Renormalization in the Golden-Mean Semi-Siegel H\'enon Family: Universality and Non-Rigidity

TL;DR

This paper demonstrates the universality of the asymptotic form of renormalizations in the golden-mean semi-Siegel Hénon family and proves the non-rigidity of the Siegel disk boundary.

Contribution

It shows that the two-dimensional renormalizations are universal and parameterized by the average Jacobian, extending previous results on convergence and rigidity.

Findings

Renormalizations converge super-exponentially to a fixed point.

The asymptotic form is universal and depends on the average Jacobian.

The boundary of the Siegel disk is proven to be non-rigid.

Abstract

It was recently shown by Gaidashev and Yampolsky that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel H\'enon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal, and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalization in the H\'enon family considered by de Carvalho, Lyubich and Martens. As an application of our result, we prove that the boundary of the golden-mean Siegel disk of a dissipative H\'enon map is non-rigid.