Absence of absolutely continuous diffraction spectrum for certain S-adic tilings

Quasiperiodic tilings are often considered as structure models of quasicrystals. In this context, it is important to study the nature of the diffraction measures for tilings. In this article, we investigate the diffraction measures for S-adic tilings in R^d, which are constructed from a family of geometric substitution rules. In particular, we firstly give a sufficient condition for the absolutely continuous component of the diffraction measure for an S-adic tiling to be zero. Next, we prove this sufficient condition for"almost all"binary block-substitution cases and thus prove the absence of the absolutely continuous diffraction spectrum for most of the S-adic tilings from a family of binary block substitutions.


Introduction
A tiling is a cover of R d by its countably many subsets (tiles) T with the property that T = T • (i.e., each tile is the closure of its interior). There exist tilings T that are non-periodic (meaning that T + x = T holds for x = 0 only) but still admit repetitions of patterns: for example, T may be repetitive [6,Definition 5.8], or almost periodic in a sense, such as in [7,Chapter 5] and in [16], to name a few. For this reason, such tilings are often considered as structure models of quasicrystals. The diffraction measures defined for these tilings then correspond to physical diffraction patterns. In this context, it is important to study the nature of diffraction measures for tilings. Especially, it is interesting to know when a diffraction measures is pure point (a sum of point or Dirac measures).
There are several ways to construct interesting non-periodic tilings. One of the most common approaches is via substitution (or inflation) rules. (There are "symbolic" substitution rules and "geometric" ones, the spectrum of which are related [15], but in this article we only deal with "geometric" ones.) Given a substitution rule ρ in R d , it gives rise to self-affine tilings, which are often repetitive and almost-periodic. The class of self-affine tilings is included in the class of S-adic tilings, which are tilings that are generated by a finite family of substitution rules. Concerning the spectral properties of self-affine tilings, a key conjecture is the Pisot substitution conjecture, which states that self-affine tilings obtained from substitution rules of Pisot type are pure point diffractive, that is, their diffraction measures are pure point. This is still an open problem, but there are several partial positive answers. Here, we just mention that the binary one-dimensional case, in which there are only two tiles up to translation, is solved [24]. The definition of Pisot type for substitution includes irreducibility, but for some reducible cases, in the setting where the substitution is binary block-substitution, Mañibo [17,18] and Baake-Grimm [8] proved the absence of absolutely continuous components in the diffraction pattern.
In this article, we study the diffraction spectrum for S-adic tilings in R d , which generalizes the single substitution (self-affine) case [8,17,18]. In particular, we generalize the method from [5,18] to prove that (I) an inequality for Fourier matrices is sufficient for the absence of an absolutely continuous component in the diffraction measure, for quite a general class of S-adic tilings (including, but not only, the binary case), and (II) the sufficient condition in (I) is satisfied for "almost all" binary block-substitution cases, and so, for such an Sadic tiling, the absolutely continuous part of the diffraction measure is zero. The precise statement for claim (I) is found in Theorem 3.2, in the setting specified in Setting 3.1. The special case for claim (II) is elaborated on below, and the precise statement for claim (II) is Theorem 3.30, where the setting for this result is detailed in Setting 3. 19. The key ingredients are renormalization technique developed by [3,4,8,9,17,18] and Furstenberg-Kesten and Oseledets theorems. To elaborate on the claim (II), let us consider two substitutions, ρ 1 in Figure 1(a) and ρ 2 in Figure 1(b). Such substitutions (one with prototiles with support [0, 1] d ) are called block substitutions. For arbitrary sequence i 1 , i 2 , . . . in {1, 2} N , by choosing an appropriate increasing sequence n 1 < n 2 < · · · and appropriate patches P k , k = 1, 2, . . ., we have a convergence and T is a tiling. (For details, see page 6.) Such T is called an S-adic tiling belonging to the sequence (i n ) n for ρ 1 , ρ 2 . The special case for the main result of this paper (Theorem 3.30) is as follows.
Theorem 1.1 (A special case of Theorem 3.30). Let p 1 , p 2 be two positive real numbers with p 1 + p 2 = 1. Endow {1, 2} N the product probability measure µ for the probability measure on {1, 2} defined by (p 1 , p 2 ). Then, for µ-almost all (i n ) n ∈ {1, 2} N , the S-adic tilings belonging to (i n ) n for ρ 1 , ρ 2 have zero absolutely continuous diffraction spectrum, that is, the absolutely continuous part of the diffraction measure is zero.
We can replace {1, 2} N with its subshift X, as follows: Theorem 1.2 (A special case of Theorem 3.30). Let X be a subshift of {1, 2} N which admits shift-invariant ergodic Borel probability measure µ X . Assume the shift map on X is surjective. Then, for µ X -almost all (i n ) n ∈ X, the S-adic tilings belonging to (i n ) n for ρ 1 , ρ 2 have zero absolutely continuous diffraction spectrum.
Note that this is not included in Theorem 1.1 because µ(X) might be zero. We can replace ρ 1 , ρ 2 with arbitrary finite family of binary block substitutions, with a mild assumption on substitution matrices, with possibly different expansion maps. Furthermore, the dimension can be arbitrary: for any d = 1, 2, 3 . . . and block substitutions in R d , we have similar results.
