Geodesic
Anosov flows and dynamical zeta functions
Paolo Giulietti1 ; Carlangelo Liverani2 ; Mark Pollicott3
1 Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil
2 Università di Roma <em> Tor Vergata </em> 00133 Roma, Italy
3 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Annals of mathematics, Tome 178 (2013) no. 2, pp. 687-773.

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We study the Ruelle and Selberg zeta functions for $C^r$ Anosov flows, $r > 2$, on a compact smooth manifold. We prove several results, the most remarkable being (a) for $C^\infty$ flows the zeta function is meromorphic on the entire complex plane; (b) for contact flows satisfying a bunching condition (e.g., geodesic flows on manifolds of negative curvature better than $\frac 19$-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.
DOI : 10.4007/annals.2013.178.2.6

Paolo Giulietti 1 ; Carlangelo Liverani 2 ; Mark Pollicott 3

1 Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil
2 Università di Roma <em> Tor Vergata </em> 00133 Roma, Italy
3 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom 
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Paolo Giulietti; Carlangelo Liverani; Mark Pollicott. Anosov flows and dynamical zeta functions. Annals of mathematics, Tome 178 (2013) no. 2, pp. 687-773. doi : 10.4007/annals.2013.178.2.6. http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.2.6/

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