Continuously many bounded displacement non-equivalences in substitution tiling spaces
CCONTINUOUSLY MANY BOUNDED DISPLACEMENTNON-EQUIVALENCES IN SUBSTITUTION TILING SPACES
YAAR SOLOMON
Abstract.
We consider substitution tilings in R d that give rise to point sets that are notbounded displacement (BD) equivalent to a lattice and study the cardinality of BD p X q ,the set of distinct BD class representatives in the corresponding tiling space X . We provea sufficient condition under which the tiling space contains continuously many distinctBD classes and present such an example in the plane. In particular, we show here forthe first time that this cardinality can be greater than one. Introduction
Let
X, Y Ă R d be two discrete sets, i.e. sets with no accumulation points. We say that X is bounded displacement (BD) equivalent to Y , and denote X BD „ Y , if there exists abijection φ : X Ñ Y that satisfies sup x P X } x ´ φ p x q} ă 8 . Such a mapping φ is calleda BD-map . In a similar manner we consider tilings of R d by tiles of bounded diameterand inradius that is bounded away from zero. We say that such tilings S and T are BD-equivalent , and denote S BD „ T , if there are point sets X S and Y T , which are obtainedby placing a point in each tile of S and T respectively, so that X S and Y T are BD-equivalent. Note that since the tiles have bounded diameter, the question whether S and T are BD-equivalent or not does not depend on the choice of the point sets.The BD-equivalence relation for general discrete point sets was studied in [DO1, DO2,L, DSS], where the main focus was on point sets that are BD-equivalent to a lattice. Werefer to such point sets as uniformly spread , following Laczkovich, who gave an importantcriterion to check whether a point set is uniformly spread or not. We say that a tilingis uniformly spread if its corresponding point set is such. Note that an application ofthe Hall’s marriage theorem shows that every two lattices of the same co-volume areBD-equivalent, see [DO2] or [HKW] for a proof.Recall that a point set X Ă R d is a Delone set if there exists
R, r ą X intersects every ball of radius R and every ball of radius r contains at most one point of X . There are two fundamental families of Delone sets, which are on one hand non-periodicand on the other hand are often of finite local complexity and are repetitive, and hence areimportant objects of study in the theory of mathematical quasicrystals. One is the familyof cut-and-project sets and the other is point sets that arise from substitution tilings, see[BG] for further details. The question of which cut-and-project sets are uniformly spreadwas studied in [HKW], where in [HK, HKK] the BD-equivalence of cut-and-project sets islinked to the notion of bounded remainder sets . For substitution tilings (see § a r X i v : . [ m a t h . M G ] A p r YAAR SOLOMON
Questions regarding BD-equivalence and non-equivalence between two Delone sets, noneof which is a lattice, were considered recently in [FSS]. Using similar ideas to those ofLaczkovich, a sufficient condition for BD-non-equivalence has been established.This paper studies the BD-equivalence relation on the tiling space X (cid:37) of a fixed primitivesubstitution rule (cid:37) in R d . In particular we are interested in the quantity | BD p X (cid:37) q| , whichis the cardinality of the quotient set X (cid:37) { BD „ . In particular, we show here for the first timethat this cardinality can be greater than one.We denote by 2 ℵ the cardinality of R . Let (cid:37) be a primitive substitution rule onthe prototiles t T , . . . , T n u in R d , with substitution matrix M (cid:37) , whose eigenvalues are λ ą | λ | ě . . . ě | λ n | , see §
2. For a legal patch P , we denote by v p P q P R n the vectorwhose i ’th coordinate is the number of tiles of type i in P . The notation stands for thevector all of whose coordinates are equal to 1, and W K denotes the orthogonal complementof a subspace W , with respect to the standard inner product in C n . When dim p W q “ v K the orthogonal complement of span t v u . Our main results are: Theorem 1.1.
