18 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…

Let $P_\theta(z)=e^{2i\pi\theta}z+z^2$ for $\theta\in\mathbb R$. Let $R(P_\theta)$ be the conformal radius of the Siegel disk of $P_\theta$, or $0$ if no Siegel disk exists, and le…

Let $E_{\kappa}$ be an exponential map with a Siegel disk $U$ whose rotation number is of bounded type. The singular value is $\kappa$, and $\infty$ denotes the point at infinity i…

Let $\theta^*=(\sqrt{5}-1)/2$ be the inverse golden mean, and consider holomorphic maps with a fixed point at the origin having multiplier $e^{2\pi i\theta^*}$. For such a map, let…

Quadratic Siegel-disk radius conjecture. The correct order is conjectured to be equal to $2$.

Reduced conjecture.

Buff's conjecture. For every degree $d\geq 2$, there exists $C_d>0$ such that for every Bruno number $\theta$ and every polynomial $P\in\mathcal{P}[\theta]$,

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 the corresponding Siegel disk be parametrized by $(Z(x),W(x))$. Bounded-type graph conjecture. The Siegel disk is a graph: it…

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 $C^1$.

Douady–Sullivan conjecture. The boundary of every Siegel disk of every rational map of degree at least two is a Jordan curve.

Invariant-class and renormalization conjecture. A variation of the invariant class and renormalization may be defined for $N=1$.

Boundary critical point conjecture. The boundary $\partial\Delta$ contains a critical point of $f$.

Marmi–Moussa–Yoccoz conjecture. The function $\Upsilon$ is $1/2$-Hölder continuous as a function of $\theta$.

The golden mean Siegel disk is the Siegel disk associated with the quadratic polynomial $e^{2\pi i\theta}z+z^2$, where $\theta=(\sqrt{5}-1)/2$. Consider the closest backward return…

Let $f$ be a rational map of degree at least $2$ with a Siegel disk $\Delta$, and let its rotation number belong to the Herman class $\mathcal H$. Critical-point conjecture. The bo…

A Siegel disk of a rational map $f$ is a maximal domain on which $f$ is holomorphically conjugate to an irrational rotation. Douady–Sullivan conjecture. The boundary of every Siege…