Fan Yang, Wake Forest University

Title: Expanding foliations: Margulis measures, transversal unique ergodicity, and exponential equidistribution

Abstract: Given a diffeomorphism preserving a one-dimensional expanding foliation with homogeneous exponential growth, we construct a family of reference measures on each leaf of the foliation with constant Jacobian and a certain Gibbs property. We then use this family of measures to investigate the transversal unique ergodicity and equidistribution of the foliation leaves. In particular, for a large family of partially hyperbolic diffeomorphisms, including automorphisms on 3-nilmanifolds and perturbations of geodesic flows on negative curvature surfaces, we will show that the horocycle flow is uniquely ergodic, and the Lebesgue measures on the forward images of unstable disks will equidistribute to the unique invariant measure of the horocycle flow (instead of the SRB measure) at an exponential rate. We will also present several conjectures on the unique ergodicity and (exponential) equidistribution for the strong unstable foliations of Anosov systems, and discuss their relations with the work of Eskin-Mirzakhani.