The absolute continuous spectrum of skew products of compact Lie groups

Abstract

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 productshave purely absolutely continuous spectrum in the subspace associated to π if π • φ has a Dinicontinuous Lie derivative along the flow and if a matrix multiplication operator related to the topological degree of π • 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 F1 is an ergodic translation on T d and X × G = T d × T d ′ , 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.

Commutator criteria for strong mixing

“…In all of these examples, we obtain new results, or re-obtain known results, with very simple and short proofs. In the case of skew products of compact Lie groups, our results extend previous results of K. Fraczek, A. Iwanik, M. Lemańzyk, D. Rudolph and the author [7,8,12,20]. In the case of horocycle flows, we obtain a very simple proof that all time changes of horocycle flows of class C 1 on compact surfaces of constant negative curvature are strongly mixing.…”

“…Accordingly, one can rephrase the result of Theorem 3.1(a) in the following more intuitive terms : If U is regular enough with respect to the auxiliary operator A, and if the topological degree D exists, then the flow {U N } N∈Z is strongly mixing in the subspace where D is nonzero. This result extends to a completely general hilbertian setting some related result in the framework of skew products of compact Lie groups (see [20,Rem. 3.6] and references therein).…”

“…Then, U φ is strongly mixing in the subspace (m,n)∈R n j =0 H (ρ2m−n⊗π (n) ) j ⊂ H. This result on strong mixing complements [20,Thm. 4.9], where it is proved that U φ has countable Lebesgue spectrum in the subspace (m,n)∈R n j =0 H (ρ2m−n⊗π (n) ) j if L Y (η 1 ) and L Y (η 2 ) satisfy an additional regularity assumption of Dini-type.…”

“…Then, a calculation using the expression of the matrix elements π (n) j k (see [20,Sec. 4.2]) shows that…”

“…Then, using results of [20,Sec. 4.3] one can prove the existence of the limit D := s -lim N→∞ D N and apply Theorem 3.1(a) for some indices (m, n) ∈ Z × N and some values of b 1 , b 2 ∈ Z d .…”