Anosov flows and dynamical zeta functions

• 成果类型: Article
• 署名作者: Giulietti, P.; Liverani, C.; Pollicott, M.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2013.178.2.6
• 发表日期: 2013
• 页码: 687-773
• 关键词: fixed-point formula; fredholm determinants; transfer operators; Axiom; SPECTRA; decay; MAPS; REGULARITY; Orbits; SPACES

摘要

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-infinity 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 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.