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

TL;DR

This paper investigates the spectral properties and mixing behavior of skew product transformations with cocycles taking values in compact Lie groups, introducing a degree concept that generalizes previous notions and analyzing its implications for ergodicity and spectrum.

Contribution

It defines a new notion of degree for cocycles in compact Lie groups, explores its transformation properties, and links it to mixing and spectral characteristics of the associated skew products, extending previous theories.

Findings

No nonzero degree skew product is uniquely ergodic for connected semisimple compact Lie groups.

Under certain conditions, the skew product exhibits mixing in the orthogonal complement of a specific subspace.

The spectrum of the associated unitary operator is purely absolutely continuous in that orthogonal complement.

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.