Paternain, Gabriel P. 　著

内容

目次

0 Introduction.- 1 Introduction to Geodesic Flows.- 1.1 Geodesic flow of a complete Riemannian manifold.- 1.1.1 Euler-Lagrange flows.- 1.2 Symplectic and contact manifolds.- 1.2.1 Symplectic manifolds.- 1.2.2 Contact manifolds.- 1.3 The geometry of the tangent bundle.- 1.3.1 Vertical and horizontal subbundles.- 1.3.2 The symplectic structure of TM.- 1.3.3 The contact form.- 1.4 The cotangent bundle T*M.- 1.5 Jacobi fields and the differential of the geodesic flow.- 1.6 The asymptotic cycle and the stable norm.- 1.6.1 The asymptotic cycle of an invariant measure.- 1.6.2 The stable norm and the Schwartzman ball.- 2 The Geodesic Flow Acting on Lagrangian Subspaces.- 2.1 Twist properties.- 2.2 Riccati equations.- 2.3 The Grassmannian bundle of Lagrangian subspaces.- 2.4 The Maslov index.- 2.4.1 The Maslov class of a pair (X, E).- 2.4.2 Hyperbolic sets.- 2.4.3 Lagrangian submanifolds.- 2.5 The geodesic flow acting at the level of Lagrangian subspaces.- 2.5.1 The Maslov index of a pseudo-geodesic and recurrence.- 2.6 Continuous invariant Lagrangian subbundles in SM.- 2.7 Birkhoff’s second theorem for geodesic flows.- 3 Geodesic Arcs, Counting Functions and Topological Entropy.- 3.1 The counting functions.- 3.1.1 Growth of n(T) for naturally reductive homogeneous spaces.- 3.2 Entropies and Yomdin’s theorem.- 3.2.1 Topological entropy.- 3.2.2 Yomdin’s theorem.- 3.2.3 Entropy of an invariant measure.- 3.2.4 Lyapunov exponents and entropy.- 3.2.5 Examples of geodesic flows with positive entropy.- 3.3 Geodesic arcs and topological entropy.- 3.4 Manning’s inequality.- 3.5 A uniform version of Yomdin’s theorem.- 3.5.1 Another proof of Theorem 3.32 using Theorem 3.44.- 4 Mañé’s Formula for Geodesic Flows and Convex Billiards.- 4.1 Time shifts that avoid the vertical.- 4.2 Mañé’s formula for geodesic flows.- 4.2.1 Changes of variables.- 4.2.2 Proof of the Main Theorem.- 4.3 Manifolds without conjugate points.- 4.4 A formula for the topological entropy for manifolds of positive sectional curvature.- 4.5 Mañé’s formula for convex billiards.- 4.5.1 Proof of Theorem 4.30.- 4.6 Further results and problems on the subject.- 4.6.1 Topological pressure.- 5 Topological Entropy and Loop Space Homology.- 5.1 Rationally elliptic and rationally hyperbolic manifolds.- 5.1.1 The characteristic zero homology of H-spaces.- 5.1.2 The radius of convergence.- 5.2 Morse theory of the loop space.- 5.2.1 Serre’s theorem.- 5.2.2 Gromov’s theorem.- 5.3 Topological conditions that ensure positive entropy.- 5.3.1 Growth of finitely generated groups.- 5.3.2 Dinaburg’s Theorem.- 5.3.3 Arbitrary fundamental group.- 5.3.4 Proof of Theorem 5.20.- 5.4 Entropies of manifolds.- 5.4.1 Simplicial volume.- 5.4.2 Minimal volume.- 5.5 Further results and problems on the subject.- Hints and Answers.- References.