On the Julia set of a typical quadratic polynomial with a Siegel disk. (English) Zbl 1069.37038

Petersen, C. L.; Zakeri, S.

Ann. Math. (2) 159, No. 1, 1-52 (2004).

The paper studies quadratic polynomials of the form \(P(z) = z^2 + rz\), where \(r=e^{2\pi i \theta}\) for some irrational number \(0<\theta < 1\). A growth condition is assumed on the integers \(a_n\) making up the continued fraction expansion of \(\theta\), namely that \(\log (a_n) = {\mathcal O}(\sqrt{n})\). The set of \(\theta\)’s satisfying this condition has full measure.
The main result is that for such a \(\theta\), the Julia set of \(P\) is locally connected and has Lebesgue measure 0. It follows that there is a set of \(\theta\)’s of full measure for which the corresponding polynomials \(P\) have Siegel disks which are Jordan domains whose boundary contains the critical point of \(P\). A major part of the proof involves area and length estimates which enables one to apply a theorem of David concerning the integrability of Beltrami differentials whose dilatation is unbounded. This allows one to reduce the problem to a somewhat simpler one, which the authors solve, using an approach due to Yoccoz.

Reviewer: Mike Hurley (Cleveland)

MSC:

37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F50	Small divisors, rotation domains and linearization in holomorphic dynamics

Keywords:

quadratic polynomials; growth condition; Julia set; Siegel disks; Jordan doamins