This paper is organized as follows. In Section 2, we introduce our notation and some necessary background. Section 3 contains our main results; in particular, we state and prove claims (I) and (II) given above. The first claim is proved in Section 3.1, while the second claim is proved in Section 3.2. We defer the proofs of some of our claims in Section 3.2 to an appendix.

General background
2.1. Notations. For a finite set F , we denote its cardinality by #F . In this article, µ L denotes the Lebesgue measure on R d . The symbol T refers to the one-dimensional torus {z ∈ C | |z| = 1}. For a natural number n, we will identify T n measure-theoretically with [0, 1) n , on which the Lebesgue measure µ L is the complete rotation-invariant probability measure. Let π : R n → T n be defined via π(s 1 , s 2 , . . . , s n ) = (e 2πis 1 , e 2πis 2 , . . . , e 2πisn ). In R d , for x ∈ R d and R > 0, the closed ball {y ∈ R d | x − y ≦ R} is denoted by B(x, R). If x = 0, we use the symbol B R for B(0, R).

2.2.
A generality for tilings and substitutions. In this section, we sketch a generality for the theory of tilings. For a detailed exposition, we refer to [6]. Let d be a natural number and we consider tilings in R d .
Let L be a finite set. A labelled tile is a pair T = (S, ℓ) consisting of a compact set S in R d with S • = S (the closure of the interior coincides with the original S) and an element ℓ ∈ L. The set S is called the support of T and denoted by supp T . The element ℓ is called the label of T .
Alternatively, we can consider "unlabeled" tiles, that is, a compact subset T of R d such that T • = T . For an unlabeled tile T , we denote the space it covers (that is, T itself) by supp T , called the support of T , in order to cover the theory for labeled and for unlabeled tiles by the same notation. Both labelled tiles and unlabelled tiles are called tiles. We deal with both cases simultaneously by the above notation. Note that each of the cases are required because (1) we often have to distinguish two tiles with the same support by assigning them different labels, as in Example 2.1, and because (2) we often meet situations where tiles have different support and labels are hence redundant, as in Example 2.2.
A set P of tiles in R d is called a patch if (supp T ) • ∩ (supp S) • = ∅ for each distinct S and T in P. The support of a patch P is the subset T ∈P supp T of R and is denoted by supp P. (Sometimes we take the closure after taking the union in this definition, but in this article we only deal with situations where the union is already a closed set. We use the same notation as the support of a tile, but there is no possibility of confusion.) A patch P is called a tiling if supp P = R d .
For an unlabelled tile S, S +x denotes the usual translation. For a labelled tile T = (S, ℓ) and x ∈ R d , we set T + x = (S + x, ℓ). For a patch P (with either labelled or unlabelled tiles) and x ∈ R d , we define the translate of P by x via A tiling T is said to be non-periodic if x = 0 is the only element in R d that satisfies T + x = T . In this article, we are mainly interested in non-periodic tilings.
There are several ways to construct interesting non-periodic tilings. In this article, we consider tilings constructed via substitution rules. First, for a finite set A of tiles in R d , let A * be the set of all patches of which tiles are translates of elements of A. A substitution rule (or an inflation rule) is a triple σ = (A, φ, ρ) where • A is a finite set of tiles, called the alphabet of σ, The map ρ itself is also often referred to as a substitution (or inflation) rule. Usually, the expansion map is defined as a linear map whose eigenvalues are greater than 1 in modulus, but for a technical reason, we use a stronger definition. The third condition (on the supports) means that the map ρ gives the result of first expanding the tile T by the expansion map φ and then subdividing it to obtain a patch ρ(T ). The following examples will illustrate this point. The Thue-Morse substitution is a substitution ρ TM of which alphabet is {T 1 , T 2 }, expansion map is R ∋ x → 2x ∈ R and the rule is given by The final condition in the definition of a substitution rule is indeed satisfied for this rule, since supp ρ TM (T i ) = [0, 2] and 2 supp T i = [0, 2] for i = 1, 2. 2 , the golden ratio. Let Again, with an expansion map R ∋ x → τ x ∈ R, the final condition in the definition of a substitution rule is satisfied since τ 2 = τ + 1. For a substitution rule ρ, one can define a displacement matrix and Fourier matrix, which will play important roles in the study of diffraction, as follows. Let A = {T 1 , T 2 , . . . , T na } be the alphabet for the substitution ρ in R d . For each i and j, there is a digit set T i,j ⊂ R d for ρ, which is determined by The substitution matrix of ρ is the matrix whose (i, j)-element is #T i,j . We then define the Fourier matrix, B, which is a n a × n a matrix function on R d . We need to specify its value B(t) for each t ∈ R d . We define the (i, j) component of B(t), denoted by B i,j (t), as where ·, · is the standard inner product in R d . Let us consider an explicit example to illustrate these definitions.
Given a geometric substitution rule ρ in R d , one can construct a tiling in R d by iterating the map ρ. To be more precise, for a given geometric substitution rule ρ, we can define a map ρ : A * → A * (denoted by the same symbol), as follows. First, for T ∈ A and x ∈ R d , set ρ(T + x) = ρ(T ) + φ(x) (φ being the expansion map). Then we define ρ(P), where P is a patch consisting of translates of elements of A (that is, an element of A * ), via Since now the domain and the range of the new map ρ are the same, we can iterate it. We can often take the limit lim n→∞ ρ kn (P) (2) to obtain a tiling, for a suitable k > 0 and an initial patch P. The convergence in equation (2) is with respect to the local matching topology, in which two patches P and Q are "close" if there are small displacements x, y ∈ R d such that P + x and Q + y agree inside B R for some large R > 0. (See, for example, [6, p.129].) Example 2.5. For the Thue-Morse substitution ρ TM , define P via Then, ρ 2 TM (P) is (if we write it symbolically) 1001.0110, where . denotes the place of origin. We obtain ρ 2 TM (P) ⊃ P, and this in turn means that ρ 2 TM (P) ⊂ ρ 4 TM (P) ⊂ ρ 6 TM (P) · · · . The patches obtained by iteration "grow" in R, and in the limit they form a tiling lim n→∞ ρ 2n TM (P) = n>0 ρ 2n TM (P), which is called a Thue-Morse tiling.
