  Making use of a unified approach to certain classes of induced
representations, we establish here a number of detailed spectral theoretic
decomposition results. They apply to specific problems from non-commutative
harmonic analysis, ergodic theory, and dynamical systems. Our analysis is in
the setting of semidirect products, discrete subgroups, and solenoids. Our
applications include analysis and ergodic theory of Bratteli diagrams and their
compact duals; of wavelet sets, and wavelet representations.
