Spectra of Cantor measures

By Xin-Rong Dai. 2014

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The Cantor measures $$\mu _{q, b}$$μq,b with $$2 = q, b/q \in {\mathbb Z}$$2=q,b/q∈Z are few of known singular continuous measures that admits a spectrum. The spectra of Cantor measures $$\mu _{q, b}$$μq,b are not unique and their structure is not well-studied. It has been observed that maximal orthogonal sets of the Cantor measure $$\mu _{q, b}$$μq,b have tree structure and they can be built by selecting maximal tree appropriately. A challenging problem is when a maximal orthogonal set becomes a spectrum. In this paper, we introduce a measurement on the tree structure associated with a maximal orthogonal set $$\Lambda $$Λ of the Cantor measure $$\mu _{q, b}$$μq,b, and we use boundedness and linear increment of that measurement to justify whether $$\Lambda $$Λ is a spectrum or not. As applications of our justification, we provide a characterization for the expanding set $$K\Lambda $$KΛ of a spectrum $$\Lambda $$Λ to be a spectrum again. Furthermore, we construct a spectrum $$\Lambda $$Λ such that for some integer K, the shrinking set $$\Lambda /K$$Λ/K is a maximal orthogonal set but not a spectrum.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1df09fb663382a64a65afaf49c9063bf3d4fbd49