If the holomorphic map $f:\mathbb{C}\to\mathbb{C}$
has a fixed point $p$, and the derivative $\lambda := f'(p)$ equals $e^{2\pi i \theta}$ (with irrational $\theta$), one can ask if $f$ is conjugate to $z\mapsto\lambda\cdot z$ in a neighborhood of $p$. If it exists, the largest domain of conjugacy is called a 'Siegel disk'. Two properties to keep in mind are:

 - A Siegel disk cannot contain a critical point in its interior (boundary is ok).
 - The boundary of a Siegel disk belongs to the Julia set of $f$.

Quadratic maps can have Siegel disks, but not for just any $\theta$; the number theoretical properties of this 'rotation number' are relevant. However, if $\theta$ is Diophantine, the boundary of the disk is well behaved (Jordan curve, quasi-circle...)

... But the boundary of a Siegel disk can also be wild; for instance, it can be non-locally connected. The **outrageous conjecture** that has been floating around is that:

 - There is a quadratic polynomial with a Siegel disk whose boundary equals the Julia set.

Since a quadratic Julia set is symmetric with respect to the critical point, a quadratic Siegel disk would have a symmetric preimage whose boundary *also* equals the Julia set, but the unbounded component of the complement (Fatou set) also has boundary equal to the Julia set, so our conjectured Siegel disk would form part of a 'lakes of Wada' configuration.