Extremal distributions of partially hyperbolic systems: the Lipschitz threshold
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908988880191488 |
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| author | Leguil, Martin Xu, Disheng Zhang, Jiesong |
| author_facet | Leguil, Martin Xu, Disheng Zhang, Jiesong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_01100 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extremal distributions of partially hyperbolic systems: the Lipschitz threshold Leguil, Martin Xu, Disheng Zhang, Jiesong Dynamical Systems 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. |
| title | Extremal distributions of partially hyperbolic systems: the Lipschitz threshold |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2604.01100 |