An S-adic tiling is a tiling obtained by replacing each of kn ρ's in ρ kn (P) in equation (2) with a substitution rule from a finite set of geometric substitution rules. To be precise, consider a finite set {ρ 1 , ρ 2 , . . . , ρ ma } of geometric substitution rules in R d that share the same alphabet A but do not necessarily share the same expansion map. We call sequences i 1 , i 2 , . . . of elements of {1, 2, . . . , m a } directive sequences. Given a directive sequence i 1 , i 2 , . . ., any tiling T of the form where n 1 < n 2 < · · · and where the P l are patches that are included in some ρ j 1 • ρ j 2 • · · · • ρ jm (P )(m > 0, j 1 , j 2 , . . . , j m ∈ {1, 2, . . . , m a } and P ∈ A), is called an S-adic tiling belonging to the directive sequence i 1 , i 2 , · · · for the family {ρ 1 , ρ 2 , . . . , ρ ma }. This is a geometric version of the symbolic S-adic sequences (see for example [11]) and the order of ρ i j in (3) comes from the symbolic counterpart. The convergence in (3) is assured by the following finiteness condition. In general, the family {ρ 1 , ρ 2 , . . . , ρ ma } is said to have finite local complexity (FLC) if for each compact K ⊂ R d the set is finite up to translation, where the symbol ⊓ is defined via for a patch P in R d and an S ⊂ R d . If the set of substitutions {ρ 1 , ρ 2 , . . . , ρ ma } have FLC, given an arbitrary directive sequence i 1 , i 2 , . . ., we can find some n 1 < n 2 < · · · and some patches P l such that the limit in (3) converges, because the patches after the lim symbol in (3) are included in a compact set. This is seen by the fact that a space X of patches in . We can start with a sequence (Q n ) n of patches and the sequence admits a convergent subsequence.
In the discussion of symbolic S-adic sequences, we can consider cases where the symbolic substitution rules do not share the same alphabet, but in this article we only deal with the case where substitutions are geometric and share a common alphabet. This is a strong assumption but all the block substitutions, which we mainly deal with in this paper, are included in our scope. Often, given a directive sequence i 1 , i 2 , . . ., we use the notation for two positive integers k < l.
Given an S-adic tiling of the form (3), the sequence ρ i[2,n l ) (P l ) l>0 admits a convergent subsequence, again by a diagonalization argument as above. We can take a subsequence n (2) l , P (2) l l of the sequence (n l , P l ) so that that the limit lim l ρ [2,n (2) l ) (P (2) l ) converges. We can further take a subsequence n for each k > 0 with common (m l ) l and (Q l ) l . This implies that, for each k, we have ρ k (T (k+1) ) = T (k) . These "de-substituted tilings" T (2) , T (3) , . . . of the given T = T (1) will be useful later; such an inverse-limit structure enables us to construct renormalization scheme, by which we can use ergodic theory to study the diffraction spectrum for T (1) .

Patch frequencies.
In order to discuss the diffraction of tilings, we use the concept of the frequency of patches. In general, if T is a tiling in R d , if P is a (usually finite) non-empty patch and if the limit converges, this limit is called the frequency of P in T and denoted by freq T P or freq P.
(Here, we consider averaging with respect to B R | R > 0 , but we can also consider averaging along van Hove sequences.) If T is an S-adic tiling, often the following uniform patch frequency holds.
Theorem 2.6. Let ρ 1 , ρ 2 , . . . , ρ ma be (geometric) substitution rules in R d that share a common alphabet. Let A i be the substitution matrix for ρ i . Take a directive sequence i 1 , i 2 , . . . ∈ {1, 2, . . . , m a } and an S-adic tiling T belonging to this directive sequence. Assume the following four conditions: (1) there are n 0 > 0 and i 0,1 , i 0,2 , . . . , i 0,n 0 ∈ {1, 2, . . . , m a } such that all entries in the product matrix are greater than 0; (2) for any n > 0 there is k > n such that Then, for any finite non-empty patch P, there is c P ∈ R such that Sketch of proof. This is the "geometric" version of the argument in [11, Section 5.2] and the proof is similar.
Note that the uniform convergence on the orbit {T + t | t ∈ R d } implies the uniform convergence on the continuous hull, the closure of the orbit with respect to the local matching topology. Note also that for the single substitution case (m a = 1), the above conditions (1)-(3) for the convergence of patch frequency are satisfied if the substitution matrix is primitive.
2.4. Fourier transform, diffraction and the Lebesgue decomposition. The diffraction measures associated with tilings are physically important. They model the results of diffraction experiments. Mathematically, the diffraction measure of a tiling is the Fourier transform of the autocorrelation measure associated with the tiling, described as follows.
