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

TL;DR

This paper investigates the solutions to the linear cohomological equation over quasi-periodic cocycles in t^d  SU(2), establishing conditions for distributional unique ergodicity and confirming a conjecture linking smooth solutions to Diophantine rotations.

Contribution

It proves distributional unique ergodicity for generic cocycles and confirms Katok's conjecture relating smooth solutions to Diophantine rotations in tori.

Findings

Distributional unique ergodicity holds for a dense set of functions.

Generic functions do not admit smooth solutions.

The results support Katok's conjecture on smooth solutions and Diophantine rotations.

Abstract

We study the linear cohomological equation in the smooth category over quasi-periodic 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.