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We study the linear cohomological equation in the smooth category overquasi-periodic cocycles in $\mathbb{T} ^{d} \times SU(2)$. We prove that, undera full measure condition on the rotation in $\mathbb{T} ^{d}$, for a genericcocycle in an open set of cocycles, the equation admits a solution for a denseset of functions on $\mathbb{T} ^{d} \times SU(2)$ of zero average with respectto the Haar measure. This property is known as Distributional Unique Ergodicity(DUE). We then show that given such a cocycle, for a generic function no suchsolution exists. We thus confirm in this context a conjecture by A. Katokstating that the only dynamical systems for which the linear cohomologicalequation admits a smooth solution for all $0$-average functions with respect toa smooth volume are Diophantine rotations in tori. The proof is based on acareful analysis of the K.A.M. scheme of Krikorian (1999) and Karaliolios(2015), inspired by Eliasson (2002), which also gives a proof of the localdensity of cocycles which are reducible via finitely differentiable ormeasurable transfer functions.

Invariant Distributions and local theory of quasiperiodic cocycles in $\mathbb{T} ^{d} \times SU(2)$}