Anosov flows and dynamical zeta functions

被引：73

作者：

Giulietti, P. ^{[1
]}

Liverani, C. ^{[2
]}

Pollicott, M. ^{[3
]}

机构：

[1] Univ Fed Rio Grande do Sul, Porto Alegre, RS, Brazil

[2] Univ Roma Tor Vergata, Rome, Italy

[3] Univ Warwick, Coventry CV4 7AL, W Midlands, England

来源：

ANNALS OF MATHEMATICS | 2013年 / 178卷 / 02期

基金：

英国工程与自然科学研究理事会;

关键词：

FIXED-POINT FORMULA; FREDHOLM DETERMINANTS; TRANSFER OPERATORS; AXIOM; SPECTRA; DECAY; MAPS; REGULARITY; ORBITS; SPACES;

D O I：

10.4007/annals.2013.178.2.6

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

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.

引用

页码：687 / 773

页数：87