The absolute continuous spectrum of skew products of compact Lie groups

Abstract

Let $X$ and $G$ be compact Lie groups, $F_1:X\to X$ the time-one map of a $C^\infty$ measure-preserving flow, $\phi:X\to G$ a continuous function and $\pi$ a finite-dimensional irreducible unitary representation of $G$. Then, we prove that the skew products $$ T_\phi:X\times G\to X\times G,\quad(x,g)\mapsto\big(F_1(x),g\;\phi(x)\big), $$ have purely absolutely continuous spectrum in the subspace associated to $\pi$ if $\pi\circ\phi$ has a Dini-continuous Lie derivative along the flow and if a matrix multiplication operator related to the topological degree of $\pi\circ\phi$ has nonzero determinant. This result provides a simple, but general, criterion for the presence of an absolutely continuous component in the spectrum of skew products of compact Lie groups. As an illustration, we consider the cases where $F_1$ is an ergodic translation on $\T^d$ and $X\times G=\T^d\times\T^{d'}$, $X\times G=\T^d\times\SU(2)$ and $X\times G=\T^d\times\U(2)$. Our proofs rely on recent results on positive commutator methods for unitary operators.