Douady's conjecture on exotic Siegel disks for rational maps

Let $f$ be a polynomial or rational function of degree $d\ge 2$. An irrationally indifferent periodic point is contained in the Fatou set exactly when it is linearizable; the corresponding Fatou component is a simply connected Siegel disk. A Siegel disk is exotic if its rotation number is not a Brjuno number, where, for the convergents $p_n/q_n$ of an irrational $\alpha$, the Brjuno numbers are

$${\mathcal B}=\left\{\alpha\in\mathbb R\setminus\mathbb Q:\sum q_n^{-1}\log q_{n+1}<\infty\right\}.$$

Douady's conjecture. Polynomials and rational functions of degree $d\ge 2$ do not have exotic Siegel disks.

Brjuno and Rüssmann proved that Brjuno rotation numbers imply linearizability, while Yoccoz proved sharpness for quadratic polynomials. The conjecture asks whether this sharpness holds for all polynomials and rational functions of degree at least two; it remains open in the source.