Renormalization and Siegel disks for complex H\'enon maps

TL;DR

This paper proves that the boundaries of Siegel disks in certain complex Hénon maps are topological circles, using hyperbolicity of renormalization, with results extending to various rotation numbers under specific conditions.

Contribution

It establishes the topological circularity of Siegel disk boundaries in dissipative quadratic complex Hénon maps using hyperbolic renormalization techniques, extending to a broader class of rotation numbers.

Findings

Siegel disk boundaries are topological circles in certain Hénon maps.

Hyperbolicity of renormalization is key to the proof.

Results depend on an assumed renormalization hyperbolicity property.

Abstract

We use hyperbolicity of golden-mean renormalization of dissipative H\'enon-like maps to prove that the boundaries of Siegel disks of sufficiently dissipative quadratic complex H\'enon maps with golden-mean rotation number are topological circles. Conditionally on an appropriate renormalization hyperbolicity property, we derive the same result for Siegel disks of H\'enon maps with all eventually periodic rotation numbers.