Let (cid:37) be a primitive substitution rule in R d and let t ě be the minimalindex for which λ t has an eigenvector whose sum of coordinates is non-zero. Assume that | λ t | ą λ d ´ d and that there exist two legal patches P, Q , such that (1) supp p P q and supp p Q q differ by a translation. (2) v p P q ´ v p Q q R v K t , where v t is an eigenvector of M (cid:37) in K , whose eigenvalue isequal in modulus to | λ t | .Then | BD p X (cid:37) q| “ ℵ . Observe that every uniformly discrete set in R d , with separation constant δ ą
0, isBD-equivalent to a subset of the lattice δ Z d , hence the upper bound | BD p X (cid:37) q| ď ℵ is trivial. We also remark that the primitivity assumption implies that the space X (cid:37) isminimal with respect to the action of R d by translations, see [Q]. In particular, thosecontinuously many tilings that we find, which are pairwise BD-non-equivalent, belong tothe R d -orbit closure of any one of the tilings in space. Corollary 1.2.
There exists a primitive substitution rule (cid:37) on a set of two prototiles inthe plane such that | BD p X (cid:37) q| “ ℵ . This corollary strengthen a similar result in the context of mixed substitution thatwas obtained in [FSS], that is where more than one substitution rule on the prototiles isallowed. In view of Theorem 2.4 below, we also conjecture the following.
Conjecture 1.3.
Let t ě be as in Theorem 1.1, and assume that | λ t | ą λ d ´ d , then | BD p X (cid:37) q| “ ℵ . Remark 1.4.
Conjecture 1.3 says that other than the case of equality, where | λ t | “ λ d ´ d ,and under certain regularity assumptions of the tiles, the following dichotomy holds: ‚ | λ t | ă λ d ´ d ùñ | BD p X (cid:37) q| “ ‚ | λ t | ą λ d ´ d ùñ | BD p X (cid:37) q| “ ℵ .In view of Theorem 2.4 below, for tiles that are biLipschitz homeomorphic to closed balls,the former implication is clear, since in this case every tiling in X (cid:37) is uniformly spread.We also remark that in the case of equality both implications fail. Indeed, as shown in[FSS], there are examples where equality holds and every T P X (cid:37) is uniformly spread and ONTINUOUSLY MANY BD NON-EQUIVALENCES IN SUBSTITUTION TILING SPACES 3 there are such examples where every T P X (cid:37) is not uniformly spread. In the latter, onecan repeat the arguments of our Lemma 3.1 and Corollary 3.2 below, with the examplein [FSS], showing that | BD p X (cid:37) q| ą bi-Lipschitz equivalence relation , inwhich Delone sets are equivalent if there exists a bi-Lipschitz bijection between them. It isnot hard to verify that BD-equivalence of Delone sets implies bi-Lipschitz equivalence. Itwas shown in [Mag] that the cardinality of the set of Delone sets in R d modulo bi-Lipschitzequivalence is 2 ℵ . Nonetheless, as all point sets that arise from primitive substitutiontilings are bi-Lipschitz equivalent to a lattice (see [S1]), all the distinct BD-equivalenceclass representatives that we find here belong to the same bi-Lipschitz equivalence class. Acknowledgments.
The author thanks Jeremias Epperlein, Dirk Frettl¨oh, Scott Schmied-ing, Yotam Smilansky and Barak Weiss for useful discussions and remarks.2.
Background and definitions
We use bold figures to denote vectors in C n . The notation x¨ , ¨y stands for the standardinner product in C n , } v } : “ a x v , v y , and M T (resp. u T ) denotes the transpose of amatrix M (resp. vector u ). This chapter contains preliminaries on tilings that are neededfor our discussion. For further reading see [BG].A tile T is a compact subset of R d . A large variety of additional regularity assumptionson tiles appears in the literature. We assume here that H d ´ pB T q P p , for every tile T , where H s p A q stands for the s -dimensional Hausdorff measure of the set A Ă R d , see[Mat, Chap. 4].A tiling of a set S Ă R d is a collection of tiles, with pairwise disjoint interiors, suchthat their union is equal to S . A tiling P of a bounded set B Ă R d is called a patch ,and we denote the set B , which is the support of P , by supp p P q . In particular, by ourassumption on the tiles, for any scaling constant t ą P one has H d ´ p t ¨ B supp p P qq “ t d ´ H d ´ pB supp p P qq P p , , (2.1)see [Mat, p. 57]. This assumption is used in (4.11).Tiles are called translation equivalent if they differ by a translation and representativesof this equivalence relation are called prototiles . The set of prototiles is denoted by F ,and F ˚ is the set of representatives of patches. Finally, given a tiling T of R d , a boundedset B Ă R d and a finite set S , we denote by S : “ the cardinality of S and by r B s T : “ the patch of T that consists of all tiles that intersect B. (2.2)2.1. Substitution tilings.
