In this course we will study the existence of dominated splitting and possible existence of invariant foliation for diffeomorphism on compact manifolds with certain hyperbolicity. This includes the study of the C^r section theorem, the local theory of normally hyperbolic invariant compact manifolds and the existence of center manifolds. We will also study certain bunching conditions and discuss their relationship with the regularity of the invariant foliations.
1-M. W. Hirsch, C.C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, New York, 1977. MR0501173 (58:18595).
2-C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), 517–546.