Digit sets for connected tiles via similar matrices I: Dilation matrices with rational eigenvalues

DIGIT SETS F OR CONNECTED TILES VIA SIMILAR MA TRICES I: DILA TION MA TRICES W ITH RA TIONAL EIGENV ALUES A VRA S. LAARAKKER AND EV A CURR Y Abstract. Given any m -dimensional dilation matrix A with rational eigen- v alues, we demonstrate the existence of a digit s et D such that the attractor T ( A, D ) of the i terated function system generated by A and D is connected. W e give an easily v erified sufficient condition on A for a sp ecific digit set, which we call the cen tered canonical digit set for A , to give rise to a conne cted attractor T ( A, D ). 1. In tr oduction A dilation matrix is a matrix A ∈ M m ( Z ) that is expanding in the sense that all eigenv a lue s λ of A satisfy | λ | > 1. Let A be a dila tio n matrix. Let D b e a co mplete set of coset represe ntatives of Z m / A ( Z m ), with exactly o ne representativ e from e ach coset. Note that the num ber of dig its is equal to | A | [3 ]. The set D is called a digit set (or, more sp ecifically , a b asic digit set [9 ]) for A , and the elements d ∈ D are called digits . If F is congruent to R m / Z m , then A ( F ) ∩ Z m is a (ba s ic) digit set for A [2]. F o r exa mple, if F = [0 , 1) m , the standard fundament al domain for the la ttice Z m , then A ( F ) ∩ Z m corres p onds to the usual base b digit set for b ≥ 2 a dilation in one dimension. When F is a transla ted fundamental doma in centered a t the o rigin (whic h we will refer to as the “centered fundamental doma in” ), F =  − 1 2 , 1 2  m , we will ca ll D A := A ( F ) ∩ Z m the c enter e d c anonic al digit set for A . Let A b e a dilation ma tr ix and D any digit set for A . F or each d ∈ D , set f d ( x ) := A − 1 ( x + d ) , x ∈ R m . The collec tio n of maps { f d : d ∈ D } is an iterated function system (IFS). L e t T ( A, D ) b e the attractor of the IFS. Then [3] T ( A, D ) = { ∞ X j =1 A − j d j : d j ∈ D } . Note that T ( A, D ) is self-affine, that is , T ( A, D ) = [ d ∈ D A − 1 ( T ( A, D ) + d ) = [ d ∈ D f d ( T ( A, D )) . It is a ls o known that T ( A, D ) tiles R m by transla tion either by Z m or by a sub- lattice of Z m [8]. In fact, if all singular v alues σ o f A satisfy | σ | > 2, then ther e is a digit s et D such that T ( A, D ) tiles R m by transla tion by the full lattice Z m [2]. In this pap er, w e co nsider dila tion matrice s A having only rational eigenv alues, λ ∈ Q . W e demonstrate the existence of a digit set D ∗ A such tha t T ( A, D ∗ A ) is 1 2 A VRA S. LAARAKKER AND EV A CURR Y connected for any suc h dilatio n matrix A . The digit set D ∗ A will b e derived from the c e ntered ca nonical dig it s e t for the Jordan form J of the matrix A , relying o n the fact that J is a similar matr ix to A . Our approach thus c o nt ras ts with that of Kirat a nd Lau [6], who have studied connec tedness of sets T ( A, D ) by generalizing the usual digit sets for ba se b num b er systems in one dimension to consecutive colinear digit sets, which ar e not required to b e ba sic digit s ets in gener al. Since the question of connectedness for the sets T ( A, D ) that w e co nsider is of interest to res earchers from div erse backgrounds, including in the wav elet and measureable dynamical sys tems comm unities as well as computational and algebr a ic nu mber theory , we include a brief review of the Jorda n for m. See [4] for a more complete discussion. Definition 