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We focus on the irreducibility of wavelet representations. We present someconnections between the following notions: covariant wavelet representations,ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators andsolutions of refinement equations. We investigate the irreducibility of thewavelet representations, in particular the representation associated to theCantor set, introduced in \cite{DuJo06b}, and we present several equivalentformulations of the problem.

Irreducible wavelet representations and ergodic automorphisms on solenoids