Semi-classical zeta functions for geodesic flows on negatively curved manifolds.
Masato Tsujii (The University of Kyushu, Fukuoka)

Semi-calssical zeta function (or Gutzwiller-Voros zeta) function is a zeta function defined for non-singular (hyperbolic) flows in terms of  their periodic orbits. In the case of geodesic flows on manifolds with negative constant (say $-1$) curvature, this coincides with the classical dynamical zeta function defined by Smale (up to translation). And the well-known result of Selberg gives a precise description  on its singularities (zero and poles), which resembles the Riemann hypothesis, and the zeros on the critical line $¥Re(s)=1/2$ are related to the eigenvalues of the Laplacian on the underlying manifold. In this talk, I would like to discuss to what extend such results can be extended to more general cases of variable curvature. The method in the proof of the Selberg's result (based on his trace formula) breaks down in such cases, so that we need to develop a more dynamical method using transfer operators.

Colloquium, Department of Mathematics, Rice University
March 19, 2009, 4-5PM, in HB 227


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