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Positive topological entropy of Reeb flows on spherizations

Published online by Cambridge University Press:  08 April 2011

LEONARDO MACARINI  and
FELIX SCHLENK
LEONARDO MACARINI
Affiliation:
Universidade Federal do Rio de Janeiro, Instituto de Matemática, Cidade Universitária, CEP 21941-909, Rio de Janeiro, Brazil. e-mail: [email protected]
FELIX SCHLENK
Affiliation:
Institut de Mathématiques, Université de Neuchâtel, Rue Émile Argand 11, CP 158, 2009 Neuchâtel, Switzerland. e-mail: [email protected]
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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 ϕt be the restriction to Σ of the Hamiltonian flow of H.

Theorem 1. The topological entropy of ϕt 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 Σq = Σ ∩ Tq*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 ϕt from Σq to Σq′ 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 S2, ℝP2, the torus and the Klein bottle. Then ϕt carries a horseshoe. In particular, the number of geometrically distinct closed orbits grows exponentially in time.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 1 , July 2011 , pp. 103 - 128
Copyright
Copyright © Cambridge Philosophical Society 2011

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