On spectra and spectral eigenmatrix problems of the planar Sierpinski measures

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

“…Our first result establishes the following characterization of maximal orthogonal set of the measure µ A,D via maximal tree mapping (see Definition 2.1). It extends the result of An, Dong and He [1], which is about the Sierpiński type self-similar spectral measure (i.e., n = m), to self-affine case (i.e., n = m). The original idea is due to Dutkay et al [21] who characterized the maximal orthogonal set of µ 4,{0,2} by using the so-called tree labeling method.…”

“…a (2) . We claim that |a (1) | < q 1 and |a (2) | < q 2 . In fact, if i 1 = 1, when q 1 = 1, then τ (I0 n i 1 ) = 0 and thus |a (1)…”

“…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. In this paper, we continue the line and consider the Sierpiński-type self-affine spectral measures, which are usually far more subtle than self-similar spectral measures.…”

“…Now we return to the proof of Theorem 3.1 and the rest argument is standard. 1) . It follows from Theorem 3.2 and Lemma 3.3 that λ∈Λα n+s |μ n+s (ξ + λ)| 2 = 1.…”

On the quasi-Beurling dimensions of the spectra for planar Moran-type Sierpinski spectral measures

“…Many spectral measures have been found, see, for example [1,3,7,8,9,10,11,12,15,22,25,27,33,34,36,35,38] and references therein. For a singular spectral measure, generally speaking, its spectrum containing 0 is not unique, see [2,8,24,21,21,27,31,32] and references therein. However, it is a big challenge to determine all the spectra for a given singular spectral measure.…”