1. Two matric es A and B ar e s imilar if t her e exists an invertible matrix P such that A = P B P − 1 . Definition 2. L et A b e any matrix in M m ( C ) . L et λ 1 , . . . , λ r b e t he eigenvalues of A , wher e e ach λ i c orr esp onds to a distinct, irr e ducible eigensp ac e. • F or e ach eigenvalue λ i of A , the Jorda n blo ck c orr esp onding to λ i , J i , is define d as fol lows: – if the gener alize d eigensp ac e of λ i is one-dimensional, then the Jor dan blo ck c orr esp onding to λ i is also one-dimensional, with J i = λ i ; – if the gener alize d eigensp ac e of λ i has dimension k > 1 , then the Jor- dan blo ck J i is a k × k matrix with λ i in every ent ry on the diagonal, 1 in every entry on the sup er diagonal, and 0 in every other entry, J i =       λ 1 0 . . . . . . . . . 1 0 λ       . • The Jordan form of A is t he matrix J := diag { J i } . Note t hat J is un ique up to r e or dering of the eigenvalues λ 1 , . . . , λ r . It can b e shown that A and it’s Jor da n form J are similar matrice s . As well, the matrix P such that A = P J P − 1 is the matrix whose columns are the eig env ec to rs and genera lized eigenv ecto rs for the eigenv alues of A , listed in the appro priate order. When A has entries in any subset of C , even Z , Q , or R , it may s till hav e ir rational or complex eig env alues, with J, P , P − 1 ∈ M m ( C ) in general. How ever, in this pap er we consider only matrices with r ational eigenv alue s . Thu s for o ur dilation matric e s A , we will ha ve J ∈ M m ( Q ). It follows that P , P − 1 ∈ M m ( Q ) as well. 2. Methods 2.1. Using the Jordan F orm. E x per iment al evidence indicates that an attra c tor T ( A, D ) for the iterated function sy stem asso ciated with a dilation matrix A a nd centered canonical digit set D fails to b e co nnected prima r ily when A includes to o large of a skew comp onent[7], [6]. Thus our basic approach to finding digit sets for which T ( A, D ) is connected is to consider the Jor dan form of A , where as muc h of the skew comp onent as po ssible has b een re mov ed. W e can generate dig it sets for A from dig it sets for the Jordan for m J , or any matrix that is s imilar to A , a s follows. DIGIT SETS F OR CONNECTED TILES VIA SIMILAR M A TRICES I 3 Let A b e a dilation matrix, let J b e any ma tr ix tha t is similar to A (for ex- ample, the Jo rdan for m o f A ), and let P be a ny matrix that gives a s imilarity transformatio n A = P J P − 1 . Supp ose that D J is a basic digit s et for J , and write D J = { g 1 , . . . , g q } , where q = | det( J ) | = | det( A ) | . Since P is a n in vertible matrix (and thus a contin uous linear tr ansformation), D A = P D J will b e a basic digit set for A , with digits d i = P g i for i = 1 , . . . , q . Likewise, the a ttractors of the corres p onding iterated function systems are rela ted: T ( A, D A ) = ∞ X i =1 A i d i for d i ∈ D A = ∞ X i =1 ( P J P − 1 ) i d i = ∞ X i =1 P J i P − 1 d i = P ∞ X i =1 J i ( P − 1 d i ) ! = P ( T ( J, D J )) . Since the mapping defined b y P is a contin uous homeomorphism betw een T ( A, D A ) and T ( J, D J ), we may conclude the following. Lemma 2.1. The set T ( A, D A ) is c onne cte d if and only if the set T ( J, D J ) is c onne cte d. While a n y consta nt m ultiple of P will define a similarity transforma tio n betw een A and J , P is gene r ally taken to hav e determinant 1 for the Jorda n decomp osition of a ma trix A . This means, how ever, that ev en if J has in teger entries, the entries in P are not necessar ily int egra l, a nd thus the new digit set D A = P D J will no t necessarily b e in Z m . F or ex a mple, consider the matrix A and its Jordan form J , A =  3 10 0 3  , J =  3 1 0 3  . Then the matrix P 1 with determinant 1 that gives the simila rity transforma tion A = P 1 J P − 1 1 , P 1 =  √ 10 √ 10 0 1 √ 10  , is not in M 2 ( Z ), nor is the res ulting digit set P 1 D J . How ever, P = √ 10 P 1 ∈ M 2 ( Z ) also satisfies A = P J P − 1 , and will yield a digit se t D A = P D J that is in Z 2 . T o ensure that D A ⊂ Z m , w e will consider o nly matrices J ∈ M m ( Z ) and will take the sma lle st m ultiple o f P 1 such that P ∈ M m ( Z ). The columns of P are eigenvectors and generaliz ed eigenv ectors of A , sa tisfying ( A − λI ) s v for so me eigenv a lue λ of A and m ultiplicit y s ( s ≥ 2 when v is not itse lf an eigen vector), so we see that P = min { r P 1 : r > 0 , r P 1 ∈ M m ( Z ) } is well-defined. Notice that the mea sure of the set T ( A, D A ) is det P = r m times the measure of the set T ( J, D J ), so the new tiles T ( A, D A ) that we find fro m the Jordan form for A will not have measure 1 in g eneral. 4 A VRA S. LAARAKKER AND EV A CURR Y W e co njecture that the ce ntered canonica l digit set D J for J will always yield a connected a ttractor T ( J, D J ), a nd thus allow us to find a digit set D A = P D J for A that gives a connected attrac tor T ( A, D A ). In this pa pe r , we prov e the r esult in the case that A has rationa l eig e n v alues . 2.2. Tili ng Considerations. Let A b e a dilation matr ix and D a dig it set for A . By Theorem 1.1 of [8], there is a lattice Γ for which T ( A, D ) gives a la ttice tiling of R m . That is, Γ is a sublattice of Z m satisfying (1) T ( A, D ) ∩ ( T ( A, D ) + γ ) has m -dimensional Leb esgue measure 0 for all nonzero γ ∈ Γ, and (2) ∪ γ ∈ Γ ( T ( A, D ) + γ ) = R m . W e ca ll such a lattice Γ a lattic e of tr anslations for T ( A, D ). Lemma 2.2 . Ther e ex ist s a lattic e of tr anslations Γ A for T ( A, D ) with D ⊂ Γ A . Pr o of. If Γ = Z m , such as when A yields a ra dix repr esentation for Z m with digit set D [2], then the result is immediate. In this case, Γ A is the unique lattice o f translations for T ( A, D ). In g eneral, let Γ b e any lattice of tr anslations for T ( A, D ). Let { b 1 , . . . , b m } be a lattice basis for Γ. Then every γ ∈ Γ ca n be written a s a linear combination of the basis elements with integer co efficients, and R m = [ c 1 ,...,c m ∈ Z ( c 1 b 1 + · · · + c m b m + T ( A, D )) . Using the self-a ffine prop erty o f T ( A, D ), R m = A R m = [ c 1 ,...,c m ∈ Z ( c 1 Ab 1 + · · · + c m Ab m + AT ( A, D )) = [ c 1 ,...,c m ∈ Z [ d ∈ D ( c 1 Ab 1 + · · · + c m Ab m + d + T ( A, D )) . Thu s Γ A = A Γ + D is also a lattice of tr anslations for T ( A, D ). The la ttice Γ must contain 0, ther efore b y construction D ∈ Γ A .  