Geodesic Flows

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Springer Science & Business Media, Sep 1, 1999 - Mathematics - 149 pages
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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
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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.
Page 140 - Math. 117 (1994), 403 - 446. 3. K. Burns and M. Gerber, Real analytic Bernoulli geodesic flows on S2, Ergod.
Page 144 - GP Paternain, On the topology of manifolds with completely integrable geodesic flows, Ergod.

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