Let ξ ą F “ t T , . . . , T n u be a set of tiles in R d . Definition 2.1. A substitution rule on F is a fixed way to tile each one of the elementsof ξ F by the tiles in F . By applying (cid:37) on a tile T i P F we mean first scaling T i by ξ and then substitute ξT i by its fixed given tiling. Formally, it is a mapping (cid:37) : F Ñ F ˚ satisfying supp p ξT i q “ supp p (cid:37) p T i qq for every i . The number ξ is the inflation factor of (cid:37) .The function (cid:37) can naturally be extended to F ˚ , and to tilings by tiles of F , by applying (cid:37) separately to each tile. Definition 2.2.
Given a substitution rule (cid:37) on F in R d , consider the patches: L (cid:37) “ t (cid:37) m p T q : m P N , T P F u . YAAR SOLOMON
A patch is called legal if it is a sub-patch of an element of L (cid:37) . The tiling space X (cid:37) is thecollection of tilings of R d with the property that every patch in them is legal. A tiling T P X (cid:37) is called a substitution tiling that corresponds to (cid:37) . Definition 2.3.
The substitution matrix M (cid:37) “ p a ij q of (cid:37) is defined by a ij “ t tiles of type i in (cid:37) p T j qu .(cid:37) is called primitive if M (cid:37) is a primitive matrix. Namely, if there exists an m P N suchthat all entries of M m(cid:37) are positive.2.2. Further notations and properties.
We assume throughout that (cid:37) is primitive.Perron-Frobenius theorem then implies that the eigenvalues of M (cid:37) can be ordered suchthat λ ą | λ | ě . . . ě | λ n | . We denote by p v , . . . , v n q a Jordan basis of M (cid:37) , v i corre-sponds to λ i .Given a patch P in a tiling T P X (cid:37) , let v p P q P Z n denote the vector whose i ’thcoordinate is the number of tiles of type i in P . Denote by e i the i ’th element of thestandard basis of R n , so e i “ v p T i q and for every m P N one has M m(cid:37) e i “ v p (cid:37) m p T i qq .Observe that λ “ ξ d , and that M T(cid:37) u “ λ u , where u is the vector whose i ’thcoordinate is vol p T i q . This implies that for every patch P we havevol p supp p P qq “ x u , v p P qy and P “ x , v p P qy , (2.3)where : “ p , , . . . , q T and F denotes the cardinality of a finite set F .2.3. Bounded displacement equivalence.
Given a tiling T of R d by tiles of boundeddiameter, let Λ T Ă R d be a point set with a point in each tile of T (taken with multiplicityin the case that a point x P T X T was chosen for tiles T , T in T ). T is called uniformlyspread if there is a BD-map φ : Λ T Ñ α Z d , for some α ą
0. Namely, a map that satisfiessup x P Λ T } x ´ φ p x q} ă 8 . Theorem 2.4. [S2, Theorem 1.2]
Let (cid:37) be a primitive substitution rule on prototiles thatare biLipschitz-homeomorphic to closed balls in R d , and let T P X (cid:37) . Let t ě be theminimal index for which the eigenvalue λ t has an eigenvector v t R K . (I) If | λ t | ă λ d ´ d then T is uniformly spread. (II) If | λ t | ą λ d ´ d then T is not uniformly spread. Given x “ p x , . . . , x d q P R d , following [FSS], we denote by C p x q the axis-parallel unitcube Ś di “ r x i ´ , x i ` q centered at x . Denote by Q d the set (cid:32) C p x q | x P Z d ( of latticecentered unit cubes, and let Q ˚ d be the collection of all finite subsets of Q d . Theorem 2.5. [FSS, Theorem 1.1]
Let Λ , Λ be two Delone sets in R d and suppose thatthere is a sequence p A m q m P N of sets A m P Q ˚ d for which lim m Ñ8 | p Λ X A m q ´ p Λ X A m q| µ d ´ pB A m q “ 8 . (2.4) Then there is no BD-map φ : Λ Ñ Λ . Remark 2.6.
