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Spectral measures and Cuntz algebras

We consider a family of measures $μ$ 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 $L^2(μ)$ consisting of complex exponentials, i.e., a Fourier basis corresponding to a discrete subset $Γ$ in $\br^d$. Here we offer two computational devices for understanding the interplay between the possibilities for such sets $Γ$ (spectrum) and the measures $μ$ themselves. Our computations combine the following three tools: duality, discrete harmonic analysis, and dynamical systems based on representations of the Cuntz $C^*$-algebras $\mathcal O_N$.