In what follows, we have to deal with objects such as t∈D c t δ t , where D ⊂ R d , c t ∈ C and δ t is the Dirac (point) measure at t. We consider them as complex measures in the sense of [12] and call them Radon measures. For a Radon measure µ on R d and a function ϕ ∈ L 1 (µ), we use a notation Let C c (R d ) denote the vector space of all complex-valued, continuous, compactly supported functions on R d . According to [2], a Radon measure µ on R d is said to be Fourier transformable if there is another Radon measure ν on R d such that, for each ϕ, ψ ∈ C c (R d ), the inverse Fourier transformφ * ψ of the convolution of ϕ and ψ is in L 1 (ν) and If such a ν exists, it is unique, called the Fourier transform of µ and denoted byμ. It is known that if µ is positive definite, that is, if for each ϕ ∈ C c (R d ) we have µ, ϕ * φ ≧ 0, then µ is Fourier transformable and the Fourier transformμ is positive [7,Theorem 4.11.5].
Given a Radon measure µ, we define its diffraction measure as follows. First, assume the following limit, the autocorrelation measure, exists: where, for a Radon measure µ and a subset S ⊂ R, the restriction µ| S is a Radon measure (5) is nothing but a Radon measure that sends ϕ to By construction, this limit µ ⊛μ is positive definite, and so its Fourier transform exists and is positive. We call this Fourier transform the diffraction measure for µ [6, Definition 9.2].
In general, given a finite set A = {T 1 , T 2 , . . . , T na } of tiles and a tiling T in R d whose tiles are translates of elements of A, we set D i via We then take complex numbers w 1 , w 2 , . . . , w na , and consider a Radon measure The diffraction measure for µ T is called the diffraction measure for T . It is easy to prove that the autocorrelation measure is The Lebesgue decomposition [13, §5] of a Radon measure is fundamental in the theory of diffraction. In general, a Radon measure µ on R d is pure point if its total variation |µ| is pure point, that is, a sum of Dirac measures. If |µ|({x}) = 0 for each x ∈ R d , then µ is said to be continuous. Any Radon measure µ is uniquely decomposed into its pure point part µ pp and continuous part µ c . The continuous part is further decomposed into the singular continuous component µ sc , which is mutually singular with the Lebesgue measure on R d , and absolutely continuous component µ ac , which is absolutely continuous with respect to the Lebesgue measure. Thus we have a decomposition This decomposition of µ into its pure point, its continuous and singular, and its continuous and absolutely continuous part is unique. That µ ac is absolutely continuous means that there is a locally integrable function f ∈ L 1 loc (R d ) such that where the right-hand side is the integral with respect to the Lebesgue measure µ L . This f is called the Radon-Nikodym derivative of the Radon measure µ.

3.1.
A relation between the asymptotic behavior of Fourier matrices and absolutely continuous spectrum for S-adic tilings. In this section, we prove a sufficient condition for the absence of the absolutely continuous part of the diffraction measure for S-adic tilings. The following setting is assumed for the whole section.
Setting 3.1. In this section, we take finite set {ρ 1 , ρ 2 , . . . , ρ ma } of substitution rules in R d that share the same arbitrary (not We assume the family {ρ 1 , ρ 2 , . . . , ρ ma } has FLC (page 7). (Note that each substitution here is a "geometric" one and not a "symbolic" one.) The existence of such common tiles and alphabet is the assumption which we start with. Let φ i be the expansion map for the substitution ρ i . (For different i and j, the maps φ i and φ j may be different.) The Fourier matrix for k,j (t)) k,j . We consider a directive sequence (i j ) j=1,2,··· in {1, 2, . . . , m a } N and let T (1) be an S-adic tiling that belongs to (i j ) j . As we have seen on page 8, we have an increasing sequence n 1 < n 2 < · · · of natural numbers and patches P l consisting of translates of alphabets such that converges for each k = 1, 2, · · · . Note that we do not assume recognizability here, but we do assume that, for each T (k) , the patch frequencies converge.
Since each ρ i , regarded as a map that sends a patch P to another patch ρ i (P), is continuous with respect to the local matching topology, we see that for each k = 1, 2, . . .. This can be used to "compare" the autocorrelation measure for T (k) and one for T (k+1) .
The fundamental idea to study the diffraction spectrum is to use renormalization equations [3,4,8,9,17,18]. The above "de-substitution" or inverse-limit structure gives us a renormalization scheme, which in turn gives us a sufficient condition for zero absolutely continuous spectrum in terms of an asymptotic of norms of Fourier matrices (Theorem 3.2). Such an asymptotic behavior can be checked by ergodic theory as in Section 3.2. The special case of Theorem 3.2 for the substitution case (the case where m a = 1) was proved by Mañibo [17,18]. Below we adapt Mañibo's idea to the general S-adic case.
The goal of this section is to prove the following theorem.
for Lebesgue-a.e. t ∈ R, where * denotes the adjoint, then the diffraction spectrum of T (1) has zero absolutely continuous part.
