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.

中文翻译：

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康托集上的Baumslag-Solitar群的统一表示

Cantor集支持Borel概率测度，即Hutchinson测度，它满足众所周知的定点关系（Hutchinson在印第安纳大学数学J 30（5）：713–747，1981年）。以前，Jorgensen和Dutkay已经证明，可以在实线的子集上将Cantor集扩展为膨胀的Cantor集\（{\ mathcal {R}} \），它支持扩展的Hutchinson测度\（ {\ bar {\ mu}} \）（Dutkay和Jorgensen在Rev. Mat。Iberoamericana 22（1）：131–180，2006中）。可以在满足Baumslag-Solitar组关系的\（L ^ 2（{\ mathcal {R}}，{\ bar {\ mu}}）\）上定义扩张和平移运算符，从而产生对希尔伯特空间\（L ^ 2（{\ mathcal {R}}，{\ bar {\ mu}}）\）称为多分辨率分析（Dutkay and Jorgensen 2006）。对应于此构造的低通滤波器函数可用于在称为3-电磁噪声的紧凑拓扑组上生成度量m，表示为\（{\ mathcal {S}} _ 3 \）（Trans Am Math Soc中的Dutkay 358（12）：5271-5291，2006）。希尔伯特空间\（L ^ 2（{\ mathcal {S}} _ 3，m）\）也允许Baumslag–Solitar群的unit表示，并且\（L ^ 2（{ mathcal {R}}，{\ bar {\ mu}}）\）和\（L ^ 2（{\ mathcal {S}} _ 3，m）\）（Dutkay 2006）。在本文中，我们以Jorgensen和Dutkay的工作为基础，证明\（L ^ 2（{\ mathcal {S}} _ 3，m）\）上的unit算子 上面提到的这些是通过一系列满足等价于Cuntz关系的部分等距相互关联的。