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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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