The absolute continuous spectrum of skew products of compact Lie groups

TL;DR

This paper establishes a general criterion for the presence of absolutely continuous spectrum in skew products of compact Lie groups, based on the properties of the associated representations and derivatives, with applications to various group cases.

Contribution

The paper introduces a simple criterion for absolutely continuous spectrum in skew products of compact Lie groups, extending spectral analysis techniques using positive commutator methods.

Findings

Purely absolutely continuous spectrum under certain derivative conditions.

Applicable to skew products involving tori, SU(2), and U(2).

Provides a unified approach for spectral analysis of these systems.

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.