Spectra of Self-Similar Measures

This paper is devoted to the characterization of spectrum candidates with a new tree structure to be the spectra of a spectral self-similar measure μN,D generated by the finite integer digit set D and the compression ratio N−1. The tree structure is introduced with the language of symbolic space and widens the field of spectrum candidates. The spectrum candidate considered by Łaba and Wang is a set with a special tree structure. After showing a new criterion for the spectrum candidate with a tree structure to be a spectrum of μN,D, three sufficient and necessary conditions for the spectrum candidate with a tree structure to be a spectrum of μN,D were obtained. This result extends the conclusion of Łaba and Wang. As an application, an example of spectrum candidate Λ(N,B) with the tree structure associated with a self-similar measure is given. By our results, we obtain that Λ(N,B) is a spectrum of the self-similar measure. However, neither the method of Łaba and Wang nor that of Strichartz is applicable to the set Λ(N,B).


Introduction
Let µ be a probability measure on R d with compact support K. We say that µ is a spectral measure if there exists a countable set Λ ⊂ R d such that the set of exponential functions E Λ := {exp 2πi λ, x : λ ∈ Λ} is an orthogonal basis of L 2 (µ). In this case, Λ is called a spectrum of µ and (µ, Λ) is called a spectral pair. In particular, if µ is the normalized Lebesgue measure restricted on K, we say K is a spectral set.
In [1], Fuglede introduced the notion of a spectral set in the study of the extendability of the commuting partial differential operators and raised the famous conjecture: K is a spectral set if and only if K is a translational tile. Although the conjecture was finally disproven for the case that K ⊂ R d with d ≥ 3 and is still open for R d with d ≤ 2, it has led to the development of harmonic analysis, operator theory, tiling theory, convex geometry, etc.
Consider the iterated function system (IFS) {φ j } q j=1 given by where N is an integer with |N| > 1 and D = {d j } q j=1 is a finite subset of R. It is well known (see [22] or [23]) that there exists a unique probability measure µ N,D satisfying where δ a is the Dirac measure at a. Write Using the dominated convergence theorem, Strichartz [24] proved that µ N,D is a spectral measure with a spectrum Λ(N, S) under the conditions that (δ 1 N D , S) is a spectral pair with 0 ∈ S and the Fourier transform of δ 1 N D does not vanish on T(N, S). By using the Ruelle transfer operator, Łaba and Wang in [3] removed the condition that the Fourier transform of δ 1 N D does not vanish on T(N, S). Furthermore, they obtained the following conclusion: It is well known that to prove the spectrality of the invariant measure µ N,D , the first key step is to construct a suitable spectrum candidate. In this process, the set Λ(N, S) = S + NS + N 2 S + · · · (finite sum) is the natural spectrum candidate to be considered. Form Theorem 1, we conclude that Λ(N, S) is not a spectrum of µ N,D if and only if there is a periodic orbit {η j } m−1 j=0 ⊂ Z\{0} under the dual IFS {ψ i (x) = 1 N (x + s i ) : s i ∈ S}. The following example implies that the natural spectrum candidate has a weak point. When D = {0, 1}, the invariant measure µ 2,D is just the Lebesgue measure on the unit interval with the unique spectrum Z. However, Λ(2, {0, 1}) = N = Z in this case. In other words, the natural candidate Λ(2, {0, 1}) is not a spectrum of µ 2,D . Actually, any set with form S + 2S + 2 2 S + · · · (finite sum) is not a spectrum of µ 2,D . In this case, one needs to consider the spectrum candidate with a more general form S 1 + NS 2 + N 2 S 3 + · · · (finite sum), where ( 1 N D, S i ) are compatible pairs. Moreover, it is well known that a spectral self-similar (or self-affine) measure has more than one spectrum in general. The results in [7,[9][10][11] show that one may consider spectrum candidates with a tree structure. It is worth mentioning that Li [16] obtained a simplified form of Theorem 1. To the best of our understanding, partial results have been obtained in the case of a higher-dimensional space. Developing the method in [3], Dutkay and Jorgensen [14] obtained a sufficient condition for the spectral pair of self-affine measures, and Li [19] obtained a necessary condition for the natural spectrum candidate to be a spectrum of a self-affine measure. Motivated by the above results, we considered a class of spectrum candidates with a tree structure (defined in Section 2) and obtained three necessary and sufficient conditions for such spectrum candidates not to be the spectra of µ N,D (Theorem 2), which generalizes Łaba and Wang's result.
The most difficult part of the proof of Theorem 2 is that the first statement implies the second. For this purpose, we show a new criterion for Λ to be a spectrum of µ N,D . As an application, we give an example involving a self-similar measure µ and a spectrum candidate Λ(N, B) with a tree structure in Section 4. By Theorem 2, we obtain (µ, Λ(N, B)) is a spectral pair. However, neither the criterion of Łaba and Wang (Theorem 1) nor that of Strichartz [24] is applicable to this set Λ(N, B).

