Dynamical zeta functions and resonances for infinite area surfaces

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This proposal deals with complex functions first introduced by the famous norwegian mathematician and Fields medalist Atle Selberg in 1956, and subsequently called Selberg zeta functions. These were originally associated to compact surfaces of constant negative curvature.Their definition was by analogy with the famous Riemann zeta function, except that the role of the prime numbers is replaced by the lengths of closed geodesics on the surface. The striking fact is that in this setting the zeros lie on specific lines, which is very similar to the famous Riemann Hypothesis, both one of the problems from Hilbert's famous list of 23 problems and the Clay Institute's Millennium Problems. However, by contrast, in the case of many examples of open surfaces, or infinite area surfaces, the zeros of the associated Selberg zeta functions are much more complicated. These individual zeros are often called "resonances" and play a role similar to that of the eigenvalues of the laplacian for the compact case, and are important geometric and dynamical invariants for the surfacesWith the development of better computational methods and computer hardware over recent years a much clearer picture of the patterns of these zeros has emerged in some interesting cases. Somewhat surprisingly, the plots of these zeros had strikingly beautiful patterns. They appear to lie on very delicately defined curves in shapes reminiscent of lace embroidery. These plots of the zeros have their simplest structures when the underlying surface has more symmetries.This work will help to understand these patterns of zeta function zeros and the information that it gives on both the zeta function and the associated surface.