Research Problems in Function Theory — Problem 2.83

Definitions and mathematical statement

Let $f$ be a rational map of degree at least two with a fixed point at zero and $f'(0)=e^{2\pi i\alpha}$ for irrational $\alpha\in\mathbb R$. Assume zero is analytically linearizable: a univalent local map $h$ with $h(0)=0$ conjugates multiplication by $e^{2\pi i\alpha}$ to $f$. Let $S$ be the associated Siegel disc, the maximal connected rotation domain containing zero. The source asks the following three questions.

1. Must $\alpha$ satisfy the Brjuno condition $\sum_{n\ge0}\log q_{n+1}/q_n<\infty$, where $p_n/q_n$ are its continued-fraction convergents? Ask the same for a nonlinear entire map with such a fixed point.
2. Is the restriction of $f$ to the boundary $\partial S$ injective?
3. Does $\partial S$ contain no periodic point of $f$, of any positive period?

Boundaries and periodic points are taken on the Riemann sphere for rational maps. A Siegel disc need not be assumed to have a Jordan boundary, so injectivity on its boundary is meaningful without choosing a boundary parametrization. The source records the affirmative Brjuno necessity for quadratic polynomials; the first question concerns the larger classes, while the boundary questions retain their full scope.

Short English statement

What arithmetic restrictions follow from a Siegel disc, and must its boundary carry an injective map with no periodic points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302083 (AMR-022-2083), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.83. Original source indexed by AMR-022-2083. https://arxiv.org/abs/1809.07200.