Preliminaries
In this section, we shall recall some basic properties of spectral measures and introduce the tree structure using symbolic space.
Let µ be a probability measure on R. The Fourier transform of µ is defined bŷ By using the Parseval identity, Jorgenson and Pederson ( [2]) obtained the following basic criterion for the orthogonality of E Λ in L 2 (µ). Proposition 1. The exponential function set E Λ is an orthogonal set of L 2 (µ) if and only if Q Λ (ξ) ≤ 1 for all ξ ∈ R, and E Λ is an orthogonal basis of L 2 (µ) if and only if Q Λ (ξ) = 1 for all ξ ∈ R.
Given a finite set D ⊂ R, we call is a unitary matrix.
The following conclusion is well known.

Lemma 1.
For two finite subsets D and S of R with the same cardinality m, the following statements are equivalent: (i). (D, S) is a compatible pair; (ii). m D (s 1 − s 2 ) = 0 for any s 1 = s 2 ∈ S; (iii). ∑ s∈S |m D (ξ + s)| 2 = 1 for any ξ ∈ R.
In other words, (D, S) is a compatible pair if and only if S is a spectrum of the uniform probability measure on D.
Let N be an integer with |N| > 1 and D = {d j } q j=1 a finite subset of Z with 0 ∈ D. We denote by µ N,D the unique invariant measure with respect to the IFS {φ j (x) = 1 N (x + d j ) : 1 ≤ j ≤ q} with equal probability weights, i.e., In the sequel, we write µ = µ N,D for simplicity. Thus, we havê For k ≥ 1, we writeμ Now, we introduce the tree structure. First, we recall some basic notation of symbolic space. Given a positive integer q > 1, write stand for the set of all finite words, where Σ 0 q = {ϑ} denotes the set of empty words. The length of a finite word σ is the number of symbols it contains and is denoted by |σ|. The concatenation of two finite words σ and σ is written as σσ . We say σ is a prefix of σσ . Given σ = σ 1 σ 2 · · · σ n ∈ Σ * and 1 ≤ k ≤ n, write σ| k = σ 1 · · · σ k . The following definition will bring convenience to us.

Definition 2.
A sequence of finite words {I n } n≥1 ⊂ Σ * is called increasing if for any n ≥ 1, I n is a prefix of I n+1 and |I n+1 | = |I n | + 1.
Let C be a mapping from Σ * to Z satisfying C(ϑ) = 0 and C(I) = 0 if I ends with the symbol 0. It induces a family of mapping F = {F I } I∈Σ * defined by where I J| i is the concatenation of I and J|i for 1 ≤ i ≤ |J|. We write F(J) = F ϑ (J) for convenience. By a simple deduction, we have the following consistency: for any I, J, K ∈ Σ * , =C(I J| 1 ) + · · · + N |J|−1 C(I J) + N |J| C(I JK| 1 ) + · · · + N |JK|−1 C(I JK) =F I (JK). Definition 3. We say a countable set Λ ⊂ R has a (C, F ) tree structure if there exists a mapping C and an associated family of mappings F defined in the above paragraph such that Entropy 2022, 24,1142 For I ∈ Σ * , let S I = {C(Ii) : i ∈ Σ q }. According to the definition of the mapping C, we have C(I0) = 0 ∈ S I . Remark 1. Given a sequence of finite sets S = {S n } n≥1 , if S I = S n+1 for any I ∈ Σ n (n ≥ 0), we obtain In particular, if S n = S for n ≥ 1, we obtain which is just the case considered by Łaba and Wang in [3].
In this paper, we consider a countable set Λ as a spectrum candidate satisfying the following three conditions: (C1). Λ has a (C, F ) tree structure. (C2). For any I ∈ Σ * , ( 1 N D, S I ) is a compatible pair. (C3). The set S = I∈Σ * S I is bounded.

Remark 2.
Since we only assume that ( 1 N D, S I ) is a compatible pair with S I = {C(Ii) : i ∈ Σ q }, the map C may not be a maximal mapping defined in [8] (Definition 2.5) even if D = {0, 1, · · · , q − 1}. Now, we exploit some basic properties of Λ satisfying the conditions (C1), (C2), and (C3). The first one is the uniqueness of the tree representation.

