The absolute continuous spectrum of skew products of compact Lie groups

Abstract

Let X and G be compact Lie groups, F ₁: X → X the time-one map of a C ^∞ measure-preserving flow, ϕ: X → G a continuous function and π a finite-dimensional irreducible unitary representation of G. Then, we prove that the skew products

$${T_\phi }:X \times G \to X \times G,(x,g) \mapsto ({F_1}(x),g\phi (x))$$

, have purely absolutely continuous spectrum in the subspace associated to π if π po ϕ has a Dini-continuous Lie derivative along the flow and if a matrix multiplication operator related to the topological degree of πpoϕ 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 × G = T ^d × T ^dʹ, X × G = T ^d × SU(2) and X × G = T ^d × U(2). Our proofs rely on recent results on positive commutator methods for unitary operators.