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arXiv · 2401.13559

Renormalization of Unicritical Diffeomorphisms of the Disk

Sylvain Crovisier·Mikhail Lyubich·Enrique Pujals·Jonguk Yang
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Abstract

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 H\'enon-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 H\'enon-like maps of bounded type belong to our class.

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Sylvain Crovisier, Mikhail Lyubich, Enrique Pujals, Jonguk Yang. 2024-01-24. Renormalization of Unicritical Diffeomorphisms of the Disk. https://arxiv.org/abs/2401.13559

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