We study the Ruelle and Selberg zeta functions for Cr Anosov ows, r > 2, on a compact smooth manifold. We prove several results, the most remarkable being (a) for C∞ ows the zeta function is meromorphic on the entire complex plane; (b) for contact ows satisfying a bunching condition (e.g., geodesic ows on manifolds of negative curvature better than 1/9 -pinched), the zeta function has a pole at the topological entropy and is analytic in a strip to its left; (c) under the same hypotheses as in (b) we obtain sharp results on the number of periodic orbits. Our arguments are based on the study of the spectral properties of a transfer operator acting on suitable Banach spaces of anisotropic currents. © 2013 Department of Mathematics, Princeton University.

Anosov flows and dynamical zeta functions

Giulietti P.;Liverani C.;
2013-01-01

Abstract

We study the Ruelle and Selberg zeta functions for Cr Anosov ows, r > 2, on a compact smooth manifold. We prove several results, the most remarkable being (a) for C∞ ows the zeta function is meromorphic on the entire complex plane; (b) for contact ows satisfying a bunching condition (e.g., geodesic ows on manifolds of negative curvature better than 1/9 -pinched), the zeta function has a pole at the topological entropy and is analytic in a strip to its left; (c) under the same hypotheses as in (b) we obtain sharp results on the number of periodic orbits. Our arguments are based on the study of the spectral properties of a transfer operator acting on suitable Banach spaces of anisotropic currents. © 2013 Department of Mathematics, Princeton University.
2013
Giulietti, P.; Liverani, C.; Pollicott, M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/1128846
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