## Global density of reducible quasi-periodic cocycles on $$\mathbb T^1\times \text{SU}(2)$$.(English)Zbl 1030.37003

In the study of linear quasi-periodic differential equations of the form \begin{aligned} X'&=U(\theta)X(\theta)\\ \theta'&=\omega,\end{aligned} where $$\omega\in \mathbb{R}^{d}$$, and $$U:T^{d}\to g$$ is a real analytic map from the $$d$$-dimensional torus $$T^{d}$$ to the Lie algebra $$g$$, of the Lie group $$G$$, an important role is played by the quasi periodic cocycles, i.e. the set of diffeomorphisms of $$T^{d}\times G$$ of the form: \begin{aligned} (\alpha, A):T^{d}\times G&\rightarrow T^{d}\times G\\ (\theta,y)&\mapsto (\theta+\alpha, A(\theta)\cdot y),\end{aligned} where $$\alpha\in T^{d}$$ and $$A\in C^\infty(T^{d},G)$$. In the set of cocycles corresponding to a fixed $$\alpha$$, a cocycle which is conjugated in some sense with the constant cocycle $$(\alpha, A)$$, i.e. $$A$$ is a constant function, is called reducible. Reducibility appears as an extension of the Floquet theory to the quasi-periodic case.
In this paper, the author deals with the case $$d=1$$, and $$G=\text{SU}(2)$$, the compact Lie group of $$2\times 2$$ unitary matrices. The main result states that there exists a subset $$\Sigma$$ of $$T^{1}$$ of the full Haar measure such that for any fixed $$\alpha\in\Sigma$$ the set of $$A\in C^\infty(T^{1}, \text{SU}(2))$$ for which the cocycle $$(\alpha, A)$$ is reducible is dense for the $$C^\infty$$ topology.
The proof is based on the renormalization of abelian $$\mathbb{Z}^{2}$$ actions on $$\mathbb{R}\times \text{SU}(2)$$.

### MSC:

 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems 37C55 Periodic and quasi-periodic flows and diffeomorphisms 37H05 General theory of random and stochastic dynamical systems
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