Renormalization of Unicritical Diffeomorphisms of the Disk
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914651411841024 |
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| author | Crovisier, Sylvain Lyubich, Mikhail Pujals, Enrique Yang, Jonguk |
| author_facet | Crovisier, Sylvain Lyubich, Mikhail Pujals, Enrique Yang, Jonguk |
| contents | We introduce a class of infinitely renormalizable, unicritical diffeomorphisms of the disk (with a non-degenerate "critical point"). In this class of dynamical systems, we show that under renormalization, maps eventually become Hénon-like, and then converge super-exponentially fast to the space of one-dimensional unimodal maps. We also completely characterize the local geometry of every stable and center manifolds that exist in these systems. The theory is based upon a quantitative reformulation of the Oseledets-Pesin theory yielding a unicritical structure of the maps in question comprising regular Pesin boxes co-existing with "critical tunnels" and "valuable crescents". In forthcoming notes we will show that infinitely renormalizable perturbative Hénon-like maps of bounded type belong to our class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13559 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Renormalization of Unicritical Diffeomorphisms of the Disk Crovisier, Sylvain Lyubich, Mikhail Pujals, Enrique Yang, Jonguk Dynamical Systems 37D25, 37E20, 37E30 (Primary), 37C05, 37C70, 37E05 (Secondary) We introduce a class of infinitely renormalizable, unicritical diffeomorphisms of the disk (with a non-degenerate "critical point"). In this class of dynamical systems, we show that under renormalization, maps eventually become Hénon-like, and then converge super-exponentially fast to the space of one-dimensional unimodal maps. We also completely characterize the local geometry of every stable and center manifolds that exist in these systems. The theory is based upon a quantitative reformulation of the Oseledets-Pesin theory yielding a unicritical structure of the maps in question comprising regular Pesin boxes co-existing with "critical tunnels" and "valuable crescents". In forthcoming notes we will show that infinitely renormalizable perturbative Hénon-like maps of bounded type belong to our class. |
| title | Renormalization of Unicritical Diffeomorphisms of the Disk |
| topic | Dynamical Systems 37D25, 37E20, 37E30 (Primary), 37C05, 37C70, 37E05 (Secondary) |
| url | https://arxiv.org/abs/2401.13559 |