On the degree of cocycles with values in the group SU(2)
Let \(T: S^1\to S^1\) be an irrational rotation of the circle. Every measurable function \(\varphi: S^1\to \text{SU}(2)\) defines a skew product \(T_\varphi(x, g)= (Tx, g\varphi(x))\) which is determined by the cocycle \[ \varphi^{(n)}(x)= \begin{cases} \varphi(x)\cdot\varphi(Tx)\cdots \varphi(T^{n-1}x)\quad &\text{for }n> 0,\\ e\quad &\text{for }n= 0,\\ (\varphi(T^n x)\cdots \varphi(T^{n+1} x)\cdots \varphi(T^{-1} x))^{-1}\quad &\text{for }n< 0.\end{cases} \] If \(\varphi\) is of class \(C^1\) then the degree of \(\varphi\) is the unique number \(d(\varphi)> 0\) such that \[ {1\over n}\| D\varphi^n(x)(\varphi^n(x))^{-1}\|\to d(\varphi) \] for almost every \(x\in S^1\), where \(\|\;\|\) is the usual norm on the Lie algebra of \(\text{SU}(2)\) induced by the Killing form. The main purpose of the paper is to relate ergodic properties of the skew-product defined by a function \(\varphi\) of class \(C^2\) to the degree of \(\varphi\). Extending earlier results of the same author [Monatsh. Math. 131, 279--307 (2001; Zbl 0988.37005)] and using techniques due to \textit{R. Krikorian} [Ann. Math. (2) 154, 269--326 (2001; Zbl 1030.37003)], it is shown that \(d(\varphi)\in 2\pi\mathbb{N}\) for every \(C^2\)-cocycle \(\varphi: S^1\to \text{SU}(2)\). Moreover, \(C^2\)-cocycles \(\varphi_1,\varphi_2: S^1\to \text{SU}(2)\) have the same degree if they are measurably cohomologous. If the rotation number \(r\) for the irrational rotation \(T\) satisfies a suitably chosen Diophantine condition (which is valid for a subset of \(S^1\) of full Lebesgue measure) then a smooth cocycle \(\varphi\) with \(d(\varphi)= r> 0\) is smoothly conjugate to a cocycle of the form \[ x\to \begin{pmatrix} e^{2\pi i(rx+ w)}\\ & e^{-2\pi i(rx+ w)}\end{pmatrix}\in \text{SU}(2), \] where \(w\in\mathbb{R}\).
scientific article; zbMATH DE number 2114533