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Closed geodesics in homology classes on surfaces of variable negative curvature. (English) Zbl 0732.53035
Let $$M^ n$$ be a compact Riemannian manifold of negative curvature. It is known that there exist countably many closed geodesics in $$M^ n$$. I. G. Sinai was the first to investigate the asymptotic behaviour of the number N(t) of closed geodesics with length t. Namely, if $$-K^ 2_ 1<K^ 2_ 2$$ denote the bounds of the curvature, then $$\exp (t(n-1)K_ 2)\leq N(t)\leq \exp (t(n-1)K_ 1)$$ as $$t\to \infty.$$
G. A. Margulis [Funct. Anal. Appl. 3, 335-336 (1969); translation from Funkts. Anal. Prilozh. 3, No.4, 89-90 (1969; Zbl 0207.203)] has established the following asymptotic formula: $$N(t)\sim e^{ht}/ht$$ as $$t\to \infty$$, where $$h>0$$ is the topological entropy of the geodesic flow. R. Phillips and P. Sarnak [Duke Math. J. 55, 287-297 (1988; Zbl 0642.53050)] and A. Katsuda and T. Sunada [Am. J. Math. 110, No.1, 145-155 (1988; Zbl 0647.53036)] have investigated the asymptotic behaviour of N(t;m) - the number of closed geodesics in the homology class m with lengths $$\leq t$$- for the manifolds of constant negative curvature: $$N(t;m)\sim c\cdot e^{ht}/t^{1+g}$$, where 2g is the rank of the homology group $$H_ 1M.$$
In the paper under review the author extends the last result to manifolds of variable negative curvature. More precisely, let $$[\omega_ 1],...,[\omega_{2g}]$$ be the base in the cohomology space $$H^ 1M$$. For each form $$\omega_ j$$ let $$W_ j$$ denote the function on the unit tangent bundle SM, $$W_ j(x^ j,v)=<\omega_ j(x),v>$$. For $$\xi \in R^{2g}$$ define -$$\Gamma$$ ($$\xi$$) to be the maximum entropy of an invariant probability measure $$\lambda$$ on SM satisfying $$\int W_ j d\lambda =\xi_ j (j=1,2,...,2g)$$. The main result is the following asymptotic formula, where $$\xi =t^{-1}(m_ 1,...,m_{2g}):$$ $N(t;m)\sim \exp (-t\Gamma (\xi))\cdot t^{-g-1}(2\pi)^{-g}(\det \nabla^ 2\Gamma (\xi))^{1/2}(<\nabla \Gamma (\xi),\xi >-\Gamma (\xi))^{-1}$ uniformly for $$\xi$$ in some neighborhood of 0.
The proof uses the symbolic dynamics for geodesic flows and certain aspects of Ruelle’s “thermodynamic formalism”.

##### MSC:
 53C22 Geodesics in global differential geometry 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53D25 Geodesic flows in symplectic geometry and contact geometry 37E99 Low-dimensional dynamical systems
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