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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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