Center foliation: absolute continuity, disintegration and rigidity (Q2793108)

The author studies volume-preserving, partially hyperbolic diffeomorphisms that are isotopic to linear Anosov automorphisms on the 3-dimensional torus \({\mathbf T}^3\) (called their linearizations). It is assumed that the diffeomorphisms are of class \(C^{1+\alpha}\), \(\alpha>0\).NEWLINENEWLINELet us mention several main results of the paper:NEWLINENEWLINE1. There exist volume-preserving, partially hyperbolic Anosov diffeomorphisms with non-absolutely continuous center foliation for which the disintegration of volume (by conditional measures) on center leaves is neither Lebesgue nor atomic.NEWLINENEWLINE2. There exist volume-preserving, partially hyperbolic Anosov diffeomorphisms with center foliation of class \(C^1\) that are not \(C^1\) conjugate to their linearizations.NEWLINENEWLINE3. If the center foliation is of class \(C^1\) and is transversely absolutely continuous with bounded Jacobians, then \(f\) is \(C^1\) conjugate to its linearization (and hence, it is Anosov).