×

On the degree of cocycles with values in the group SU(2). (English) Zbl 1061.37004

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 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 \(w\in\mathbb{R}\).

MSC:

37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
37E10 Dynamical systems involving maps of the circle
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cornfeld, I. P.; Fomin, S. W.; Sinai, J. G., Ergodic Theory (1982), Berlin: Springer-Verlag, Berlin · Zbl 0493.28007
[2] Frączek, K., On cocycles with values in the group SU(2), Monatshefte für Mathematik, 131, 279-307 (2000) · Zbl 0988.37005 · doi:10.1007/s006050070002
[3] Gabriel, P.; Lemańczyk, M.; Liardet, P., Ensemble d’invariants pour les produits croisés de Anzai, Mémoires de la Société Mathématique de France, 47, 1-102 (1991) · Zbl 0754.28011
[4] M. R. Herman,Non-topological conjugacy of skew products on SU (2), manuscript (1989).
[5] Iwanik, A.; Lemańczyk, M.; Rudolph, D., Absolutely continuous cocycles over irrational rotations, Israel Journal of Mathematics, 83, 73-95 (1993) · Zbl 0786.28011 · doi:10.1007/BF02764637
[6] Krikorian, R., Global density of reducible quasi-periodic cocycles onT^1 ×SU(2), Annals of Mathematics, 154, 269-326 (2001) · Zbl 1030.37003 · doi:10.2307/3062098
[7] Rychlik, M., Renormalization of cocycles and linear ODE with almost-periodic coefficients, Inventiones Mathematicae, 110, 173-206 (1992) · Zbl 0771.58013 · doi:10.1007/BF01231330
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.