Determining pure discrete spectrum for some self-affine tilings

By the algorithm implemented in the paper [2] by Akiyama-Lee and some of its predecessors, we have examined the pure discreteness of the spectrum for all irreducible Pisot substitutions of trace less than or equal to $2$, and some cases of planar tilings generated by boundary substitutions due to the paper [17] by Kenyon.


Introduction
Self-affine tilings are often studied for the research of tiling dynamics. Especially to pure discrete spectrum of the tiling dynamics, many equivalent properties are known. Akiyama and Lee [2] have given a computable algorithm to check one of these properties. Here we use this algorithm to check the pure discrete spectrum of the tiling dynamics for special cases of self-affine tilings. One of these cases is on 1-dimension irreducible Pisot substitution tilings. There is a long-standing conjecture [4] that these tilings have pure discrete spectrum. For the cases of two tiles, it is known that the conjecture is true. But not much is known for the cases of more than three tiles. Already the case of three tiles is computationally large. So we first compute the case of T r(M ) = 1, i.e., the trace of the corresponding incidence matrix M is 1 and the case of T r(M ) = 2 with the unit property of constant term of the characteristic polynomial. The case of T r(M ) = 2 with non-unit substitutions seem to be much harder in computation and we could not finish the computation by our algorithm.
The other case of tilings we consider is the self-affine tilings that are constructed from the endomorphisms of free groups by Kenyon [12]. We have looked at the quadratic polynomials whose coefficients are all less than 3 except for two examples whose computation is beyond our computer capability. All other examples turn out to be pure discrete.
We also give some examples of the self-affine tilings which arise from 4 interval exchanges studied by Arnoux-Ito-Furukado. Interestingly they are not pure discrete. We provide the computational algorithm in [3].

Pisot substitutions with small trace
In this section, we wish to computationally confirm the Pisot substitution conjecture to be true for some special case of irreducible Pisot substitutions with three tiles.
Consider a monoid A * over finite alphabets A equipped with concatenation and write the identity as λ, the empty word. A symbolic substitution σ is a non-erasing homomorphism of A * , defined by σ(a) ∈ A + = A * \ {λ} for a ∈ A. The set A Z of two sided sequences is compact by the product topology of the discrete topology on A. The substitution σ acts naturally to A Z by σ(. . . a −1 a 0 a 1 a 2 . . . ) = . . . σ(a −1 )σ(a 0 )σ(a 1 )σ(a 2 ) . . . . Let M σ be the incidence matrix (|σ(j)| i ) ij where i, j ∈ A. Here |w| j is the cardinality of j appearing in a word w ∈ A * . Denote by χ σ the characteristic polynomial of M σ . The substitution σ is primitive if M σ is primitive and it is irreducible if χ σ is irreducible 1 . A Pisot number is an algebraic integer λ > 1 whose all the other algebraic conjugates of λ lie strictly inside the unit circle. If the Perron-Frobenius root of M σ is a Pisot number then we say that σ is a 1 We always assume the irreducibility of Mσ in the sense of Perron-Frobenius theory. So the irreducibility in this article is for χσ. Pisot substitution. A word w ∈ A * is admissible if there exist k ∈ N and a ∈ A such that w is a subword of σ k (a). Let X σ = {(a n ) n∈Z ∈ A Z | a k a k+1 . . . a is admissible for all k, with k < }.
Then (X σ , s) forms a topological dynamical system where s is the shift map defined by s((a n ) n∈Z ) = (a n+1 ) n∈Z . By the definition, X σ is the orbit closure of an arbitrary chosen σ-periodic point, that is, ω ∈ A Z such that σ n (ω) = ω for some integer n > 0. We say that ω ∈ A Z is a fixed point if it is σ-periodic with n = 1, i.e., σ(ω) = ω. Further by the primitivity, the system is minimal and uniquely ergodic with the unique invariant measure µ. Therefore we can discuss the spectrum of the unitary operator U σ acting on L 2 (X σ , µ) for which (U σ (f ))(x) = f (s(x)). The substitution σ has pure discrete dynamical spectrum if the spectral measure associated to U σ consists only of point spectra, or equivalently, the linear span of eigenfunctions is dense in L 2 (X σ , µ). It is conjectured [4] that this Z-action by U σ is pure discrete if σ is an irreducible Pisot substitution -so called Pisot substitution conjecture. For the primitive substitution σ, we can also discuss a natural suspension of (X σ , s) by associating to each letter the length determined by the associated entries of the left eigenvector of M σ . Then it gives a repetitive tiling on R with an inflation matrix Q = (β) where β is the Perron-Frobenius root of M σ . This gives a tiling dynamical system (X T , R) which is also minimal and uniquely ergodic. It is known [6] that if σ is an irreducible Pisot substitution, this R-action on (X T , R) is pure discrete if and only if Z-action on (X σ , s) is pure discrete.
