Renormalization for Bruin-Troubetzkoy ITMs
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| Název: | Renormalization for Bruin-Troubetzkoy ITMs |
|---|---|
| Autoři: | Artigiani, Mauro, Hubert, Pascal, Skripchenko, Alexandra |
| Zdroj: | Discrete and Continuous Dynamical Systems. 47:519-547 |
| Publication Status: | Preprint |
| Informace o vydavateli: | American Institute of Mathematical Sciences (AIMS), 2026. |
| Rok vydání: | 2026 |
| Témata: | FOS: Mathematics, 37E05 (Primary) 37A05, 37A44, 11J70 (Secondary), Dynamical Systems (math.DS), Dynamical Systems |
| Popis: | We study a class of interval translation mappings introduced by Bruin and Troubetzkoy, describing a new renormalization scheme, inspired by the classical Rauzy induction for this class. We construct a measure, invariant under the renormalization, supported on the parameters yielding infinite type interval translation mappings in this class. With respect to this measure, a.e. transformation is uniquely ergodic. We show that this set has Hausdorff dimension between 1.5 and 2, and that the Hausdorff dimension coincides with the affinity dimension. Finally, seeing our renormalization as a multidimensional continued fraction algorithm, we show that it has almost always the Pisot property. We discover an interesting phenomenon: the dynamics of this class of transformations is often (conjecturally: almost always) weak mixing, while the renormalizing algorithm typically has the Pisot property. 30 pages, 7 figures; Final version, according to the Referees' reports, minor changes. Comments are welcome! To appear in Discrete and Continuous Dynamical Systems |
| Druh dokumentu: | Article |
| ISSN: | 1553-5231 1078-0947 |
| DOI: | 10.3934/dcds.2025127 |
| DOI: | 10.48550/arxiv.2412.07928 |
| Přístupová URL adresa: | http://arxiv.org/abs/2412.07928 |
| Rights: | CC BY NC ND |
| Přístupové číslo: | edsair.doi.dedup.....1a6d90e0fa75aff0ee8609ebc2234b08 |
| Databáze: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | We study a class of interval translation mappings introduced by Bruin and Troubetzkoy, describing a new renormalization scheme, inspired by the classical Rauzy induction for this class. We construct a measure, invariant under the renormalization, supported on the parameters yielding infinite type interval translation mappings in this class. With respect to this measure, a.e. transformation is uniquely ergodic. We show that this set has Hausdorff dimension between 1.5 and 2, and that the Hausdorff dimension coincides with the affinity dimension. Finally, seeing our renormalization as a multidimensional continued fraction algorithm, we show that it has almost always the Pisot property. We discover an interesting phenomenon: the dynamics of this class of transformations is often (conjecturally: almost always) weak mixing, while the renormalizing algorithm typically has the Pisot property.<br />30 pages, 7 figures; Final version, according to the Referees' reports, minor changes. Comments are welcome! To appear in Discrete and Continuous Dynamical Systems |
|---|---|
| ISSN: | 15535231 10780947 |
| DOI: | 10.3934/dcds.2025127 |