Strong coincidence and Overlap coincidence

We show that strong coincidences of a certain many choices of control points are equivalent to overlap coincidence for the suspension tiling of Pisot substitution. The result is valid for degree $\ge 2$ as well, under certain topological conditions. This result gives a converse of the paper by Akiyama-Lee and elucidates the tight relationship between two coincidences.


Introduction
Self-affine tiling dynamical system in R d is a generalization of substitution dynamical system on letters, which gives a nice model of self-inducing structures appear in dynamical systems, number theory and the mathematics of aperiodic order. Pure discreteness of self-affine tiling dynamics is long studied from many points of views. The idea of coincidence 1 appeared firstly in Kamae [8], and then in a comprehensive form in Dekking [6] for constant length substitution (see also [16]). Generalizing a pioneer work of Rauzy [17], Arnoux-Ito [3] gave a geometric realization of irreducible Pisot unit substitution of degree d. They defined strong coincidence, which ensures that their geometric substitution gives rise to a domain exchange of R d−1 , which is also semi-conjugate to the toral rotation of T d . It is remarkable that in many cases, it is even conjugate to the total rotation, which immediately implies that the system in pure discrete (see [4,7,5,18] for further developments). On the other hand, overlap coincidence introduced by Solomyak [19] is an equivalent condition for pure discreteness of a given self-affine tiling dynamical system. This is also described as a geometric/combinatorial condition which guarantees that the tiling and its translation by return vectors become exponentially close if we iteratively enlarge return vectors by substitution. Lee [11] showed deeper characterizations that overlap coincidence is equivalent to algebraic coincidence, and the fact that the corresponding point set is an inter-model set.
Until now, the relation between strong coincidence and overlap coincidence is not fully understood.
Motivated by the claim of Nakaishi [15], Akiyama-Lee [1] generalized the notion of strong coincidence to R d and showed that overlap coincidence implies strong coincidence, and moreover simultaneous coincidence, provided that the associated point set is admissible and its height group is trivial. In this paper we shall give a converse statement for the suspension tiling of Pisot substitution at the expense of assuming many strong coincidences at a time, that is, strong coincidences on a certain many choices of control points imply overlap coincidence and vice versa. If every tile is connected and the tiling is not a collection of unbounded connected identical colored patches, then the same result holds for d ≥ 2 (Theorem 3.2). This result elucidates the tight relationship between two coincidences.
1991 Mathematics Subject Classification. Primary: 52C23. This research was supported by the Japanese Society for the Promotion of Science (JSPS), Grant in aid 21540012. 1 In their notation, the column number is one.

Terminologies
2.1. Tiles and tilings. We shall briefly recall basic definitions used in this paper. A tile in R d is defined as a pair T = (A, i) where A is a compact set in R d which is the closure of its interior, and i = ℓ(T ) ∈ {1, . . . , m} is the color of T . We call A the support of T and denote supp(T ) = A. The translate of T is defined by g + T = (g + A, i) for g ∈ R d . Let A = {T 1 , . . . , T m } be a finite set of tiles in R d such that T i = (A i , i); we will call them prototiles. A tiling T is a collection of translates of prototiles which covers R d without interior overlaps. A finite collection of tiles which appear in T is called a patch. A generalized patch is a collection of tiles in T whose cardinality is not necessarily finite. Its support is defined to be the union of the supports of tiles. The diameter of a generalized patch is the supremum of Euclidean distance of two points lie within the support of the patch. A map Ω from A to the set of patches is called a substitution with a d × d expansive matrix Q if there and the last union has mutually disjoint interiors. The substitution (2.1) extends to all translates of prototiles and patches in a natural way. A substitution tiling of Ω is a tiling T that all the patches of T is a sub-patch of Ω n (T ) for some n ∈ N and T ∈ T . A substitution tiling T is a fixed point of Ω if Ω(T ) = T holds. We say that a substitution tiling is primitive if the corresponding substitution matrix M = (♯D ij ) is primitive, and irreducible if the characteristic polynomial of M is irreducible. We say that T has finite local complexity (FLC) if for any R there are only finitely many patches of diameter less than R up to translation. A tiling T is repetitive if every patch is relatively dense in T . A FLC substitution tiling of a primitive substitution is called a self-affine tiling. Every self-affine tiling is repetitive, which follows from the primitivity of substitution. Let λ > 1 be the Perron-Frobenius eigenvalue of the substitution matrix M and D be the set of eigenvalues of Q. By the tiling criterion of Lagarias-Wang [9], λ is the element of D of maximum modulus. We say that Q fulfills Pisot family condition if every algebraic conjugate µ of an element of D with |µ| ≥ 1 is The set of all substitution tilings of Ω forms a tiling space. By using a fixed point T of Ω, we can describe this space as the orbit closure of T under the translation action: the closure is taken by 'local topology'. The FLC assumption implies X T is compact and we get a topological dynamical system (X T , R d ) where R d acts by translations. This system is minimal and uniquely ergodic ( [19,12]), and we are interested in the spectra of self-affine tiling dynamical systems.
