Dynamics on Surfaces

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This is a project to investigate aspects of low-dimensional dynamical systems. The proposed research addresses the connection between algebraic properties of a group that acts on a surface in an area-preserving way and the possible topological nature of the dynamics of the action. One theme is to try to find global fixed points for a smooth group action and use the induced representation of the group into the automorphisms of the tangent space at the fixed point to conclude information about the action. Anticipated results from this project will advance our knowledge of dynamical systems and will explore new relationships between dynamics and algebra.This proposal concerns transformations of surfaces as they evolve in time. The study of such transformations, especially area-preserving ones, has a long history going back to work of Henri Poincare and G. D. Birkhoff, work that was motivated by problems in celestial mechanics. For such a system a "state" is a point on a surface and the objective is to understand how the collection of all states evolves in time. Time can be considered as either continuous (represented by a real number) or discrete (represented by an integer). The present project deals with the discrete case and generalizes it to consider evolutions where the analogue of time is represented, for example, by a matrix rather than an integer. This is part of a long-term program to understand the relationship between the algebraic nature of dynamical systems and the geometric or topological behavior they exhibit. There are numerous applications of results in this area to broader fields of science, especially to classical mechanics, celestial mechanics, and more modern chaos theory.