Invariant Distributions and local theory of quasiperiodic cocycles in $\mathbb{T} ^{d} \times SU(2)$}
Abstract
We study the linear cohomological equation in the smooth category over quasiperiodic cocycles in $\mathbb{T} ^{d} \times SU(2)$. We prove that, under a full measure condition on the rotation in $\mathbb{T} ^{d}$, for a generic cocycle in an open set of cocycles, the equation admits a solution for a dense set of functions on $\mathbb{T} ^{d} \times SU(2)$ of zero average with respect to 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 such solution exists. We thus confirm in this context a conjecture by A. Katok stating that the only dynamical systems for which the linear cohomological equation admits a smooth solution for all $0$average functions with respect to a smooth volume are Diophantine rotations in tori. The proof is based on a careful analysis of the K.A.M. scheme of Krikorian (1999) and Karaliolios (2015), inspired by Eliasson (2002), which also gives a proof of the local density of cocycles which are reducible via finitely differentiable or measurable transfer functions.
 Publication:

arXiv eprints
 Pub Date:
 July 2014
 arXiv:
 arXiv:1407.4763
 Bibcode:
 2014arXiv1407.4763K
 Keywords:

 Mathematics  Dynamical Systems;
 Primary 37C55
 EPrint:
 The paper spent more than 3 years under review in a respected journal of the field. After 3 years of superficial and unconstructive reports, we decided to withdraw the paper from that journal. We do not intend to resubmit the paper in the near future. This version addresses the wellfounded part of the referee's criticism, regarding exposition. The core of the paper remains the same