Proposition 2.
Let N ∈ Z with |N| > 1 and D ⊂ Z with 0 ∈ D and gcd(D) = 1. Assume that a countable set Λ satisfies the conditions (C1), (C2), and (C3). Then, for any I ∈ Σ * q and J, K ∈ Σ n q with n 0, we have F I (J) = F I (K) if and only if J = K.
Proof. We just prove the necessity. Suppose there exist I ∈ Σ * q and J = K ∈ Σ n q with n 1 such that F I (J) = F I (K). Let l be the smallest integer with J| l = K| l . From F I (J) = F I (K), it follows that is not a compatible pair, which is a contradiction to the condition (C2).
Proof. Given α = β ∈ Λ, there exist two finite words I, J ∈ Σ * such that If |I| = |J|, we add symbol 0 in the end of I or J to obtain |I| = |J|. Without loss of generality, we assume that I, J ∈ Σ n q for some integer n. Let l be the smallest positive integer satisfying I| l = J| l . Recall that F(I| l ) = C(I| 1 ) + NC(I| 2 ) · · · + N l−1 C(I| l ). Then, there exists an integer z 0 such that By virtue of the condition (C2), we know that ( 1 N D, S I| l−1 ) is a compatible pair. Noting that both C(I| l ) and C(J| l ) belong to S I| l−1 , we obtain This leads tô

=0.
For any I ∈ Σ * q and k 1, define We write Λ k := Λ k ϑ for simplicity. It is clear that From the condition (C2) and Lemma 1(ii), it follows that E(Λ k I ) is an orthogonal set of L 2 (µ k ). By (2), we obtain #Λ k I = q k . Noting the fact that dim(L 2 (µ k )) = q k , we conclude that E(Λ k I ) is an orthogonal basis of L 2 (µ k ). In other words, Λ k I is a spectrum of µ k . By Lemma 1, we have In fact, we have the following conclusion.

Proposition 4.
Let N ∈ Z with |N| > 1 and D ⊂ Z with 0 ∈ D and gcd(D) = 1. Assume that a countable set Λ satisfies the conditions (C1), (C2), and (C3). Then, Q Λ (ξ) ≡ 1 if and only if Q Λ I (ξ) ≡ 1 for any I ∈ Σ * , Proof. By virtue of Λ ϑ = Λ, the sufficiency is obvious. Next, we prove the necessity. Given n ≥ 1 and I ∈ Σ n q , write It is easy to see that both B I and B I are compact sets. Noting the fact thatμ n can be extended to be an entire function on the complex plane, µ n has at most finitely many zero points in B I . On the other hand, recall that Noting the fact that every integer is a period of m D , we haveμ n (ξ + F(I J)) =μ n (ξ + F(I)) for any I ∈ Σ n q and J ∈ Σ * q . Hence, In combination with (3), this means Q Λ I (ξ) takes 1 on except at most finitely many points in B I , which implies Q Λ I (ξ) ≡ 1 by using the continuity of Q Λ I (ξ).
In the end of this section, we define the dual IFS {Φ s (x) = 1 N (x + s) : s ∈ S}, which plays an important role in what follows. Let T be the invariant set of the IFS, i.e.,

