On the spectra of a class of self-affine measures on R2

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Abstract

For i{1,2}, let 0<|ρi|<1. For an expanding real matrix M=diag[ρ11,ρ21] and a three-element digit set DZ2 with cardinality |D|, let μM,D be the self-affine measure defined by μM,D()=1|D|dDμM,D(M()d). Let F32:=13{(l1,l2)t:l1,l2N+,0<l1,l22} and Z(mD):={xR2:dDe2πid,x=0}. If aF32, Z(mD)={±a}(modZ2) and {x[0,1)2:dDe2πid,x=|D|}={(0,0)t}, then L2(μM,D) has an exponential orthonormal basis if and only if ρi13Z{0} for i{1,2}.

Introduction

Let μ be a probability measure on Rn with compact support. We call the measure μ a spectral measure if there exists a discrete set ΛRn such that EΛ:={e2πiλ,x:λΛ} is an orthonormal basis for L2(μ) 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 R2. For i{1,2}, let 0<|ρi|<1. Let M=diag[ρ11,ρ21]M2(R) be an 2×2 expanding real matrix (that is, all the eigenvalues of M have moduli >1), and DZ2 be a finite subset with cardinality |D|. Let {ϕd(x)}dD be an iterated function system (IFS) defined by ϕd(x)=M1(x+d)(xR2,dD). Then there exists a unique probability measure μ satisfyingμ=μM,D=1|D|dDμϕd1. Such a measure μM,D is supported on the attractor T(M,D) of the IFS {ϕd}dD [19]. In particular, if M=ρ1 with 0<|ρ|<1 and D={0,1,,m1}Z with m2, we denote the measure μM,D as μρ,m.

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 1/k-Cantor measure μ1/k on R is a spectral measure if k is even (Strichartz provided a simplified proof in [27]). The more general Bernoulli convolution μρ was considered by Hu and Lau in [18], and was completed by Dai [4], who proved that μ1/(2k) is the only spectral measure among the μρ, 0<ρ<1. 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 R3 is a spectral measure. If M=13kI2 and D={(0,0)t,(1,0)t,(0,1)t}, Dutkay and Jorgensen [13] showed that μM,D is a spectral measure, where I2 is the 2×2 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 μM,D be defined by (1.1). If ρ1=ρ2 and D={(0,0)t,(1,0)t,(0,1)t}, then L2(μM,D) admits an infinite orthogonal set of exponential functions if and only if |ρ1|=(p/(3q))1r for some p,q,rN+ with gcd(p,3q)=1.

Theorem B Theorem 1.2, [9]

Let μM,D be defined by (1.1). If ρ1=ρ2 and D={(0,0)t,(1,0)t,(0,1)t}, then μM,D is a spectral measure if and only if |ρ1|=13q for some qN+.

Encouraged by the above results, in this paper, we determine the spectrality of self-affine measure μM,D on R2. The main results of this paper are the following two theorems.

For a finite digit set BR2, letmB(x)=1|B|bBe2πib,x(xR2) andZ(mB)={xR2:mB(x)=0}.

For convenience, letA={±(pq)1/r:p,q,rN+,(p,q)=1,p<q,3|q} andF32:=13{(l1,l2)t:l1,l2N+,0<l1,l22}.

Theorem 1.1

For i{1,2}, let 0<|ρi|<1. Let M=diag[ρ11,ρ21]M2(R) be an 2×2 expanding real matrix, and DZ2 be a three-element digit set. Let μM,D,Z(mD),A,F32 be defined by (1.1), (1.3), (1.4) and (1.5), respectively. If aF32 and Z(mD)=±a(modZ2), then L2(μM,D) admits an infinite orthogonal set of exponential functions if and only if for ρiA for i{1,2}.

Theorem 1.2

For i{1,2}, let 0<|ρi|<1. Let M=diag[ρ11,ρ21]M2(R) be an 2×2 expanding real matrix, and DZ2 be a three-element digit set. Let μM,D,Z(mD),F32 be defined by (1.1), (1.3) and (1.5), respectively. If aF32, Z(mD)=±a(modZ2) and {x[0,1)2:mD(x)=1}=(0,0)t, then μM,D is a spectral measure if and only if ρi13Z{0} for i{1,2}.

