Mathematical Sciences: Problems of Complex Analysis Arising in Complex Dynamics

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Abstract Bedford 9706818 The research of this proposal applies to the methods of complex analysis to several problems of dynamical systems in the complex domain. The dynamics of polynomial maps of 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 principal emphasis is a two-part investigation of polynomial diffeomorphisms of C^2. The first part involves mappings for which the Julia set is connected: to analyze the structure of the dynamical set J and the structure of parameter space (a higher-dimensional Mandelbrot space). The second part involves diffeomorphisms of C^2 which are quasi-hyperbolic: these mappings are not uniformly expanding on the infinitesimal level but have uniform expansion and bounded geometry at a fixed (finite) level . This work involves several mathematical techniques from complex analysis: harmonic measure, pluri-potential theory, and the geometry of currents. Many physical processes are modeled as dynamical systems: systems which evolve over time as the iteration (repetition) of a specific rule. An interesting discovery has been that even if the defining rule is rather simple, the resultant behavior can show surprising complex behavior over time. One of the most intriguing features of this model has been the way the behavior of the system depends on parameters (the period-doubling cascade to chaotic behavior). This is more clearly seen when we pass to the (seemingly more complicated) complex system. The behavior in the complex leads to an interplay between dynamic space and parameter space. The resulting parameter (bifurcation) space is the Mandelbrot set, which has been shown to have "universal" properties. We consider the dynamics of polynomial mappings of two complex variables, which are what is obtained when the analogous behavior is considered in two variables. Our research seeks to identify the analogue of the Mandelbrot in two complex dimensions and to ident ify its associated dynamical properties.