Theorem 2.5 originally includes the additional assumption that the Delonesets have box diameter ě
1. Namely that for every x P R d the cube C p x q contains atmost one element of each of the Delone sets. As also mentioned in [FSS], this additionalassumption is unnecessary since one may replace the sets Λ i and A m by a mutual rescalingof them, with a suitable constant, so that this assumption holds. ONTINUOUSLY MANY BD NON-EQUIVALENCES IN SUBSTITUTION TILING SPACES 5
Theorem 2.5 can also be stated in the language of tilings, where this new formulationfollows directly from Theorem 2.5, using the notation in (2.2). We say that two giventilings are
BD-non-equivalent if there is no BD-map between their corresponding Delonesets.
Corollary 2.7.
Let T , T be two tilings of R d . Suppose that there is a sequence of sets A m P Q ˚ d for which lim m Ñ8 ˇˇ r A m s T ´ r A m s T ˇˇ µ d ´ pB A m q “ 8 . (2.5) Then the tilings T and T are BD-non-equivalent. Proof of Corollary 1.2
We begin with an example proving Corollary 1.2, relying on Theorem 1.1. Considerthe following substitution rule (cid:37) on a set of two tiles T and T in the plane, where T isa 1 ˆ T is a 2 ˆ Figure 1.
The substitution rule (cid:37) on a square and rectangle.Note that the corresponding substitution matrix here is M (cid:37) “ ` ˘ , whose eigenvaluesare λ “ , λ “ v “ ` ˘ , v “ ` ´ ˘ respectively.For k P N we denote by R p k q a translated copy of the patch (cid:37) k p T q and by S p k q atranslated copy of the patch supported on two adjacent patches of the form (cid:37) k p T q , sothat supp ` S p k q ˘ is equal to supp ` R p k q ˘ , up to a translation. To indicate that these patchesare centered at the origin we use the notations R p k q and S p k q . Lemma 3.1.
For every k P N we have ˇˇ R p k q ´ S p k q ˇˇ “ k .Proof. By the definition of the substitution matrix, the number of tiles in the patch R p k q (resp. S p k q ) is given by the sum of the coordinates of the vector M k(cid:37) ` ˘ (resp. 2 M k(cid:37) ` ˘ ).So the required quantity is the sum of the coordinates of the vector M k(cid:37) ` ˘ ´ M k(cid:37) ` ˘ “ M k(cid:37) ` ´ ˘ . Since ` ´ ˘ “ v we obtain M k(cid:37) ` ´ ˘ “ k ` ´ ˘ , and hence ˇˇ R p k q ´ S p k q ˇˇ “ ˇˇˇˇ k x ˆ ´ ˙ , ˆ ˙ y ˇˇˇˇ “ k . (cid:3) Observe that R p q “ (cid:37) p R p q q (resp. S p q ) contains a copy of a centered R p q (resp. S p q ).Thus the sequences ´ R p k q ¯ k P N and ´ S p k q ¯ k P N are nested sequences that define two fixedpoints of (cid:37) in X (cid:37) by R : “ ď k P N R p k q S : “ ď k P N S p k q . Since µ pB supp ` R p k q ˘ q “ µ pB supp ` S p k q ˘ q “ ¨ k , Corollary 3.2 below follows from Corol-lary 2.7 with A m : “ supp ´ R p m q ¯ , and from Lemma 3.1. YAAR SOLOMON
Corollary 3.2.
The tilings R and S are BD-non-equivalent. The substitution rule in Figure 1 with the patches R p q and S p q also provide a prooffor Corollary 1.2. Proof of Corollary 1.2.
