Decomposition of wavelet representations and Martin boundaries

We study a decomposition problem for a class of unitary representations associated with wavelet analysis, wavelet representations, but our framework is wider and has applications to multi-scale expansions arising in dynamical systems theory for non-invertible endomorphisms. Our main results offer a direct integral decomposition for the general wavelet representation, and we solve a question posed by Judith Packer. This entails a direct integral decomposition of the general wavelet representation. We further give a detailed analysis of the measures contributing to the decomposition into irreducible representations. We prove results for associated Martin boundaries, relevant for the understanding of wavelet filters and induced random-walks, as well as classes of harmonic functions. Our setting entails representations built from certain finite-to-one endomorphisms $r$ in compact metric spaces $X$, and we study their dilations to automorphisms in induced solenoids. Our wavelet representations are covariant systems formed from the dilated automorphisms. They depend on assigned measures $\mu$ on $X$. It is known that when the data $(X, r, \mu)$ are given the associated wavelet representation is typically reducible. By introducing wavelet filters associated to $(X, r)$ we build random walks in $X$, path-space measures, harmonic functions, and an associated Martin boundary. We construct measures on the solenoid $(X_\infty, r_\infty)$, built from $(X, r)$. We show that $r_\infty$ induces unitary operators $U$ on Hilbert space \H and representations $\pi$ of the algebra $L^\infty(X)$ such that the pair $(U, r_\infty)$, together with the corresponding representation $\pi$ forms a cross-product in the sense of $C^*$-algebras. We note that the traditional wavelet representations fall within this wider framework of $(\H, U, \pi)$ covariant crossed products.