Hertz–Hertz–Ures Ergodicity Conjecture

Establish that any C^r (r>1) conservative partially hyperbolic diffeomorphism f of a closed 3-manifold M that does not admit any embedded 2-torus tangent to the joint bundle E^s ⊕ E^u is ergodic with respect to the preserved volume.

Background

Ergodicity in partially hyperbolic dynamics is substantially more delicate than in uniformly hyperbolic settings due to neutral behavior in the center direction. Building on Hopf, Anosov, and Sinai, the accessibility property emerged as a decisive factor for ergodicity in partially hyperbolic systems, motivating the search for topological obstructions.

The Hertz–Hertz–Ures Ergodicity Conjecture posits that the only obstruction to ergodicity for conservative partially hyperbolic diffeomorphisms in dimension three is the presence of an embedded 2-torus tangent to E^s ⊕ E^u (an su-torus). This conjecture has been verified in several manifold classes and broader settings, but remains open in full generality. The present paper confirms it for systems with quasi-isometric center under non-wandering assumptions and no su-tori.