The substitution (cid:37) in Figure 1 is primitive and we have d “ t “ | λ t | “ ą “ λ p d ´ q{ d . The patches P “ R p q and Q “ S p q clearly satisfy the assumptionsof Theorem 1.1 and hence | BD p X (cid:37) q| “ ℵ . (cid:3) Proof of Theorem 1.1
This chapter contains the proof of Theorem 1.1. Throughout this chapter, (cid:37) is aprimitive substitution rule defined on the set of prototiles F “ t T , . . . , T n u in R d , λ ą| λ | ě . . . ě | λ n | are the eigenvalues of M (cid:37) , p v , . . . , v n q is a corresponding Jordan basis,and t ě Lemma 4.1.
Suppose that
P, Q are two legal patches of (cid:37) and assume that ‚ vol p P q “ vol p Q q . ‚ p ´ q R v K t , where p “ v p P q , q “ v p Q q .Then there exist a constant c ą that depend on P, Q and (cid:37) such that ˇˇ (cid:37) k p P q ´ (cid:37) k p Q q ˇˇ ě c | λ t | k . (4.1) Proof.
Recall that u denotes the first eigenvector of M T(cid:37) , thus u K “ span t v , . . . , v n u .Since u can be taken to be the vector of volumes of the prototiles, as in (2.3), x u , p y “ vol p P q “ vol p Q q “ x u , q y , and thus p ´ q P u K “ span t v , . . . , v n u . (4.2)In addition, for every k P N we have (cid:37) k p P q ´ (cid:37) k p Q q “ x , M k(cid:37) p p qy ´ x , M k(cid:37) p q qy “ x , M k(cid:37) p p ´ q qy . (4.3)By (4.2), M k(cid:37) p p ´ q q “ α λ k v ` . . . ` α n λ kn v n , for some constants α , . . . , α n P C . But by the definition of t , for any j ă t we have x , v j y “ x , M k(cid:37) p p ´ q qy “ n ÿ j “ t x , α j λ kj v j y “ n ÿ j “ t α j λ kj x , v j y . (4.4)Note that by assumption p ´ q R p span t v t uq K , then α t ‰
0. Combining (4.3) and (4.4)we see that ˇˇ (cid:37) k p P q ´ (cid:37) k p Q q ˇˇ “ ˇˇˇˇˇ n ÿ j “ t α j λ kj x , v j y ˇˇˇˇˇ , and since α t ‰
0, the assertion follows. (cid:3)
ONTINUOUSLY MANY BD NON-EQUIVALENCES IN SUBSTITUTION TILING SPACES 7
Let P and Q be two legal patches and write P “ (cid:37) a p T i q and Q “ (cid:37) a p T j q with a , a P N and i, j P t , . . . , n u . For a patch P and a point x P supp p P q we use the notation P x : “ the translated copy of P in which x is at the origin . (4.5)The primitivity of (cid:37) is used for the simple observation that is given in the following lemma. Lemma 4.2.
