Quadratic rational maps with a $2$-cycle of Siegel disks

TL;DR

This paper studies quadratic rational maps with a 2-cycle of Siegel disks, proving boundary properties related to critical points and describing the structure of parameter space regions where certain boundary conditions hold.

Contribution

It establishes boundary critical point containment properties and characterizes the parameter locus as a Jordan curve for maps with a 2-cycle of Siegel disks.

Findings

Boundaries of Siegel disks contain at most one critical point.

The locus with two critical points on the boundary is a Jordan curve.

Results apply to maps with bounded type Siegel disks.

Abstract

For the family of quadratic rational functions having a $2$-cycle of bounded type Siegel disks, we prove that each of the boundaries of these Siegel disks contains at most one critical point. In the parameter plane, we prove that the locus for which the boundaries of the $2$-cycle of Siegel disks contain two critical points is a Jordan curve.