Received 6 November 1998; revised 11 March 1999. 2000 Mathematics Subject Classification 52C20, 52C22 (primary), 28A80 (secondary). J. London Math.
ON THE CONNECTEDNESS OF SELF-AFFINE TILES IBRAHIM KIRAT KA-SING LAU
A Let T be a self-affine tile in n defined by an integral expanding matrix A and a digit set D. The paper gives a necessary and sufficient condition for the connectedness of T. The condition can be checked algebraically via the characteristic polynomial of A. Through the use of this, it is shown that in #, for any integral expanding matrix A, there exists a digit set D such that the corresponding tile T is connected. This answers a question of Bandt and Gelbrich. Some partial results for the higher-dimensional cases are also given.
1. Introduction Throughout the paper, we use Mn() to denote the set of nin matrices with entries in , and A is an expanding integral matrix in Mn(), that is, all its eigenvalues have modulus 1. Let Qdet AQ l q and let D l od , … , dqq 7 n be a set of q distinct " vectors, called a q-digit set. If we let Sj(x) l A−"(xjdj), 1 j q, then they are contractive maps under a suitable norm in n [9, p. 29] and it is well known that there is a unique non-empty compact set T satisfying T l qj= Sj(T ) [3, 8], which is " explicitly given by
(
_
*
T B T(A, D) l A−idj : dj ? D . i i i=" T is called the attractor of the system oSjqqj= , and is called a self-affine tile if its " Lebesgue measure µ(T ) is positive. In particular, a self-affine tile T(A, D) with D 7 n is called an integral self-affine tile. A simple sufficient condition for µ(T ) 0 is that D is a complete set of coset representatives of n\An [1]. One of the very interesting aspects of the self-affine tiles is the connectedness. This property has been considered by Bandt and Gelbrich [2], Gro$ chenig and Haas [5] and Hacon et al. [6]. So far, the knowledge is still very limited, and it is our main purpose in this paper to consider this property. It is almost trivial to see that in , for A l [q] and D l o0, d, … , (qk1) d q with q, d ? +, T is an interval. The converse is also true (see Corollary 4.4). Hence it is natural to consider an extension of such tiles in n in regard to the connectedness. We give a criterion for such an attractor to be a tile by the following theorem. T 1.1. Suppose that A ? Mn() is an expanding matrix with Qdet AQ l q, where q 2 is a prime. Let D l od , … , dq q with ? nBo0q, for di ? . Then T is a " self-affine tile (that is, µ(T ) 0) if and only if o, A, … , An−"q is a linearly independent set and od , … , dqq l qlod , … , dqq, where l is a nonnegatie integer with od , … , dqq, is a " " " complete set of coset representaties of q.
Received 6 November 1998 ; revised 11 March 1999. 2000 Mathematics Subject Classification 52C20, 52C22 (primary), 28A80 (secondary). J. London Math. Soc. (2) 62 (2000) 291–304
-
292
In particular, if ? nBo0q, then o, A, … , An−"q is automatically a linearly independent set. Hence T is a self-affine tile if and only if the aboe od , … , dqq is a " complete set of coset representaties of q. We call the digit set of the above form a collinear digit set. In Theorem 1.1, if q is not a prime, then we need to assume additional conditions on A and to ensure that µ(T ) 0 (Theorem 3.3). It follows easily from the theorem that if Qdet AQ l 2, then for any 2-digit set with integer entries, T has positive measure. It is also connected, as was proved in [6]. Our approach to attack the connectedness is the following general criterion. T 1.2. Let T be an attractor defined by A ? Mn() and a digit set D 7 n. Define l o(di, dj) : (Tjdi)E(Tjdj) 6, di, dj ? Dq. Then T is connected if and only if for any pair of distinct di, dj ? D, there exists a finite set odj , … , dj q 7 D such that dj l di, dj l dj and (dj , dj ) ? for 1 l kk1. "
k
"
k
l
l+"
It is easy to check that (di, dj) ? if and only if _
dikdj l A−kk, k ? DkD. k=" We show that in many cases the expression can be checked by using the characteristic polynomial of A (Proposition 5.3). This, together with Theorem 1.1 and the other results in Section 3, leads us to the following theorem. T 1.3. Let A ? M () be an expanding matrix with Qdet AQ l q (not # necessarily prime). Then there exists a digit set D l od , … , dqq 7 # such that T is a " connected tile. This answers a question mentioned in [2] (see also [5]). By using a similar technique, we also obtain some partial results in $. A more satisfactory solution in n depends on a conjecture on the characteristic polynomial of A, which we will discuss at the end of the paper. We organize the paper as follows. In Section 2, we start with some elementary facts on the tiles, and an algebraic result on the irreducible polynomials. We prove Theorem 1.1 in Section 3, and the connectedness criterion, Theorem 1.2, in Section 4. The technique of checking the criterion is discussed in Section 5 (Proposition 5.3). By using this, we prove Theorem 1.3 and some partial results in higher dimensions. In Section 6, we make some concluding remarks and pose some open problems. 2. Preliminaries Some of the results in this section are probably known. However, we will include the proofs for completeness. We begin with a simple result for the attractors. P 2.1. Let A ? Mn() be an expanding matrix with Qdet AQ l q, and let D , D be two q-digit sets with D l x jD for some x ? n. Then the two attractors " # # ! " ! T and T corresponding to (A, D ) and (A, D ) are related by " # " # T l T j(AkI )−" x . # " !
