Margulis Measures on Expanding Foliations: Construction and Rigidity

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-Jacobian of $f$ must be cohomologous to a constant via a measurable function. We provide 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.