International Journal of Mathematics ( IF 0.604 ) Pub Date : 2021-01-06 , DOI: 10.1142/s0129167x2150004x
Ming-Liang Chen; Zhi-Hui Yan

In this paper, we study the spectral property of the self-affine measure $μR,𝒟$ generated by an expanding real matrix $R=diag(b,b)$ and the four-element digit set $𝒟={00,10,01,−1−1}$. We show that $μR,𝒟$ is a spectral measure, i.e. there exists a discrete set $Λ⊆ℝ2$ such that the collection of exponential functions ${e−2πi〈λ,x〉:λ∈Λ}$ forms an orthonormal basis for $L2(μ)$, if and only if $b=2k$ for some $k∈ℕ$. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math.231(3) (2012) 1681–1693], over which $b=2k$. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of $μR,𝒟$ by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math.242 (2013) 187–208]. We also extend these results to the more general self-affine measures.

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