Wavelets and spectral triples for fractal representations of Cuntz algebras

C. Farsi, E. Gillaspy, A. Julien, S. Kang, J. Packer

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

7 Scopus citations


In this article we provide an identification between the wavelet decompositions of certain fractal representations of C -algebras of directed graphs, as introduced by M. Marcolli and A. Paolucci (2011), 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 Cuntz C -algebras OD. In particular, in this setting we use results of J. Pearson and J. Bellissard (2009), and A. Julien and J. Savinien (2011), 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 OD, and in the uneven weight case for certain representations of O2, and show how the eigenspaces and wavelet subspaces at different levels (first constructed in C. Farsi, E. Gillaspy, S. Kang, and J. Packer) are related.

Original languageEnglish
Title of host publicationContemporary Mathematics
PublisherAmerican Mathematical Society
Number of pages31
StatePublished - 2017

Publication series

NameContemporary Mathematics
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627


This work was partially supported by a grant from the Simons Foundation (#316981 to Judith Packer).

FundersFunder number
Simons Foundation316981


    • Laplace beltrami operators
    • Spectral triples
    • Ultrametric cantor set
    • Wavelets
    • Weighted bratteli diagrams


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