There is a similar result for the one-dimensional case for the corresponding dynamical spectrum in a recent paper by Bufetov and Solomyak [14,Corollary 4.5]. They proved a sufficient condition for the absence of absolutely continuous dynamical spectrum for suspension flows for S-adic sequences. This covers some cases which Theorem 3.2 does not cover, but Theorem 3.2 deals with some cases which Bufetov and Solomyak did not. The sufficient condition in [14] is similar to Theorem 3.2, but they replace 1 2k log λ i 1 λ i 2 · · · λ i k with 1 2k log A i 1 A i 2 · · · A i k (A i is the substitution matrix for ρ i ) and assume that the limit of these as k → ∞ is convergent. Moreover, they assume the recognizability for the directive sequence i 1 , i 2 , . . .. Therefore, Theorem 3.2 covers some cases that Bufetov and Solomyak did not cover, since the dimension d of the tiling is arbitrary and the directive sequence is arbitrary in this theorem. On the other hand, Bufetov and Solomyak deal with arbitrary suspension flows for S-adic sequences, whereas in this paper, for the one-dimensional cases, we only deal with tile lengths that come from Perron-Frobenius eigenvector.
For the rest of this section we will prove Theorem 3.2. The readers may skip the proof for the first reading and move to an application of this theorem in Section 3.2.
where the limit is convergent since the patch frequencies converge. For each k = 1, 2, · · · , each i, j = 1, 2, . . . , n a and each z ∈ R d , we set Lemma 3.5. For each k, we have the following equation: Proof. The proof is essentially the same as the substitution case (that is, the case where m a = 1) in [18], but since the proof in [18] appears to use recognizability, we here give an outline of the proof without using recognizability. Take i, j, k and z ∈ D j , since T i + t ∈ T (k) and T j + t + z ∈ T (k) , there are m, n ∈ {1, 2, . . . , n a }, s m ∈ D (k+1) m and s n ∈ D (k+1) n such that T i + t ∈ ρ i k (T m + s m ) and T j + t + z ∈ ρ i k (T n + s n ). (These m, n, s m , s n are unique.) By computation, it follows that i,m and y ∈ T (i k ) j,n , and so we obtain a map that sends to where means taking the union while regarding the sets as disjoint. We can show that this map is a bijection.
We then see that m,n≦na x∈T where C ′ is the maximal norm of vectors in all digits for ρ 1 , ρ 2 , . . . , ρ ma and C is the maximal diameter for the elements of the alphabet. By dividing these by µ L (B R ) and taking the limit as R → ∞, we obtain the desired equation.
Definition 3.6. For each k = 1, 2, · · · , i, j = 1, 2, . . . , n a , we set By FLC, for each compact set K the set K ∩ (D i ) is a finite set, and so the infinite sum z∈D (k) j −D (k) i (with respect to the vague topology) is well-defined.
Note that, by equation (6), it suffices to investigate these Υ (k) i,j in order to understand the diffraction measure for T (k) . The Fourier transforms of the Υ (k) i,j will generate the diffraction measures for T (k) . In order to investigate the nature of these diffraction measures, it is useful to use the relation between Υ (k) i,j (i, j = 1, 2, . . . , n a ) and Υ (k+1) m,n (m, n = 1, 2, . . . , n a ) stated in Proposition 3.8 below. In order to give a statement, we first introduce two symbols.
Definition 3.7. If a Radon measure µ on R d and a homeomorphism g : R d → R d are given, define another Radon measure g.µ via Let µ be a Radon measure on R d and let ξ be a continuous function on R d . The new Radon measure ξµ is defined via In what follows, we use convolutions of Radon measures [7, Definition 4.9.18].
Proof. This can be proved using Lemma 3.5 by a direct computation.
This renormalization equation gives rise to an equation between Fourier transforms, as in Proposition 3.9. In what follows, for each t ∈ R d , the symbol exp t denotes the exponential function defined via exp t (s) = e 2πi s,t for each s ∈ R d , where ·, · is the standard inner product.
i,j is Fourier transformable and we have where * denotes the adjoint map.
Proof. To prove that each Υ (k) i,j is Fourier transformable, fix k and set ω i = x∈D (k) i δ x for each i = 1, 2, . . . , n a . By computation we have which is the sum of four positive definite Radon measures by the polarization identity. Since each positive definite Radon measure is Fourier transformable, the sum Υ (k) i,j is also Fourier transformable. The formula (13) is a direct consequence of Proposition 3.8.
We can decompose each Υ The Radon-Nikodym derivative of the absolutely continuous part (Υ . By the uniqueness of the Lebesgue decomposition and by Proposition 3.9, we have the following result.
We will investigate when h (1) i,j are zero and so the absolutely continuous spectrum for T (1) is zero. It is convenient to define the following Radon-Nikodym matrix.
Definition 3.11. For each k, define the Radon-Nikodym matrix H (k) for the tiling T (k) , which is a matrix-valued function on R d , via i,j (t)) i,j . Then, Proposition 3.10 now translates into the following result.
In Section 3.2, we will prove that these H (k) (t) are in fact zero for certain choices of {ρ 1 , ρ 2 , . . . , ρ ma }. For this purpose, the following lemma is useful, because a positive definite matrix is zero if its trace is zero. Proof. By the definition of a positive definite matrix, we have to prove that, whenever we take w 1 , w 2 , . . . , w na ∈ C, we have For w j which are arbitrarily taken from C, set By (6), the autocorrelation measure for γ is positive definite and coincides with Since the Fourier transform of this Radon charge is positive, so is its Radon-Nikodym derivative, which is by the uniqueness of the Lebesgue decomposition.
