Mathematical Sciences: Topology and Dynamics in Dimension Two

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Handel will work on two topics in topology and dynamics. One is a continuation of previous research concerning applications of Nielsen theory to surfaces of infinite Euler characteristic, with emphasis on the isotopy classification of homeomorphisms whose action on the ends of the surface is (homotopically) periodic. The second part of the project concerns invariant train tracks (in the sense of Thurston) for surface diffeomorphisms. Very little is known in this area, and Handel will begin with a detailed study of those diffeomorphisms that arise from the standard horseshoe map by blowing up a finite set of periodic orbits. The dynamics of surface transformations is clearly a physical concept, and it is surprising how much information can be obtained from purely topological considerations.