Geometric properties of partially hyperbolic measures and applications to measure rigidity

Abstract

We give a geometric characterization of the quantitative non-integrability, introduced by Katz, of strong stable and unstable bundles of partially hyperbolic measures and sets in dimension 3. This is done via the use of higher order templates for the invariant bundles. Using the recent work of Katz, we derive some consequences, including the measure rigidity of $uu$-states and the existence of physical measures.

52 pages. Substantial revision thanks to input from several colleagues. Added some flowcharts to describe the structure of the proof and included some toy versions to convey the structure of the proof of some points

Cited by in corpus (2)

• Uniqueness of $u$-Gibbs measures for hyperbolic skew products on ${\mathbb{T}}^4$
• Rigidity of $\text{U}$-Gibbs measures near conservative Anosov diffeomorphisms on ${\mathbb{T}}^3$