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Wavelets and spectral triples for fractal representations of Cuntz algebras

In this article we provide an identification between the wavelet decompositions of certain fractal representations of $C^*-$algebras of directed graphs of M. Marcolli and A. Paolucci, and the eigenspaces of Laplacians associated to spectral triples constructed from Cantor fractal sets that are the infinite path spaces of Bratteli diagrams associated to the representations, with a particular emphasis on wavelets for representations of $\mathcal{O}_D$. In particular, in this setting we use results of J. Pearson and J. Bellissard, and A. Julien and J. Savinien, to construct first the spectral triple and then the Laplace Beltrami operator on the associated Cantor set. We then prove that in certain cases, the orthogonal wavelet decomposition and the decomposition via orthogonal eigenspaces match up precisely. We give several explicit examples, including an example related to a Sierpinski fractal, and compute in detail all the eigenvalues and corresponding eigenspaces of the Laplace Beltrami operators for the equal weight case for representations of Cuntz algebras, and in the uneven weight case for certain representations of $\mathcal{O}_2$, and show how the eigenspaces and wavelet subspaces at different levels are related.