Periodic point on a Siegel-disk boundary

Definitions and mathematical statement

For a rational map $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$ of degree at least two, the Fatou set is the largest open set on which the iterates $\{R^n:n\ge0\}$ form a normal family in the spherical metric. A Fatou component $U$ of least period $p\ge1$ is a Siegel disk if there are a conformal bijection $h:\mathbb D\to U$ and an irrational $\theta$ such that \[ R^p(h(z))=h(e^{2\pi i\theta}z)\qquad(z\in\mathbb D). \] Its boundary $\partial U$ is taken in the Riemann sphere.

The problem asks whether there can exist such an $R,U$ and a point $a\in\partial U$ which is periodic for $R$: \[ R^q(a)=a\qquad\text{for some integer }q\ge1. \] The period $q$ need not initially be specified, and the map is allowed to have any finite degree. A construction would have to prove both that $U$ is the full Siegel Fatou component and that the periodic point lies on its actual boundary.

The question does not concern the center $h(0)$, which is already a periodic point inside $U$. It also does not assume that $h$ extends continuously to the circle, that the boundary is a Jordan curve, or that the rotation number has bounded continued-fraction entries. Those regularity hypotheses can exclude the proposed phenomenon and would restrict the general question. Accumulation of periodic points near the boundary is different from a periodic point on the boundary.

Short English statement

Can the boundary of a rational map’s Siegel disk contain a periodic point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5300036 (AMR-052-0036), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Stony Brook - Open Problems in Dynamical Systems, 1992, Milnor local P11. Original source indexed by AMR-052-0036. https://www.math.stonybrook.edu/open-problems-dynamical-systems.
• S3 Milnor, Problems on local connectivity, Problem11, in Bielefeld–Lyubich (1992). https://www.math.stonybrook.edu/preprints/ims92-7.pdf.