The following assertion may be known but we did not find it in a literature. It gives a bound for the number of irreducible primitive substitutions over m letters.
The cardinality of primitive substitutions over m letters, whose Perron Frobenius root is less than or equal to B, is less than a bound m m 4 B 2(m−1) 2 +2 .
Proof. Let σ be the substitution over m letters whose incidence matrix is M σ . Then there is a positive integer k that M k σ = M σ k is a positive matrix. In fact, one can take k ≤ (m−1) 2 +1 for all primitive matrix M σ (see [10], [16]). Denote the characteristic polynomial i β ki , because other roots of Φ σ k are less than β k in modulus. Our aim is to show that there are only finitely many matrices M k σ = (a ij ). Indeed this implies that all entries of M σ are bounded by max{a ij | i, j ≤ m}, since otherwise there is an entry of M σ k larger than this bound. From 0 ≤ i a ii = c m−1 ≤ mβ k , we have a ii < mβ k and it suffices to show that a ij is bounded for i = j. Using Since a ji ∈ N, we have the bound a ij ≤ m 2 B 2k . Thus we have b ij ≤ m 2 B 2k where M σ = (b ij ) and the number of possible σ(i)'s for each i is bounded by the multinomial coefficient: which gives the required bound.
Although Lemma 2.1 gives the termination of computation by the bound, it is too large to execute. In the sequel, we deduce a practical estimate by the property of the Pisot number to narrow the range of computation. If σ is a Pisot substitution of degree d whose incidence matrix has a Pisot number β as the Perron-Frobenius root, we have β − (d − 1) < Tr(M σ ) < β + (d − 1). Thus it is meaningful to check the Pisot conjecture for irreducible substitutions whose incidence matrix M σ has small trace with a fixed degree. Our first result is the following proposition.
Proposition 2.2. Let σ be a cubic Pisot substitution having a fixed point. If Tr(M σ ) = 1, then σ is pure discrete. Furthermore, if σ is a unit substitution with Tr(M σ ) = 2, then σ is pure discrete.
Proof. Note that Tr(M σ ) > 0, since σ has a fixed point. Let be the characteristic polynomial of σ. It is the minimal polynomial of a cubic Pisot number if and only if be the incidence matrix of the primitive substitution σ. The coefficients p, q, r of the characteristic polynomial (2.2) can be written as We claim that a, b, c, d, e, f are not greater than L, where using the idea of Lemma 2.1. Note that by the primitivity, none of six vectors (a, b), (c, d) , . By symmetry, we only prove this bound (2.3) for a. Consider q and r as a linear polynomial on a. Then the leading coefficients are c and de − ck 3 and we see that either c = 0 or de − ck 3 = 0 holds. If c = 0, then the formula for q gives a ≤ ac ≤ q + k 1 k 2 + k 2 k 3 + k 1 k 3 . If c = 0 then, the formula for q implies be + df ≤ q + k 1 k 2 + k 2 k 3 + k 1 k 3 and the formula for r gives So the claim follows. Since k 1 , k 2 , k 3 are non-negative integers, from (2.3) one can also deduce a, b, c, d, e, f ≤ r + p(q + p 2 ). Changing the order of letters, for p = 1 we may assume that the incidence matrix M σ is of the form:   Table 1. The numbers in the columns of M σ and σ imply the number of substitutions and the additions + means the sums of the number of (k 1 , k 2 ) = (2, 0) type and the number of (1, 1) type.
We may assume that the substitution σ has a fixed point because the substitution dynamical systems for σ and σ n are identical. However taking power of σ leads to heavy computation. For our computation, the fixed point assumption is necessary to control execution time for substitutions uniformly.