The paper deals with a class of self-affine measures that admit orthogonal Fourier series, which is basic for the spectral measure theory. Following earlier papers, starting with Jorgensen and Pedersen, we focus on the case of self-affine measures. This paper considers the planar case, more precisely to a class of Sierpinski configurations. By studying this class of measures, we give the sufficient and necessary condition for a spectral measure. The above two theorems extend the known results on the spectrality of the self-affine measure, which is supported on the two-dimensional Sierpinski gasket (see Theorem A and Theorem B). It is said that Dai et al. have investigated the spectral property of μM,D, where M=diag[ρ11,ρ21] and D={(0,0)t,(1,0)t,(0,1)t}.Theorem 1.1, Theorem 1.2 include this situation in the paper. The proof of Theorem 1.1 is through [30] and some algebraic identities induced by the orthogonality. The sufficiency of Theorem 1.2 follows from [12] by producing a compatible pair. For the necessity, we show that μM,D is not a spectral measure in the following cases:
  • (i)

    ρiA for some i{1,2} (Lemma 3.2);

  • (ii)

    There exists i0{1,2} such that ρi0A and |ρi0|Q (Lemma 4.2);

  • (iii)

    |ρi|=piqi with pi<qi, gcd(pi,qi)=1 and 3|qi for i{1,2}, but there exists i0{1,2} such that pi0>1 (Lemma 5.1).

The main difficulty in the proof of Theorem 1.2 is to show that the spectrum has a special expression. The way to find a special expression of the spectrum is of independent interest, which can be used for the study of the spectrality of more general singular measures.

By Theorem 1.2, we can decide whether a measure μ is a spectral measure, i.e., whether the functions in L2(μ) 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 V(x) is an almost periodic function; namely, V(x) can be expanded in a Fourier series: V(x)=ωΩνωe2iπωx, 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 RMn(R) be an n×n expanding real matrix, BRn be a finite subset with cardinality |B|. Let μR,B be defined by (1.1). The Fourier transform of μR,B is defined as usual,μˆ(ξ)=e2πiξ,xdμ(x). It follows from [13] thatμˆR,B(ξ)=j=1mB(Rjξ),(ξRn) where R denotes the transposed conjugate of R, and mB is given by (1.2). For any λ1,λ2Rn, λ1λ2, the orthogonality condition0=e2πiλ1,x,e2πiλ2,xL2(μR,B)=e2πiλ1λ2,xdμR,B(x)=μˆR,B(λ1λ2) relates to the zero set Z(μˆR,B) directly. It is

Proof of Theorem 1.1

We introduce the following lemma for later use. It was proved in [30].

Lemma 3.1

Theorem 1.1, [30]

Let 0<|ρ|<1 and m2 be an integer. Let μρ,m be defined by (1.1). Then L2(μρ,m) contains an infinite orthogonal set of exponential functions if and only if ρ=±(p/q)1/r for some p,q,rN+ with gcd(p,q)=1 and gcd(q,m)>1.

Lemma 3.2

Under assumptions of Theorem 1.1. If L2(μM,D) contains an infinite orthogonal set of exponential functions, then ρiA for i{1,2}.

Proof

If L2(μM,D) contains an infinite orthogonal set of exponential functions EΛ,

Spectrality for irrational contraction rate

In this section, we prove the necessity of Theorem 1.2 when there exists i0{1,2} such that ρi0A and |ρi0|Q.

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 βA admits a minimal integer polynomial qxr±p and satisfies a1βk+a2βj=a3βl, where p,q,rN+, k,j,l0 and a1,a2,a3Z\{0}. Then kjl(modr).

Lemma 4.2

Under assumptions of Theorem 1.2. If there exists i0{1,2} such that ρi0A and |ρi0|Q, then

Spectrality for rational contraction rate

In this section, we consider the case that |ρi|=piqi with pi<qi, gcd(pi,qi)=1 and 3|qi for i{1,2}, but there exists i0{1,2} such that pi0>1.

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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The research is supported in part by the NNSF of China (No. 11831007) and a project supported by Scientific Research Fund of Hunan Provincial Education Department (No. 19C0579).

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