Margulis Measures on Expanding Foliations: Construction and Rigidity
arXiv:2607.13556
The paper constructs special reference measures on the leaves of a one‑dimensional expanding foliation preserved by a diffeomorphism and shows that any measure with maximal u‑entropy must be equivalent to these reference measures, leading to rigidity results for the Jacobian when the measure is a Gibbs state.
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 controlled Jacobian and a Gibbs property. We then prove that for any measure of maximal -entropy, its conditional measures on each leaf must be equivalent to the reference measures. When the measure of maximal -entropy is a Gibbs -state (i.e., when the reference measures are equivalent to the leafwise Lebesgue measure), we prove that the log-Jacobian of 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.