Absolute continuity of the rotation number for quasi-periodic co-cycles in \(\mathrm{SL}(2,\mathbb R)\) (Q471823)

The author considers the cocycle \[ x_{n+1}= A(E,\theta+\omega)x_n = (A(E)+F(E,\theta+n\omega))x_n, \] where \(x_n\in \mathbb{R}^2\), \(n\in \mathbb{Z}\), \(E\in \mathbb{R}\), \(A(E), A(E,\theta+\omega)\in \mathrm{SL}(2,\mathbb{R})\), \(A(E,\theta+\omega)\) is homotopic to the identity and \(\theta \in \mathbb{T}^d\), the \(d\)-dimensional torus. For any \(E\), \(F(E,\cdot)\) is assumed to be analytic on \(\mathbb{T}^d\), and both \(A\) and \(F\) are assumed to be of class \(C^2\). The frequency vector \(\omega \in \mathbb{R}^d\) is assumed to be Diophantine, that is, there exist \(\tau>0\) and \(K >0\) such that for all \(n\in \mathbb{T}^d \setminus \{0\}\) \[ \inf_{j\in \mathbb{Z}} \big|\frac{1}{2}\langle n, \omega \rangle-j \pi\big| \geq K |n|^{-\tau}. \] For this type of cocycle, the rotation number is well defined. Here, it is proven that, under smallness conditions on \(F\), the rotation number is absolutely continuous. The proof relies on the KAM scheme developed in [the author, Commun. Math. Phys. 287, No. 2, 565--588 (2009; Zbl 1201.37066)].