Minimality of the unstable foliation in partially hyperbolic systems

Determine when the strong unstable foliation is minimal (i.e., every strong unstable leaf is dense) for partially hyperbolic diffeomorphisms, beyond the known results for Anosov diffeomorphisms on the 3-torus under the “some hyperbolicity” condition on the center bundle.

Background

The paper discusses recent progress proving minimality of the strong unstable foliation for Anosov diffeomorphisms on the 3-torus under a weak expanding condition on the center bundle (the “some hyperbolicity” or SH condition).

Outside this specific Anosov setting, the authors note that establishing minimality of the unstable foliation remains unresolved in the broader class of partially hyperbolic diffeomorphisms. This problem is relevant to rigidity phenomena and measure-theoretic properties (e.g., uniqueness and structure of u-Gibbs states) explored throughout the paper.

References

Meanwhile, in the general partially hyperbolic setting, it remains open to determine when the unstable foliation is minimal.

— Extremal distributions of partially hyperbolic systems: the Lipschitz threshold  (2604.01100 - Leguil et al., 1 Apr 2026) in Introduction, Subsection “Applications to the rigidity of u-Gibbs states”