Renormalization in the Golden-Mean Semi-Siegel H\'enon Family: Non-Quasisymmetry

TL;DR

This paper demonstrates that, unlike quadratic polynomials, quadratic Hénon maps with golden-mean semi-Siegel dynamics do not have boundaries that are quasicircles, revealing a fundamental difference in their geometric structure.

Contribution

It establishes that the boundary of the semi-Siegel Hénon map's domain is not a quasicircle, contrasting the behavior observed in one-dimensional quadratic polynomials.

Findings

Quadratic polynomials have quasicircle Siegel disk boundaries.

Quadratic Hénon maps do not have quasicircle boundaries in the semi-Siegel case.

The geometric structure of Hénon maps differs fundamentally from one-dimensional polynomials.

Abstract

For quadratic polynomials of one complex variable, the boundary of the golden-mean Siegel disk must be a quasicircle. We show that the analogous statement is not true for quadratic H\'enon maps of two complex variables.