Entropy and closed geodesics. (English) Zbl 0525.58027

Katok, A.

Ergodic Theory Dyn. Syst. 2, 339-365 (1982).

Author’s abstract: We study asymptotic growth of closed geodesics for various Riemannian metrics on a compact manifold which carries a metric of negative sectional curvature. Our approach makes use of both variational and dynamical description of geodesics and can be described as an asymptotic version of lengtharea method. We also obtain various inequalities between topological and measuretheoretic entropies of the geodesic flows for different metrics ’on the same manifold. Our method works especially well for any metric conformally equivalent to a metric of constant negative curvature. For a surface with negative Euler characteristics every Riemannian metric has this property due to a classical regularization theorem. This allows us to prove that every metric of non-constant curvature has strictly more close geodesics of length at most \( T \) for sufficiently large \( T \) than any metric of constant curvature of the same total area. In addition the common value of topological and measure-theoretic entropies for metrics of constant negative curvature with the fixed area separates the values of two entropies for other metrics with the same area.

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MSC:

37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25	Geodesic flows in symplectic geometry and contact geometry
58E10	Variational problems in applications to the theory of geodesics (problems in one independent variable)
53C22	Geodesics in global differential geometry
28D20	Entropy and other invariants
37A99	Ergodic theory

Keywords:

asymptotic growth of closed geodesics; metric of negative sectional curvature; topological and measure-theoretic entropies; classical regularization theorem