Spectral measures and Cuntz algebras

We consider a family of measures $\mu$ supported in $\br^d$ and generated in the sense of Hutchinson by a finite family of affine transformations. It is known that interesting sub-families of these measures allow for an orthogonal basis in $L^2(\mu)$ consisting of complex exponentials, i.e., a Fourier basis corresponding to a discrete subset $\Gamma$ in $\br^d$. Here we offer two computational devices for understanding the interplay between the possibilities for such sets $\Gamma$ (spectrum) and the measures $\mu$ themselves. Our computations combine the following three tools: duality, discrete harmonic analysis, and dynamical systems based on representations of the Cuntz $C^*$-algebras $\mathcal O_N$.