Problems of Complex Analysis Arising in Complex Dynamics

Indiana University, Bloomington IN

Investigators

Abstract

Abstract. Problems of Complex Analysis Arising in Complex Dynamics The research of this proposal applies the methods of complex analysis to several problems of dynamical systems in the complex domain. The dynamics of a polynomial map of the complex plane C has been studied in depth and has produced a theory which is both a beautiful and compelling model for many systems in nature. The dynamics of polynomial mappings of two-dimensional complex space C^2 should produce a theory with equal beauty and utility. The work of this project includes three parts: 1. The first objective is to study mappings which are quasi-hyperbolic, a condition which is more general and will be more easily verified than the more traditional condition of (uniform) hyperbolicity. At the same time, the quasi-hyperbolic mappings have expansion/contraction properties which makes them understandable in a way similar to hyperbolic mappings. The corresponding dynamical sets will be shown to have bounded geometry. 2. A second objective is to apply these complex methods to the study of polynomial difffeomorphisms of the real plane. This will give a new approach that will give results not easily obtained with purely real methods. 3. The third objective is to understand the nature of parameter space, the space of all mappings. A particular focus will be on the connectedness locus, which is a higher-dimensional analogue of the Mandelbrot set. In each of these cases, the methodology involves the further development of mathematical techniques from complex analysis: analytic functions and varieties, harmonic measure, pluri-potential theory, and the geometric theory of currents.