On the spectra of a class of self-affine measures on ☆
Introduction
Let μ be a probability measure on with compact support. We call the measure μ a spectral measure if there exists a discrete set such that is an orthonormal basis for and call Λ a spectrum of μ. For the special case that the spectral measure is the restriction of the Lebesgue measure on a bounded Borel set Ω, we call Ω a spectral set.
The study of spectrum has received much attention since the famous spectral set conjecture raised by Fuglede [17] in 1974. Although the spectral set conjecture has been disproved by Tao [28] and others [21], [22] in dimension three and higher, it is still suggestive in the study of spectral measure.
Spectral measures have surprising connections with harmonic analysis, number theory, dynamical system, and fractal geometry [7], [11], [14], [20]. The first singular, non-atomic, spectral measure was constructed by Jorgensen and Pedersen in 1998, which opened up a new field in researching harmonic analysis on fractals [20]. Harmonic analysis has a wide application in number theory, statistics, medicine, geophysics, and quantum physics [29]. In recent years, it is found that harmonic analysis on fractals can be applied to problems in image compression and physics [3], [15].
This paper is intended to study the spectrality of a class of self-affine measures on . For , let . Let be an expanding real matrix (that is, all the eigenvalues of M have moduli >1), and be a finite subset with cardinality . Let be an iterated function system (IFS) defined by . Then there exists a unique probability measure μ satisfying Such a measure is supported on the attractor of the IFS [19]. In particular, if with and with , we denote the measure as .
The study on the spectral property of self-affine measures dates back to the work of Jorgensen and Pedersen [20], and many spectral properties of self-affine measures have been found in [2], [4], [7], [10], [18] and references cited therein. Among the many results, Jorgensen and Pedersen [20] proved that the -Cantor measure on is a spectral measure if k is even (Strichartz provided a simplified proof in [27]). The more general Bernoulli convolution was considered by Hu and Lau in [18], and was completed by Dai [4], who proved that is the only spectral measure among the , . This is generalized further to the N− Bernoulli convolutions [6], [7]. In [20] (see also [27]), the authors showed that the canonical Hausdorff measure on the Sierpinski gasket is not a spectral measure, but such measure on the Sierpinski tower in is a spectral measure. If and , Dutkay and Jorgensen [13] showed that is a spectral measure, where is the identical matrix. In [1], [23], [24], [25], [26], Li, An and He proved some further results of the exponential orthogonal sets for the case that M is an expanding integral matrix. Recently, Deng and Lau [9] gave the following two sufficient and necessary conditions:
Theorem A Theorem 1.1, [9] Let be defined by (1.1). If and , then admits an infinite orthogonal set of exponential functions if and only if for some with .
Theorem B Theorem 1.2, [9] Let be defined by (1.1). If and , then is a spectral measure if and only if for some .
Encouraged by the above results, in this paper, we determine the spectrality of self-affine measure on . The main results of this paper are the following two theorems.
For a finite digit set , let and
For convenience, let and
Theorem 1.1 For , let . Let be an expanding real matrix, and be a three-element digit set. Let be defined by (1.1), (1.3), (1.4) and (1.5), respectively. If and , then admits an infinite orthogonal set of exponential functions if and only if for for .
Theorem 1.2 For , let . Let be an expanding real matrix, and be a three-element digit set. Let be defined by (1.1), (1.3) and (1.5), respectively. If , and , then is a spectral measure if and only if for .
- (i)
for some (Lemma 3.2);
- (ii)
There exists such that and (Lemma 4.2);
- (iii)
with , and for , but there exists such that (Lemma 5.1).
By Theorem 1.2, we can decide whether a measure μ is a spectral measure, i.e., whether the functions in have Fourier expansions. From [3], we see that many physical systems can be described via a Schrödinger equation with an almost periodic potential. This is the case for linear organic conducting chains, electronic properties of crystals in a uniform magnetic field, and the Ginzburg-Landau theory of filamentary superconductors. In all these systems, the problem is reduced to a one-dimensional Schrödinger equation, in which the potential is an almost periodic function; namely, can be expanded in a Fourier series: , where Ω is a countable subset of the real numbers. Thus, spectral measures have a close relationship with physical applications.
This paper is organized as follows. We devoted Section 2 to introduce some basic definitions and lemmas. Section 3 is devoted to prove Theorem 1.1. Theorem 1.2 is proved in Section 4 and Section 5.
Added in proof. During the proof reading of the paper, the author realized similar results by X.R. Dai et al. have just appeared in [5, ACHA 2020]. The technique used here is independent and different from their, and the result is more general.
Section snippets
Preliminaries
Let be an expanding real matrix, be a finite subset with cardinality . Let be defined by (1.1). The Fourier transform of is defined as usual, It follows from [13] that where denotes the transposed conjugate of R, and is given by (1.2). For any , , the orthogonality condition relates to the zero set directly. It is
Proof of Theorem 1.1
We introduce the following lemma for later use. It was proved in [30].
Lemma 3.1 Let and be an integer. Let be defined by (1.1). Then contains an infinite orthogonal set of exponential functions if and only if for some with and .Theorem 1.1, [30]
Lemma 3.2 Under assumptions of Theorem 1.1. If contains an infinite orthogonal set of exponential functions, then for . Proof If contains an infinite orthogonal set of exponential functions ,
Spectrality for irrational contraction rate
In this section, we prove the necessity of Theorem 1.2 when there exists such that and .
Before stating our main result, we first introduce and not include the proof of the following lemma, since it follows readily from Lemma 2.5 in [8].
Lemma 4.1 Assume that admits a minimal integer polynomial and satisfies , where , and . Then .
Lemma 4.2 Under assumptions of Theorem 1.2. If there exists such that and , then
Spectrality for rational contraction rate
In this section, we consider the case that with , and for , but there exists such that .
In order to prove the author's main result, the author's main efforts are to reduce the two-dimensional case to one dimension, and then with the help of methods and techniques in [9] and [16] to establish the following lemmas suitable for two-dimensional case. In particular, Lemma 5.5, Lemma 5.6, Lemma 5.7 refer to the proof methods of Lemma 4.6–4.8 in
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