Singular values and bounded Siegel disks

Let $f$ be an entire transcendental function of finite order and $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$ 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.