-
293
Proof. We observe that AT l (T jd ) l (T jdkx ) " " " ! ? ? d D"
d D#
l (T jdk(AkI ) (AkI )−" x ) ! " ? d D#
l ((T jdj(AkI )−" x ))kA(AkI )−" x . ! ! " ? d D#
By putting the right-most term to the left and by using the uniqueness of the attractor corresponding to (A, D ), we have T l T j(AkI )−" x . # # " ! We next consider a case with A replaced by kA. P 2.2. Let A ? Mn() be an expanding matrix with Qdet AQ l q, and let D l od , … , dqq 7 n. Suppose that there exists a ector ? n such that D l kD. " Then T(kA, D) l T(A, D)k(AkA−")−" . Proof. We first consider the case D lkD ; then
( * l ( A ((k1) d ) : d ? D* " _
T(kA, D) l (kA)−i dj : dj ? D i i i=" _
−i
i=
i
ji
ji
l T(A, D). For the case D l kD, we let Dh l "kD. Then Dh lkDh. A direct application # of the above and Proposition 2.1 yields the proposition. R 2.3. From Proposition 2.1, we see that the Lebesgue measure and the connectedness of the attractor T are invariant under a translation of the digit set. Hence we can assume without loss of generality that 0 ? D in all the cases. Also, in Proposition 2.2, the condition D l kD on the digit set D is satisfied in most of the cases in this paper. Let A ? Mn() be expanding with Qdet AQ l q, and let Λ 7 n be a full rank Ainariant lattice, that is, Λ has rank n and A(Λ) 7 Λ. Then the quotient group Λ\A(Λ) has q distinct cosets. A q-digit set D l od , … , dqq 7 Λ is called a complete " set of coset representaties of Λ\A(Λ) if Λ l qi= (dijA(Λ)) and (dijA(Λ))E " (djjA(Λ)) l 6 for i j. Using the fact that affine transformations do not affect the property of having positive Lebesgue measure, [1, Theorem 1] can be put in the following form, which we will need later. T 2.4. Let A ? Mn() be expanding, and let Λ 7 n be a full rank Ainariant lattice. Suppose that D is a complete set of coset representaties of Λ\A(Λ). Then the unique non-empty compact set T(A, D) has positie Lebesgue measure. It is easy to see that µ(T(A, D)) 0 does not imply that D is a complete set of coset representatives. For example, we can take A l [4] and D l o0, 1, 8, 9q ; then we have T(A, D) l [0, 1]D[2, 3]. We conclude this section with a proposition on the irreducible polynomials. We use [x] to denote the set of polynomials with rational coefficients, and [x] to denote the set of polynomials with integral coefficients.