Note that Γ = A Γ + D if and only if Γ is A -inv aria nt . This implies that if T ( A, D ) has a lattice of tra ns lations Γ tha t is not A -inv a riant, then Γ will not b e unique. Also, if Γ is the unique lattice of transla tio ns for T ( A, D ), then Γ = A Γ + D m ust b e A -inv ariant. As a partial conv erse, if Γ is A -in v a riant, then Γ A = Γ is the unique A -inv aria nt lattice of tra nslations for T ( A, D ). W e cannot determine from the a bove discussion whether T ( A, D ) may a lso have a non- A -in v a riant lattice of translations in this case, how ever. A tile T ( A, D ) need not hav e an A -inv ariant lattice of tra nslations in gener al; Lagar ia s and W ang give a n example o f a matrix A and dig it s e t D such that no lattice of tra nslations Γ for T ( A, D ) can b e A -in v ar iant ([8], equations (1.5 )). Insp e ction of the pro o f of their Theorem 1.1 shows that this can only o ccur fo r so-called “ stretched tiles” , howev er [8]. W e conjecture that centered canonical digit sets do not give rise to s tretched tiles. F or the remainder of the pap er, we fix a lattice o f translatio ns Γ A with D ⊂ Γ A . In subsequent sections, we will also assume that Γ A is A -inv ar ia nt . The following lemma holds more generally , how ever. DIGIT SETS F OR CONNECTED TILES VIA SIMILAR M A TRICES I 5 Lemma 2.3. If D is the c enter e d c anonic al digit set for A and F is the c enter e d fundamental domain for Γ A , then D = AF ∩ Γ A . Pr o of. By definition, D = A  − 1 2 , 1 2  m \ Z m . F rom Lemma 2 .2, we know tha t D ⊂ Γ A . As well, Γ A ⊆ Z m implies that ( − 1 / 2 , 1 / 2] m ⊆ F , so D ⊆ AF ∩ Γ A . Ther e are only q = | det A | po ints in the set AF ∩ Γ A , how ever, whic h is equal to the cardinality of the digit set D . Thus D = AF ∩ Γ A .  The transla tes of T = T ( A, D ) that are adjacent to the origina l tile T play an impo rtant role in determining whether T ( A, D ) is connected or disconnected. W e use the following definition fr o m Scheic her and Th uswaldner [10]. Definition 3. F or s ∈ Γ A , let B s = T ∩ ( T + s ) . The set of neig h b ours of T is the set S := { s ∈ Γ A \{ 0 } : B s 6 = ∅} . 2.3. Lev el Sets and the Iterated Approac h to Connectedness. T o show con- nectedness of a n attr a ctor T ( A, D ), we will ta ke adv ant age o f the iterated function system structure. Kira t and Lau fir st prov ed the following useful re s ult. Lemma 2. 4. [6] Supp ose that T n is a se qu enc e of c omp act, c onn e cte d subset s of R m , and t hat, in the H au s dorff metric, T = lim n →∞ T n . Then T is c onne cte d. Note that this theorem holds for a r bitrary sets T a nd T n satisfying the hypothe- ses. In our case, we will set T 0 = [0 , 1] m and re c ursively define T n := A − 1 [ d ∈ D ( T n − 1 + d ) ! . W e no te that T ( A, D ) = lim n →∞ T n in the Hausdorff metric [1]. W e then see that the tile T = T ( A, D ) is co nnected if a nd only if ther e exists an N suc h that T n is connected for all n > N . Kirat a nd Lau used Lemma 2 .4 to pr ov e a criterion for connectednes s, which we present a refinement of. Fir st, consider a finite subset B of Z m . The s e t B will generate a sublattice of Z m , which we may consider as a gra ph. Definition 4 . We say that a s et S is B -connected if S forms a c onne cte d sub gr aph of the lattic e gener ate d by B . If Γ is any su blattic e of Z m , we wil l also say that the set S is Γ-connected if S forms a c onne cte d sub gr aph of Γ . W e may now state the following cr iter ion for connec tednes s of T ( A, D ). Prop ositio n 2. 5. L et S b e the set of neighb ours of T = T ( A, D ) , and B a b asis of Γ A such that B ⊂ S . If D is B -c onne cte d, then T is c onne cte d. Our pro o f follows the ba sic metho d of Kir at and Lau. Pr o of. W e note that Le mma 2.2 g ua rantees that D ⊂ Γ A , s o that the hypo thesis that D is B -connected makes sense. T o show tha t T is connected, we first show that if any subset Q o f R m that is congruent to R m / Γ A is connected, then A − 1 ( Q + D ) is connected. 