Ergodic Theory and Negative Curvature: CIRM Jean-Morlet Chair, Fall 2013 (Lecture Notes in Mathematics, 2164)

Focussing on the mathematics related to the recent proof of ergodicity of the (Weil–Petersson) geodesic flow on a nonpositively curved space whose points are negatively curved metrics on surfaces, this book provides a broad introduction to an important current area of research. It offers original textbook-level material suitable for introductory or advanced courses as well as deep insights into the state of the art of the field, making it useful as a reference and for self-study. The first chapters introduce hyperbolic dynamics, ergodic theory and geodesic and horocycle flows, and include an English translation of Hadamard's original proof of the Stable-Manifold Theorem. An outline of the strategy, motivation and context behind the ergodicity proof is followed by a careful exposition of it (using the Hopf argument) and of the pertinent context of Teichmüller theory. Finally, some complementary lectures describe the deep connections between geodesic flows in negative curvature and Diophantine approximation. Front Matter ....Pages i-vii Introduction to Hyperbolic Dynamics and Ergodic Theory (Boris Hasselblatt)....Pages 1-124 On Iteration and Asymptotic Solutions of Differential Equations by Jacques Hadamard (Boris Hasselblatt, (Translator))....Pages 125-128 Dynamics of Geodesic and Horocyclic Flows (Barbara Schapira)....Pages 129-155 Ergodicity of the Weil–Petersson Geodesic Flow (Keith Burns, Howard Masur, Amie Wilkinson)....Pages 157-174 Ergodicity of Geodesic Flows on Incomplete Negatively Curved Manifolds (Keith Burns, Howard Masur, Carlos Matheus, Amie Wilkinson)....Pages 175-208 The Dynamics of the Weil–Petersson Flow (Carlos Matheus)....Pages 209-291 A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature (Jouni Parkkonen, Frédéric Paulin)....Pages 293-326 Back Matter ....Pages 327-328