294
-
P 2.5. Let p(x) l xmjam− xm−"j…ja , where ai ? and Qa Q l q is " ! ! a prime. Suppose that all the roots α hae modulus 1 ; then p(x) is irreducible in [x] and all the roots are simple. Proof. It is well known that p(x) is irreducible in [x] if and only if it is irreducible in [x] (see, for example, [7, p. 163]). Suppose that p(x) is not irreducible ; then p(x) l g(x) h(x) for some non-constant g(x), h(x) ? [x]. Since q is a prime, it follows that the constant term of one of the g(x) or h(x) must be p1, so that at least one of the roots has modulus less than or equal to 1. This contradicts the hypothesis on p(x), and the irreducibility is proved. To show that all the roots must be simple, we assume that p(x) l (xkα)% g(x) where % ? , for % 1. Let ph(x) l %(xkα)%−" g(x)j(xkα)% gh(x) be the formal derivative of p. Then ph(x) l mxm−"j(mk1) am− xm−#j…ja is in " " [x] also, and p(x) and ph(x) have a common factor (xkα). On the other hand, the division algorithm implies that there exist s(x), t(x) ? [x] such that p(x) s(x)j ph(x) t(x) l gcd ? [x]. This implies that gcd () p(x)) has a factor (xkα), and hence gcd does not equal 1 ; this contradicts the statement that p(x) is irreducible in [x] just proved. 3. Integral self-affine tiles In the following discussion we consider the collinear digit sets D l od , … , dq q. " We let Dh l od , … , dqq. " T 3.1. Suppose that A ? Mn() is an expanding matrix with Qdet AQ l q, where q 2 is a prime. Let D l od , … , dq q with ? nBo0q, for di ? . Then T is a self" affine tile (that is, µ(T ) 0) if and only if o, A, … , An−"q is a linearly independent set and Dh l qhod , … , dqq, where l is a nonnegatie integer and od , … , dqq is a complete set " " of coset representaties of q. In particular, if ? nBo0q, then o, A, … , An−"q is always a linearly independent set. Hence T is a self-affine tile if and only if the aboe od , … , dqq is a complete set of " coset representaties of q. Proof. We first prove the necessity. Since T l o_ β A−j : βj ? Dhq and µ(T ) 0, j=" j n− " it follows that o, A, … , A q is a linearly independent set. To prove the second part, we let p(x) be the characteristic polynomial of A, and C be the companion matrix of p(x). C has elements of 1 below the diagonal, the negatives of the coefficients of p(x) in the last column, and zeros at all other entries ; it is the matrix representation of the map A with respect to basis l o, A, … , An−"q, and can be represented as e l [1 0 … 0]t ? n. Then D l od e , … , dq e q 7 n, and µ(T(A, D)) 0 " " " " if and only if µ(T(C, D)) 0. Let
0
1
1 1 q d hV k(x) B exp(2πifC −kd, xg) l exp 2πi j f(C †)k e , xg , " q d?D q j= qk " where C † l qC " is an integral matrix. By [10, Theorem 2.1(iii)], µ(T(C, D)) 0 is equivalent to the following. For each m ? nBo0q, there exists a nonnegative integer
-
295
k l k(m) such that hV k(m) l 0. It follows that o1\(qk−") f(C †)k e , mg dj : dj ? Dhq is a " complete set of coset representatives of q for each m ? nBo0q (see the proof of [10, Theorem 4.1], assuming that 0 ? D by Proposition 2.1). In particular, for k l k(m), we must have qk−" Q f(C †)k e , mg dj and qk H f(C †)k e , mg dj " " for each dj ? DBo0q. Thus Dh l qlod , … , dqq with l a nonnegative integer and " od , … , dqq a complete set of coset representatives of q. " To prove the sufficiency, we can assume without loss of generality that l l 0. Let Λ be the lattice generated by the basis l o, A, … , An−"q. It is easy to show that D is a complete set of coset representatives of Λ\A(Λ) and it follows from Theorem 2.4 that µ(T ) 0. To prove the second part, we need only to show that no non-zero vector with rational entries is contained in a proper A-invariant subspace ; this will imply that is a basis of n. Suppose that has rational entries and that o, A, … , An−"q is contained in a proper A-invariant subspace of n ; hence it is a linearly dependent set. " c Aj l 0 for some c ? , not all zero. Since A and have rational We have n− j=! j j " c xj ; then h(A) l 0. It entries, we can choose cj to be rationals. Let h(x) l n− j=! j follows that h(λ) l 0 for some eigenvalue λ of A. This contradicts the assertion that p(x) is irreducible by Proposition 2.5. C 3.2. Suppose that A ? Mn() is an expanding matrix with Qdet AQ l 2. Let D l o0, q for any ? nBo0q ; then always µ(T ) 0. Proof. We can apply the theorem with q l 2, d l 0, d l 1. " #
If q is not assumed to be a prime in Theorem 3.1, then we have the modifications as in the following theorem and in Proposition 3.5. T 3.3. Suppose that A ? Mn() is an expanding matrix with Qdet AQ l q (not necessarily prime) and it has n distinct eigenalues. Let l ni= ci ui ? n, where " the ui are the corresponding eigenectors and ci 0 for all i ? o1, … , nq. Let D l od , d , … , dq q and Dh l od , d , … , dqq 9 . Then T is a self-affine tile (that is, " # " # µ(T ) 0) if Dh is a complete set of coset representaties of q. Moreoer, the aboe can be chosen to be in n. Proof. We need to show that l o, A, … , An−"q is a basis of n, and then " a Ai, where apply the same idea as in the proof of Theorem 3.1. For n− i=! i a , a , … , an− ? are not all zero, we let f(x) B a ja xj…jan− xn−" ; then we can ! " " ! " " " a Ai l n c f(λ ) u where the λ are the eigenvalues of A. Since f(x) write n− i=! i i i i i=" i can have at most nk1 roots and the λi are distinct, we have at least one non-zero " a Ai 0. Hence is a linearly independent set, and thus a basis ci f(λi) so that n− i=! i of n. To prove the last part, we consider l ni= ci ui ? n with ci ? (or in " depending on the eigenvectors ui) in the matrix form l Uc. Observe that the set C l oc : ci l 0 for some 1 i nq is the union of n (real or complex) hyperplanes and ! U −"(n) is a full-rank lattice (in n or n respectively). Hence U −"(n) ; C and there ! exist ? n, c @ C such that l Uc, that is, we can find l ni= ci ui ? n with ! " ci 0.
