Mathematical Sciences: Planar Dynamical Systems

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The principal investigator will continue his research on dynamical systems. In particular he will study the dynamics of entire functions, maps associated with Newton's method, and some antiholomorphic maps. The dynamics of these maps will be applied to the problem of trying to understand the automorphisms of the shift map. These techniques will also be applied to other families of interest in real dynamics including the Henon map, the standard family, and a family of maps which arise in scattering theory. Dynamical systems are playing an increasing role in various subfields of mathematics and in mathematical modeling. One of the applications to be studied in this project is an application to scattering theory. Scattering theory is the branch of mathematics which attempts to reconstruct an operator from a description of the action of the operator on functions. It is a basic tool in physics. The amalgamation of different methods is playing an important role in the solution of many mathematical and scientific problems; this project is one example of cross-disciplinary research at work.