Definition 3.14. For each k ≧ 1, set and set λ (0) = 1. For each k ≧ 1 and t ∈ R d , define By iterating the equation (15), we have: for Lebesgue-a.e. t ∈ R. Hence we have the following inequality for the traces: By Proposition 3.15, we have the following strategy to prove that Tr(H (1) (t)) = 0 and hence H (1) (t) = 0 (H (1) (t) is a positive definite matrix) and the absence of an absolutely continuous component of the diffraction measure for T (1) .
Lemma 3.16. There are A > 0, k 0 > 0, R 0 > 0 such that, for any natural number k ≧ k 0 and any real number R > R 0 , we have Proof. By [7,Corollary 4.9.12], there is an If k is large enough, z = 0 is the only element z in D such that φ (k−1) (z) ∈ supp f * f . For any k > 0 which is large enough in this sense, i = 1, 2, . . . , n a and t ∈ R d , we have i . This is finite since A is fixed. For an arbitrary choice of R > 0, if n is a natural number with n ≦ R < n + 1, we have: By taking A > 2 d n a f * f (0)B, we have the conclusion.
We now prove a sufficient condition for H (1) to be zero. As we stated before, the condition is the exponential convergence of B (k) (t) 2 λ (k) → 0 as k → ∞. We will prove that this sufficient condition is actually satisfied for a class of S-adic tilings, in Section 3.2, and thus prove the absence of absolutely continuous part of the diffraction spectrum for this class.
Proof. Take an arbitrary positive real number R. For each k ≧ 1, set The sets E 1 , E 2 , . . . are increasing and k≧1 E k has full measure (2R) d .
For each k ≧ 1 and t ∈ E k , we have and Here, if R is large enough, the first term tends to zero as k → ∞, since there exists A > 0 as in Lemma 3.16 and since The second term also tends to zero because k≧1 E k has full measure (2R) d and We see that which, by Lemma 3.13, implies that Tr(H (1) (t)) = 0 a.e., which in turn implies that H (1) (t) = 0 a.e., since all the eigenvalues are zero and H (1) (t) is diagonalizable.
The proof for Theorem 3.2. The absence of the absolutely continuous part of the diffraction spectrum for T (1) follows from equation (6).
Remark 3.18. If the patch frequency converges uniformly, the corresponding dynamical system for T (1) is uniquely ergodic, and so by [10,Theorem 5], the absence of absolutely continuous diffraction spectrum holds for any tiling in the continuous hull of T (1) .

3.2.
The absence of absolutely continuous spectrum for binary blocksubstitution case. In this section, we deal with S-adic tilings with binary block substitution rules. The following setting is assumed for the whole section. We take a finite family of substitutions ρ 1 , ρ 2 , . . . , ρ ma with alphabet A. We assume these are block substitutions. In other words, the expansion map φ i for ρ i is defined by a diagonal matrix with integer diagonal entries. B (i) (t) is the Fourier matrix for ρ i .
We assume that, for each i, there is a t ∈ R d such that the Fourier matrix B (i) (t) is not singular. This means that B (i) (t) is not singular for almost every t ∈ R d , since the determinant is a sum of exponential functions.
Next, we take a measure-preserving system (X, B, µ, S 0 ), where (X, B, µ) is a standard probability space and S 0 : X → X is a measure-preserving transformation. We assume that the map S 0 is ergodic and surjective.
We choose a decomposition X = ma j=1 E j of X into pairwise disjoint subsets E j ∈ B, j = 1, 2, . . . , m a .
The goal of this section is to prove Theorem 3.30. In Theorem 3.30, we will show the absence of the absolutely continuous component in the diffraction spectrum for Sadic tilings constructed from ρ 1 , ρ 2 , . . . , ρ ma obtained from "almost all" directive sequences given by the coding of the system (X, S 0 ), by using Theorem 3.2. Specifically, we prove that there is some E ⊂ X with measure 1 such that, for each x ∈ E, the directive sequence i 1 , i 2 , . . . defined by for each n, gives rise to an S-adic tiling with zero absolutely continuous diffraction spectrum. (We prove that any S-adic tiling belonging to such a directive sequence has this property.) For example, if X ⊂ {1, 2, . . . , m a } N is a subshift with surjective shift-map and E j = {x ∈ X | x 1 = j}, then the directive sequence defined by x ∈ X is x itself. The theorem says that for almost all x ∈ X, the S-adic tilings belonging to x have zero absolutely continuous spectrum. Special cases for the theorem are stated in Introduction. The readers can skip the detail of this section and jump to Theorem 3.30 and Example 3.31 for the first reading.
Note that since each ρ i is a block substitution, the digit T (i) k,l for ρ i is inside Z d , and moreover for each k and i, We denote this set by F i . Theorem 3.2 applies to the Fourier matrices B (i) (t), but we will replace B (i) (t) with the following matrices C (i) (z), where z ∈ T d : the latter has the advantage of a compact domain.
Definition 3.20. For each i = 1, 2, . . . , m a and t ∈ R d , set (π : R d → T d is the quotient map.) Since the elements in the digits are all in Z d , this is well-defined.
For a technical reason, we replace the system (X, B, µ, S 0 ) with its natural extension. Let (Y, C, ν, S 1 ) be a natural extension of (X, B, µ, S 0 ). In other words, (Y, C, ν) is a separable and complete probability space and S 1 : Y → Y is an invertible measure-preserving transformation, and there exists a factor map f : Y → X. Such a system and a factor map exist since S 0 is surjective, and S 1 is ergodic [20, Section 1.6.3].