6 A VRA S. LAARAKKER AND EV A CURR Y Let d, d ′ ∈ D . W e wan t to show tha t there exists a sequence from Q + d to Q + d ′ . That is, a sequence d = d 1 , . . . , d r = d ′ such that ( Q + d j ) ∩ ( Q + d j +1 ) 6 = ∅ for j = 1 , . . . , r − 1. This means tha t the pa th from Q + d to Q + d ′ is a path through neighbour translates (translates b y e le men ts of S ) of our tile T . W e know that fo r any d, d ′ ∈ D there exists a path d = d 1 , . . . , d r = d ′ such that d j +1 − d j ∈ B s inc e D is B -connected. Also, B ⊂ S implies that Q ∩ ( Q + b ) 6 = ∅ for all b ∈ B . Thus Q ∩ ( Q + d j +1 − d j ) 6 = ∅ , a nd so ( Q + d j ) ∩ ( Q + d j +1 ) 6 = ∅ . W e see that this same s equence gives the path that we need to c o nnect Q + d and Q + d ′ . Thu s Q + D is connected whenever Q is connected, and therefor e A − 1 ( Q + D ) is connected. Let T 0 be the standar d fundamen tal domain for Γ A , a co nnected set, and let T n +1 = A − 1 ( T + D ) for n ≥ 0 as ab ov e. B y induction, with Q = T n for each n ≥ 0, we see that each T n is connected. Then, b y Lemma 2.4, T = lim n →∞ T n is connec ted a s well.  As the propo sition hints at, ev en though out g oal is to show connectedness of a set in R m , it is more practical to w ork in the dis crete setting. Thus instead of the approximating sets T n defined in the pro of ab ov e , we wish to co nsider discr ete sets D n , the level sets of the digit set D . Reca ll that the sets D n are defined as D n = { k ∈ Z m : k = n − 1 X i =0 A i d i , with d i ∈ D } , and note tha t the sets can a lso be defined recur s ively: D n := A ( D n − 1 + D ) , D 1 = D . Lemma 2.6. S upp ose that D n ⊂ Γ A . L et D b e t he c enter e d c anonic al digit set for A . The set T n is c onne cte d with no finite cut sets if and only if the set D n is Γ A -c onne cte d. Pr o of. A clos er loo k at the proo f of Lemma 2 .2 reveals that, for any fixed integer n ≥ 1, we ca n in fa c t find a lattice of tra nslations Γ A,n for T ( A, D ) such that D n ⊂ Γ A,n . Note that in o rder to have D n ⊂ Γ A for every n ≥ 1, Γ A needs to b e A -inv aria nt , how ever. The condition that D n ⊂ Γ A is required for the hypo thesis that D n is Γ A -connected to make sense. Let F b e the centered fundamental domain for the lattice Γ A (that is, a translated fundamen tal domain for Γ A that is cen tered at the o rigin). Note that F is congruent to R m / Γ A , and that the Leb esgue mea sure of F is equal to the Leb e s gue mea s ure of T . Supp ose that D is Γ A -connected. DIGIT SETS F OR CONNECTED TILES VIA SIMILAR M A TRICES I 7 Let T 0 = F T 1 = A − 1 ( T 0 + D ) = A − 1 ( F + D ) T 2 = A − 1 ( T 1 + D ) = A − 1 ( A − 1 ( F + D ) + D ) = A − 2 F + A − 2 D + A − 1 D . . . T n = A − n F + A − n D + · · · + A − 2 D + A − 1 D . Multiplying T n by A n we hav e the following equality A n T n = F + D + · · · + A n − 2 D + A n − 1 D = F + D n . By our definition o f F , this implies that A n T n = F + D n is connected with no finite cut sets (that is, no finite subset o f points s uch that, if we remov e thos e p oints, the resulting set w ould b e disconnected) if and o nly if D n is Γ A -connected. The lemma follows, since A n T n is co nnected (or has a finite cut set) if and only if T n is connected (resp ectively , has a finite cut set).  