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