The main tool to prove Theorem 3.30 is Furstenberg-Kesten theorem and Oseledets ergodic theorem. For these theorems, see for example [25]. We will use these theorems to the following cocycle and obtain an estimate (8). We define a map C : (I is the identity matrix, but this does not matter since C (i) (z) is almost surely invertible.) We set R : for t ∈ R d and x ∈ f −1 (E i ), for each i = 1, 2, . . . , m a .
The following lemma will be useful when we use ergodic theorems later. Here, µ L is the Lebesgue measure on T d , which is measure theoretically identified with [0, 1) d . It is in the proof of this lemma where we use the fact that S 1 is invertible and ergodic.
Lemma 3.21. The transformation R is µ L × ν-preserving and ergodic.
Proof. This will be proved in the appendix.
We apply the Furstenberg-Kesten theorem and the multiplicative ergodic theorem by Oseledets to T d × Y , R and C. In order to invoke these theorem, we first have to confirm the following lemma.
Lemma 3.22. The two maps that send (z, x) ∈ T d × Y to log + C(z, x) and to log + C(z, x) −1 , respectively, are both integrable with respect to µ L × ν (the product measure of µ L and ν).
Proof. It suffices to prove that the maps that send z ∈ T d to log C (i) (z) ±1 are integrable for each i.
We take i and fix it. Since C (i) (z) is bounded for z ∈ T d , we see that log + C (i) (z) is integrable. To prove the integrability of log where ad denotes the adjugate matrix. It suffices to show that log | det C (i) (z)| is integrable, and this is proved by [23,p.223,Lemma 2].
In what follows, we set C n (ω) = C(R n−1 (ω)) C(R n−2 (ω)) · · · C(ω), for each ω ∈ T d × Y and n ∈ Z >0 . (It is customary to denote this by C n , but to clearly distinguish this from C (i) , we prefer to use this notation.) Proposition 3.23. There are χ + , χ − ∈ R such that for almost all ω ∈ T d × Y . We also have converges for almost all ω and the limit is either χ + or χ − .
Proof. This is a direct consequence of the Furstenberg-Kesten theorem [25,Theorem 3.12] and the Oseledets theorem [25, Theorem 4.1 and Section 4.3.3], if we modify the latter to the case of flags inside C 2 , not R 2 .
We then analyze χ + and χ − to obtain the estimate (8). These are related to logarithmic Mahler measures for certain polynomials, as follows. (The logarithmic Mahler measure m(p) for a complex-coefficient, d-variable polynomial p is defined via where the integral is with respect to the (normalized) Lebesgue measure. For Mahler measures, see for example [23, p.224].) First, we define polynomials associated with each substitution ρ 1 , ρ 2 , . . . , ρ ma . Definition 3.24. For each i = 1, 2, . . . , m a and k, l = 1, 2, we set In other words, S Define polynomials q (i) k,l (for i = 1, 2, . . . , m a and k, l = 1, 2) as follows. First set . Then, we obtain the following results, Lemma 3.27, Lemma 3.28 and Lemma 3.29, which are modifications of results from [17,18].
Moreover, for each i and z, the vector (1, −1) ⊤ is an eigenvector of C (i) (z) with eigenvalue q Note that the assumption (Setting 3.19) of non-singularity of the B (i) (t), and hence of the C (i) (z), implies that, for all i, the polynomials q To analyze χ + and χ − , we first prove the following.
Lemma 3.28. We have χ + = 0 and Proof. By equation (22), Lemma 3.21 and Birkhoff's ergodic theorem, for almost all ω ∈ T d × Y , we have Note that we used the fact that the logarithmic Mahler measure for f ∈F i z f vanishes by Jensen's formula [1, p.207].
On the other hand, by Lemma 3.27, setting again by Birkhoff's ergodic theorem. By Proposition 3.23, the last value is either χ + or χ − . Since χ + ≧ χ − and m(q Proof. By Jensen's inequality and Hölder's inequality, for each i, 2,1 is not a monomial, its absolute value is not a constant function and so the Jensen's inequality is strict. If the polynomial q (i) 1,2 − q (i) 2,1 is a monomial we have also a strict inequality.
We now prove the main result of this section. We consider S-adic tilings belonging to a directive sequence i 1 , i 2 , . . . that is obtained by a coding of S 0 starting at some x ∈ X. In other words, i 1 , i 2 , . . . is the sequence of {1, 2, . . . , m a } that satisfies S n−1 0 (x) ∈ E in for each n = 1, 2, . . .. Theorem 3.30. Suppose that the substitution matrix A i for each ρ i has only strictly positive entries, or more generally, suppose that (1) there are i 0,1 , i 0,2 , . . . , i 0,n 0 ∈ {1, 2, . . . , m a } such that the substitution matrix for ρ i 0,1 • ρ i 0,2 · · · • ρ i 0,n 0 has only strictly positive entries, and that Then, there is a set E ∈ B of full measure such that, if we take x ∈ E and define a directive sequence i 1 , i 2 · · · as the coding of S 0 starting at x with respect to the decomposition X = E i , and if T is an S-adic tiling which belongs to this directive sequence for the family {ρ 1 , ρ 2 , . . . , ρ ma }, then T has zero absolutely continuous diffraction spectrum.
(These are equal as words.) The patch frequencies converge by Theorem 2.6.
The statement, the absence of absolutely continuous spectrum, is proved by using Theorem 3.2.
