The principal investigator will relate dynamical systems to topology and geometry using zeta functions. This project will extend his previous contributions which clarify relationships between periodic trajectory structure and the topology of the underlying space. Besides broadening this connection, the principal investigator will study analogous problems for a complex manifold which supports a holomorphic flow. The study of the relationship between shapes of surfaces and types of periodic orbits has its origins with Poincare at the turn of the century. For example, any vector field on the sphere must have a fixed point. Thus the wind cannot blow everywhere on the surface of the earth at the same time; there must be still air somewhere. The principal investigator will extend current theory which explains which kinds of periodic behavior on surfaces and higher-dimensional manifolds are permitted.
