Orthonormal dilations of Parseval wavelets

We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag–Solitar group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$BS(1, 2) = \langle{u, t\,|\, utu^{-1} = t^2}\rangle.$$\end{document}We give a precise description of this representation in some special cases, and show that for wavelet sets, it is related to symbolic dynamics (Theorem 3.14). We prove that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics (Theorem 3.24). We give concrete examples of Parseval wavelets for which we compute the orthonormal dilations in detail; we construct Parseval wavelet sets which have infinitely many non-isomorphic orthonormal dilations.