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Dynamics and zeta functions on conformally compact manifolds. (English) Zbl 1346.37040

Summary: In this note, we study the dynamics and associated Zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces, we obtain optimal meromorphic extensions of weighted dynamical Zeta functions and asymptotic counting estimates for the number of weighted closed geodesics. A meromorphic extension of the standard dynamical Zeta function and the prime orbit theorem follow as corollaries. Finally, we investigate interactions between the dynamics and spectral theory of these spaces.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C22 Geodesics in global differential geometry
53D25 Geodesic flows in symplectic geometry and contact geometry
35P15 Estimates of eigenvalues in context of PDEs
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