Seminar Algebra and Geometry: Rafael Tiedra (Pontificia Universidad Católica de Chile)

The absolute continuous spectrum of skew products of compact Lie groups

Let X and G be compact Lie groups, F₁: 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_φ:X x G → X x G,  (x,g) → (F₁(x), gφ(x)),

have purely absolutely continuous spectrum in the subspace associated to π if ποφ has a Dini-continuous Lie derivative along the flow and if a matrix multiplication operator related to the topological degree of ποφ 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.

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