Tiling dynamical system X T has pure discrete spectrum if the eigenfunctions for the R d -action forms a complete orthonormal basis of L 2 (X T , µ) [19].

Control points.
A Delone set is a relatively dense and uniformly discrete subset of R d . We and there exist an expansive matrix Q and finite sets D ij for i, j ≤ m such that where the union on the right side is disjoint.
Given a fixed point T of Ω, we can associate a substitution Delone multi-color set Λ T = (Λ i ) i≤m of T by taking representative points of tiles in the relatively same positions for the same color tiles in the tiling. There is a canonical way to choose representative points, called control points. A tile map γ = γ Ω is a map from T to itself which sends a tile T to the one in Ω(T ) such that γ(T 1 ) and γ(T 2 ) are located in the same relative position in Ω(T 1 ) and Ω(T 2 ) whenever ℓ( In section 3, we have to assume a lot of strong coincidences by changing control points for a given tiling T . If we change control points of tiles of T by Λ ′ i = Λ i − g i , then the set equation will be shifted like The corresponding tile equation becomes To avoid heavy notation, we do not distinguish such changes of control points and use the same symbols Λ i and T i .
We say that a self-affine tiling T admits overlap coincidence if there exists ℓ ∈ Z + such that for each overlap O in T , Ω ℓ O contains a coincidence. Two overlaps (U, y, V ) and (U 1 , The equivalence class is denoted by (U, y, V ). Hereafter we assume an important condition that Ξ(T ) forms a Meyer set.
This condition is equivalent to the Pisot family condition for Q, if Q is diagonalizable and all its eigenvalues are algebraic conjugate with the same multiplicity [13]. The number of equivalence classes of overlaps is finite, by the Meyer property of Ξ(T ). The action of Ω is well-defined on equivalence classes of overlaps. An overlap graph with multiplicity is a finite directed graph whose vertices are the equivalence classes of overlaps. Multiplicities of the edge from (U, y, V ) to (A, z, B) is given by the number of overlaps in Ω((U, y, V )) equivalent to (A, z, B) (c.f. [2]). Overlap coincidence is confirmed by checking whether from each vertex of this graph there is a path leading to a coincidence. Overlap coincidence is equivalent to pure discreteness of self-affine tiling dynamical system X T [19].
Strong coincidence on letter substitution is naturally generalized to self-similar tiling in R d in [1].
We adapt this definition to control points. Let T be a self-affine tiling in R d and A = {T 1 , · · · , T m } be the prototile set of T . We say that the set of the control points is admissible if ∩ i≤m (supp(T i )−c(T i )) has non-empty interior.
Let T be the fixed point of Ω. Let c(T i ) (i = 1, . . . , m) be the admissible control points and Λ be an associated substitution Delone multi-color set for which If for any 1 ≤ i, j ≤ m, there is a positive integer L that then we say that Λ admits strong coincidence. In other words, strong coincidence means that for every pair of tiles (U, V ) ∈ T 2 , Ω L (U − c(U )) and Ω L (V − c(V )) share a common tile in the same position for some L.

Strong coincidence and Overlap coincidence
The set of eventually return vectors is defined by which is independent of the choice of i, by primitivity of Ω. The tiling dynamical system is invariant under replacement of the substitution rule Ω by Ω n . We consider control points of Ω n as well.
Hereafter we put Λ = m i=1 Λ i for Λ = Λ T = (Λ i ) to distinguish the multi-color set and its union. Let G be the additive subgroup of R d generated by G. We say that T satisfies multiple strong coincidence of level n if all multi-color Delone set Λ's generated by admissible control points of Ω n with Λ − Λ ⊂ G admit strong coincidence.