Lemma 2.4 tog ether with the above lemma yield the fo llowing cor ollary . Corollary 2.7. Supp ose that Γ A is A -invariant. If t he level sets D n ar e Γ A - c onne cte d for al l sufficiently lar ge n , then T ( A, D ) is c onne cte d. The condition tha t Γ A is A -inv ar iant is req uir ed for D n ⊂ Γ A for every n ≥ 1 . W e can simplify the c riterion tha t the D n are Γ A -connected for all sufficiently large n to a sufficient condition for connectedness of T ( A, D ) that is only slightly less genera l, but significantly easier to chec k. Let F b e the c e n tered fundamental domain for Γ A , and let S AF be the set of e dge neighbours of the par allelepip ed AF in Γ A . Tha t is, S AF consists of the neighbours g in Γ A of the set AF such that AF ∩ ( AF + g ) is not just a sing le p oint. No te that, since AF is a parallelepip ed, S AF consists of 2 m po int s, S AF = {± g 1 , . . . , ± g m } , where if { b 1 , . . . , b m } is a ba sis for Γ A consisting of neighbour s of F , then we can set g i = Ab i for each i = 1 , . . . , m . That is, S + AF := { g 1 , . . . , g m } is a basis for A (Γ A ). Theorem 2. 8. S u pp ose t hat Γ A is A -invariant. L et D b e the c enter e d c anonic al digit set for A . If ( AF ∪ ( g + AF )) ∩ Γ A is Γ A -c onne cte d for e ach g ∈ S AF , then T ( A, D ) is c onne cte d. Pr o of. Supp ose that ( AF ∪ ( g + S AF )) ∩ Γ A is Γ A -connected for each g ∈ S AF . W e will show inductively that D n is Γ A -connected for each n ≥ 1. F or the base 8 A VRA S. LAARAKKER AND EV A CURR Y F F+b F+b 2 1 AF AF+g AF+g 1 2 Figure 1. The edge neighbo ur s o f AF . Here Γ A = Z 2 ; b 1 = (1 , 0), b 2 = (0 , 1), g 1 = Ab 1 = (3 , 0), and g 2 = Ab 2 = (4 , 3). 2 1 AF+g AF AF+g (a) The hypotheses are satisfied. AF+g 1 AF+g 2 AF (b) The hypotheses are not sat isfied. Figure 2. E xamples illustr ating the hypo thesis of Theorem 2.8. In case (b), ( AF + ( g 2 + AF )) ∩ Z 2 is lattice- dis connected. case, note that D 1 = D = AF ∩ Γ A by Lemma 2.3. Thus, by hypothesis, D 1 is Γ A -connected. F or the inductive step, supp ose that D n − 1 is Γ A -connected. Then T n − 1 is connected (with no finite cut s et). Recall from the pro of of Le mma 2.6 that A n − 1 T n − 1 = D n − 1 + F . Thus A n T n − 1 = AD n − 1 + AF is connected (with no finite cut sets). Since AF is a cen tered fundament al domain for A (Γ A ), this implies that AD n − 1 is A (Γ A )-connected, equiv a lent ly , tha t AD n − 1 is S AF -connected. Since AF + g is Γ A -connected to AF for each g ∈ S AF by hypothesis, we may thus c onclude that ( AD n − 1 + AF ) ∩ Γ A is Γ A -connected. Spec ific a lly , to find a connected path in Γ A betw een a ny tw o p oints Ak 1 + d k 1 and Ak 2 + d k 2 in ( AD n − 1 + AF ) ∩ Γ A , we first find the S AF -connected pa th g 0 = Ak 1 , . . . , g r − 1 = Ak 2 betw een k 1 and k 2 . F or each step g i , we find the Γ A -connected path fr o m the o r igin 0 to g i (guaranteed to exist by the hypothesis that ( AF + ( g + AF )) ∩ Γ A is Γ A -connected for ea ch g ∈ S AF ); denote this path by d ( i ) 1 = DIGIT SETS F OR CONNECTED TILES VIA SIMILAR M A TRICES I 9 0 , . . . , d ( i ) s i = g i . Set ℓ i + j = d k 1 + g i + d ( i ) j for i = 0 , . . . , r − 1 and j = 1 , . . . s i . Then { ℓ i + j } forms a Γ A -connected pa th from Ak 1 + d k 1 to Ak 2 + d k 2 . By hypo thesis, D is Γ A -connected, so we a ppend to this the Γ A -connected path from d k 1 to d k 2 translated by Ak 2 . This gives the require d pa th in Γ A connecting our t w o p oints. Note also that ( AD n − 1 + AF ) ∩ Γ A = AD n − 1 + D = D n . Therefore we hav e shown that D n is Γ A -connected, implying that T n is connected (with no finite c ut sets).  