-
296
We do not have a complete answer on the collinear digit sets when q is not a prime and the eigenvalues of A have higher multiplicities. In #, we have a substitute for the collinear digit sets (Proposition 3.5 below), which we will use to prove Theorem 1.3. We say that two matrices A and B are -similar if there exists a unimodular matrix P ? GLn() such that P−"AP l B. L 3.4. Suppose A ? M () is an expanding matrix with repeated eigenalue λ. # Then λ is an integer and A is -similar to a matrix of the form
9λ0 aλ: . Proof. For such A the characteristic polynomial is p(x) l (xkλ)# ; it is easy to see that λ is an integer. Let x l y
9:
be an eigenvector ; then can be chosen as a vector with integral entries x, y such that gcd(x, y) l 1. Hence there exist m, mh ? such that xmkymh l 1. Let Pl
9xy mhm : .
Then a direct calculation shows that P−"AP l
9kym
: 9
: 9 aλ:
kmh x mh λ A l x y m 0
for some a ? .
Hence for A with repeated eigenvalues, we need only to examine the matrices of the form λ a Al . 0 λ
9 :
If a l 0, it is obvious that we can take Dl
(9 ij: : 0 i, j λk1*
and T(A, D) is a square. Otherwise, we have the following proposition. P 3.5. Let Al and let Dl Then µ(T(A, D)) 0.
9λ0 aλ: ,
a0
(9iaj: : 0 i, j λk1*.
Proof. Let l oae , e q and let Λ be the lattice generated by ; then Λ\AΛ has " # λ# distinct cosets. In basis , the matrix of the map A is
9λ0 1λ: , D l (9 ij: : 0 i, j λk1*
-
297
and A(Λ) contains elements of the form
9λmjmh λmh : with m, mh ? . Hence it is easy to see that if dkd h ? AΛ for dl
9 ij: ,
dh l
9 ihjh: ? D,
then λ Q ( jkjh) and λ Q (ikih), so that j l jh and i l ih. This implies that D is a complete set of coset representatives of Λ\A(Λ). We again apply Theorem 2.4 to complete the proof. 4. A criterion for connectedness In [6], it has been shown that if Qdet AQ l 2 and D l o0, q 7 n is a complete set of coset representatives of the quotient group n\An, then T(A, D) is a connected tile. In fact, we can modify this result as follows. P 4.1. Suppose that A ? Mn() is an expanding matrix with Qdet AQ l 2. Let D l o0, q such that ? nBo0q and o, A, … , An−"q is a linearly independent set. Then T(A, D) is a pathwise and locally connected tile. Proof. By Theorem 3.1, we conclude that T is a tile. Also, from the proof of [6], we see that there exists a continuous map from [0, 1] onto T (a space filling curve). It follows that T is path-wise and locally connected by the Hahn–Mazurkiewicz Theorem (see [11]). In the following, we give a general criterion of connectedness by using a ‘ graph ’ argument on D. Let (A, D) be given ; we define l o(di, dj) : (Tjdi)E(Tjdj) 6, di, dj ? Dq to be the set of ‘ edges ’ for the set D. We say that di and dj are -connected if there exists a finite sequence odj , … , dj q 7 D such that dj l di, dj l dj and (dj , dj ) ? , k k l l+" " " 1 l kk1. Let ∆D B DkD. P 4.2. Let A ? Mn() be expanding and let D ? n be a digit set. Then (di, dj) ? if and only if dikdj l _ A−kk, for k ? ∆D. k=" Proof. Note that for di, dj in D, (Tjdi)E(Tjdj) 6 if and only if dij _ A−kdj l djj_ A−kdj for some _ A−kdj , _ A−kdj ? T. The proposition k=" k=" k=" k=" k k k k follows by setting k l dj kdj . k
k
We now give our general criterion for connectedness. T 4.3. Let A ? Mn() be an expanding matrix with Qdet AQ l q, and let D l od , … , dqq 7 n be a q-digit set. Then T is connected if and only if any two " di, dj ? D are -connected.
