Katok–Herman conjecture on cohomological rigidity

Let $M$ be a compact manifold with volume measure $\mu$, and let $CR(M)$ denote the set of diffeomorphisms for which every smooth zero-mean function is a smooth coboundary. A translation of a torus is Diophantine when its rotation vector satisfies the relevant Diophantine lower bounds.

Katok–Herman conjecture. The only examples, up to smooth diffeomorphism and conjugacy, of $CR(M)$ diffeomorphisms are Diophantine translations in tori.

This conjecture asks whether cohomological rigidity is exclusive to Diophantine toral translations. The paper explains that the Anosov–Katok construction does not provide counterexamples in the settings studied, but the conjecture remains open.