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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”.

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
Full Text: DOI
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