Singular values and bounded Siegel disks

TL;DR

This paper investigates the dynamics of entire transcendental functions with bounded Siegel disks, establishing conditions under which the boundary contains critical points and when these points are recurrent.

Contribution

It introduces a general theorem that links the presence of singular values with bounded orbits to the structure of Siegel disk boundaries, extending previous results.

Findings

Boundary of Siegel disk contains a critical point if two singular values have bounded orbits.

Provides criteria for the recurrence of critical points on the boundary.

Generalizes earlier theorems with less restrictive assumptions.

Abstract

Let $f$ be an entire transcendental function of finite order and $\Delta$ be a forward invariant bounded Siegel disk for $f$ with rotation number in Herman's class $\mathcal{H}$. We show that if $f$ has two singular values with bounded orbit, then the boundary of $\Delta$ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow.