Dynamics on Two-Dimensional Surfaces

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This is a proposal to investigate aspects of low dimensional dynamicalsystems. The most important long term project is to betterunderstand a topological description of C^r generic area preservingdiffeomorphisms of surfaces where r 1. In addition the proposerhopes to investigate the action of more general groups than the integers or the reals on surfaces, and in particular area preserving actions.Professor Franks will continue his investigation of thedynamics of surface maps. The study of such transformations,especially area preserving ones, has a long history going back toPoincare and G. D. Birkhoff. There are numerous applications ofresults in this area to classical mechanics as well as more modernchaotic dynamics. The proposed research will address questionsconcerning the existence of periodic behavior in area preservingtwo-dimensional discrete dynamical systems and a topologicaldescription of the dynamics of such systems.