The mo st obvious (p erha ps o nly) examples of dilation matrices A that s atisfy the hypotheses of Theor em 2.8 a re matrices that do not have to o la rge a s kew comp onent, as illustra ted in Figure 2. If the eig env alues of A are of sufficiently large magnitude, so that all singular v alues σ of A satisfy σ > 2, then D contains the standar d bas is for Z m , and thus Γ A = Z m . 3. Centered Canonical Digit Sets for J Let A b e a dilation matrix with r ational eigenv alues. Note that since A ∈ M m ( Z ), the eigenv alues ar e algebr aic inte gers , that is, ro ots o f a monic p olyno mial with int eger co efficients (the characteristic po lynomial for A ). Algebraic integers have bee n studied extensively in connection with other alg ebraic and n umber theoretic questions. F o r our purp os es, the importa nt result to note is that, if we deno te the set of all alge braic integers by A , then A ∩ Q = Z (se e , for example, Theorem 6 .1.1 of [5]). Thus we are consider ing dilatio n ma tr ices A with eig env alues { λ 1 , . . . , λ r } (for some 1 ≤ r ≤ m ), w ith λ i ∈ Z for i = 1 , . . . , r . Then the Jo rdan form J = diag { J 1 , . . . , J r } is in M m ( Z ) as well. Let m i be the size o f the Jo rdan blo ck corresp onding to the eigenv alue λ i , and J i ∈ M m i ( Z ) be the Jo rdan blo ck cor resp onding to λ i . W e will consider each Jordan blo ck s e parately . Set F i :=  − 1 2 , 1 2  m i . Note that the hypotheses | λ i | > 1 and λ i ∈ Z together imply that | λ i | ≥ 2. Thu s J i F i is a pa rallelepip ed in R m i with co rners J i        ǫ 1 2 ǫ 2 2 . . . ǫ m i − 1 2 ǫ m i 2        = 1 2        λ i ǫ 1 + ǫ 2 λ i ǫ 2 + ǫ 3 . . . λ i ǫ m i − 1 + ǫ m i λ i ǫ m i        , where ǫ j = ± 1 for j = 1 , . . . , m i , and J i F i is the conv ex h ull of this set of points, excluding the faces where ǫ j = − 1 for any j . Examples with λ i = 2 in dimensions 1, 2 , and 3 ar e shown in Fig ure 3 . Examples with λ i = 3 in dimens ions 1 , 2, and 3 are shown in Figure 4. F or a n y λ i and m i , we no te that J i F i is contained in the cube C outer :=  − λ i 2 − 1 2 , λ i 2 + 1 2  m i − 1 ×  − λ i 2 , λ i 2  , 10 A VRA S. LAARAKKER AND EV A CURR Y 0 1 −1 (a) 1 1 (b) (3/2,1/2,−1) (−1/2,1/2,−1) (1/2,−3/2,−1) (−3/2,−3/2,−1) (−1/2,3/2,1) (−3/2,1/2,1) (3/2,3/2,1) (1/2,−1/2,1) (c) Figure 3. The parallelepip ed J i F i for λ i = 2 in dimensions (a) m i = 1, (b) m i = 2, a nd (c) m i = 3. 0 1 −1 (a) 1 1 (b) (2,1,−3/2) (−1,1,−3/2) (1,−2,−3/2) (−2,−2,−3/2) (2,2,3/2) (−1,2,3/2) (−2,−1,3/2) (1,−1,3/2) (c) Figure 4. The parallelepip ed J i F i for λ i = 3 in dimensions (a) m i = 1, (b) m i = 2, a nd (c) m i = 3. and J i F i contains the cube C inner :=  − λ i 2 + 1 2 , λ i 2 − 1 2  m i − 1 ×  − λ i 2 , λ i 2  . Thu s C inner \ Z m i ⊂ J i F i \ Z m i ⊂ C outer \ Z m i . Being a cub e, C inner ∩ Z m i is lattice-connected. Observe tha t the s e t of p oints ( C outer ∩ Z m i ) \ ( C inner ∩ Z m i ) for m a one-p oint-wide shell aro und C inner ∩ Z m i DIGIT SETS F OR CONNECTED TILES VIA SIMILAR M A TRICES I 11 in a ll ex cept the m th i -dimension. Thus each p oint in this shell is lattice-adjacent to C inner ∩ Z m i with the pos sible exceptio n of the corners, or extremal p oints. That is, if C outer was truly the m i -dimensional cube o ne unit larg er than C inner in ea ch direction, the corners would not b e lattice-adjac en t to C inner , but would instead be at dis tance 2 in the taxicab metric (see Figure 5). Instead, C outer coincides 1 1 C inner Figure 5. A cub e aro und C inner extending one unit in e ach di- rection - not C outer ! with C inner in the m th i -dimension, a s shown in Figur e 6. The extremal p oints of 1 1 C outer C inner (a) λ i = 2 1 1 C inner C outer (b) λ i = 3 Figure 6. C outer and C inner in dimension m i = 2. 