Moscow Mathematical Journal

Volume 22, Issue 3, July–September 2022 pp. 451–491.

Golden Mean Siegel Disk Universality and Renormalization

Authors: Denis Gaidashev (1) and Michael Yampolsky (2)
Author institution:(1) Uppsala University, Uppsala, Sweden
(2) University of Toronto, Toronto, Canada

Summary:

We provide a computer-assisted proof of one of the central open questions in one-dimensional renormalization theory — universality of the golden-mean Siegel disks. We further show that for every function in the stable manifold of the golden-mean renormalization fixed point the boundary of the Siegel disk is a quasicircle which coincides with the closure of the critical orbit, and that the dynamics on the boundary of the Siegel disk is rigid.

Furthermore, we extend the renormalization from one-dimensional analytic maps with a golden-mean Siegel disk to two-dimensional dissipative Hénon-like maps and show that the renormalization hyperbolicity result still holds in this setting.

2020 Math. Subj. Class. 37E20, 37F25, 37F50, 37F80.

Keywords: Renormalization, universality, Siegel disk, Henon-like map.

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