Low Dimensional Dynamics

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Abstract for proposal DMS-0555463:This project investigates aspects of low dimensional dynamicalsystems. The study of such transformations, especially areapreserving ones, has a long history going back to Poincare andG. D. Birkhoff. In particular the project considers smooth groupactions on surfaces and the relation between the algebraic propertiesof the group and the dynamics the action exhibits. A related questionaddressed by this proposal is the question of the existence of globalfixed points for two-dimensional dynamical systems and how thisrelates to the algebraic properties of the system.This project investigates aspects of dynamical systems onsurfaces. There are numerous applications of results in this area tobroader fields of science, especially to classical mechanics and moremodern chaos theory. The novelty of the proposed research is that itaddresses the relationship between the algebraic nature of dynamicalsystems called ``group actions'' and the geometric or topologicalbehavior they exhibit. Anticipated results from this proposal willadvance our knowledge of dynamical systems and will explore newrelationships between dynamics and algebra.