Date of Award
Doctor of Philosophy (PhD)
In 1981, Hutchinson showed that for each iterated function system on the real line, there exists a unique probability measure, called a Hutchinson measure or invariant measure, whose support is the attractor of the of the iterated function system. A common question that has been asked about measures among this class is whether or not they are spectral. That is, for a given Hutchinson measure µ does there exist a set of complex exponential functions which forms an orthonormal basis for L2(µ)? It has since been observed that such measures are capable of having infinitely many spectra. These multiple spectra give rise to canonically-defined unitary operators on the L2-spaces of Hutchinson measures. Moreover, one of these operators found on the L2-space of a Hutchinson measure known as the 1/4-Cantor measure was shown to have a unique “fractal-like” construction and was dubbed the “Operator Fractal” by Jorgensen, Kornelson, and Shuman in 2012. In this publication, we wish to determine if other operator fractals exist on the L2-spaces of Hutchinson measures. Furthermore, acknowledging that it is not known in general precisely which Hutchinson measures are spectral, we divulge an if and only if condition which determines whether or not a particular class of Hutchinson measures associated to iterated function systems containing 3 elements is spectral. We demonstrate that such a measure is spectral if and only if its components are “well-spaced.” We then find an explicit spectrum of each such spectral measure.
Long, Ian, "Spectral Hutchinson-3 Measures and Their Associated Operator Fractals" (2017). Mathematics Graduate Theses & Dissertations. 53.