A new construction of wavelet sets.

Abstract

We show that the class of (dyadic) wavelet sets is in one-to-one correspondence to a special class of Lebesgue measurable isomorphisms of \([0,1)\) which we call {\it wavelet induced} maps. We then define two natural classes of maps \(\fwi\) and \(\swi\) which, in order to simplify their construction, retain only part of the characterization properties of a wavelet induced map. We prove that each wavelet induced map appears from the Schröder-Cantor-Bernstein construction applied to some \(u\in \mathcal{WI}_1\) and \(v\in \mathcal{WI}_2\). Consequently, the construction of a wavelet set is basically equivalent to the easier construction of two maps \(u\in \mathcal{WI}_1\) and \(v\in \mathcal{WI}_2\). Some older results on wavelet sets are recovered using this new point of view. The connectivity result of Speegle (\cite{sp}) is recaptured and the completeness in the natural metric of the class of wavelet sets is reestablished. Although these ideas seem to generalize to more than one dimension, specific examples are given only in the one dimensional case.