Spectral measures and Cuntz algebras (Q2840012)

Letting \(\mu\) denote the Hutchinson measure which is supported on the attractor of an iterated function system \(\Phi\) consisting of \(n\) affine contractions, in the current article, under a separation condition on the pair \((\Phi, \mu)\), an algorithmic approach for constructing an orthonormal set of complex exponentials (a so-called Fouier bases) of \(\mathcal{L}^{2}(\mu_{B})\) is given. In order to recursively construct and exhibit such an orthonormal family of complex exponentials, the authors use a representation of the Cuntz algebra \(\mathcal{O}_{n}\), which they construct using the transfer operator associated to \(\Phi\). Through this work, the authors demonstrate that key results in the representation theory of \(\mathcal{O}_{n}\) have important implications on the spectral theory of affine fractals and vice versa.