-
298
Proof. The necessity is obvious. To prove the sufficiency, we let Sj(x) l A−"(xjdj),
j l 1, … , q
and for J l ( j , … , jk), we let SJ l Sj @ … @ Sj . There exists a large ball B such that " k " T 7 B, Sj(B) 7 B, j l 1, 2, … , q, and _
q
0
1
T l Sj(T ) l SJ(B) . k=" QJ Q=k j=" It is clear that SJ(B) is also connected. We claim that QJ Q=k SJ(B) is connected. For k l 1, we note that Sj(T ) 7 Sj(B) and that the -connectedness property on D implies that qj= Sj(B) is connected. For the inductive step kj1, we write "
0
1
0
1
SJ(B) l S SJ(B) D … DSq SJ(B) . " Q Q QJ Q=k J =k "
QJ Q=k+
By the induction hypothesis, QJ Q=k SJ(B) is connected and we can employ the argument above to conclude that QJ Q=k+ SJ(B) is connected. This proves the claim. " To show that T is connected, we assume that there exist two components T and " T such that dist(T , T ) ε 0. Since oQJ Q=k SJ(B)q_ converges to T in the k=" # " # Hausdorff metric, then for k large enough, we have dist(T, QJ Q=k SJ(B)) ε\4 and diam(SJ(B)) ε\4 for all QJ Q l k. Let
(
i l J : QJ Q l k, dist(Ti, SJ(B))
*
ε , i l 1, 2. 4
Then from dist(T , T ) ε, we have (J? SJ(B))E(J? SJ(B)) l 6. This contra" # " # dicts the assertion that QJ Q=k SJ(B) is connected. Note that in the above theorem, we do not assume that µ(T ) 0. Hence the criterion holds for all attractors T(A, D). For the one-dimensional case, we have the following simple corollary. C 4.4. Let q ? , for QqQ 2. Suppose A l [q] and D 7 is a QqQ-digit set. Then T(A, D) is a connected tile if and only if, up to a translation, D l o0, a, 2a, … , (QqQk1) aq for some a 0. Proof. The sufficiency is easy : for such D, we have T l [0, a] if q 0, and by Proposition 2.2, T l [qa\(QqQj1), a\(QqQj1)] if q 0. To prove the necessity, we assume without loss of generality that q 0 and D l o0, d , … , dq− q with 0 d … dq− . Suppose that D is not of the required form ; " " " " then there exists at least one dj such that djkdj− (dq− )\(qk1). If (dj, dj− ) ? , then " " " by Proposition 4.2 _ _ d d djkdj− l k q−" l q−" , k k " q q qk1 k=" k=" which is a contradiction. Hence (dj, dj− ) @ . In view of the increasing property of the " digits, we see that dj and dj− are not -connected and T is not connected, by the above " theorem.
-
299
5. More on connectedness α Am−j, where αj l 0, In this section, we will consider expressions sm B m j=" j p1, … , p(qk1). We will call such sm a polynomial for convenience. Also, we define pk to be the shift operator associated with a polynomial sk l kj= αj Ak−j, for αj l 0, " p1, … , p(qk1), given by k pk(sm) l A smjsk. T 5.1. Let A ? Mn() be expanding and let D l o0, , … , (qk1) q with ? nBo0q. Then T(A, D) is connected if there exist a monic polynomial sm and a shift operator pk such that pk(sm) l sm. Proof. To prove the theorem, we need to show that (dj, dj+ ) ? for each j. By " Proposition 4.2, it suffices to construct a sequence o βjq_ with β l 1 and βj l 0, j= ! ! p1, … , p(qk1) for j 1 such that _ β A−j l 0. By the hypothesis, we have j=! j plk(sm) l pk @ pk @ … @ pk(sm) JMMKMML l times l−" l Aklsmj Akjsk j=! l sm.
Let tl B A−kl( plk(sm)) l smjlj= A−kjsk and t B liml _ tl. Then t l 0 since tl l A−klsm " 0. Write A−m+"(t) l _ β A−j by inserting the full expressions for sm and sk. j=! j This series converges since starting from j l m, o βjq_ is periodic (repeated coefficients j=! −j l 0. We have β l 1 since s is monic. Therefore, o β q_ is of sk). Also, _ β A j=! j m j j=! ! the required sequence. For convenience in the later construction, we include the following corollary. C 5.2. The shift operator pk in the aboe theorem can be replaced by qk defined by qk(sm) lkAksmjkj= αj Ak−j. " Proof. Suppose there exist qk as above and a monic polynomial sm such that qk(sm) l sm ; then we can take p k(sm) l A#ksmjs k with s k lkkj= αj A#k−jj # # # " #k α A#k−j, which yields j=k+ " j−k k #k p k(sm) l A#ksmk αj A#k−j j αj−k A#k−j # j=" j=k+"
0
1
k
lkAk(qk(sm))j αj Ak−j j=" k
lkAksmj αj Ak−j j=" l sm. It follows from Theorem 5.1 that T(A, D) is connected.
The above discussion leads to the following algebraic method to check the connectedness.
