Quadratic Siegel Disks with Rough Boundaries

Abstract

In the quadratic family (the set of polynomials of degree 2), Petersen and Zakeri proved the existence of Siegel disks whose boundaries are Jordan curves, but not quasicircles. In their examples, the critical point is contained in the curve. In the first part, we prove the existence of quadratic examples that do not contain the critical point. In the second part, using a more abstract point of view (suggested by Avila), we show that we can control quite precisely the degree of regularity of the boundary of the quadratic Siegel disks we create by perturbations. For instance there exists examples where the boundary is C^n but not C^{n+1}.