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

Abstract

We consider skew products $$T_{\varphi }:X\times G\to X\times G,(x,g)\mapsto (F_1(x),g\varphi (x)),$$ where $X$ is a compact manifold with probability measure, $G$ a compact Lie group with Lie algebra $\mathfrak{g}$, $F_1:X\to X$ the time-one map of a measure-preserving flow, and $\varphi \in C^1(X,G)$ a cocycle. Then, we define the degree of $\varphi$ as a suitable function $P_{\varphi }{M}_{\varphi }:X\to \mathfrak{g}$, 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 ${\mathfrak{g}}_{\pi }$ the Lie algebra of $\pi (G)$, we define in an analogous way the degree of $\pi \circ \varphi$ as a suitable function $P_{\pi \circ \varphi }{M}_{\pi \circ \varphi }:X\to {\mathfrak{g}}_{\pi }$. If $F_1$ is uniquely ergodic and the functions $\pi \circ \varphi$ diagonal, or if $T_{\varphi }$ is uniquely ergodic, then the degree of $\varphi$ reduces to a constant in $\mathfrak{g}$ given by an integral over $X$. As a by-product, we obtain that there is no uniquely ergodic skew product $T_{\varphi }$ with nonzero degree if $G$ is a connected semisimple compact Lie group. Next, we show that $T_{\varphi }$ is mixing in the orthocomplement of the kernel of $P_{\pi \circ \varphi }{M}_{\pi \circ \varphi }$, and under some additional assumptions we show that $U_{\varphi }$ has purely absolutely continuous spectrum in that orthocomplement if $(i{P}_{\pi \circ \varphi }{M}_{\pi \circ \varphi }{)}^2$ is strictly positive. Summing up these results for each $\pi$, one obtains a global result for the mixing and the absolutely continuous spectrum of $T_{\varphi }$. 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.

arXiv comment38 pages