-
300
P 5.3. Let A ? Mn() be an expanding matrix with Qdet AQ l q and characteristic polynomial p(x). Let D l o0, , … , (qk1) q with ? nBo0q. Suppose that there exists a polynomial g(x) ? [x] such that h(x) l g(x) p(x) l xkjak− xk−"jak− xk−#j…ja xpq, " # " with QaiQ qk1, for 1 i kk1. Then T(A, D) is connected. Proof. Let s l . We have two cases. " (i) h(0) lkq. We take pk(s ) l Akj(ak− Ak−"jak− Ak−#j…ja Ak(qk1) I ) . " # " " Then pk(s ) l h(A) j l s since h(A) l 0. By Theorem 5.1, T(A, D) is connected. " " (ii) h(0) l q. We take qk(s ) lkAkj(kak− Ak−"kak− Ak−#k…ka Ak(qk1) I ) , " " " # so that qk(s ) lkh(A) j l s . In this case, Corollary 5.2 implies that T(A, D) is " " connected. We need the following lemma for the characteristic polynomial of A ? M () # which can easily be proved by using the quadratic formula [2]. L 5.4. Let f(x) l x#jaxpq, where a ? , 1 q ? . Then f(x) is expanding if and only if QaQ q for f(0) l q, and QaQ qk2 for f(0) lkq. We can now apply Proposition 5.3 to conclude the following theorem. T 5.5. Let A ? M () be an expanding matrix with Qdet AQ l q (not # necessarily prime). Then there exists a digit set D l od , … , dqq 7 # such that T is a " connected tile. Proof. That T is a tile follows from Theorem 3.3 and Proposition 3.5 for an appropriate choice of D. To show the connectedness of T, we let p(x) l x#jaxpq be the characteristic polynomial of A. We divide our consideration into two cases. Case (i) : A has simple eigenvalues. By Theorem 3.3, we can take D l o0, , … , (qk1) q for some . By Lemma 5.4, we see that all the characteristic polynomials p(x) satisfy the condition in Proposition 5.3 with g(x) 1, except for the case p(x) l x#jqxjq. For that, we let g(x) l xk1 ; then h(x) l (xk1) p(x) l (xk1) (x#jqxjq) l x$j(qk1) x#kq again satisfies the condition in Proposition 5.3. Hence T is connected. Case (ii) : A has a repeated eigenvalue λ. We cannot apply Proposition 5.3 because the digit set D given in Proposition 3.5 is not a collinear digit set. By Lemma 3.4, we only need to examine the matrices of the form Al If a l 0, then we choose Dl
9λ0 aλ: .
(9 ij: : 0 i, j λk1*
-
301
as our digit set. Then T(A, D) l [0, 1]# is a connected tile. If a 0, we take ia : 0 i, j λk1 Dl j
(9 :
*
as in Proposition 3.5. We observe that
9:
It follows that
9:
9:
_ a a 1 a l (AkI )−" l . A−j 0 0 (λk1) 0 j="
9
for 0 α λk2 and
: 9 : 9:
9
:
_ (αj1) a αa a (λk1) a k l l A−j 0 0 0 0 j="
βa a 9βaγ :k9γk1 : l 901: l " A 9λk1 : _
−j
j=
for 0 β λk1, 1 γ λk1. Then Proposition 4.2 and Theorem 4.3 imply that T(A, D) is connected. In the rest of this section, we apply the criterion in Proposition 5.3 to consider some special connected tiles in $. The following lemma, combined with Proposition 2.2, enables us to consider only the characteristic polynomials with positive constant terms, that is, det A 0. L 5.6. Let p(x) and pg (x) denote the characteristic polynomials of A, kA ? Mn() respectiely. Then pg (x) l (k1)n p(kx). P 5.7. Let A ? M () be expanding with Qdet AQ l q l 3, 4 or 5 ; then $ there exists a q-digit set D such that T(A, D) is a self-affine connected tile. Proof. We first consider the case with det A lk3 in $. For D l o0, , 2q as in Theorem 3.1, the corresponding T is a tile. We use Proposition 5.3 to check the connectedness. There are 25 expanding characteristic polynomials p(x) with p(0) l 3 ; 17 of them are already in the form of h(x) in Proposition 5.3. The rest of them can be reduced to the form of h(x) by multiplying a polynomial g(x) as shown in Table 1. T 1. g(x) for the case det A l 3. Characteristic polynomial l p(x)
g(x)
h(x) l g(x) p(x)
x$k3xj3 x$jx#k4xj3 x$jx#k3xj3 x$j2x#j3xj3 x$j3x#j2xj3 x$j3x#j3xj3 x$j3x#j4xj3 x$j4x#j5xj3
