Physical measures and absolute continuity for one-dimensional center direction

Abstract

For a class of partially hyperbolic $C^k$, $k>1$ diffeomorphisms with circle center leaves we prove the existence and finiteness of physical (or Sinai–Ruelle–Bowen) measures, whose basins cover a full Lebesgue measure subset of the ambient manifold. Our conditions hold for an open and dense subset of all $C^k$ partially hyperbolic skew-products on compact circle bundles.

Our arguments blend ideas from the theory of Gibbs states for diffeomorphisms with mostly contracting center direction together with recent progress in the theory of cocycles over hyperbolic systems that call into play geometric properties of invariant foliations such as absolute continuity. Recent results show that absolute continuity of the center foliation is often a rigid property among volume preserving systems. We prove that this is not at all the case in the dissipative setting, where absolute continuity can even be robust.