Degree, mixing, and absolutely continuous spectrum of cocycles with values in compact Lie groups

Abstract

We consider skew products $$T_\phi:X\times G\to X\times G,~~(x,g)\mapsto(F_1(x),g\;\!\phi(x)),$$ where $X$ is a compact manifold with probability measure, $G$ a compact Lie group with Lie algebra $\frak g$, $F_1:X\to X$ the time-one map of a measure-preserving flow, and $\phi\in C^1(X,G)$ a cocycle. Then, we define the degree of $\phi$ as a suitable function $P_\phi M_\phi:X\to\frak g$, we show that it transforms in a natural way under Lie group homomorphisms and under the relation of $C^1$-cohomology, and we explain how it generalises previous definitions of degree of a cocycle. For each finite-dimensional irreducible representation $\pi$ of $G$, and $\frak g_\pi$ the Lie algebra of $\pi(G)$, we define in an analogous way the degree of $\pi\circ\phi$ as a suitable function $P_{\pi\circ\phi}M_{\pi\circ\phi}:X\to\frak g_\pi$. If $F_1$ is uniquely ergodic and the functions $\pi\circ\phi$ diagonal, or if $T_\phi$ is uniquely ergodic, then the degree of $\phi$ reduces to a constant in $\frak g$ given by an integral over $X$. As a by-product, we obtain that there is no uniquely ergodic skew product $T_\phi$ with nonzero degree if $G$ is a connected semisimple compact Lie group. Next, we show that $T_\phi$ is mixing in the orthocomplement of the kernel of $P_{\pi\circ\phi}M_{\pi\circ\phi}$, and under some additional assumptions we show that $U_\phi$ has purely absolutely continuous spectrum in that orthocomplement if $(iP_{\pi\circ\phi}M_{\pi\circ\phi})^2$ is strictly positive. Summing up these results for each $\pi$, one obtains a global result for the mixing and the absolutely continuous spectrum of $T_\phi$. As an application, we present four explicit cases: when $G$ is a torus, $G=SU(2)$, $G=SO(3,\mathbb R)$, and $G=U(2)$. In each case, the results we obtain are new, or generalise previous results. Our proofs rely on new results on positive commutator methods for unitary operators.