(x#j1) (xj1) (x#j1) (xj1) xj1 xk1 xk1 xk1 xk1 (x#kxj1) (xk1)
x'jx&k2x%jx$j3 x'j2x&k2x%jx$kxj3 x%j2x$k2x#j3 x%jx$jx#k3 x%j2x$kx#jxk3 x%j2x$k3 x%j2x$jx#kxk3 x'j2x&kx%jxk3
For the case det A lk4, a direct check shows that all the characteristic polynomials have simple eigenvalues ; hence Theorem 3.3 implies that for a suitable
-
302
, D l o0, , 2, 3q will give a tile T. To check the connectedness, we see that 35 of the p(x) are already in the form of h(x) in Proposition 5.3. The rest can also be shown to satisfy such form by the multiplication of suitable polynomials g(x). (There are 16 of them ; see Table 2.) T 2. g(x) for the case det A l 4. Characteristic polynomial l p(x)
g(x)
h(x) l g(x) p(x)
x$k4xj4 x$jx#k4xj4 x$jx#k5xj4 x$j2x#k6xj4 x$j2x#k5xj4 x$j2x#k4xj4 x$j2x#j4xj4 x$j3x#j4xj4 x$j3x#j5xj4 x$j4x#j2xj4 x$j4x#j3xj4 x$j4x#j4xj4 x$j4x#j5xj4 x$j4x#j6xj4 x$j5x#j6xj4 x$j5x#j7xj4
(x#j1) (xj1) xj1 (x#j1) (xj1) (x#j1) (xj1) xj1 xj1 xk1 xk1 xk1 xk1 xk1 xk1 xk1 xk1 (xk1)# (x#j1) (xk1)#
x'jx&k3x%jx$j4 x%j2x$k3x#j4 x'j2x&k3x%jx$kxj4 x'j3x&k3x%jx$k2xj4 x%j3x$k3x#kxj4 x%j3x$k2x#j4 x%jx$j2x#k4 x%j2x$jx#k4 x%j2x$j2x#kxk4 x%j3x$k2x#j2xk4 x%j3x$kx#jxk4 x%j3x$k4 x%j3x$jx#kxk4 x%j3x$j2x#k2xk4 x&j3x%k3x$k3x#k2xj4 x(j3x'kx&k2x%k3x$kx#kxj4
For the case det A lk5, we basically go through the same process as in the case det A lk3. There are 85 expanding characteristic polynomials with constant coefficient 5 ; 26 of them are non-trivial, as shown in Table 3. T 3. g(x) for the case det A l 5. Characteristic polynomial l p(x)
g(x)
h(x) l g(x) p(x)
x$k5xj5 x$jx#k5xj5 x$jx#k6xj5 x$j2x#j5xj5 x$j2x#k5xj5 x$j2x#k6xj5 x$j2x#k7xj5 x$j3x#j6xj5 x$j3x#j5xj5 x$j3x#k5xj5 x$j3x#k6xj5 x$j3x#k7xj5 x$j3x#k8xj5 x$j4x#j7xj5 x$j4x#j6xj5 x$j4x#j5xj5 x$j5x#j8xj5 x$j5x#j7xj5 x$j5x#j6xj5 x$j5x#j5xj5 x$j5x#j4xj5 x$j5x#j3xj5 x$j5x#j2xj5 x$j6x#j9xj5 x$j6x#j8xj5 x$j6x#j7xj5
(x#j1) (xj1) xj1 (x#j1) (xj1) xk1 xj1 xj1 (x#j1) (xj1) xk1 xk1 xj1 xj1 xj1 (x#j1) (xj1) xk1 xk1 xk1 xk1 xk1 xk1 xk1 xk1 xk1 xk1 (x#j1) (xk1)# (xk1)# (x#j1) (x$j1) (xk1)#
x'jx&k4x%jx$j5 x%j2x$k4x#j5 x'j2x&k4x%jx$kxj5 x%jx$j3x#k5 x%j3x$k3x#j5 x%j3x$k4x#kxj5 x'j3x&k4x%jx$k2xj5 x%j2x$j3x#kxk5 x%j2x$j2x#k5 x%j4x$k2x#j5 x%j4x$k3x#kxj5 x%j4x$k4x#k2xj5 x'j4x&k4x%jx$k3xj5 x%j3x$j3x#k2xk5 x%j3x$j2x#kxk5 x%j3x$jx#k5 x%j4x$j3x#k3xk5 x%j4x$j2x#k2xk5 x%j4x$jx#kxk5 x%j4x$k5 x%j4x$kx#jxk5 x%j4x$k2x#j2xk5 x%j4x$k3x#j3xk5 x(j4x'kx&k3x%k3x$k2x#kxj5 x"!j4x*k2x)kx'k2x&k3x%k2xj5 x&j4x%k4x$k3x#k3xj5
-
303
6. Remarks We have seen in the last section that in many cases the characteristic polynomial can be reduced to the form in Proposition 5.3 so that the connectedness holds. We do not know whether this will hold in general. With regard to this, we give the following result of Garsia [4, Lemma 1.6] (modified into the present setting). If A ? Mn() is an expanding matrix with Qdet AQ l q (not necessarily prime), and if λ is a root of the characteristic polynomial of A, then λ satisfies an integral polynomial with coefficients 0, p1, … , pq. There is another interesting question related to connectedness : the determination of whether the tile is disk-like. Some partial results can be found in [2]. They show that in #, for A ? M () with Qdet AQ l 3, some of the connected tiles obtained with # collinear digit sets are not disk-like. In fact, there is no disk-like tile for A with det A lk3 and tr(A) lp1. We do not know much about this in general. However, in #, we have the following partial result. P 6.1. Suppose that A ? M () is an expanding matrix with Qdet AQ l q # and characteristic polynomial p(x) l x#pq. Then there exists a q-digit set D l od , … , dqq 7 # such that T(A, D) is a disk-like tile. " Proof. Let Al
9aa"$ aa#%: .
