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Homology of closed geodesics in a negatively curved manifold. (English) Zbl 0618.58028
W. Klingenberg has shown [Ann. Math., II. Ser. 99, 1-13 (1974; Zbl 0272.53025)] that every compact manifold M whose geodesic flow on the unit tangent bundle is Anosov has a unique geodesic in every free homotopy class of closed paths. The authors use combinatorial arguments and Ruelle’s infinite dimensional Frobenius-Perron theorem [D. Ruelle, Thermodynamic formalism (1978; Zbl 0401.28016)] to prove that there are infinitely many geodesics with primitive period in each homology class in $$H_ 1(M,Z)$$ in the sense that for the number N(x,$$\alpha)$$ of such geodesics of given homology class $$\alpha$$ and length $$\leq x$$ we have $$\lim_{x\to \infty} x^{-1} \log N(x,\alpha)=the$$ (positive) topological entropy of the geodesic flow.
Reviewer: H.Guggenheimer

##### MSC:
 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53D25 Geodesic flows in symplectic geometry and contact geometry 53C22 Geodesics in global differential geometry
##### Keywords:
geodesic flow; Anosov
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