Blaschke-type models for multimodal circle maps
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915987387842560 |
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| author | de Faria, Edson de Melo, Welington Salomão, Pedro A. S. Vargas, Edson |
| author_facet | de Faria, Edson de Melo, Welington Salomão, Pedro A. S. Vargas, Edson |
| contents | For each integer $m \geq 1$, we construct a finite-dimensional family of rational maps, given by Blaschke-type products, whose restriction to the unit circle consists of $2m$-multimodal maps. We show that every post-critically finite $2m$-multimodal circle map satisfying natural dynamical conditions is topologically conjugate to a map in this family. Moreover, we prove that this realization is unique up to rotation: two maps in the family that are topologically conjugate on the circle differ by a rigid rotation. In particular, the family provides a canonical model realizing all post-critically finite combinatorics in this class. The proofs combine a detailed description of the critical geometry of these Blaschke-type maps with a Thurston-type fixed point argument for a pull-back operator on the parameter space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05823 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Blaschke-type models for multimodal circle maps de Faria, Edson de Melo, Welington Salomão, Pedro A. S. Vargas, Edson Dynamical Systems Complex Variables 37F10 37F20 37E10 For each integer $m \geq 1$, we construct a finite-dimensional family of rational maps, given by Blaschke-type products, whose restriction to the unit circle consists of $2m$-multimodal maps. We show that every post-critically finite $2m$-multimodal circle map satisfying natural dynamical conditions is topologically conjugate to a map in this family. Moreover, we prove that this realization is unique up to rotation: two maps in the family that are topologically conjugate on the circle differ by a rigid rotation. In particular, the family provides a canonical model realizing all post-critically finite combinatorics in this class. The proofs combine a detailed description of the critical geometry of these Blaschke-type maps with a Thurston-type fixed point argument for a pull-back operator on the parameter space. |
| title | Blaschke-type models for multimodal circle maps |
| topic | Dynamical Systems Complex Variables 37F10 37F20 37E10 |
| url | https://arxiv.org/abs/2605.05823 |