Spectra of Self-Similar Measures

Abstract

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 aba 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 aba 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 aba and Wang nor that of Strichartz is applicable to the set Λ(N,B).

Beurling dimension of a class of spectra of the Sierpinski-type spectral measures

“…There are many papers studying the maximal orthogonal sets and spectral structure of different kind of fractal spectral measures by extending the method of tree labelling method, see [1,13,7,18,46]. In particular, An, Dong and He [1] recently obtained a complete characterization of the maximal orthogonal sets of Sierpiński-type self-similar spectral measure and gave a sufficient condition for a maximal orthogonal set to be a spectrum.…”

“…Furthermore, they obtained more general criteria for maximal orthogonal set of exponentials in L 2 (µ b,{0,1,...,q−1} ) to be spectra. Recently, the spectral structure of more general self-similar measures, some self-affine measures and Moran measures have been studied [15,7].…”