Spectral measures and Cuntz algebras

We consider a family of measures $\mu$ supported in ${\text{br}}^d$ 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 $\mathrm{\Gamma }$ in ${\text{br}}^d$. Here we offer two computational devices for understanding the interplay between the possibilities for such sets $\mathrm{\Gamma }$ (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 $C^{\ast }$-algebras ${\mathcal{O}}_N$.