We first assume that a 0 ; then we have a lka , a lk(a#pq)\a . # % " $ " # Case (i). If Qdet AQ l q is an odd number, let l
901:
and
D l o0, p, … , pqk1\2 q.
Then T(A, D) l o(x, ka \a xjt) : QxQ Qa Q\2, QtQ "q is a parallelogram. " # # # Case (ii). If Qdet AQ l q is an even number, then we let l and l
9k10 : , D l o, … , qq if det A l q,
901: , D l o0, , … , (qk1) q if det A lkq.
Then it follows from direct calculation that 1
T(A, D) l 4
3
2
(0x, kaa"# xjt1 : 0 x a#, 0 t 1* (0x, kaa"# xjt1 : a# x 0, 0 t 1*
is again a parallelogram. It remains to consider the case a l 0. Let # 1 k1 0 Pl if a l 0 and P l $ 0 1 1
9
:
if a 0, # if a
#
9 10: if a$ 0.
0
-
304
Let Dh l PD, where D is as in cases (i), (ii), and Ah l P −"AP. Then T(A, Dh) l PT(Ah, D), and T(Ah, D) is again a parallelogram. The proof can easily be implemented on Matlab by checking the coordinates of the corner points. Acknowledgements. The authors would like to thank Professor Y. Wang for introducing this topic while he was visiting the Institute of Mathematical Sciences at the Chinese University of Hong Kong. They would also like to thank the referee for many helpful suggestions and comments about revising the paper. Thanks are also extended to Professor K. W. Leung for clarifying some algebraic facts, and to Dr S. M. Ngai and Dr H. Rao for modifying the connectedness criterion. The first author was supported by a grant from Sakarya University in Turkey and the Institute of Mathematical Sciences at the Chinese University of Hong Kong. The second author was supported by an RGC grant from Hong Kong. Note added in proof, April 2000. We have proved that Proposition 5.7 is true for any expanding A ? M (). The detail will appear elsewhere. $ References 1. C. B, ‘ Self-similar sets. 5 : Integer matrices and fractal tilings of n ’, Proc. Amer. Math. Soc. 112 (1991) 549–562. 2. C. B and G. G, ‘ Classification of self-affine lattice tilings ’, J. London Math. Soc. 50 (1994) 581–593. 3. K. J. F, Fractal geometry : mathematical foundations and applications (John Wiley, 1990). 4. A. G, ‘ Arithmetic properties of Bernoulli convolutions ’, Trans. Amer. Math. Soc. 102 (1962) 409–432. 5. K. G$ and A. H, ‘ Self-similar lattice tilings ’, J. Fourier Anal. Appl. 1 (1994) 131–170. 6. D. H, N. C. S and J. J. P. V, ‘ Remarks on self-affine tilings ’, Experiment. Math. 3 (1994) 317–327. 7. T. W. H, Algebra, 2nd edn (Springer, New York, 1996). 8. J. E. H, ‘ Fractals and self-similarity ’, Indiana Uni. Math. J. 30 (1981) 713–747. 9. J. C. L and Y. W, ‘ Self-affine tiles in n ’, Ad. Math. 121 (1996) 21–49. 10. J. C. L and Y. W, ‘ Integral self-affine tiles in n. I : Standard and nonstandard digits sets ’, J. London Math. Soc. 54 (1996) 161–179. 11. S. W, General topology (Addison–Wesley, 1970).
Department of Mathematics Uniersity of Pittsburgh Pittsburgh PA 15260 USA
Department of Mathematics Chinese Uniersity of Hong Kong Hong Kong
ibkstj!pitt.edu
kslau!math.cuhk.edu.hk
