# Geodesic Flows

Springer Science & Business Media, 1999 - Mathematics - 149 pages
Geodesic flows are of considerable current interest since they are, perhaps, the most remarkable class of conservative dynamical systems. They provide a unified arena in which one can explore numerous interplays among several fields, including smooth ergodic theory, symplectic and Riemannian geometry, and algebraic topology. The work begins with a concise introduction to the geodesic flow of a complete Riemannian manifold, emphasizing its symplectic properties and culminating with various applications, such as the non-existence of continuous invariant Lagrangian subbundles for manifolds with conjugate points. Subsequent chapters develop the relationship between the exponential growth rate of the average number of geodesic arcs between two points in the manifold and the topological entropy of the geodesic flow. A complete proof of Mane's formula relating these two quantities is presented. A final chapter explores the link between the topological entropy of the geodesic flow and the homology of the loop space of a manifold. This self-contained monograph will be of interest to graduate students and researchers of dynamical systems and differential geometry. Numerous exercises and examples as well as a comprehensive bibliography and index make the work an excellent self-study resource or useful text for a one-semester course or seminar. Series: Progress in Mathematics, Vol. 180 Table of Contents Preface 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 tangent bundles 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 Langrangian subspaces 2.4 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 subbundels 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 Yodim's theorem 3.5.1 Another proof of Theorem 3.32 using Theorem 3.44 4 Mane's Formula for Geodesic Flows and Convex Billiards 4.1 Time shifts that avoid the vertical 4.2 Mane's formula for geodesic flow 4.2.1 Changes of variables 4.2.2 Proof of the Main Theorem 4.3 Manifolds without conjugate points 4.4 Entropy for positive curvature 4.5 Mane'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 Index

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### Contents

 Introduction to Geodesic Flows 7 11 Geodesic flow of a complete Riemannian manifold 8 12 Symplectic and contact manifolds 9 122 Contact manifolds 10 13 The geometry of the tangent bundle 11 132 The symplectic structure of TM 14 133 The contact form 15 14 The cotangent bundle TM 19
 34 Mannings inequality 69 35 A uniform version of Yomdins theorem 73 351 Another proof of Theorem 332 using Theorem 344 75 Manes Formula for Geodesic Flows and Convex Billiards 77 42 Manes formula for geodesic flows 82 421 Changes of variables 83 422 Proof of the Main Theorem 88 43 Manifolds without conjugate points 90

 15 Jacobi fields and the differential of the geodesic flow 20 16 The asymptotic cycle and the stable norm 21 162 The stable norm and the Schwartzman ball 26 The Geodesic Flow Acting on Lagrangian Subspaces 31 21 Twist properties 32 22 Riccati equations 37 23 The Grassmannian bundle of Lagrangian subspaces 38 24 The Maslov index 39 241 The Maslov class of a pair X E 41 242 Hyperbolic sets 42 243 Lagrangian submanifolds 43 25 The geodesic flow acting at the level of Lagrangian subspaces 44 251 The Maslov index of a pseudogeodesic and recurrence 45 26 Continuous invariant Lagrangian subbundles in SM 48 27 Birkhoffs second theorem for geodesic flows 50 Geodesic Arcs Counting Functions and Topological Entropy 51 311 Growth of nT for naturally reductive homogeneous spaces 56 32 Entropies and Yomdins theorem 58 322 Yomdins theorem 60 323 Entropy of an invariant measure 61 324 Lyapunov exponents and entropy 62 325 Examples of geodesic flows with positive entropy 63 33 Geodesic arcs and topological entropy 64
 44 A formula for the topological entropy for manifolds of positive sectional curvature 92 45 Manes formula for convex billiards 93 451 Proof of Theorem 430 97 46 Further results and problems on the subject 102 461 Topological pressure 104 Topological Entropy and Loop Space Homology 109 511 The characteristic zero homology of Hspaces 112 512 The radius of convergence 114 52 Morse theory of the loop space 115 521 Serres theorem 116 522 Gromovs theorem 117 53 Topological conditions that ensure positive entropy 119 532 Dinaburg s Theorem 120 533 Arbitrary fundamental group 121 534 Proof of Theorem 5 20 122 54 Entropies of manifolds 126 542 Minimal volume 127 55 Further results and problems on the subject 130 Hints and Answers 133 References 139 Index 147 Copyright

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