Past lectures

Abstract

Partial hyperbolicity theory originated more than 30 years ago in works of Shub, Pugh and Brin, Pesin. It served as a mathematical model for chaotic systems with periodic forcing. In Riemannian geometry this theory was used to describe ergodic properties of frame flows on manifolds of negative curvature. Partial hyperbolicity theory has recently gained its momentum due to rapid development of stable ergodicity theory (Shub, Pugh, Dologopyat, Wilkinson, Hertz), study of systems with dominated splitting (Mane, Viana, Bochi), description of topological foliations on three-manifolds (Ruelle, Wilkinson and Brin, Burago, Ivanov) as well as its applications to systems with chaotic behavior (Burns, Dolgopyat, Pesin). I will present a good account of partial hyperbolicity theory, describe its basic concepts, discuss major results and examples. Although no preliminary knowledge of dynamical systems theory is necessary some familiarity with ergodic theory and dynamics will be quite helpful. Lecture notes will be distributed to guide through the course.

Content

1. Complete and Partially Hyperbolicity

2. Basic Examples

3. Mather Spectrum Theory

4. Integrability and Absolute Continuity

5. Accessibility

6. Stable Ergodicity