2010/12/02 by Marcelo Viana, Jiagang Yang, Viana, Marcelo +1 · 1 citation
Mathematics · Physics and Astronomy · #37A99 #37C99 #37D25 #37D30 #Advanced Differential Geometry Research #Advanced Mathematical Theories and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematics and Applications #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1012.0513
openalex publication_date 2010/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a class of partially hyperbolic Ck, k>1 diffeomorphisms with circle center leaves we prove existence and finiteness of physical (or Sinai-Ruelle-Bowen) measures, whose basins cover a full Lebesgue measure subset of the ambient manifold. Our conditions contain an open and dense subset of all Ck 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.