Extremal distributions of partially hyperbolic systems: the Lipschitz threshold

We prove a sharp phase transition in the regularity of the extremal distribution $E^s\oplus E^u$ for $C^{\mathrm{\infty }}$ volume-preserving partially hyperbolic diffeomorphisms on closed $3$-manifolds: if $E^s\oplus E^u$ is Lipschitz, then it is automatically $C^{\mathrm{\infty }}$. 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 $\mathrm{\ell }$-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 $C^{\mathrm{\infty }}$ classification results for partially hyperbolic diffeomorphisms on $3$-manifolds under various assumptions.