Hereafter when we speak about a topological/metrical property (connected, bounded, diameter) of a generalized patch, it refers to the corresponding property of its support. A rod is an unbounded connected generalized patch of T whose tiles have an identical color. A rod tiling is a tiling that every tile belongs to a rod. For ease of negation, a non-rod tiling is a tiling which is not a rod tiling.  However we do not know an example of non-periodic self-affine rod tiling.  Indeed, 1 × 1 matrix Q = (β) satisfies Pisot family condition, tiles are intervals and the suspension tiling can not be a rod tiling, since it has at least two translationally inequivalent tiles in R.
Remark 3.4. Multiple strong coincidence of level n requires many strong coincidences at a time for a fixed tiling T even when n = 1. In dimension one, the claim of Nakaishi [15] reads a single strong coincidence implies overlap coincidence. Theorem 3.2 covers general cases but the requirement is much stronger. It would be interesting is to make smaller the constant n in Theorem 3.2. For e.g., can we take n = 1 ?
We prepare a lemma.
Lemma 3.5. Let G be a strongly connected finite directed graph and C be a set of cycles of G. Then there is a subgraph G(C) of G with the following property.
• The set of vertices of G(C) is equal to that of G.
• Every vertex has exactly one outgoing edge.
• The set of cycles of G(C) is equal to C.
Proof. Put H 0 = C. We inductively construct H i for i = 0, 1, . . . which satisfies: • Every vertex has exactly one outgoing edge.
• The set of cycles of H i is equal to C.
Assume that the induced graph G \ H i is non empty and take a vertex v from G \ H i . Since G is strongly connected, there is a path from v leading to H i . So there is a vertex u ∈ G \ H i and an edge from u to a vertex of H i . We define H i+1 by adding this u and the outgoing edge. Then H i+1 clearly satisfies above two conditions. Since G is finite, we find m that G \ H m is empty, i.e., the set of vertices of G and H m are the same. We finish the proof by taking G(C) = H m .
Proof of Theorem 3.2. Theorem 4.3 of [1] shows that overlap coincidence of T implies multiple strong coincidence of level n for any n ≥ 1. We prove that there is a constant n such that multiple strong coincidence of level n implies overlap coincidence.
Assume that T does not admit overlap coincidence. Construct the overlap graph G of T with multiplicity. Since T does not admit overlap coincidence, there is a strongly connected component 2 S of G such that its spectral radius is equal to | det(Q)| and from each overlap of S there is no path leading to a coincidence in G. Without loss of generality, we may assume that the incidence matrix of S is primitive 3 . Thus we can find a positive integer n 0 such that for every overlap (U, y, V ), Ω n0 (U, y, V ) contains an overlap equivalent to (U, y, V ). Since Ξ(T ) is a Meyer set, number of equivalence classes of overlaps is finite and bounded by a constant which depends only on T . Thus there is an upper bound of n 0 which depends only on T . We further assume multiple strong coincidence of level n = n 0 on T and derive a contradiction.
We claim that in the component S there is an overlap (U, y, V ) with ℓ(U ) = ℓ(V ) for any nonrod self-affine tiling by connected tiles. Assume on the contrary that all overlaps in S are of the 2 In this assertion, one can take either usual overlaps or potential overlaps as we like. 3 If the incidence matrix of S is irreducible but not primitive, then take a suitable power of Ω by Perron-Frobenius theory.
form (A, z, B) with ℓ(A) = ℓ(B). Since S does not contain a coincidence, z = 0 for these overlaps.
Taking k-th inflated overlap of (A, z, B), we obtain of patches P and Q, both contain large balls, say B p (r) and B q (r), that the tiles of P close to p and the tiles of Q close to q are in multiple correspondence in the following sense. Putting x = Q k z, for a tile U ∈ P close to p there are several (at least two) tiles V ∈ Q that (U, x, V ) are overlaps in 4 S and supp(U ) is contained in the union of supp(V − x), and the same statements hold after interchanging the role of U and V . Take a tile U with p ∈ supp(U ) ⊂ B p (r). Then overlaps (U, x, V ) with V ∈ Q give rise to a patch V 1 = V that supp(U ) supp(V 1 ) − x. By assumption, ℓ(V 1 ) = ℓ(U ) for every V 1 ∈ V 1 . By using path connectedness of tiles 5 , the patch V 1 is path connected. If supp(V 1 ) ⊂ B q (r), then there is a patch U 1 = U 1 where U 1 ∈ P are taken from all overlaps of the form (U 1 , x, V 1 ) with some V 1 ∈ V 1 . This patch is also path connected and satisfies supp(V 1 ) − x supp(U 1 ) and each tile of U 1 has the same color as U . In this manner, by taking large r, we obtain a long sequence of path connected patches The number of tiles strictly increases and all tiles appear in this sequence has the same color ℓ(U ).
