## Geodesic FlowsGeodesic 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 |

147 | |

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### Common terms and phrases

A(SM algebra Anosov asymptotic cycle Borel Chapter codimension compact conjugate points contact form contact manifolds Corollary counting functions Cr-size curve define Definition denote diffeomorphism differential dimensional ergodic Exercise exists a constant finite geodesic arcs geodesic flow Geometry given Gromov h,op Hence homology homotopy horizontal subbundle htop htop(g hyperbolic set implies isomorphism Jacobi equation Jacobi fields JMxM Lagrangian subspaces Lemma lim sup liminf Liouville measure loop space Mane's formula Maslov class Maslov cycle Maslov index metric g nN(x nT(x obtain open set orthogonal positive integer probability measure proof of Theorem Proposition prove pseudo-geodesic rationally elliptic restricted Riemannian manifold Riemannian metric Sasaki metric sectional curvature sequence simply connected simply connected manifold smooth stable norm subbundle subset Suppose symmetric symplectic form TgSM TgTM topological entropy twist property vector field vertical subbundle volume y)dxdy Yomdin's theorem

### Popular passages

Page 140 - On the relations among various entropy characteristics of dynamical systems. Math. USSR Izv.

Page 141 - MH Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17 (1982), 357-453.