Cohomological rigidity and the Anosov-Katok construction

Abstract

We provide a general argument for the failure of Anosov-Katok-like constructions (as in [AFK15] and [Kar14]) to produce Cohomologically Rigid diffeomorphisms in manifolds other than tori. A C ∞ smooth diffeomorphism f of a compact manifold M is Cohomologically Rigid iff the equation, known as Linear Cohomological one,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 ϕ. 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.