Unitary Representations of the Baumslag–Solitar Group on the Cantor Set

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.