Spectra of Cantor measures

The Cantor measures with are few of known singular continuous measures that admits a spectrum. The spectra of Cantor measures are not unique and their structure is not well-studied. It has been observed that maximal orthogonal sets of the Cantor measure 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 of the Cantor measure , and we use boundedness and linear increment of that measurement to justify whether is a spectrum or not. As applications of our justification, we provide a characterization for the expanding set of a spectrum to be a spectrum again. Furthermore, we construct a spectrum such that for some integer K, the shrinking set is a maximal orthogonal set but not a spectrum.