Spectrality of a class of planar self-affine measures with three-element digit sets
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Archiv der Mathematik
Spectrality of a class of planar self-affine measures with three-element digit sets Yan Chen, Xin-Han Dong, and Peng-Fei Zhang
Abstract. Let μM,D be the self-affine measure generated by an expanding integer matrix M ∈ M2 (Z) and an integer three-element digit set D = {(0, 0)T , (α, β)T , (γ, η)T }. In this paper, we show that if 3 | det(M ) and 3 αη − βγ, then L2 (μM,D ) has an orthogonal basis of exponential functions if and only if M ∗ u ∈ 3Z2 , where u = (η − 2β, 2α − γ)T . Mathematics Subject Classification. 28A80, 42C05. Keywords. Orthogonal exponential functions, Self-affine measures, Spectral measures, Non-spectral measures.
1. Introduction. Let M ∈ Mn (Z) be an expanding integer matrix (all the eigenvalues of M have moduli > 1) and D ⊆ Zn be a finite digit set with cardinality #D. Let {φd (x)}d∈D be the iterated function system (IFS) defined by {φd (x) = M −1 (x + d)}d∈D . It is known [9] that there exists a unique probability measure μ := μM,D satisfying 1 μ ◦ φ−1 (1.1) μ= d . #D d∈D
We say that μ is a spectral measure if there exists a countable set Λ ⊆ Rn such that EΛ := {e2πi : λ ∈ Λ} forms an orthogonal basis of L2 (μ), the set Λ is called a spectrum for μ. Otherwise, we say that μ is a non-spectral measure. Non-spectral measures can be divided into two classes: (1) there are at most a finite number of orthogonal exponential functions in L2 (μ), at this time, the main question is to find the best number of orthogonal exponential functions in L2 (μ); (2) there are infinitely many orthogonal exponential functions in L2 (μ), but none of them form an orthogonal basis of L2 (μ). In 1998, Jorgensen and Pedersen [10] initially discovered that the 1/4-Cantor measure is a spectral The research is supported by NSFC (No. 11831007).
Y. Chen et al.
Arch. Math.
measure. Recently, the spectrality of fractal measures have attracted the attention of many scholars [3–5,7,8]. One of the most important measures is the plane self-affine measure, we focus our attention on it. J.L. Li [12] studied the spectrality of the generalized planar Sierpinski type measure μM,D , which is generated by M ∈ M2 (Z) with 3 | det(M ) and D = {(0, 0)T , (1, 0)T , (0, 1)T }. An and He [1] removed the condition 3 | det(M ) in [12], and gave a necessary and sufficient condition such that μM,D is a spectral measure. More generally, if M ∈ M2 (Z) with 3 | det(M ) and D = {(0, 0)T , (α, β)T , (γ, η)T } ⊆ Z2 with 3 αη −βγ, J.J. Li [11] obtained a sufficient condition for μM,D to be a spectral measure. For the case of 3 det(M ) and αη − βγ = 0, Chen and Liu [2] proved that the number of mutually orthogonal exponential functions in L2 (μM,D ) is finite. In this paper, we always assume that the expanding integer matrix M and the three-element digit set D satisfy the following conditions: M ∈ M2 (Z), 3 | det(M ),
(1.2)
D = {(0, 0) , (α, β) , (γ, η) }, 3 αη − βγ. T
T
T
(1.3)
∗
Let M be the conjugate transposed matrix of M , then it can be decomposed into the following form: ab ∗ + M , (1.4) := 3M M = 3M + cd ∈ M2
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