Spectra of Cantor measures

Abstract

The Cantor measures \(\mu _{q, b}\) with \(2 = q, b/q \in {\mathbb Z}\) are few of known singular continuous measures that admits a spectrum. The spectra of Cantor measures \(\mu _{q, b}\) are not unique and their structure is not well-studied. It has been observed that maximal orthogonal sets of the Cantor measure \(\mu _{q, b}\) 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 \(\Lambda \) of the Cantor measure \(\mu _{q, b}\), and we use boundedness and linear increment of that measurement to justify whether \(\Lambda \) is a spectrum or not. As applications of our justification, we provide a characterization for the expanding set \(K\Lambda \) of a spectrum \(\Lambda \) to be a spectrum again. Furthermore, we construct a spectrum \(\Lambda \) such that for some integer K, the shrinking set \(\Lambda /K\) is a maximal orthogonal set but not a spectrum.