The term conformal dynamics has recently gained some currency and is meant to cover real and complex dynamics. This merging of sub-areas makes sense because their methods are showing more overlap than ever. The project suggests a variety of goals. The first is the study of boundaries of connectedness loci for families of polynomials. This generalizes the study of the boundary of the Mandelbrot set. Problems include the metric structure of the boundary, including the boundary behavior of the Riemann map of the complement and the distribution of the harmonic measure. This part of the problem continues the on-going work by J. Graczyk and the proposer. The second goal is the study of boundaries of Siegel disks. The main problem in this area is deciding whether such boundaries are Jordan curves. The proposed approach is based on gaining information on the Riemann map of the Siegel disk by means of a cohomological equation. The third problem is in real dynamics and concerns the existence of wild attractors with additional properties. The forth goal is to estimate the measure and Hausdorff dimension of sets invariant under certain iterated function systems.
To understand the meaning of this research in the broad perspective of science, one has first to realize the role played by one-dimensional, or conformal, dynamics. Systems which appear as models in natural sciences, such as physics, astronomy, meteorology, economics or ecology, involve multiple parameters. However, frequently one dominant parameter emerges which controls long time-behavior of the system. In such a way the logistic family appears in the study of many complicated systems. The logistic maps are quadratic polynomials. It is useful to study them on the complex plane, and thus complex quadratic polynomials enter the picture. This is the first object of our study. In other situations the system begins to show quasi-periodic or ``rotational'' behavior. This is situation is modeled by Siegel disks which are another subject of our proposed study. Wild attractors, which are our third subject, refer to a very unusual chaotic regime for a system in which two completely different limiting modes of long-term behavior co-exist side by side. The attractor for an iterated function system which we will also investigate appears as a projection of a certain fractal set. This kind of problem appears in applications where one often has a multi-dimensional fractal set which can be studied by projections onto lower-dimensional spaces.
