Conformal Dynamics

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DMS-0072312The term conformal dynamics has recently gained some currency and ismeant to cover real and complex dynamics. This merging of sub-areasmakes sense because their methods are showing more overlap than ever. The project suggests a variety of goals. The first is the study ofboundaries of connectedness loci for families of polynomials. Thisgeneralizes the study of the boundary of the Mandelbrot set. Problemsinclude the metric structure of the boundary, including the boundarybehavior of the Riemann map of the complement and the distribution ofthe harmonic measure. This part of the problem continues the on-goingwork by J. Graczyk and the proposer. The second goal is the study ofboundaries of Siegel disks. The main problem in this area is decidingwhether such boundaries are Jordan curves. The proposed approach isbased on gaining information on the Riemann map of the Siegel disk bymeans of a cohomological equation. The third problem is in realdynamics and concerns the existence of wild attractors with additionalproperties. The forth goal is to estimate the measure and Hausdorffdimension of sets invariant under certain iterated function systems. To understand the meaning of this research in the broad perspective ofscience, one has first to realize the role played by one-dimensional,or conformal, dynamics. Systems which appear as models in naturalsciences, such as physics, astronomy, meteorology, economics orecology, involve multiple parameters. However, frequently one dominant parameter emergeswhich controls long time-behavior of the system. In such a way thelogistic family appears in the study of many complicated systems. Thelogistic maps are quadratic polynomials. It is useful to study them onthe complex plane, and thus complex quadratic polynomials enter thepicture. This is the first object of our study. In other situations the system begins to show quasi-periodicor ``rotational'' behavior. This is situation is modeled by Siegeldisks which are another subject of our proposed study. Wildattractors, which are our third subject, refer to a very unusualchaotic regime for a system in which two completely different limitingmodes of long-term behavior co-exist side by side. The attractorfor an iterated function system which we will also investigate appearsas a projection of a certain fractal set. This kind of problem appearsin applications where one often has a multi-dimensional fractal setwhich can be studied by projections onto lower-dimensional spaces.