There exists an a P N such that (1) The patch (cid:37) a p P q contains a patch P , which is a translated copy of P whosesupport is disjoint from the boundary of the support of (cid:37) a p P q . In particular, thereis a (unique) point x p P q P supp p P q so that the copy P in (cid:37) a p P x p P q q coincideswith the patch P x p P q . (2) The patch (cid:37) a p P q also contains a translated copy of Q .Proof. By the primitivity of (cid:37) , for a large integer b , (cid:37) b p P q contains copies of all tile types,and also tiles of all types that are disjoint from B supp ` (cid:37) b p P q ˘ . Hence there exists some a so that for every a ě a the patch (cid:37) a p P q contains translated copies of both P and Q , which are disjoint from B supp p (cid:37) a p P qq . Fix a copy of P in (cid:37) a p P q , whose support isdisjoint from B supp p (cid:37) a p P qq , and denote it by P .The point x p P q can be defined as follows. Repeating the above argument one finds apatch P inside (cid:37) a p P q , a patch P inside (cid:37) a p P q , etc. Each P m ` is a copy of P thatsits inside (cid:37) a p P m q , thus the nested intersection Ş m P N ξ ´ ma P m is a point that satisfiesthe requirements. (cid:3) To prove Theorem 1.1 we explicitly construct continuously many distinct tilings, whereeach one of them is defined as an increasing union of a certain nested sequence of patches.To define these patches, we set the following notations.Let P and Q be two legal patches whose supports differ by a translation. We fix markedpoints x p P q P supp p P q and x p Q q P supp p Q q as in Lemma 4.2. For any scaled copy βP of P (resp. Q) we set x p βP q : “ β ¨ x p P q . We also fix the number a to be the maximumbetween the values of a that are obtained when applying Lemma 4.2 with P and with Q . Then the patch (cid:37) a p P q contains a copy of P and the patch (cid:37) a p Q q contains a copy of Q , as in Lemma 4.2. We refer to these particular patches as ‚ the centered copy of P in (cid:37) a p P q . ‚ the centered copy of Q in (cid:37) a p Q q .Lemma 4.2 can be applied repeatedly. For integers k ă m , the notions of ‚ the centered copy of (cid:37) ka p P q in (cid:37) ma p P q‚ the centered copy of (cid:37) ka p Q q in (cid:37) ma p Q q ,play an important role in the proof of Proposition 4.3 below, which is the core of the proofof Theorem 1.1. We use the notation σ i P t
P, Q u i for a finite sequence of length i , where σ i p (cid:96) q denotes the (cid:96) ’th letter and σ i r . . . (cid:96) s is the prefix of length (cid:96) of σ i . Finally, relyingon the assumption | λ t | ą λ d ´ d of Theorem 1.1, we fix h P a ¨ N to be the smallest multipleof a that satisfies λ h ă | λ t | λ p d ´ q{ d and set k i : “ h i ´ . (4.6) Proposition 4.3.
For every i P N and every sequence σ i P t
P, Q u i of length i there existsa legal patch P p k i q σ i such that for i “ we have P p q P “ P x p P q , P p q Q “ Q x p Q q , and so thatthe following properties hold for every i P N and every σ i P t
P, Q u i : YAAR SOLOMON (1) P p k i q σ i is a translated copy of (cid:37) k i p P q , if σ i p i q “ P(cid:37) k i p Q q , if σ i p i q “ Q . (2) If σ i is a prefix of σ i ` then P p k i ` q σ i ` contains a copy of P p k i q σ i as a sub-patch, whosesupport contains the origin and is disjoint from the boundary of supp ´ P p k i ` q σ i ` ¯ . (3) ››› x ´ P p k i ` q σ i ` ¯››› ď c ¨ λ k i { d , where c “ λ a { d diam p supp p P qq .Proof. The proof is by induction on i . For i “ P p q P “ P x p P q , P p q Q “ Q x p Q q .Suppose that the patches P p k i q σ i were defined and that the above properties hold for every σ i P t
P, Q u i , we define the patches P p k i ` q σ i ` as follows. Fix some σ i ` P t
