Problems of Complex Analysis Arising in Complex Dynamics

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Abstract. Problems of Complex Analysis Arising in Complex DynamicsThe research of this proposal applies the methods of complex analysis toseveral problems of dynamical systems in the complex domain. The dynamicsof a polynomial map of the complex plane C has been studied in depth andhas produced a theory which is both a beautiful and compelling model formany systems in nature. The dynamics of polynomial mappings oftwo-dimensional complex space C^2 should produce a theory with equalbeauty and utility. The work of this project includes three parts:1. The first objective is to study mappings which are quasi-hyperbolic, acondition which is more general and will be more easily verified than themore traditional condition of (uniform) hyperbolicity. At the same time,the quasi-hyperbolic mappings have expansion/contraction properties whichmakes themunderstandable in a way similar to hyperbolic mappings. The correspondingdynamical sets will be shown to have bounded geometry.2. A second objective is to apply these complex methods to the study ofpolynomial difffeomorphisms of the real plane. This will give a newapproach that will give results not easily obtained with purely real methods.3. The third objective is to understand the nature of parameter space, thespace of all mappings. A particular focus will be on the connectednesslocus, which is a higher-dimensional analogue of the Mandelbrot set.In each of these cases, the methodology involves the further development ofmathematical techniques from complex analysis: analytic functions andvarieties, harmonic measure, pluri-potential theory, and the geometrictheory of currents.