Spectrality of self-affine Sierpinski-type measures on ${\mathbb{R}}^2$

Abstract

In this paper, we study the spectral property of a class of self-affine measures ${\mu }_{R,\mathcal{D}}$ on ${\mathbb{R}}^2$ generated by the iterated function system ${\{{\varphi }_d(\cdot )=R^{-1}(\cdot +d)\}}_{d\in \mathcal{D}}$ associated with the real expanding matrix $R=\left(\begin{array}{cc}\hfill b_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill b_2\hfill \end{array}\right)$ and the digit set $\mathcal{D}=\{\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{c}1\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 1\end{array}\right)\}$. We show that ${\mu }_{R,\mathcal{D}}$ is a spectral measure if and only if $3|b_i$, $i=1,2$. This extends the result of Deng and Lau [J. Funct. Anal., 2015], where they considered the case $b_1=b_2$. And we also give a tree structure for any spectrum of ${\mu }_{R,\mathcal{D}}$ by providing a decomposition property on it.

Introduction

Let μ be a probability measure with compact support on ${\mathbb{R}}^n$. We say that μ is a spectral measure if there exists a countable discrete set $\mathrm{\Lambda }\subseteq {\mathbb{R}}^n$ such that $E(\mathrm{\Lambda }):=\{e^{2\pi i〈\mathit{\lambda },x〉}:\mathit{\lambda }\in \mathrm{\Lambda }\}$ forms an orthonormal basis for $L^2(\mu )$. In this case, we call Λ a spectrum of μ.

A basic problem in harmonic analysis is to classify which kind of measures are spectral measures. This problem could date back to Fuglede [21], who initiated to determine when a Lebesgue measure restricted on a measurable set Ω is spectral, and proposed a far-reaching spectral set conjecture [29], [34], [39]. The first singularly continuous spectral measure was discovered by Jorgensen and Pedersen [28]. Since then, various techniques were developed to characterize the spectrality of measures, such as Ruelle operator, operator algebras and Hadamard matrix. The readers may see [2], [3], [8], [11], [12], [13], [14], [19], [30], [31], [36] and the references therein for recent advances. Also various new phenomena different from spectral theory for the Lebesgue measure were discovered. For instance, a singularly continuous spectral measure may admit many spectra with different Beurling dimensions [4], [6], [9], and the Fourier series corresponding to different spectra could have completely different convergence property [10], [37]. Following the discoveries, the theory of spectral measures has been extensively studied, which reached out to several research areas, such as the theory of quasi-crystals [22], the p-adic fields ${\mathbb{Q}}_p$ [17], [18], and Gabor and wavelet bases/frames [9], [15], [23], [25], [33], [38].

In this paper, we consider the spectral properties of a class of self-affine measures ${\mu }_{R,\mathcal{D}}$, which satisfies

$${\mu }_{R,\mathcal{D}}(\cdot )=\frac{1}{\mathrm{\#}\mathcal{D}}\sum _{d\in \mathcal{D}}{\mu }_{R,\mathcal{D}}(R(\cdot )-d),$$

where $R=\left(\begin{array}{cc}\hfill b_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill b_2\hfill \end{array}\right)$ with $b_1,b_2>1$, is an expanding matrix, and $\mathcal{D}=\{\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{c}1\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 1\end{array}\right)\}$. We call such ${\mu }_{R,\mathcal{D}}$ the Sierpinski-type measure, which plays an important role in fractal geometry and geometric measure theory [16], [27].

Furthermore, we would like to mention the study of the spectrality of N-Bernoulli measure ${\mu }_{\rho ,N}$, the self-similar measure generated by the iterated function system (IFS) ${\{{\varphi }_j(\cdot )=\rho (\cdot +j)\}}_{j=0}^{N-1}$. The spectral property of the Bernoulli convolution ${\mu }_{\rho ,2},0<\rho <1$, was considered by Hu and Lau in [26], and was completed by the first author [2]. These results are generalized further to the N-Bernoulli measures [5], and later to the Riesz product measures and some classes of Moran measures [1], [6], [19], [20], [24], [35]. Unlike the one dimensional situation, the study on the spectrality of measures in higher dimensions is seldom addressed. The main result of this paper is as follows.

Theorem 1.1

Let ${\mu }_{R,\mathcal{D}}$ be the Sierpinski-type measure defined as in (1.1). Then ${\mu }_{R,\mathcal{D}}$ is a spectral measure if and only if $3|b_i$, $i=1,2$.

