Complex Dynamics and Polygonal Billiards

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Abstract for Proposal DMS-0601299 Complex Dynamics and Polygonal BilliardsThe aim of the proposed research is to bring new methods and concepts to bear on the problem of understanding two dimensional dynamics and polygonal billiards. In the complex dynamics direction, the aim is to make a connection between some of the ideas behind the recent advances in dynamics of one complex variable and two dimensional dynamics. The PI proposes a scheme for using methods from dynamics in one complex variable and computer tools to construct symbolic codings which could lead to understanding two dimensional diffeomorphisms. He also proposes methods to explore structure in parameter space through the monodromy of Julia sets. The PI also deals with the classical dynamical behavior of trajectories on rational polygonal tables. The questions involving the dynamics of the billiard flow translate into questions involving an action of the special linear group on the moduli space of translation structures. The problem is to understand and geometrically characterize orbits of this action.Mathematical models that describe how systems change with time provide essential insights in many areas of science. They are used to describe the rhythms of the heart, the pulsing of lasers and the spread of disease. When the models are nonlinear the long term behavior, or dynamics, of these models can be chaotic and the dependence of the dynamics on the parameters of the model can be extraordinarily delicate. This is true even in very simple models. The potential impact of a deeper understanding of nonlinear dynamics is profound and could be felt in many areas of science. One aim of this proposal is to determine how computational and theoretical methods interact in exploring families of dynamical systems. Answers to such questions could have broad impact as questions about how best to use computers arise in many scientific areas. The study of polygonal billiards ties together some very simple dynamical systems with intuitive appeal with some deep questions about the structure of certain moduli spaces that are the focus of interest from many viewpoints including algebraic geometry algebraic topology and string theory. This link vividly illustrates the connectedness of mathematics and raises the possibility of the cross fertilization between very different fields.