Nassiri Conjecture on Transitivity Implying Ergodicity

Determine whether every C^{1+α} transitive, volume-preserving partially hyperbolic diffeomorphism f: M^3 → M^3 of a closed 3-manifold is ergodic with respect to the preserved volume.

Background

A long-considered conjecture suggests a deep link between topological transitivity and measure-theoretic ergodicity in partially hyperbolic dynamics, particularly compelling in the case of one-dimensional center bundles. In manifolds with non-virtually solvable fundamental groups, this conjecture would follow from the Hertz–Hertz–Ures Ergodicity Conjecture.

The paper demonstrates that, under the quasi-isometric center hypothesis, transitivity and ergodicity coincide for conservative C^2 partially hyperbolic diffeomorphisms, thus supporting the conjecture in this setting. The general statement, however, remains open beyond the quasi-isometric center framework.