On the spectrality of self-affine measures with four digits on ℝ2

In this paper, we study the spectral property of the self-affine measure [Formula: see text] generated by an expanding real matrix [Formula: see text] and the four-element digit set [Formula: see text]. We show that [Formula: see text] is a spectral measure, i.e. there exists a discrete set [Formula: see text] such that the collection of exponential functions [Formula: see text] forms an orthonormal basis for [Formula: see text], if and only if [Formula: see text] for some [Formula: see text]. 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 [Formula: see text]. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of [Formula: see text] 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.

A LOWER BOUND OF TOPOLOGICAL HAUSDORFF DIMENSION OF FRACTAL SQUARES

Given an integer [Formula: see text] and a digit set [Formula: see text], there is a self-similar set [Formula: see text] satisfying the set equation [Formula: see text]. This set [Formula: see text] is called a fractal square. By studying the line segments contained in [Formula: see text], we give a lower estimate of the topological Hausdorff dimension of fractal squares. Moreover, we compute the topological Hausdorff dimension of fractal squares whose nontrivial connected components are parallel line segments, and introduce the Latin fractal squares to investigate the question when the topological Hausdorff dimension of a fractal square coincides with its Hausdorff dimension.

THE NON-SPECTRAL PROPERTY OF A CLASS OF PLANAR SELF-SIMILAR MEASURES

Let the self-similar measure [Formula: see text] be generated by an expanding real matrix [Formula: see text] and a digit set [Formula: see text] in space [Formula: see text]. In this paper, we only consider [Formula: see text] and the case [Formula: see text] is similar. We show that there exists an infinite orthogonal set of exponential functions in [Formula: see text] if and only if [Formula: see text] for some [Formula: see text] with [Formula: see text]. Furthermore, for the cases that [Formula: see text] does not admit any infinite orthogonal set of exponential functions, the exact cardinality of orthogonal exponential functions in [Formula: see text] is given.

Orthogonal exponentials of self-affine measures on ℝn

Let the self-affine measure [Formula: see text] be generated by an expanding matrix [Formula: see text] and a finite integer digit set [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] for an integer [Formula: see text], then [Formula: see text] admits an infinite orthogonal set of exponential functions if and only if there exists [Formula: see text] such that [Formula: see text] for some [Formula: see text] with [Formula: see text] and [Formula: see text].

CONNECTEDNESS OF A CLASS OF SELF-AFFINE CARPETS

Let [Formula: see text], where [Formula: see text] are integers and [Formula: see text] be a digit set. Then the pair [Formula: see text] generates a fractal set [Formula: see text] satisfying [Formula: see text] which is a unit square. However, if we remove one digit from [Formula: see text], then the structure of [Formula: see text] will become very interesting. A well-known example is the Sierpinski carpet. In this paper, we study the resulting self-affine sets of moving a digit in [Formula: see text] to a different place. That is, we consider a digit set [Formula: see text], where [Formula: see text]. We give a complete characterization for the connectedness of self-affine carpet [Formula: see text] in terms of the domains of [Formula: see text] and [Formula: see text].

Spectral property of the planar self-affine measures with three-element digit sets

Let the self-affine measure {\mu_{M,D}} be generated by an expanding real matrix {M=\operatorname{diag}(\rho_{1}^{-1},\rho_{2}^{-1})} and an integer digit set {D=\{(0,0)^{t},(\alpha_{1},\alpha_{2})^{t},(\beta_{1},\beta_{2})^{t}\}} with {\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0}. In this paper, the sufficient and necessary conditions for {L^{2}(\mu_{M,D})} to contain an infinite orthogonal set of exponential functions are given.

A CLASS OF SPECTRAL MORAN MEASURES ON ℝ

Let [Formula: see text] be an arithmetic digit set for each [Formula: see text], where [Formula: see text], and let [Formula: see text] be a sequence of integers larger than 1. In this paper, we prove that the Moran measure [Formula: see text] generated by infinite convolution of finite atomic measures [Formula: see text] is a spectral measure if [Formula: see text] and [Formula: see text].

ORTHOGONAL EXPONENTIAL FUNCTIONS OF THE PLANAR SELF-AFFINE MEASURES WITH FOUR DIGITS

For the self-affine measure [Formula: see text] generated by an expanding matrix [Formula: see text] and an integer digit set [Formula: see text] with [Formula: see text], Su et al. proved that if [Formula: see text], then [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if [Formula: see text] [J. Su, Y. Liu and J. C. Liu, Non-spectrality of the planar self-affine measures with four-element digit sets, Fractals (2019), https://doi.org/10.1142/S0218348X19501159 ]. In this paper, we show that the above conclusion also holds for [Formula: see text]. So, a complete characterization of [Formula: see text] containing an infinite orthogonal set of exponential functions is given.