This shows for any M > 0, there exists a ball of radius R that each tile U in the ball belongs to a connected patch in T having diameter greater than M , whose tiles have an identical color ℓ(U ).
Therefore by using FLC, among X T we can choose a rod tiling. Being a rod tiling is invariant under translation and closure operation, using minimality of X T we see that every tiling in X T is a rod tiling. This gives a contradiction, which finishes the proof of the claim.
Consider a directed graph V over {1, . . . , m} whose edge i → j is given if there are U, V ∈ S that V ∈ Ω n (U ) with i = ℓ(U ) and j = ℓ(V ). Clearly V is strongly connected as well. Pick one overlap (U, y, V ) from S that ℓ(U ) = ℓ(V ) and select one of the overlaps equivalent to (U, y, V ) in Ω n (U, y, V ). We select a tile map γ = γ Ω n which sends γ(U ) to this U in (U, y, V ), and γ(V ) to the V in (U, y, V ), which correspond to two cycles ℓ(U ) → ℓ(U ) and ℓ(V ) → ℓ(V ) on V. Let C be the set of these two cycles and take V(C) by Lemma 3.5. The tile map γ = γ Ω n is chosen so that ℓ(U ) → ℓ(γ(U )) for U ∈ {T 1 , . . . , T m } forms the set of edges of V(C). By the choice of the subgraph, every path of length m on this subgraph must fall into one of the two cycles. Note that by this choice of γ, the control points of U and V − y are exactly matching, because both of them are equal to a . We claim that by this γ, we have Λ − Λ ⊂ G . In fact, since every overlap in the overlap graph is of the form (A, z, B) with z ∈ m i=1 (Λ i − Λ i ), and control points of U and V − y are matching on (U, y, V ), i.e., c(U ) = c(V ) − y, we have c(U ) − c(V ) ∈ G. By construction of V for any x, y ∈ Λ, we have Q m x, Q m y ∈ Λ ℓ(U) ∪ Λ ℓ(V ) . For instance, if Q m x ∈ Λ ℓ(U) and Q m y ∈ Λ ℓ(V ) , then Q m x = c(U ) + f, Q m y = c(V ) + g hold with f ∈ Λ ℓ(U) − Λ ℓ(U) , g ∈ Λ ℓ(V ) − Λ ℓ(V ) . Therefore we have Λ − Λ ⊂ G .
We also see that the set of control points Λ = (Λ i ) associated to γ is admissible. In fact, since (U, y, V ) is an overlap, supp(U ) ∩ supp(V − y) has an inner point. Since y = c(V ) − c(U ), we have (supp(U − c(U ))) • ∩ (supp(V − c(V ))) • = ∅. The admissibility follows from Q m x ∈ Λ ℓ(U) ∪ Λ ℓ(V ) for any x ∈ Λ. 4 We say that an overlap belongs to S if its equivalence class does. 5 Connectedness and path connectedness are equivalent for self-affine tiles [14].
Summing up, from (U, y, V ) ∈ S, we have chosen a tile map γ Ω n which produces a substitution Delone multi-color set of admissible control points with Λ − Λ ⊂ G . By the assumption of multiple strong coincidence of level n, we know Ω k (U − c(U )) ∩ Ω k (V − c(V )) is non empty for some k, which shows that (U, y, V ) leads to a coincidence, giving a desired contradiction.
Remark 3.6. We use the assumptions that each tile is connected and T is a non-rod tiling only to show that there is an overlap (U, y, V ) ∈ S that ℓ(U ) = ℓ(V ), which allows us to define a tile map. It is likely that these assumptions are not necessary, i.e., every non-periodic self-affine tiling that Ξ(T ) is a Meyer set, then such overlap must appear in S.