the work of Kalle, Kong, Langeveld and Li [2020]. Further, we derive a method to calculate the Hausdorff dimension of such survivor sets as well as results on certain bifurcation sets. Moreover, by taking unions of survivor sets of intermediate $β$-transformations one obtains an important class of sets arising in metric number theory, namely sets of badly approximable numbers in non-integer bases. We prove, under the assumption that the underlying symbolic space is of finite type, that these sets of badly approximable numbers are winning in the sense of Schmidt games, and hence have the countable intersection property, extending the results of Hu and Yu [2014], Tseng [2009] and Färm, Persson and Schmeling [2010]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_14584 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Intermediate $β$-shifts as greedy $β$-shifts with a hole Langeveld, Niels Samuel, Tony Dynamical Systems 11K55, 11A63, 68R15, 26A30, 28D05, 37B10, 37E05, 37E15 We show that every intermediate $β$-transformation is topologically conjugate to a greedy $β$-transformation with a hole at zero, and provide a counterexample illustrating that the correspondence is not one-to-one. This characterisation is employed to (1) build a Krieger embedding theorem for intermediate $β$-transformation, complementing the result of Li, Sahlsten, Samuel and Steiner [2019], and (2) obtain new metric and topological results on survivor sets of intermediate $β$-transformations with a hole at zero, extending the work of Kalle, Kong, Langeveld and Li [2020]. Further, we derive a method to calculate the Hausdorff dimension of such survivor sets as well as results on certain bifurcation sets. Moreover, by taking unions of survivor sets of intermediate $β$-transformations one obtains an important class of sets arising in metric number theory, namely sets of badly approximable numbers in non-integer bases. We prove, under the assumption that the underlying symbolic space is of finite type, that these sets of badly approximable numbers are winning in the sense of Schmidt games, and hence have the countable intersection property, extending the results of Hu and Yu [2014], Tseng [2009] and Färm, Persson and Schmeling [2010]. |
| title | Intermediate $β$-shifts as greedy $β$-shifts with a hole |
| topic | Dynamical Systems 11K55, 11A63, 68R15, 26A30, 28D05, 37B10, 37E05, 37E15 |
| url | https://arxiv.org/abs/2211.14584 |