P, Q u i ` .Recall that h is a multiple of a and observe that k i ` “ h i is a much larger integerthan k i ` a “ h i ´ ` a . If the i ` σ i ` is P , denote by T P the centered copy of (cid:37) p k i ` a q p P q inside (cid:37) p k i ` q p P q (respectively, if σ i ` p i ` q “ Q let T Q be the centered copyof (cid:37) p k i ` a q p Q q inside (cid:37) p k i ` q p Q q ). A key observation is that positioning T P (resp. T Q ) in R d forces the position of the much larger patch (cid:37) p k i ` q p P q (resp. (cid:37) p k i ` q p Q q ) that containsit. Consider the centered copy of (cid:37) p k i q p P q or the copy of (cid:37) p k i q p Q q inside T P , which existsby Lemma 4.2, depending on whether the i ’th letter of σ i ` is P or Q (resp. inside T Q consider the centered copy of (cid:37) p k i q p Q q or the copy of (cid:37) p k i q p P q ). One of these two patches,depending on the i ’th letter of σ i ` , is a translated copy of the patch P p k i q σ i ` r ...i s that wehave obtained from the induction hypothesis. We place T P (resp. T Q ) so that theabove particular copy of (cid:37) p k i q p P q or of (cid:37) p k i q p Q q in it coincide with P p k i q σ i ` r ...i s , seeFigure 2. The above placement fixes the position of the copy of the patch (cid:37) p k i ` q p P q or (cid:37) p k i ` q p Q q from which we have started, and we define this fixed patch to be P p k i ` q σ i ` .It is left the verify the validity of properties (1), (2) and (3). By the induction hypothesisthe origin is contained in P p k i q σ i ` r ...i s , hence properties (1) and (2) follow directly from theconstruction. Note that the support of P p k i q σ i ` r ...i s is indeed disjoint from the boundaryof supp ´ P p k i ` q σ i ` ¯ by (1) of Lemma 4.2. To see (3), note that by our definition of thenotion of a centered copy, the point x ´ P p k i ` q σ i ` ¯ belongs to T P (or to T Q , depends on σ i ` p i ` q ), which also contains the origin. Since for every m P N the diameter ofsupp p (cid:37) m p P qq is ξ m diam p supp p P qq “ p λ { d q m diam p supp p P qq (see § T P is atranslate of (cid:37) k i ` a p P q , we have ››› x ´ P p k i ` q σ i ` ¯››› ď diam ` supp ` (cid:37) k i ` a p P q ˘˘ “ λ ki ` ad diam p supp p P qq “ c ¨ λ k i { d . In case σ i ` p i ` q “ Q , since diam p supp p P qq “ diam p supp p Q qq , the same computationholds and the proof is complete. (cid:3) Lemma 4.4.
For every infinite sequence ω P t
P, Q u N there exists a tiling T ω P X (cid:37) so thatfor every i P N the tiling T ω contains the patch P p k i q ω r ...i s , defined in Proposition 4.3.Proof. Let ω P t
P, Q u N . By (2) of Proposition 4.3, the sequence of patches ´ P p k i q ω r ...i s ¯ i P N is a nested sequence and by the proof of Proposition 4.3 it exhausts the plane. Thus T ω : “ ď i P N P p k i q ω r ...i s ONTINUOUSLY MANY BD NON-EQUIVALENCES IN SUBSTITUTION TILING SPACES 9
Figure 2.
This picture corresponds to the case where the last two lettersof σ i ` are QP and it is done similarly for the other three possible options.The illustration shows how to position the patch (cid:37) p k i ` q p P q , which is laterdefined to be P p k i ` q σ i ` , providing that we know the position of P p k i q σ i ` r ...i s ,which was given to us by the induction hypothesis. In this picture, as the i ’th letter of σ i ` is Q , the patch P p k i q σ i ` r ...i s is a translated copy of (cid:37) p k i q p Q q .We place (cid:37) p k i ` q p P q such that the copy of (cid:37) p k i q p Q q inside the centered copyof (cid:37) p k i ` a q p P q in (cid:37) p k i ` q p P q (given by Lemma 4.2), coincide with P p k i q σ i ` r ...i s .is a tiling of R d and it satisfies the assertion. (cid:3) Let Ω : “ t
P, Q u N . Consider the equivalence relation on Ω in which ω „ ω if the set t i P N | w p i q ‰ w p i qu is finite. Since every equivalence class in this relation is countable,the cardinality of a set r Ω Ă Ω of equivalence class representatives is 2 ℵ . We fix such aset of representatives r Ω, then the following lemma completes the proof of Theorem 1.1.
Lemma 4.5.
