The aim of this book is to present the fundamental concepts and properties of the geodesic flow of a closed Riemannian manifold. The topics covered are close to my research interests. An important goal here is to describe properties of the geodesic flow which do not require curvature assumptions. A typical example of such a property and a central result in this work is Mane's formula that relates the topological entropy of the geodesic flow with the exponential growth rate of the average numbers of geodesic arcs between two points in the manifold. The material here can be reasonably covered in a one-semester course. I have in mind an audience with prior exposure to the fundamentals of Riemannian geometry and dynamical systems. I am very grateful for the assistance and criticism of several people in preparing the text. In particular, I wish to thank Leonardo Macarini and Nelson Moller who helped me with the writing of the first two chapters and the figures. Gonzalo Tomaria caught several errors and contributed with helpful suggestions. Pablo Spallanzani wrote solutions to several of the exercises. I have used his solutions to write many of the hints and answers. I also wish to thank the referee for a very careful reading of the manuscript and for a large number of comments with corrections and suggestions for improvement.
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.
"The main goal of the book is to present, in a self-contained way, results of the author and of Ricardo Mane about various ways to calculate or estimate the topological entropy of the geodesic flow on a closed Riemannian manifold M. The book begins with two introductory chapters on general properties of geodesic flows including a discussion of some of its properties as a Hamiltonian system acting on the tangent bundle TM of M. The third and fourth chapters present a formula for the topological entropy of the geodesic flow in terms of asymptotic growth of the average number of geodesic arcs in M connecting two given points. This, and similar other formulas for the topological entropy are obtained as an application of a fundamental result of Y. Yomdin which is also discussed, however without proof. The last chapter contains results, mainly due to the author, on topological conditions for M which guarantee that the topological entropy of the geodesic flow for every metric on M is positive. It is also shown that there are manifolds which satisfy these conditions, but for which the infimum of the entropies for metrics with normalized volume vanishes. The text is accompanied by many exercises. Many of the easier details of the material are presented in this form..."
"Unique and valuable... the presentation is clean and brisk...useful for self-study, and as a guide to the subject and its literature."
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