Renormalization of Unicritical Diffeomorphisms of the Disk
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.
Keywords
Cite
@article{arxiv.2401.13559,
title = {Renormalization of Unicritical Diffeomorphisms of the Disk},
author = {Sylvain Crovisier and Mikhail Lyubich and Enrique Pujals and Jonguk Yang},
journal= {arXiv preprint arXiv:2401.13559},
year = {2024}
}