Arbeitsgemeinschaft: Julia Sets of Positive Measure

Abstract

A polynomial $P:\\C\\to \\C$ can be considered as a dynamical system. We are interested in the sequences $(z_n)$ defined by induction: $$ z0\\in \\C\\quad\\text{and}\\quad z{n+1}=P(z_n). $$ The filled-in Julia set $K_P$ is the set of points $z_0\\in \\C$ for which the sequence $(z_n)$ is bounded. This set is compact. The Julia set $J_P$ is the boundary of $K_P$. In particular, it has empty interior. There is a small collection of polynomials, for instance $$ P(z) = z^d\\quad ,\\quad P(z) = z^2-2, $$ for which the Julia set can be fairly easily understood, but most exhibit \`\`fractal'' geometry and \`\`chaotic'' behavior, the analysis of which requires serious tools from complex analysis, dynamical systems, topology, combinatorics, $\\ldots$ This subject has a fairly long history, with contributions by Koenigs, Schr\\"{o}der, B\\"{o}ttcher in the late 19th century, and the great memoirs of Fatou and Julia around 1920. There followed a dormant period, with notable contributions by Cremer (1936) and Siegel (1942), and a rebirth in the 1960's (Brolin, Guckenheimer, Jakobson). Since the early 1980's, partly under the impetus of computer graphics, the subject has grown vigorously, with major contributions by Douady, Hubbard, Sullivan, Thurston, and more recently Lyubich, McMullen, Milnor, Shishikura, Yoccoz \\ldots Fatou found sufficient conditions for the boundary of the basin of an attracting fixed point to be a Cantor set with Lebesgue measure equal to $0$. He could not tell whether or not the measure could be positive. For some time and until the 1990's, the conjecture, reinforced by the analogy with Ahlfors's conjecture on the area of limit sets of Kleinian groups, was that no Julia set of a polynomial could have positive area. Results in this direction were obtained by Douady and Hubbard in the case of hyperbolic or subhyperbolic maps, by Branner, Hubbard and McMullen in the case of non-renormalizable cubic polynomials with an escaping critical point, by Lyubich and Shishikura in the case of finitely renormalizable quadratic polynomials without indifferent cycles, by Petersen in the case of quadratic polynomials having a Siegel disk with bounded type rotation number. In the 1990's, Douady began to catch a glimpse of a method for Julia sets of positive area: in the family of degree 2 polynomials with an indifferent Cremer fixed point. Recently, we brought Douady's method to completion. The Arbeitsgemeinschaft {\\em Julia sets of positive measure} focused on the proof of existence of quadratic polynomials having a Julia set of positive area. It was held March 30th--April 5th, 2008. It was attended by 36 participants.