Together with Xavier buff, we settled down an old question, following a plan set up by our common advisor Adrien Douady.
Let us recall that the Julia set of a rational fraction is either the whole Riemann sphere, or has empty interior.
Formerly open question: does there exist a rational fraction such that the Julia set has empty interior, but positive Lebesgue measure (i.e. of area not equal to zero) ?
Douady's conjecture: there is an irrational number θ such that the polynomial
For such a polynomial, the Julia set is the boundary of the basin of attraction of infinity. Douady's idea was to construct inductively a sequence of numbers θn whose filled-in Julia set (the complement of the basin of infinity) has non empty interior and a measure bounded below. Provided the polynomial associated to the limit of the sequence θn is not linearizable, its Julia set then has positive measure.
I had been able to reduce this conjecture to another, suggested by known analoguous results and encouraging computer experiments. It concerned the measure of the basin of infinity, seen in Fatou coordinates.
In the process, I discovered a useful tool, the control on parabolic explosion which proved fertile in other questions about the dynamics of these specific polynomials (degree 2 with an irrationally indiferent fixed point).
Among others, it enabled me to find an independent proof of a famous theorem of Yoccoz (optimality of the Brjuno condition for the quadratic family), by studying how parabolic cycles obstruct each other while exploding.
A collaboration with Xavier Buff begun. By closely studying the renormalization of Yoccoz, he found a semi-continuity with loss statement for perturbation of Siegel disks.
This yielded a simple construction of Siegel disks with smooth boundaries in the quadratic family, (proof further simplified by Artur Avila, and Lukas Geyer, and extended to wider families) (see also the work of Perez-Marco).
He then introduced another powerfull tool, the relative Schwarz lemma (we are indebted to McMullen for suggesting us the use of Ahlfors' ultrahyperbolic metrics for the proof).
With all these tools, we were able to prove a conjecture of Stefano Marmi: the continuity with respect to θ of the function
We also proved a general abstract result on the regularity of boundaries of Siegel disk, which roughly states that there are examples of boundaries of Siegel disks in the quadratic family with any prescribed regularity. These examples are Jordan curves that do not contain the critical point (the only other known polynomial examples were given by Herman by a completely different construction). Corollaries of the abstract result include: Smooth boundaries, Quasianalytic, C^n but not C^n+1 (as a curve), Non-quasicircle.
We completed the plan of Douady by proving in 2005 the remaining conjectures blocking the way to positive measure Julia set. For this we used a breakthrough of Inou and Shishikura.
Works in progress include the study of the modulus of continuity of the function Ψ+log(r), (conjectured to be 1/2-Hölder by Marmi Moussa and Yoccoz).
Last update: feb 25th 2007.