Conformal Radius Bounds for Polynomial Siegel Disks

Definitions and mathematical statement

Let $d\geq3$ be an integer, and let $f\in{\mathbb{C}}[z]$ have degree exactly $d$, with $f(0)=0$ and $f'(0)=e^{2\pi i\theta}$ of modulus one. List the critical points $c_1,\ldots,c_{d-1}$ with multiplicity. Since $f'(0)\ne0$, their minimum modulus is positive. When the fixed point has a Siegel disk $U$ and $\psi:\mathbb D\to U$ is conformal with $\psi(0)=0$, its conformal radius is $\operatorname{rad}(f)=|\psi'(0)|$; the local rotation number is $\theta\in{\mathbb{R}}/{\mathbb{Z}}$. For irrational $\theta$, one standard Brjuno-sum convention uses the denominators $q_j$ of its continued-fraction convergents and \[B(\theta)=\sum_{j\geq0}\frac{\log q_{j+1}}{q_j}\in[0,\infty].\] The exact Brjuno-function convention is not given in the source. Its conjecture asks for a real constant depending only on degree such that \[\forall d\geq3\ \exists C(d)\ \forall f:\qquad \log\operatorname{rad}(f)\leq-\frac{B(\theta)}{d-1}+\log\!\left(\min_{1\leq i\leq d-1}|c_i|\right)+C(d).\] The constant is uniform in the rotation number and in the critical-point configuration, and no monic normalization or distinct-critical-points condition is supplied.

Indifferent includes rationally indifferent or nonlinearizable fixed points, for which a Siegel-disk conformal radius is not defined as stated. One possible extended convention sets the radius to zero when no Siegel disk exists, takes $\log0=-\infty$, and assigns $B=+\infty$ where appropriate; in exponential form the bound would read $\operatorname{rad}(f)\leq e^{C(d)}\min_i|c_i|\exp(-B(\theta)/(d-1))$. These extensions are not specified by the extract and cannot silently replace its all-indifferent quantifier by an irrational Brjuno-only question. In particular the $B=+\infty$ case is substantive if a positive conformal radius is allowed. The supplied 2026-08-21 triage labels the general bound open and reports reductions and special cases rather than a proof in all degrees.

Short English statement

Ask for a degree-uniform conformal-radius bound controlled by the Brjuno sum and distance to the nearest critical point. The all-indifferent formulation needs conventions for rational rotation numbers and fixed points without a Siegel disk.

Sources

• Local UnsolvedMath registry, permanent ID 30001088, source problem catalogue entry, Conformal Radius Bounds for Polynomial Siegel Disks, proposed year 2008; no individual proposer supplied.
• Supplied citation: Trends and Developments in Complex Dynamics (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/48, pp. 2703--2762: https://doi.org/10.4171/owr/2008/48. Supplied literature label: original OWR report.
• Supplied literature label: Buff--Chéritat--Roesch, bounded-type conformal-radius maxima, https://arxiv.org/abs/2606.24833.
• Supplied literature label: current reduced-conjecture record, https://mathdb.com/p/375544/reduced-conjecture-on-conformal-radii-for-non-bruno-rotation.
• Literature assessment date: 2026-08-21. The status and progress claims are supplied dated curation, not independently verified here.