The boundaries of golden-mean Siegel disks in the complex quadratic H\'enon family are not smooth

TL;DR

This paper proves that the boundary curves of golden-mean Siegel disks in certain complex quadratic Hénon maps are topological circles but cannot be smoothly differentiable, revealing their complex geometric nature.

Contribution

It establishes that these Siegel disk boundaries are topological circles that are not $C^1$-smooth, advancing understanding of their geometric properties.

Findings

Boundaries are topological circles

Boundaries are not $C^1$-smooth

Enhances understanding of Siegel disk geometry

Abstract

As was recently shown by the first author and others, golden-mean Siegel disks of sufficiently dissipative complex quadratic H\'enon maps are bounded by topological circles. In this paper we investigate the geometric properties of such curves, and demonstrate that they cannot be $C^1$-smooth.