Expanding on Average Diffeomorphisms

We study the statistical properties of partially hyperbolic diffeomorphisms with a two-dimensional center exhibiting mixed behavior, without assuming a dominated splitting inside the center. We introduce the notion of an expanding on average for partially hyperbolic diffeomorphism and, under a center jacobian assumption, we prove the existence of finitely many ergodic hyperbolic SRB measures, and show the strong positive recurrence property, which implies that each SRB measure, up to a period, satisfies exponential decay of correlations, among other strong statistical laws. We apply our results to several examples.