Golden mean Siegel disk universality and renormalization

TL;DR

This paper proves the universality of golden-mean Siegel disks using computer-assisted methods, showing boundary rigidity and extending results to two-dimensional Hénon-like maps, advancing understanding in complex dynamics.

Contribution

It provides a computer-assisted proof of Siegel disk universality and extends renormalization results from one-dimensional to two-dimensional dissipative maps.

Findings

Boundary of Siegel disk is a quasicircle

Dynamics on the boundary is rigid

Renormalization hyperbolicity extends to Hénon-like maps

Abstract

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 H\'enon-like maps and show that the renormalization hyperbolicity result still holds in this setting.