5 problems

Let $f$ be a polynomial of degree $d$ with an indifferent fixed point at the origin. Let $c_i$ be the critical points of $f$, let $\theta$ be the rotation number at the origin, and…

Reduced conjecture.

Douady's optimality conjecture. If $\theta$ is not a Bruno number, then no polynomial of degree at least $2$ can have a Siegel disk of rotation number $\theta$.

Let $b1$ be a rotation number of bounded type, and let $\Gamma_{b1}(\theta)$ denote the parametrization of the boundary of the corresponding Siegel disk. Bounded-type nonsmoothness…

Bounded-type regularity conjecture. If $b1$ is of bounded type, then the map $\Gamma_{b1}$ is at least $C^1$.