Example 3.31. Let ρ 1 be the period-doubling substitution and ρ 2 be the Thue-Morse substitution. Let X = {1, 2} N and (X, B, µ, S 0 ) be the Bernoulli shift for some probability vector (p 1 , p 2 ) with p 1 p 2 = 0. Then, for almost all x ∈ X, the S-adic tilings belonging to the sequence x 1 , x 2 , . . . have zero absolutely continuous spectrum. We can replace (X, B, µ, S 0 ) with an arbitrary surjective ergodic measure-preserving system on a standard probability space and get the same conclusion. For example, we take some Sturmian sequences as a directive sequence and obtain the same conclusion, by replacing (X, B, µ, S 0 ) with an irrational rotation on T. The same conclusion holds when we replace ρ 1 and ρ 2 with arbitrary two binary substitution rules such that the supports of the prototiles are all [0, 1] and the substitution matrices have only strictly positive entries. The expansion factors can be different.
We can also apply Theorem 3.30 for higher-dimensional block substitutions. We can replace the above ρ 1 and ρ 2 with arbitrary two block substitutions in R d with substitution matrices with only strictly positive entries. For example, for d = 2, if ρ 1 , ρ 2 are ones that were depicted in Figure 1(a) and 1(b), we have the same conclusion as in the first paragraph in this example.
Moreover, the number of substitutions does not have to be 2. Let ρ 1 , ρ 2 , . . . , ρ ma be a finite family of block substitutions in R d . Take a space {1, 2, . . . , m a } N and endow it the product probability measure for the probability measure on {1, 2, . . . , m a } given by a probability vector (p 1 , p 2 , . . . , p ma ) with p 1 p 2 · · · p ma = 0. We can also relax the assumption on the substitution matrices and the substitution matrices may have 0 as entries, but we assume there are i 0,1 , i 0,2 , . . . , i 0,n 0 ∈ {1, 2, . . . , m a } such that the product A i 0,1 A i 0,2 · · · A i 0,n 0 has only strictly positive entries. Then, for almost all x ∈ {1, 2, . . . , m a } N , the S-adic tilings constructed by ρ 1 , ρ 2 , . . . , ρ ma belonging to the directive sequence x has zero absolutely continuous diffraction spectrum. We can also take other subshifts of {1, 2, . . . , m a } N and obtain the same conclusion. In this case, we assume the existence of i 0,1 i 0,2 · · · i 0,n 0 in the language with the same condition. For example, let X be the hull of a m a -letter primitive symbolic substitution. Endow X with a shift-invariant probability measure defined by frequencies. Then, with respect to that measure, for almost all x ∈ X, the S-adic tilings belonging to x has zero absolutely continuous diffraction spectrum.
Remark 3.32. In Theorem 3.30, the author could not prove the absence of absolutely continuous part for an arbitrary directive sequence, since this theorem is an "almost sure" result. For example, when m a = 2, we do not know what happens if we take the Fibonacci sequence as a directive sequence. It would be interesting to decide whether this theorem holds for arbitrary directive sequences, or whether for some directive sequences there may be non-vanishing absolutely continuous components in the diffraction spectrum. Lemma 3.35. If I 0 ⊂ [0, 1) d is a product of intervals, then we have lim n=(n 1 ,n 2 ,...,n d )→∞ 1 n 1 n 2 · · · n d #{I ∈ I n | I ⊂ I 0 } = µ L (I 0 ). By using Lemma 3.35, we prove the following lemma. Proof. It suffices to prove the claim for the case where E and F are products of intervals, since any general E and F are approximated by finite disjoint unions of products of intervals. Assume that E and F are products of intervals. For each n ∈ Z d >0 and I ∈ I n , we have either (1) I ⊂ F , (2) I ∩ F = ∅ and I ⊂ F , or (3) I ∩ F = ∅. In case (1), we have For each n, the number of I in I n with case (2) is at most 2 d j=1 n 1 n 2 · · · n j−1 n j+1 · · · n d .
For each such I, the measure µ L ( T n | −1 I (E) ∩ F ) is at most We also define a map T i : T d → T d for i = (i 1 , i 2 , . . . , i n ), via for each t ∈ R d .
Lemma 3.38. Let n ∈ Z >0 and i ∈ I(n). For a Borel subset F ⊂ T d and E ∈ C with E ⊂ E ′ i , we have R −n (F × E) = (T i × S n ) −1 (F × E).
This proves that R −n (F × E) ⊂ (T i × S n ) −1 (F × E). The reverse inclusion is proved in a similar way.
Lemma 3.39. The transformation R is measure preserving with respect to the product measure µ L × ν.
Proof. For E ∈ C and a Borel F ⊂ T d , by Lemma 3.38 we have Lemma 3.40. R is ergodic with respect to µ L × ν.
Proof. For Borel sets F 1 , F 2 ⊂ T d , G 1 , G 2 ∈ C and n ∈ Z >0 , we have where we used Lemma 3.38 for the second equality. By Lemma 3.36, for arbitrary ε > 0, there is n 0 > 0 such that, if n ≧ n 0 and i ∈ I(n), the number We therefore have that Since the E ′ i (with i ∈ I(n)) give a partition of X, we firstly see that |ε n | < ε, and secondly conclude that For each N > 0, we have By ergodicity of S, as N → ∞ this converges to up to an error term of absolute value less than ε. Since ε was arbitrary, we see that Since F 1 , F 2 , G 1 and G 2 are arbitrary, this proves the ergodicity of R.