Main Theorem
In this section, we will give our main results involving three equivalent statements.
To prove the most difficult part of the proof, we prepared several lemmas including a new criterion for a spectrum candidate with a tree structure to be a spectrum of a self-similar measure. At the end of this section, we show that the new criterion is just a sufficient and necessary condition, which is stated as a corollary .
Proof of Theorem 2 (iii) ⇒ (i). We shall prove Q Λ J (β 1 ) = 0. Thus, from Proposition 4, the conclusion follows. Given λ ∈ Λ I , there exists a positive integer m 1 and L ∈ Σ m q such that Since the sequence {β l } l≥1 is nonzero, the sequence of integers {C(Ji 1 · · · i l )} l 1 has infinitely many nonzero terms. Thus, there exist infinitely many terms l with i l = 0. Take an integer r > m with i r = 0. Write λ * := F J (K) ∈ Λ r J . According to Proposition 2 and i r = 0, we have λ = λ * and λ ∈ Λ m J ⊂ Λ r J . From β k+1 = N −1 (β k + C(Ji 1 · · · i k ))(k 1), it follows that The following three lemmas play key roles in the proof of Theorem 2 (i) ⇒ (ii). First, we show a new criterion for Λ to be a spectrum of µ.
To use Lemma 2, we need the following lemma, which implies that, under some conditions for any point in T, there exists a path that escapes from Z (μ, T).
According to the proof of Theorem 2(iii) ⇒ (i), we obtain Q Λ I (ξ) = 0, which is a contradiction to the condition inf ξ∈T Q Λ I (ξ) > 0 for any I ∈ Λ * . Next, we shall prove the existence of v. We write J := j 1 · · · j w and η := N −w (ξ + F I ( J)), where J = ϑ, F I ( J) = 0 and η = ξ when w = 0. In what follows, we define a sequence of sets {Y n } n≥0 by induction on n. Define Y 0 = {ϑ}, and We have the following claim.
Claim: For n ≥ 1, we have #Y n 2 n .
Let v 1 be the smallest positive integer such that |μ(N −v (η + F I J (L)))| > 0 for some L = l 1 · · · l v . By taking J = Jl 1 · · · l v , we finish the proof.
Proof. For any θ ∈ T ∩ Z, we have m D (θ) = 1. On the other hand, the mask function m D can be extended to an entire function on the complex plane. Thus, m D is uniformly continuous on any compact set. Hence, there exists a positive number c 1 such that It is clear that there exists a positive integer K > 0 such that, for k K, we have N −k c 1 < 1 2 . Note an elementary inequality: Then, we have for all x i ∈ B(θ i , N −i ). The proof is complete.
From Lemma 2, it follows that (µ, Λ) is a spectral pair, which is a contradiction to the hypothesis.
By a similar argument in the proof of Theorem 2(iii) ⇒ (i), we obtain Q Λ I (η * ) = 0, which implies a contradiction to (11). The claim is proven.
We have the following conclusion.
Thus, we obtain If we consider N −n−1 (ξ * + F I ( Jj n+1 )) as a "next generation" of N −n (ξ * + F I ( J)) for n ≥ 1, Proposition 2 implies that different points of X k have different "next generations". Thus, we obtain #X k+1 #X k , which implies Claim 2 is true. By noting the fact that X is a subset of the finite set Z (μ, T), there exists a positive integer h ∈ N such that #X h+m = #X h , m ≥ 1.
Define β 1 := N −h (ξ * + F I (J)) and It is clear β l ∈ X h+l−1 , which implies β l is nonzero. Thus, the sequence of nonzero integers {β l } l 1 and the increasing sequence of finite words {Jj h+1 · · · j h+l } l 1 with the prefix J fulfill the request.
As a corollary of Lemma 2 and Theorem 2, we obtain another necessary and sufficient condition for Λ to be a spectrum of µ.
Proof. The sufficiency follows from Lemma 2. We just prove the necessity here. Suppose that (µ, Λ) is a spectral pair. By Propositions 3 and 4, we obtain, for any I ∈ Σ * , By a similar argument in the proof of Theorem 2 (i) ⇒ (ii), for any ξ ∈ T and I ∈ T, there exists λ ξ,I ∈ Λ I such that | µ(N −|I| (ξ + F(I)) + λ ξ,I )| ≥ c.
We finish the proof.

(38)
According to Remark 1, it is clear that Theorem 1 cannot work. We shall show Λ(N, B) is a spectrum of µ by Theorem 2 in the following Theorem 3. Then, we show that Strichartz's criterion (Theorem 2.8 in [24]) is not appropriate by proving the following Theorem 4.
Let A n denote the set of coefficients of N n (n ≥ 0) in (38). Given two integers l and k with l > k ≥ 0, we write We also write Λ k := Λ k 0 for simplicity. For three integers m, n, and k with 0 ≤ m < n < k, we have Λ n m + N n−m Λ k n =A m + N A m+1 + · · · + N n−m−1 A n−1 + N n−m A n + · · · + N k−m−1 A k−1 =Λ k m .
(40) Theorem 3. Given nonzero integer sequence {β i } i≥1 , then, for any integer M > 0, there exists an integer i M such that for any a i ∈ A i .

Proof.
Suppose that there exists a positive integer M such that, for any i > M, we have β i+1 = 6 −1 (β i + a i ). Let T 0 be the self-similar set generated by the dual IFS { 1 6 (x + s) : s ∈ 4 j=1 B j }. According to the definition of the attractor T 0 , there exists a positive integer K such that, for any i ≥ K, β i belongs to a neighborhood of T 0 , i.e., .

Summary and Conclusions
In this paper, we introduced a tree structure with the language of symbolic space. The natural spectrum candidate of a self-similar measure associated with an IFS is a set with a special tree structure. We obtained three equivalent conclusions for Λ to be a spectrum of a self-similar measure. One of them implies that there exists an infinite orbit with an element of a nonzero integer associated with the dual IFS. An example involving a selfsimilar measure and a spectrum candidate Λ(N, S) = S 0 + NS 1 + N 2 S 2 · · · showed the tree structure expands essentially the field of spectrum candidates.
It is one of the most important problems to find all spectra of a spectral measure. We are not sure that every spectrum of a self-similar measure holds a tree structure. On the other hand, the self-similar µ N,D measure has another description, µ N,D = δ 1 N D * δ 1 N 2 D * · · · . It is obvious to ask if Theorem 2 holds for the Moran-type self-similar measure. As mentioned in the Introduction, the version of Theorem 1 in higher-dimensional space has not been obtained completely. It is the next research direction to prove Theorem 2 the for self-affine measures.