A natural way to characterize the spectrality of a measure is to determine whether there exists a spectrum for it. The exponential orthogonal set for ${\mu }_{R,\mathcal{D}}$ can be constructed by the spectrum of ${\delta }_{\mathcal{D}}=\frac{1}{\mathrm{\#}\mathcal{D}}{\sum }_{d\in \mathcal{D}}{\delta }_d$, where ${\delta }_d$ is the Dirac measure at point d. However, the completeness of orthogonal set is quite challenging (see [3], [4], [8], [36] for various sufficient and necessary conditions). In [7], Deng and Lau considered the special case $b_1=b_2=b$, and constructed a connection on the exponential orthogonal set between ${\mu }_{R,\mathcal{D}}$ and 3-Bernoulli measure ${\mu }_{b^{-1},3}$. We extend their result to the general case, and we get the following theorem on the existence of infinitely many orthogonal exponentials in $L^2({\mu }_{R,\mathcal{D}})$.

Theorem 1.2

Let ${\mu }_{R,\mathcal{D}}$ be the Sierpinski-type measure defined as in (1.1). Then $L^2({\mu }_{R,\mathcal{D}})$ admits an infinite orthogonal set of exponential functions if and only if $b_i=\sqrt[r_i]{\frac{3k_i}{q_i}}$ for some $r_i,k_i,q_i\in \mathbb{N}$ with $\gcd (3k_i,q_i)=1$, $i=1,2$.

According to Theorem 1.2, the technique of proving Theorem 1.1 is to show that ${\mu }_{R,\mathcal{D}}$ is NOT a spectral measure in the following cases:

• For $i=1,2$, $b_i=\sqrt[r_i]{\frac{3k_i}{q_i}}$ and $r_1>1$ or $r_2>1$;

• For $i=1,2$, $b_i=\frac{3k_i}{q_i}$ and $q_1>1$ or $q_2>1$.

Here we assume that $r_i,i=1,2$, is the smallest positive integer such that $b_i^{r_i}\in \mathbb{Q}$.

For the purpose, we first settle (i) by investigating the structure of bi-zero sets of ${\mu }_{R,\mathcal{D}}$ (Proposition 3.3, Proposition 3.4). And then we get a property of decomposition on the spectrum Λ of ${\mu }_{R,\mathcal{D}}$ in the following theorem (Theorem 4.1).

Theorem 1.3

Let ${\mu }_{R,\mathcal{D}}$ be the Sierpinski-type measure defined as in (1.1). Suppose that ${\mu }_{R,\mathcal{D}}$ is a spectral measure, and let Λ be a spectrum of ${\mu }_{R,\mathcal{D}}$. Then Λ can be decomposed as

$$\mathit{\Lambda }=\bigcup _{i=0}^2(R{\mathit{\Lambda }}_i+{\lambda }_i),$$

where ${\mathit{\Lambda }}_i,i=0,1,2$, are spectra of ${\mu }_{R,\mathcal{D}}$ with $\mathbf{0}\in {\mathit{\Lambda }}_i\subseteq {\mathbb{Z}}^2$, and ${\{{\lambda }_i\}}_{i=0}^2$ is a spectrum of ${\delta }_{R^{-1}\mathcal{D}}$.

This illustrates a tree structure on the spectrum Λ (also see Corollary 4.2). Thus, we may arrange the elements in Λ by taking the minimum on the first or the second coordinates of elements inductively according to the step-by-step decomposition of Λ, such that we can estimate the growth rate of elements in Λ (Proposition 4.3). And then we settle the case (ii) by the property of logarithmic decay of ${\hat{\mu }}_{R,\mathcal{D}}$ (Lemma 5.2).

At the end of this paper, we prove that the spectrality of a self-affine measure is invariant under an invertible affine transform, and extend Theorem 1.2 and Theorem 1.1 to more general $\mathcal{D}\subseteq {\mathbb{R}}^2$ in the form of $\mathcal{D}=\{\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{c}a\\ 0\end{array}\right),\left(\begin{array}{c}0\\ b\end{array}\right)\}$ with $0\ne a,b\in \mathbb{R}$.

The paper is organized as follows. In Section 2, we set up some notations and summarize some basic results concerning orthogonal set of exponentials and spectrum of μ which will be for later use. Then, we prove Theorem 1.2. We settle the case (i) in Section 3 and we also give a tree structure for the spectrum of ${\mu }_{R,\mathcal{D}}$ in Section 4. In Section 5, we settle the case (ii) and prove Theorem 1.1. In Section 6, we provide some further results on the spectrality of some general Sierpinski-type measures.