Abstract
The Cantor Set supports a Borel probability measure known as the Hutchinson measure which satisfies a well known fixed point relationship (Hutchinson in Indiana University Math J 30(5):713–747, 1981). Previously it has been shown by Jorgensen and Dutkay that the Cantor set can be extended to an inflated Cantor set, \({\mathcal {R}}\), on a subset of the real line, which supports an extended Hutchinson measure \({\bar{\mu }}\) (Dutkay and Jorgensen in Rev. Mat. Iberoamericana 22(1):131–180, 2006). Unitary dilation and translation operators can be defined on \(L^2({\mathcal {R}}, {\bar{\mu }})\) which satisfy the Baumslag–Solitar group relation, and give rise to a filtration of the Hilbert space \(L^2({\mathcal {R}}, {\bar{\mu }})\) called a multi-resolution analysis (Dutkay and Jorgensen 2006). The low pass filter function corresponding to this construction can be used to produce a measure, m, on a compact topological group called the 3-solenoid, denoted \({\mathcal {S}}_3\) (Dutkay in Trans Am Math Soc 358(12):5271–5291, 2006). The Hilbert space \(L^2({\mathcal {S}}_3, m)\) also admits a unitary representation of the Baumslag–Solitar group, and there exists a generalized Fourier transform between \(L^2({\mathcal {R}}, {\bar{\mu }})\) and \(L^2({\mathcal {S}}_3,m)\) (Dutkay 2006). In this paper, we build off of Jorgensen and Dutkay’s work to show that the unitary operators on \(L^2({\mathcal {S}}_3,m)\) mentioned above are related to each other via a family of partial isometries, which satisfy properties resembling the Cuntz relations.
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Acknowledgements
The author would like to thank Judith Packer (University of Colorado) for her guidance on this research. The author would also like to thank the referee for helpful suggestions which improved the manuscript.
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Communicated by Dorin Dutkay.
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Davison, T. Unitary Representations of the Baumslag–Solitar Group on the Cantor Set. J Fourier Anal Appl 26, 30 (2020). https://doi.org/10.1007/s00041-020-09730-0
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DOI: https://doi.org/10.1007/s00041-020-09730-0