Spectra of Cantor measures

The Cantor measures μ q , b with 2 = q , b / q ∈ Z are few of known singular continuous measures that admits a spectrum. The spectra of Cantor measures μ q , b are not unique and their structure is not well-studied. It has been observed that maximal orthogonal sets of the Cantor measure μ q , b have...

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Bibliographic Details
Published in:Mathematische annalen 2016-12, Vol.366 (3-4), p.1621-1647
Main Author: Dai, Xin-Rong
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:The Cantor measures μ q , b with 2 = q , b / q ∈ Z are few of known singular continuous measures that admits a spectrum. The spectra of Cantor measures μ q , b are not unique and their structure is not well-studied. It has been observed that maximal orthogonal sets of the Cantor measure μ q , 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 Λ of the Cantor measure μ q , b , and we use boundedness and linear increment of that measurement to justify whether Λ is a spectrum or not. As applications of our justification, we provide a characterization for the expanding set K Λ of a spectrum Λ to be a spectrum again. Furthermore, we construct a spectrum Λ such that for some integer K , the shrinking set Λ / K is a maximal orthogonal set but not a spectrum.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-016-1374-5