Title: Margulis-like measures on expanding foliations: construction and rigidity
Abstract: Given a diffeomorphism preserving a one-dimensional expanding foliation $\mathcal F$ with homogeneous exponential growth, we construct a family of reference measures on each leaf of the foliation with controlled Jacobian and a Gibbs property. We then prove that for any measure of maximal u-entropy, its conditional measures on each leaf must be equivalent to the reference measures. When the measure of maximal u-entropy is a Gibbs $\mathcal F$-state (i.e., when the reference measures are equivalent to the leafwise Lebesgue measure), we prove that the log-determinant of $f$ must be cohomologous to a constant. We will discuss several applications, including the strong and center foliations of Anosov diffeomorphisms, factor over Anosov diffeomorphisms, and perturbations of the time-one map of geodesic flows on surfaces with negative curvature. We will also discuss several conjectures on the unique ergodicity and (exponential) equidistribution for the strong unstable foliation of an Anosov system. Joint with J. Buzzi, Y. Shi, and J. Yang.