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.

Inhalt

Boris Hasselblatt: Preface.- Boris Hasselblatt: Introduction to Hyperbolic Dynamics and Ergodic Theory.- Jacques Hadamard: On iteration and asymptotic solutions of differential equations (translated by Boris Hasselblatt).- Barbara Schapira: Dynamics of Geodesic and Horocyclic Flows.- Keith Burns, Howard Masur, Amie Wilkinson: Ergodicity of the Weil-Petersson Geodesic Flow.- Keith Burns, Howard Masur, Carlos Matheus and Amie Wilkinson: Ergodicity of Geodesic Flows on Incomplete Negatively Curved Manifolds.-Carlos Matheus: The Dynamics of the Weil-Petersson flow.- Jouni Parkkonen, Fre de ric Paulin: A survey of some Arithmetic Applications of Ergodic Theory in Negative Curvature.