Spectra of Self-Similar Measures

This paper is devoted to the characterization of spectrum candidates with a new tree structure to be the spectra of a spectral self-similar measure μN,D generated by the finite integer digit set D and the compression ratio N-1. The tree structure is introduced with the language of symbolic space and widens the field of spectrum candidates. The spectrum candidate considered by Łaba and Wang is a set with a special tree structure. After showing a new criterion for the spectrum candidate with a tree structure to be a spectrum of μN,D, three sufficient and necessary conditions for the spectrum candidate with a tree structure to be a spectrum of μN,D were obtained. This result extends the conclusion of Łaba and Wang. As an application, an example of spectrum candidate Λ(N,B) with the tree structure associated with a self-similar measure is given. By our results, we obtain that Λ(N,B) is a spectrum of the self-similar measure. However, neither the method of Łaba and Wang nor that of Strichartz is applicable to the set Λ(N,B).