Lectures on Partial Hyperbolicity and Stable Ergodicity
The book is based on lectures given by the author in ETH Zürich in 2003. The lectures provide a unified and systematic approach to partial hyperbolicity and stable ergodicity, recently developed branches of dynamical systems. The text consists of ten chapters. After the two introductory ones, the Mather spectrum is introduced and used for investigation of stability of partially hyperbolic maps in Chapter 3. In Chapters 4, 5 and 6, various aspects of stability are discussed: constructions of invariant foliations and their stability under small perturbations, branching phenomena for intermediate foliations and criteria for integrability of central distributions. Chapters 7 and 8 are devoted to technical tools for studying ergodic properties, namely to absolute continuity and stable accessibility. The last two chapters contain basic and also recent results in the Pugh-Shub stable ergodic theory and applications to Anosov flows and, in particular, geodesic flows. These lectures are accessible to graduate students in smooth dynamical theory. Since they are written by one of the founders of the theory, also experts may find them interesting.