Quadratic Siegel Disks with Rough Boundaries.

In the quadratic family (the set of polynomials of degree 2), Pe-tersen and Zakeri [PZ] 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, as an illustration of the flexibility of the tools developped in [BC], we prove the existence of examples that do not contain the critical point. In the second part, using a more abstract point of view (suggested by Avila in [A]), we show that we can control quite precisely the degree of regularity of the boundary of the Siegel disks we create by perturbations. Notations: U is the set of complex numbers with norm = 1. Here, rU will be used as a shorthand for: the circle of center 0 and radius r. Sometimes, we will note (r, t) for r exp(i2πt). The symbol T will denote the quotient R/Z. For θ ∈ R, P θ (z) = exp(i2πθ)z + z 2. If the fixed point z = 0 is linearizable, then ∆(θ) is the Siegel disk of P θ at 0, and r(θ) its conformal radius. Otherwise ∆(θ) = ∅ and r(θ) = 0. Reminder: ∀θ ∈ R, r(θ) < 4. Let D 2 denote the set of bounded type irrational numbers. Let us recall the following Lemma 1 (Herman). For all θ ∈ D 2 , assume that U is a connected open set containing 0 and f : U → C is a holomorphic function which fixes 0 with derivative e 2πiθ. Let ∆ be the Siegel disk of f at 0 (which exists by a theorem of Siegel). If U is simply connected and f is univalent, then ∆ cannot have compact closure in U. Remark. In fact, Herman's theorem is stronger: U needs not to be simply connected , and the condition on θ is weaker (it is called the Herman condition). See [H] for a reference. An essential tool is the following Lemma 2 (Buff, Chéritat). (independently by A. Avila) For all θ Bruno and r < r(θ), there exists a sequence of D 2 numbers θ n −→ θ such that r(θ n) −→ r. Both [BC] and [A] have stronger statements. Note that we do not only require θ n to be Bruno, but also to be a bounded type number (the bound varies with n). To the …