Let ω, η P r Ω be two distinct sequences, then the tilings T ω and T η , which aredefined in Lemma 4.4, are BD-non-equivalent.Proof. Since ω and η are in r Ω, and they are distinct, they differ at infinitely many places.Let p i m q m “ be an increasing sequence so that ω p i m q ‰ η p i m q for every m . We set k i m asin (4.6) and apply Corollary 2.7 with the sequence of sets p A m q m P N defined by B m : “ supp ` (cid:37) k im p P q x p P q ˘ , A m : “ ď t C p x q P Q d | B m X C p x q ‰ ∅ u p see § . q . By (3) of Proposition 4.3 we have ››› x ´ P p k im q ω r ...i m s ¯››› , ››› x ´ P p k im q η r ...i m s ¯››› ď c ¨ λ kim ´ d . Since B m and supp ´ P p k im q ω r ...i m s ¯ differ by a translation and since x p (cid:37) k im p P q x p P q q “ B m (cid:52) supp ´ P p k im q ω r ...i m s ¯ Ă " x P R d | D y P B B m , } x ´ y } ď c ¨ λ kim ´ d * , and therefore A m (cid:52) supp ´ P p k im q ω r ...i m s ¯ Ă " x P R d | D y P B A m , } x ´ y } ď ? d ¨ c ¨ λ kim ´ d * def “ S . using e.g. [L, Lammas 2.1 & 2.2], µ d p S q ď c p d q ¨ c d ¨ λ k im ´ ¨ µ d ´ pB A m q and hence µ d ´ A m (cid:52) supp ´ P p k im q ω r ...i m s ¯¯ ď c p d q ¨ c d ¨ λ k im ´ ¨ µ d ´ pB A m q , where c p d q is a constant that depends on the dimension d . Bounding the number of tilesin a region by the volume of the region divided by the smallest volume of a prototile, weobtain a constant c ą d , a , P and (cid:37) so that ˇˇˇ r A m s T ω ´ P p k im q ω r ...i m s ˇˇˇ ď ” A m (cid:52) supp ´ P p k im q ω r ...i m s ¯ı T ω ď c ¨ λ k im ´ ¨ µ d ´ pB A m q . (4.7)The above computations hold for P p k im q η r ...i m s instead of P p k im q ω r ...i m s as well, and so we also have ˇˇˇ r A m s T η ´ P p k im q η r ...i m s ˇˇˇ ď c ¨ λ k im ´ ¨ µ d ´ pB A m q . (4.8)Combining (4.7) and (4.8) we obtain that ˇˇ r A m s T ω ´ r A m s T η ˇˇ ě ˇˇˇ P p k im q ω r ...i m s ´ P p k im q η r ...i m s ˇˇˇ ´ c ¨ λ k im ´ ¨ µ d ´ pB A m q . (4.9)Since ω p i m q ‰ η p i m q , and by Lemma 4.1 and property (1) of Proposition 4.3, we have ˇˇˇ P p k im q ω r ...i m s ´ P p k im q η r ...i m s ˇˇˇ ě c | λ t | k im . (4.10)Relying on (2.1), let c ą H d ´ pB supp p P qq times a constant that depends on d suchthat µ d ´ pB A m q ď c ` ξ k im ˘ d ´ “ c ´ λ p d ´ q{ d ¯ k im , (4.11)then by (4.10) and 4.11 we have ˇˇˇ P p k im q ω r ...i m s ´ P p k im q η r ...i m s ˇˇˇ { µ d ´ pB A m q ě c c ˜ | λ t | λ p d ´ q{ d ¸ k im . (4.12)Note that A m P Q ˚ d , thus plugging (4.9) and (4.12) into (2.5) we obtain that ˇˇ r A m s T ω ´ r A m s T η ˇˇ µ d ´ pB A m q ě c c ˜ | λ t | λ p d ´ q{ d ¸ k im ´ c λ k im ´ . (4.13)In view of (4.6), ˜ | λ t | λ p d ´ q{ d ¸ k im “ ˜ | λ t | λ p d ´ q{ d ¸ h im ´ , λ k im ´ “ ´ λ h ¯ h im ´ and λ h ă | λ t | λ p d ´ q{ d , ONTINUOUSLY MANY BD NON-EQUIVALENCES IN SUBSTITUTION TILING SPACES 11 which implies that the quantity on the right hand side of (4.13) tends to infinity with m .Then by Corollary 2.7, the proof of the lemma and hence of Theorem 1.1 is complete. (cid:3) References [ACG] J. Aliste-Prieto, D. Coronel, J. M. Gambaudo,
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