Positive topological entropy of Reeb flows on spherizations

Abstract

Abstract Let M be a closed manifold whose based loop space Ω ( M ) is “complicated”. Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. Consider a hypersurface Σ in T * M which is fiberwise starshaped with respect to the origin. Choose a function H : T * M → ℝ such that Σ is a regular energy surface of H , and let ϕ t be the restriction to Σ of the Hamiltonian flow of H . Theorem 1. The topological entropy of ϕ t is positive. This result has been known for fiberwise convex Σ by work of Dinaburg, Gromov, Paternain, and Paternain–Petean on geodesic flows. We use the geometric idea and the Floer homological technique from [ 19 ], but in addition apply the sandwiching method. Theorem 1 can be reformulated as follows. Theorem 1'. The topological entropy of any Reeb flow on the spherization SM of T*M is positive. For q ∈ M abbreviate Σ q = Σ ∩ T q * M . The following corollary extends results of Morse and Gromov on the number of geodesics between two points. Corollary 1. Given q ∈ M, for almost every q′ ∈ M the number of orbits of the flow ϕ t from Σ q to Σ q′ grows exponentially in time. In the lowest dimension, Theorem 1 yields the existence of many closed , orbits. Corollary 2. Let M be a closed surface different from S 2 , ℝP 2 , the torus and the Klein bottle. Then ϕ t carries a horseshoe. In particular, the number of geometrically distinct closed orbits grows exponentially in time.