Cohomological rigidity and the Anosov-Katok construction

We provide a general argument for the failure of Anosov-Katok-like constructions (as in \cite{AFKo2015} and \cite{NKInvDist}) to produce Cohomologically Rigid diffeomorphisms in manifolds other than tori. A $C^{\infty }$ smooth diffeomorphism $f$ of a compact manifold $M$ is Cohomologically Rigid iff the equation, known as Linear Cohomological one, \begin{equation*} \psi \circ f - \psi = \varphi \end{equation*} admits a $C^{\infty }$ smooth solution $\psi$ for every $\varphi$ in a codimension $1$ closed subspace of $C^{\infty } (M, \mathbb{C} )$. As an application, we show that no Cohomologically Rigid diffeomorphisms exist in the Almost Reducibility regime for quasi-periodic cocycles in homogeneous spaces of compact type, even though the Linear Cohomological equation over a generic such system admits a solution for a dense subset of functions $\varphi$. We thus confirm a conjecture by M. Herman and A. Katok in that context and provide some insight in the mechanism obstructing the construction of counterexamples.