Cohomological stability and the Anosov-Katok construction

We discuss the solvability of the linear cohomological equation over a smooth volume-preserving diffomorphism of a smooth compact manifold $(M,\mu )$. The central conjecture in the subject was posed by M. Herman, and states that the only diffeomorphisms for which the equation is solvable for all observables in $C^{\infty}_{\mu } (M, \mathbb{R})$ are Diophantine rotations in tori. Recent attempts to construct counterexamples to the conjecture were based on the Anosov-Katok construction, and went only as far as constructing Distributionally Uniquely Ergodic (DUE) diffomorphisms in manifolds that are not tori. For DUE diffeomorphisms, the cohomological equation is solvable for observables in a dense subspace of $C^{\infty}_{\mu } (M, \mathbb{R})$. We will discuss the problem and propose an argument showing that the Anosov-Katok method cannot provide counterexamples to the conjecture.