In the case of non-unit substitutions with Tr(M σ ) = 2, there are two cases of r = 2 and r = 3. If r = 2 (or r = 3), there are 81 + 112 (or 66 + 103) number of incidence matrices of the substitutions, respectively. For this case, the computation was big and we have met practical difficulty in computing all possible substitutions. However, the cases that we have computed so far come out to be pure point.
The above method listing up cubic Pisot substitutions readily applies to Tr(M σ ) = 3. However there are 4,881,771 substitutions to be checked, which seems beyond our reach.
For irreducible Pisot substitutions, balanced pair algorithm (BPA) has been the quickest known computational method. However, our program using potential overlap algorithm (POA) seems be equally quick. Table 1 shows that non-unit case of Tr(M σ ) = 2 is completed. For this case, it seems like that the algorithms for the non-unit case take more time than for the unit case. Remark 2.3. Interestingly POA and BPA have different computational aspects. It is not easy to compare their speed because it depends heavily on the coding of algorithms. For e.g., by POA 0 → 01, 1 → 2, 2 → 11100000 produces a potential overlap graph of size 18163, it takes very long and requires a machine equipped with large memory. But BPA finishes without problems with 4055 balanced pairs whose longest pair has length 8214. On the other hand 0 → 01, 1 → 2, 2 → 101000 finishes by POA without problem, using a potential overlap graph of 4025 vertices. However BPA may not finish since the longest balanced pair is of length 146041, which requires large space and time. 0 → 02111, 1 → 2, 2 → 10 is an example which is rather hard by both algorithms.

Substitution tilings in R 2
A tile in R 2 is defined as a pair T = (A, i) where A = supp(T ) (the support of T ) is a compact set in R 2 which is the closure of its interior, and i = l(T ) ∈ {1, . . . , m} is the color of T . We say that a set P of tiles is a patch if the number of tiles in P is finite and the tiles of P have mutually disjoint interiors. A tiling of R 2 is a set T of tiles such that R 2 = {supp(T ) : T ∈ T } and distinct tiles have disjoint interiors.
Let A = {T 1 , . . . , T m } be a finite set of tiles in R 2 such that T i = (A i , i); we will call them prototiles. Denote by P A the set of non-empty patches. A substitution is a map Ω : Here all sets in the right-hand side must have disjoint interiors; it is possible for some of the D ij to be empty.
We say that T is a substitution tiling if T is a tiling and Ω(T ) = T with some substitution Ω. We say that T has finite local complexity (FLC) if ∀ R > 0, ∃ finitely many translational classes of patches whose support lies in some ball of radius R. A tiling T is repetitive if for any compact set K ⊂ R 2 , {t ∈ R 2 : T ∩ K = (t + T ) ∩ K} is relatively dense. A repetitive fixed point of a primitive substitution with FLC is called a self-affine tiling. Let λ > 1 be the Perron-Frobenius eigenvalue of the substitution matrix S. Let D = {λ 1 , . . . , λ d } be the set of (real and complex) eigenvalues of Q. We say that Q (or the substitution Ω) fulfills a Pisot family if, for every λ ∈ D and every Galois conjugate λ of λ, λ ∈ D, then |λ | < 1.
3.1. Endomorphisms of free group. Generalizing the idea of Dekking [7,8], Kenyon [12] introduced a class of self-similar tilings generated by the endomorphisms of a free group over three letters a, b, c: where p, q ≥ 0, r ≥ 1 are integers for which x 3 − px 2 + qx + r has exactly two roots λ 1 , λ 2 whose each modulus is greater than one. 2 The letters a, b, c are identified with vectors (1, 1), (λ 1 , λ 2 ), (λ 2 1 , λ 2 2 ) ∈ R 2 respectively if λ 1 , λ 2 are real numbers, and with 1, λ 1 , λ 2 1 ∈ C if they are complex conjugates. The endomorphism θ acts naturally on the boundary word which represent three fundamental parallelograms, and gives a substitution rule on the parallelograms. The associated tile equation are where Q is either λ 1 0 0 λ 2 or λ 1 depending on whether λ 1 is real or complex, respectively.
