Geodesic flows:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston
Birkhäuser
1999
|
| Series: | Progress in mathematics
180 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xii, 149 Seiten Illustrationen |
| ISBN: | 3764341440 0817641440 9781461272120 |
Staff View
MARC
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| 100 | 1 | |a Paternain, Gabriel P. |d 1964- |e Verfasser |0 (DE-588)121598888 |4 aut | |
| 245 | 1 | 0 | |a Geodesic flows |c Gabriel P. Paternain |
| 264 | 1 | |a Boston |b Birkhäuser |c 1999 | |
| 300 | |a xii, 149 Seiten |b Illustrationen | ||
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Record in the Search Index
| _version_ | 1804127472712679424 |
|---|---|
| adam_text | Contents
Preface xi
0 Introduction 1
1 Introduction to Geodesic Flows 7
1.1 Geodesic flow of a complete Riemannian manifold 8
1.1.1 Euler Lagrange flows 8
1.2 Symplectic and contact manifolds 9
1.2.1 Symplectic manifolds 9
1.2.2 Contact manifolds 10
1.3 The geometry of the tangent bundle 11
1.3.1 Vertical and horizontal subbundles 11
1.3.2 The symplectic structure of TM 14
1.3.3 The contact form 15
1.4 The cotangent bundle T*M 19
1.5 Jacobi fields and the differential
of the geodesic flow 20
1.6 The asymptotic cycle and the stable norm 21
1.6.1 The asymptotic cycle of an invariant measure 21
1.6.2 The stable norm and the Schwartzman ball 26
2 The Geodesic Flow Acting on Lagrangian Subspaces 31
2.1 Twist properties 32
2.2 Riccati equations 37
viii Contents
2.3 The Grassmannian bundle of Lagrangian subspaces 38
2.4 The Maslov index 39
2.4.1 The Maslov class of a pair (X, £) 41
2.4.2 Hyperbolic sets 42
2.4.3 Lagrangian submanifolds 43
2.5 The geodesic flow acting at the level of Lagrangian subspaces . . 44
2.5.1 The Maslov index of a pseudo geodesic and recurrence . . 45
2.6 Continuous invariant Lagrangian subbundles in SM 48
2.7 Birkhoff s second theorem for geodesic flows 50
3 Geodesic Arcs, Counting Functions and Topological Entropy 51
3.1 The counting functions 51
3.1.1 Growth of n(T) for naturally reductive
homogeneous spaces 56
3.2 Entropies and Yomdin s theorem 58
3.2.1 Topological entropy 58
3.2.2 Yomdin s theorem 60
3.2.3 Entropy of an invariant measure 61
3.2.4 Lyapunov exponents and entropy 62
3.2.5 Examples of geodesic flows with positive entropy 63
3.3 Geodesic arcs and topological entropy 64
3.4 Manning s inequality 69
3.5 A uniform version of Yomdin s theorem 73
3.5.1 Another proof of Theorem 3.32 using Theorem 3.44 ... 75
4 Mane s Formula for Geodesic Flows and Convex Billiards 77
4.1 Time shifts that avoid the vertical 77
4.2 Mane s formula for geodesic flows 82
4.2.1 Changes of variables 83
4.2.2 Proof of the Main Theorem 88
4.3 Manifolds without conjugate points 90
4.4 A formula for the topological entropy for manifolds of positive
sectional curvature 92
4.5 Mane s formula for convex billiards 93
4.5.1 Proof of Theorem 4.30 97
4.6 Further results and problems on the subject 102
4.6.1 Topological pressure 104
5 Topological Entropy and Loop Space Homology 109
5.1 Rationally elliptic and rationally hyperbolic manifolds 109
5.1.1 The characteristic zero homology of H spaces 112
5.1.2 The radius of convergence 114
5.2 Morse theory of the loop space 115
5.2.1 Serre s theorem 116
5.2.2 Gromov s theorem 117
Contents ix
5.3 Topological conditions that ensure positive entropy 119
5.3.1 Growth of finitely generated groups 119
5.3.2 Dinaburg s Theorem 120
5.3.3 Arbitrary fundamental group 121
5.3.4 Proof of Theorem 5.20 122
5.4 Entropies of manifolds 126
5.4.1 Simplicial volume 126
5.4.2 Minimal volume 127
5.5 Further results and problems on the subject 130
Hints and Answers 133
References 139
Index 147
|
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology |
| dewey-raw | 514/.74 |
| dewey-search | 514/.74 |
| dewey-sort | 3514 274 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV012801642 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:33:54Z |
| institution | BVB |
| isbn | 3764341440 0817641440 9781461272120 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008706252 |
| oclc_num | 41711966 |
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| physical | xii, 149 Seiten Illustrationen |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Paternain, Gabriel P. 1964- Verfasser (DE-588)121598888 aut Geodesic flows Gabriel P. Paternain Boston Birkhäuser 1999 xii, 149 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 180 Flots géodésiques ram Geodesic flows Geodätischer Fluss (DE-588)4156670-1 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Geodätischer Fluss (DE-588)4156670-1 s DE-604 Riemannscher Raum (DE-588)4128295-4 s Erscheint auch als Online-Ausgabe 978-1-4612-1600-1 Progress in mathematics 180 (DE-604)BV000004120 180 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008706252&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Paternain, Gabriel P. 1964- Geodesic flows Progress in mathematics Flots géodésiques ram Geodesic flows Geodätischer Fluss (DE-588)4156670-1 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| subject_GND | (DE-588)4156670-1 (DE-588)4128295-4 |
| title | Geodesic flows |
| title_auth | Geodesic flows |
| title_exact_search | Geodesic flows |
| title_full | Geodesic flows Gabriel P. Paternain |
| title_fullStr | Geodesic flows Gabriel P. Paternain |
| title_full_unstemmed | Geodesic flows Gabriel P. Paternain |
| title_short | Geodesic flows |
| title_sort | geodesic flows |
| topic | Flots géodésiques ram Geodesic flows Geodätischer Fluss (DE-588)4156670-1 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| topic_facet | Flots géodésiques Geodesic flows Geodätischer Fluss Riemannscher Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008706252&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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