Arithmetic criterion for Jordan Siegel disks

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an irrational angle $\theta\in\mathbb R$, let $\lambda=e^{2\pi i\theta}$ and consider a quadratic polynomial with a fixed point of multiplier $\lambda$. A Siegel disk $S$ is its maximal connected domain on which the dynamics is holomorphically conjugate to the rotation $\zeta\mapsto\lambda\zeta$. Its boundary is Jordan if it is a simple closed curve. For $\gamma>0$, the indicated Diophantine bound means that there exists $C>0$ such that $|\theta-p/q|>C/q^\gamma$ for every integer $p$ and positive integer $q$.

Statement recorded in the supplied source.

Find the arithmetic condition on $\theta$ that ensures the boundary of a quadratic Siegel disk is a Jordan curve. In particular, is there a threshold $\gamma_0>2$ such that the displayed Diophantine bound suffices when $\gamma<\gamma_0$, but no longer suffices when $\gamma>\gamma_0$? [S2]

Short English statement

Find an arithmetic criterion for a quadratic Siegel disk to have a Jordan boundary, including whether a Diophantine-exponent threshold separates sufficient from insufficient bounds.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300022 (AMR-052-0022). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Lennart Carleson, Geometry of Julia sets, in Ben Bielefeld and Mikhail Lyubich (editors), Problems in holomorphic dynamics (1992). Siegel-disk arithmetic paragraph, p. 19. https://arxiv.org/pdf/math/9205209.