Lecture Notes on Ergodicity of Partially Hyperbolic Systems

These lecture notes were written to accompany a minicourse given at the School and Workshop on Dynamical Systems at IMPA (Rio de Janeiro), January 2006. Warning: These notes are unedited lecture notes and do contain errors! Proceed with caution... Let f : M → M be a (pointwise) partially hyperbolic diffeomor-phism. Recall what partially hyperbolic means. At every point in M , there exist tangent vectors that are uniformly contracted by the derivative T f and tangent vectors that are uniformly expanded by T f. The contracted vectors lie in an invariant subbundle E s of T M , called the stable subbundle, and the expanded vectors lie in an invariant subbun-dle E u , called the unstable subbundle. The sum of these subbundles E u ⊕ E s is the hyperbolic part of T M. The rest of T M is also under control: there is a T f-invariant complement E c to E u ⊕ E s in T M , called the center bundle. Tangent vectors in E c may be expanded or contracted under the action of T f , but they are neither expanded as sharply as vectors in E u , nor contracted as sharply as vectors in E s. As you learned last week, the stable subbundle is uniquely integrable and tangent to a foliation W s , and the unstable subbundle is uniquely integrable and tangent to a foliation W u. The center bundle E c is sometimes, but not always, tangent to a foliation. The stable and unstable foliations are absolutely continuous, while the center foliation (when it exists) can fail to be absolutely continuous. We will assume throughout these lectures that f is C 2 and preserves a fixed volume m (or a measure equivalent to volume) on M , normalized so that m(M) = 1. The most basic property of a measure-preserving dynamical system is ergodicity. We will investigate which partially hyperbolic diffeomorphisms are ergodic.