Cubic polynomials with a 2-cycle of Siegel disks

TL;DR

This paper studies the parameter space of cubic polynomials with a 2-cycle of Siegel disks, revealing geometric structures of loci where critical points lie on the boundaries of these disks under certain conditions.

Contribution

It characterizes the geometric structure of the locus of cubic polynomials with critical points on Siegel disk boundaries, assuming bounded type rotation numbers.

Findings

Locus consists of two arcs and a Jordan curve on the moduli space.

Arcs correspond to critical points on the same Siegel disk boundary.

Jordan curve corresponds to critical points on different Siegel disk boundaries.

Abstract

Under conjugation by affine transformations, the dynamical moduli space of cubic polynomials $f$ with a $2$-cycle of Siegel disks is parameterized by a three-punctured complex plane as a degree-$2$ cover. Assuming the rotation number of $f^2$ on the Siegel disk is of bounded type, we show that on the three-punctured complex plane, the locus of the cubic polynomials with both finite critical points on the boundaries of the Siegel disks on the $2$-cycle is comprised of two arcs, corresponding to the cases with two critical points on the boundary of the same Siegel disk, and a Jordan curve, corresponding to the cases with two critical points on the boundaries of different Siegel disks.