Dynamics, Probability, and Conformal Invariance (05w5009)

Arriving in Banff, Alberta Saturday, March 12 and departing Thursday March 17, 2005


Ilia Binder (University of Toronto)

(Yale University)

(University of Washington, USA)

Michael Yampolsky (University of Toronto)


The most basic property of the SLE trace is its Hausdorff Dimension. For $0leqkappaleq 8$ Beffara has now announced a theorem: The Dimension is a.s. $1 + kappa/8$. Earlier results of Rohde and Schramm bounded the Minkowski dimension by that value and also yield the fact that the exterior of the trace is a H{"o}lder Domain when $kappaneq 4$. We are still missing a good understanding of the exterior of the trace when $kappa$ is greater than $4$. DuPlantier's Duality Conjecture, derived by arguments from CFT, is that the "frontier" of the trace has the same dimension for $kappa$ and $16/kappa$. Indeed the two "dual" objects are supposed to be conformally equivalent. The difficulties are closely related to the study of quadratic (or more general polynomial) Julia Sets. For example, almost all (in the sense of harmonic measure) values of $c$ in the boundary of the Mandelbrot Set have the property that the corresponding basin at infinity is a H{"o}lder Domain. There is a rough correspondence to return times for the critical point (in the setting of Julia Sets) and the behavior of the Haar coefficients in the Brownian "driver" (for the SLE process). Both mechanisms control the H{"o}lder behavior of the corresponding domains, but the exact nature of the correspondence between the two processes is poorly understood at this time. In particular, it would be very helpful to have another method of generating SLE traces. For Brownian Motion one does have different methods of generating the process - perhaps these different methods could be translated into a new method for generating SLE traces. It seems possible that one could define SLE traces, at least for $kappaleq 4$, by "bending" a curve repeatedly, i.e. in a multi-scale fashion, while applying a rescaling to keep the diameter from vanishing. In both SLE and Holomorphic Dynamics, some of the most difficult problems are related to the positive Lebesgue measure of the trace or Julia set. In SLE one knows exactly when the trace has positive measure, as it becomes space filling at $kappa = 8$. The problem for SLE is the understanding of the frontier for large values of $kappa$. In the setting of Rational Dynamics there is no example known of a Julia Set of positive measure and empty interior. Indeed if no such examples exist, it follows that hyperbolic dynamics is dense in the rational family (the main conjecture in the field, known as the Fatou Conjecture). The issue of the local connectedness of the Mandelbrot set (MLC) is closely related.

We remark that if instead of iterating the same polynomial at every step, one is allowed to choose a sequence of different polynomials (one only needs three), then one can produce a positive measure ``random Julia set''. The mechanism that enters here (closely related to "parabolic implosion") is that at $c = 1/4$, the Julia Set contains a cusp, and one can then start pushing "thin cusps" into the $c = 1/4$ Julia set to get a dendrite of positive measure. These cusps are of the same type that appear at $kappa = 4$ in the SLE trace; the exterior is no longer a H{"o}lder domain. A. Ch{'e}ritat has recently been able to push through a large part of a program initiated by A. Douady to the end of constructing a quadratic Julia Set of positive measure.

The Douady's program consists in approximating the candidate quadratic polynomial by a sequence of carefully chosen quadratics with parabolic periodic orbits. Each successive approximation removes some thin cusps from the filled Julia set -- the hope is to bound the area of what is left from below. Due to the work of X. Buff and A. Ch{'e}ritat, this approach boils down to several conjectures about cylinder renormalization. The latter was introduced by M. Yampolsky for proving the hyperbolicity of renormalization of critical circle maps. The appearance of renormalization-type arguments is common for this class of problems: for example, Shishikura used a parabolic renormalization procedure to demonstrate the existence of quadratic Julia sets of Hausdorff Dimension $2$ used in his proof of $text{HDim}(partial M)=2$. The one-dimensional renormalization theory has seen a spectacular progress since the works of Douady, Hubbard, and Sullivan which related it to Holomorphic Dynamics, culminating in a proof of the Feigenbaum Universality by Sullivan, McMullen, and Lyubich. Many important problems of scaling invariance and universality still remain open, however, even in the setting of One-Dimensional Dynamics.

Most notably, an explanation of the Feigenbaum-type universality for the case of non even integer order of the critical point is still missing, and so is a proof of renormalization convergence and hyperbolicity for the case of quadratics of non-real combinatorial types. The latter problem is intimately connected with MLC.

The study of harmonic measure has played a leading role in both Holomorphic Dynamics and SLE. Starting from Makarov's theorem on the dimension of harmonic measure, one has steadily developed an understanding of the fine structure of harmonic measure, and one now knows that there is a very strong connection between its extremal behavior and holomorphic dynamical systems. For example, a recent result of Binder and Jones states that (for $alpha geq 1$) the $f(alpha)$ spectrum of an arbitrary planar domain can be majorized by the spectrum coming from the basin at infinity for a polynomial with connected Julia Set. It is expected that this phenomenon is only the beginning of a larger theory. Binder has introduced the rotation spectrum for simply connected planar domains, and he and DuPlantier produced several results for the case of SLE traces. A full understanding of this problem requires a solution of DuPlantier's conjecture plus further developments in conformal mappings and potential theory.

The study of conformally invariant geometries related to random processes has had two great successes in the past few years. First Lawler, Schramm, and Werner solved the Mandelbrot conjecture: The Brownian Frontier has dimension $4/3$. This result turns out to be essentially the same as the SLE trace having dimension $4/3$ for $kappa = 8/3$. The second great success is Smirnov's proof that, for the hexagonal grid, the scaling limit of percolation exists and is conformally invariant. One now understands these two results to be pieces of a much larger picture involving CFT, SLE, Brownian Motion, renormalization methods, conformal mappings, etc. One of the major challenges is now to put on a fully rigorous basis the machinery of Quantum Gravity in CFT. A noteworthy first step has been taken by Angel and Schramm who have found a rigorous method to produce random, infinite triangulations of the plane (which are also conformally invariant). Deeper understanding of these results is probably needed before further there is more progress on Quantum Gravity. For example one would like a better understanding of random triangulations of Riemann surfaces. It is expected that there is also a strong relation of the above methods to scaling limits in the Ising model, and there are already strong results of e.g. R. Kenyon. Most of these problems seem to have "dual" versions where one attempts to define families of random conformal mappings, often with a "temperature" that can be varied. There are versions of these problems both for planar domains and Riemann surfaces, and all involve some sort of rescaling mechanism.