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arXiv · 2604.01100

Extremal distributions of partially hyperbolic systems: the Lipschitz threshold

Martin Leguil·Disheng Xu·Jiesong Zhang
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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.

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Martin Leguil, Disheng Xu, Jiesong Zhang. 2026-04-01. Extremal distributions of partially hyperbolic systems: the Lipschitz threshold. https://arxiv.org/abs/2604.01100

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