Smooth Siegel disks everywhere

TL;DR

This paper proves that most holomorphic maps with an indifferent fixed point at zero have Siegel disks with smooth boundaries, under certain regularity and non-degeneracy conditions, expanding understanding of boundary regularity in complex dynamics.

Contribution

It establishes the existence of smooth boundary Siegel disks in broad families of holomorphic maps and characterizes degenerate families as exceptional cases.

Findings

Most families with indifferent fixed points have smooth boundary Siegel disks.

Degenerate families, where non-linearizable maps are dense, are shown to be exceptional.

The method allows for other boundary regularity conditions.

Abstract

We prove the existence of Siegel disks with smooth boundaries in most families of holomorphic maps fixing the origin. The method can also yield other types of regularity conditions for the boundary. The family is required to have an indifferent fixed point at $0$, to be parameterized by the rotation number $\alpha$, to depend on $\alpha$ in a Lipschitz-continuous way, and to be non-degenerate. A degenerate family is one for which the set of non-linearizable maps is not dense. We give a characterization of degenerate families, which proves that they are quite exceptional.