Geometry and Measure in Complex Dynamics

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DMS-9803541The focus of this project is the geometric structure and abundance of certain invariant sets which arise in non-hyperbolic dynamics. Of primary concern are the regularity of Fatou components, the Hausdorff dimension of Julia sets, holomorphic removability of Julia sets, ergodic properties of invariant measures, convergence of the Poincare series, and abundance of prototype systems. The Collet-Eckmann condition yields Holder regularity of Fatou components and HD2 for Julia sets of rational maps. This project aims to generalize results about Collet-Eckmann maps to the more flexible setting given by the so called `summability condition,' which requires only a polynomial growth (with possible tame oscillations) along critical orbits. External addresses of the boundary of the Mandelbrot set give a natural parameterization of the locus of chaotic dynamics of complex quadratic polynomials. The main objective here is to study a distribution of harmonic measure on the boundary of the Mandelbrot set, which will determine the 1outside view' of the Mandelbrot set. The last part of the project concerns elliptic dynamics. The regularity of Siegel disks and Herman rings is considered. More generally, given a non-smooth Jordan curve on the plane, one can investigate constraints imposed on the analytic dynamics on this curve.Among non-linear smooth dynamical systems, the theory of one-dimensional real and complex maps occupies a special place. Computational accessibility makes one-dimensional maps especially useful to model and study rigorously properties of non-hyperbolic systems which may be difficult to approach in higher dimensions. Computer simulations help to predict the long time behavior and stability of such systems. Particular examples come from physics, biology, and economics. Perhaps the most famous one-dimensional model, in which the evolution of an isolated population with limited food resources is described, is studied in this project (after complexification). Many other important phenomena like transition to chaos, attractors, and bifurcation loci can be described through `fractal' geometry. Fractal sets with their complicated structure are often crucial in understanding the underlying dynamics. One of the main objectives of this project is to explore chaotic regions and estimate their fractal parameters.