Herman's condition and critical points on the boundary of Siegel disks of polynomials with two critical values

TL;DR

This paper extends Herman's theorem to polynomials with two critical values, showing that under certain diophantine conditions, Siegel disks must have a critical point on their boundary.

Contribution

It generalizes Herman's condition from unicritical to bicritical polynomials, establishing boundary critical points for Siegel disks under diophantine conditions.

Findings

Siegel disks with diophantine rotation numbers have boundary critical points.

Extension of Herman's theorem to polynomials with two critical values.

Boundary critical points are guaranteed under Herman's condition.

Abstract

We extend a theorem of Herman from the case of unicritical polynomials to the case of polynomials with two finite critical values. This theorem states that Siegel disks of such polynomials, under a diophantine condition (called Herman's condition) on the rotation number, must have a critical point on their boundaries.