The absolute continuous spectrum of skew products of compact Lie groups

• Published in: Israel journal of mathematics 2015-09, Vol.208 (1), p.323-350
• Main Author: Tiedra de Aldecoa, R.
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Applications of Mathematics; Group Theory and Generalizations; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Theoretical
• ISSN: 0021-2172; 1565-8511
• DOI: 10.1007/s11856-015-1201-9

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 , 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 π po ϕ 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.