# zbMATH — the first resource for mathematics

Homology and closed geodesics in a compact negatively curved surface. (English) Zbl 0728.53031
Let 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
Full Text: