Extremal distributions of partially hyperbolic systems: the Lipschitz threshold

TL;DR

This paper establishes a sharp phase transition in the regularity of extremal distributions for volume-preserving partially hyperbolic diffeomorphisms on 3-manifolds, extending known rigidity phenomena and deriving new classification results.

Contribution

It proves that Lipschitz extremal distributions are automatically smooth, extending rigidity results from Anosov flows to partially hyperbolic systems and applying this to various rigidity problems.

Findings

Lipschitz extremal distributions are $C^ abla$-smooth.

Rigidity results for $u$-Gibbs measures are derived.

Classification results for partially hyperbolic diffeomorphisms are obtained.

Abstract

We prove a sharp phase transition in the regularity of the extremal distribution $E^s \oplus E^u$ for $C^\infty$ volume-preserving partially hyperbolic diffeomorphisms on closed $3$-manifolds: if $E^s \oplus E^u$ is Lipschitz, then it is automatically $C^\infty$. This extends the rigidity phenomenon established by Foulon--Hasselblatt for conservative Anosov flows in dimension $3$ to the partially hyperbolic setting. This gain in regularity has several applications to rigidity problems. In particular, we study the relationship between the $\ell$-integrability condition introduced by Eskin--Potrie--Zhang and joint integrability in the conservative setting, yielding rigidity results for $u$-Gibbs measures. We also obtain several $C^\infty$ classification results for partially hyperbolic diffeomorphisms on $3$-manifolds under various assumptions.