Homology and closed geodesics in a compact negatively curved surface.

*(English)*Zbl 0728.53031Let S be a compact smooth surface of genus \(g\geq 2\) and with strictly negative curvature. A well known result of G. Margulis [Funct. Anal. Appl. 3(1969), 335-336 (1970); translation from Funkts. Anal. Prilozh. 3, No.4, 89-90 (1969; Zbl 0207.203)] gives an asymptotic formula for the number of closed geodesics on S with length less than t. For a fixed homology class \(\alpha \in H_ 1(S,{\mathbb{Z}})\), the author gives an asymptotic formula for geodesics within the specified homology class. Let \(\gamma\) be a closed geodesic, let \(\lambda\) (\(\gamma\)) be its length and let \([\gamma]\in H_ 2(S,{\mathbb{Z}})\) be the associated homology class. If \(\pi\) (t,\(\alpha\)) is the number of closed geodesics \(\gamma\) for which \(\ell (\gamma)\leq t\) and \([\gamma]=\alpha\), then there exist constants C, \(h>0\) such that \(\pi (t,\alpha)/(Ce^{ht}/t^{g+1})\to 1\) as \(t\to \infty\). This generalizes a result due to A. Katsuda and T. Sunada for the special case of constant negative curvature [Am. J. Math. 110, No.1, 145-155 (1988; Zbl 0647.53036)]. The author extends their proof which depends on properties of L-functions. He does this by using a different approach to these L-functions which is based on a thermodynamic analysis of generalized zeta functions.

Reviewer: D.Hurley (Cork)

##### MSC:

53C22 | Geodesics in global differential geometry |

37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

53D25 | Geodesic flows in symplectic geometry and contact geometry |