12 A VRA S. LAARAKKER AND EV A CURR Y C outer are thus o f the for m        ǫ 1 λ i 2 + ǫ 2 1 2 ǫ 2 λ i 2 + ǫ 3 1 2 . . . ǫ m i − 1 λ i 2 + ǫ m i 1 2 ǫ m i λ i 2        , ǫ 1 , . . . , ǫ m i = ± 1 . If λ i is o dd, the last co ordinate will not b e an in teger. If λ i is even, o nly the last co ordinate will b e an integer. In both cases, the ex tremal p oints are not integer lattice p oints, and in fact ( C outer ∩ Z m i ) \ ( C inner ∩ Z m i ) consists only of p oints that are lattice-adjac e n t to C inner ∩ Z m i . J i F i ∩ Z m i will include some p o int s of this shell and not o thers, which will v a ry dep ending on whether λ i is even or o dd; how e ver in all cases we s ee that J i F i ∩ Z m i is lattice- c o nnected. Set D J i := J i F i ∩ Z m i . Note that Γ J i contains a basis for Z m i , th us Γ J i = Z m i , and is thus a J i -inv ar iant lattice. A pplying the obse r v ation ab ove together with Theorem 2.8), we obtain the follo wing. Prop ositio n 3.1. T ( J i , D J i ) is c onne ct e d. Since J = ⊕ i J i , D J := Q i D J i is a centered ca nonical dig it set for J , and T ( J, D J ) = Q i T ( J i , D J i ). Thus we conclude: Corollary 3.2 . T ( J, D J ) is c onne ct e d. Also, tak ing P ∈ M m ( Z ) and setting D A := P D J , so that T ( A, D A ) = P T ( J, D J ): Corollary 3. 3. L et A b e any dilation matrix with only r ational eigenvalues. Ther e exists a dig it set D A for A su ch that T ( A, D A ) is c onne ct e d. References [1] BibliographyM.F. B ar nsley and H. Risi ng, F r act als Everywher e (Acad emic Press Pro- fessional, Boston, 1993). [2] BibliographyE. Curr y , ‘ R adix Represen tations, Self- A ffine Tiles, and Multiv ariable W av elets’, Pr o c. AMS 1 34 (2006), 2411-2418. [3] BibliographyK. Gr¨ oc henig and W. Madyc h, ‘Mul tiresolution analysis, Haar bases, and self-simi lar tilings’, IEEE T r ans. Inform. Thy. 38 (1992), 556-568. [4] BibliographyRoger A. Hor n and Charles R. Johnson, Matrix Ana lysis (Cam bridge U ni- v ersity Press, Cambridge, 1990). [5] BibliographyK. Ireland and M. Rosen, A Classic al Intr o duction to M o dern Numb e r The ory ( Springer-V erlag, New Y ork, 1982). [6] BibliographyI. Ki rat and K.-S. J. Lau, ‘On the connecte dness of self-affine tiles’, J. L ondon Math. So c. (2) 62 (2000), 291-304. [7] BibliographyA.S. Laarakker, ‘T opological Pr operties of Tiles and Digit Sets’, Masters thesis, Acadia Universit y , 2009. [8] BibliographyJ. Lagarias and Y. W ang, ‘In tegral Self-Affine Tiles in R n Pa rt I I: Lattice Tilings’, J. F ourier Anal. and Appl. 3 (1997), 83-102. [9] BibliographyD.W. Matula, ‘Basic Digit Sets f or Radix Representa tion’, J. ACM (4) 29 (1982), 1131-1143. [10] BibliographyK. Scheic her and J.M. Thusw aldner, ‘Neighbours of self -affine tiles i n lattice tilings’, Pr o c e ed ings of the Confer enc e F ra ctals in Gr az (eds P . Grabner and W. W o ess), T rends in Mathematics (Bir khauser, Basel, 2002), pp. 241-262.