It is known in [13,14] that if the expansion map Q of a self-affine tiling in R 2 is diagonalizable and the tiling has pure discrete dynamical spectrum, then Q should fulfill the Pisot family property. So we are interested in considering self-affine tilings with the Pisot family property on the expansion map Q. For this construction, we require that two roots of the polynomial x 3 − px 2 + qx + r are greater than one, and one root is smaller than one in modulus. In this case, we can note that there are no roots on the unit circle. We adapt Schur-Cohn criterion (see [1,Theorem 2.1] or [15, Chap.10, Th.43, 1]), which says that the number of roots within the unit circle coincides with the number of sign changes of the following sequence: if all entries are non-zero. Notice that it cannot be p = q = 0, otherwise all the roots of the polynomial have the same modulus. Thus from p, q ≥ 0 and r ≥ 1, pr + q + r 2 − 1 > 0. So the signs come from The last term is not zero, since ±1 cannot be a root of x 3 − px 2 + qx + r. When r = 1, the second term vanishes and the third term is negative (because p or q is positive). Since the roots of the polynomial are continuous with respect to coefficients, the small perturbation of r does not change the number of roots inside/outside of the unit circle. Therefore we may assume that the second coefficients are non-zero and use the Schur-Cohn criterion (c.f. [1]). As a result, the number of zeroes within the unit circle is 1 when (p−q−r−1)(p+q−r+1) < 0 and it is 2 when (p − q − r − 1)(p + q − r + 1) > 0. If r > 1, then the second term is positive. In this case, applying the small perturbation argument when the third term vanishes, the number of zeroes within the unit circle is 1 when (p − q − r − 1)(p + q − r + 1) < 0 and 0 or 2 when (p − q − r − 1)(p + q − r + 1) > 0. Overall we obtain a unified conclusion that the number of zeroes within the unit circle is 1 if and only if (p − q − r − 1)(p + q − r + 1) < 0. Our tiling exists when |p − r| < q + 1. All computed systems admit overlap coincidence. Tiles appearing in the remaining two cases are very thin and the bound for the collection of initial overlaps seems beyond the capability of our program.

Examples with non pure discrete spectrum
In this section, we wish to give two intriguing examples of self-affine tilings whose dynamical spectrum is not pure discrete.
Bandt discovered a non-periodic tiling in [5] whose setting comes from crystallographic tiles and called it a fractal chair tiling. The tile satisfies the following set equation  where ω = (1 + √ −3)/2 is the 6-th root of unity. It is called 3-rep-tile, because it is a non overlapping union of three similar contracted copies, i.e., the associated iterated function system satisfies open set condition [5]. Applying the substitution rule, we obtain a nonperiodic tiling of the plane by six tiles T i = ω i A (i = 0, 1, 2, 3, 4, 5) and their translates are given as in Figure 2. We can confirm that this tiling is repetitive and the corresponding expansion map satisfies the Pisot family property. So we can apply our algorithm in [2]. Our program shows that the fractal chair tiling is not purely discrete. The overlap graph with multipicity contains a strongly connected component of spectral radius 3, being equal to the spectral radius of the substitution matrix, which does not lead to a coincidence. We call such a component non coincident component. Here we visualize the non-coincident component in Figure 3 which does not lead to coincidence. Each figure represents an overlap of two fractal chair tiles, and the support of the overlap is depicted in thick color. The destinations of outgoing arrows show that we obtain three overlaps from each overlap by substitution.
As another example, the following substitution 1 → 1241224, 2 → 1224, (4.1) 3 → 1243334, 4 → 124334 with the characteristic polynomial (x 2 − 3x + 1)(x 2 − 6x + 1) arose in the study of selfinducing 4 interval exchanges by P. Arnoux, S. Ito and M. Furukado [11]. As it is a Pisot substitution, our algorithm readily applies. Our algorithm shows that its natural suspension tiling dynamics is not pure discrete. Although there are several known reducible Pisot substitutions which are not pure discrete, this example is noteworthy in the sense that the non coincident component is much more intricate than other known examples. We describe it by another substitution on the 12 letters: The associated tile equation of this substitution gives the non-coincident component of the suspension tiling corresponding to (4.1). In other words, (4.2) is the exact analogy of Figure  3, once we associate the intervals of canonical lengths given by the left eigenvector of the substitution matrix.