Homology of closed geodesics in a negatively curved manifold
\textit{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 [\textit{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.
scientific article; zbMATH DE number 4003113