On the connectedness of self-affine attractors
Shigeki AKIYAMA ∗and

Nertila GJINI †Abstract
Let T = T (A, D) be a self-affine attractor in Rn defined by an integral
expanding matrix A and a digit set D. In the first part of this paper, inconnection with canonical number systems, we study connectedness of T when
D corresponds to the set of consecutive integers {0, 1, . . . , | det(A)| − 1}. It
is shown that in R3 and R4 , for any integral expanding matrix A, T (A, D) isconnected.
In the second part, we study connectedness of Pisot dual tiles which play
an important role in the study of β-expansions, substitutions and symbolic
dynamical systems. It is shown that each tile of the dual tiling generated bya Pisot unit of degree 3 is arcwise connected. This is naturally expected since
the digit set consists of consecutive integers as above. However surprisingly,
we found families of disconnected Pisot dual tiles of degree 4. We even givea simple necessary and sufficient condition of connectedness of the Pisot dual
tiles of degree 4. Detailed proofs will be given in [4].

1Introduction

In this paper, we shall give a brief summary of the paper [4]. Proofs given here
are representative parts of detailed ones in [4]. Let Mn (Z) be the set of n × nmatrices with entries in Z. Let A be an expanding integral matrix in Mn (Z). The
word ‘expanding’ means that all its eigenvalues have modulus greater than 1. Let| det A| = q and D = {d1 , . . . dq } ⊂ Rn be a set of q distinct vectors, called a q−digit
set. If we put Sj (x) = A−1 (x + dj ), 1 ≤ j ≤ q, then they are contractive maps
under a suitable norm in Rn (c.f. [20]).SFurthermore it is well known that there isqa unique compact set T satisfying T = j=1 Sj (T ), which is explicitly given by
T := T (A, D) =

(∞
X

)
−i

A dji : dji ∈ D.

i=1

T is the attractor of the system {Sj }qj=1 , and it is called a self-affine tile if its
Lebesgue measure µ(T ) is positive. Basic questions and detailed studies on thetiling generated by T are found for example in J. C. Lagarias and Y. Wang [20],R. Kenyon [17], C. Bandt [8], Y. Wang [31], A. Vince [30] and their references.
One of the important aspects of the self-affine attractors is connectedness. Hata [15,Theorem 4.6] has shown that if {fj }1≤j≤m is a finite set of weak contractions ofX, then the attractor K = K(f1 , · · · , fm ) is a locally connected continuum if andonly if, for any 1 ≤ i < j ≤ m, there exists a sequence {r0 , r1 , · · · , rn , rn+1 } ⊂{1, 2, · · · , m} with r0 = i and rn+1 = j such that frk (K) ∩ frk+1 (K) 6= ∅ for
0 2000 Mathematics Subject Classification: 05B45, 52C22, 54D05, 11R06, 28A80.

∗ Supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology,

Grand-in Aid for fundamental research, 14540015.
† Supported by the Japan Society for the Promotion of Science, 12000267.

1

k ∈ {0, 1, · · · , n}. Note that if a tile is connected then it must be arcwise connected. This is seen in the proof of Hata [15, Theorem 4.6]. Thus after all arcwise connectedness and connectedness are equivalent in our framework. HaconSaldanha-Veerman [14] showed that, if | det A| = 2 and D = {0, v} ⊂ Zn is a set ofcomplete representatives of the quotient group Zn /AZn , then T (A, D) is a connected tile. Gröchenig-Haas [13] have proved the existence of connected self-similar
lattice tilings for parabolic and elliptic dilations in dimension two. Kirat-Lau [18],using a graph theoretical argument on D, rediscovered Hata’s above criterion ofconnectedness. Also Kirat and Lau showed the following sufficient condition, which
will be used in the proof of Theorems 2.1 and 2.2. Afterwards we call it Kirat-Lau
Condition.
Let A ∈ Mn (Z) be an expanding matrix with | det A| = q having a characteristic
polynomial p(x). Let D = {0, v, · · · , (q − 1)v} with v ∈ Rn \ {0}. Suppose that there
exists a polynomial g(x) ∈ Z[x] (which will be called a multiplying factor) such thath(x) = g(x)p(x) = xk + ak−1 xk−1 + ak−2 xk−2 + · · · + a1 x ± q(1)with |ai | ≤ q − 1, for 1 ≤ i ≤ k − 1. Then T (A, D) is connected.
The idea of this condition is to give a concrete point on consecutive two tiles
T + kv and T + (k + 1)v and to apply Hata’s criterion mentioned above. Using this,Kirat and Lau succeeded in proving the connectedness of a tile for a suitable digit
set in dimension 2.
In the first part of this paper, we are interested in generalizing their resultsto higher dimensional cases using digit sets corresponding to consecutive integers{0, 1, . . . , | det(A)|−1}. We need to recall the classical theory to count the number ofroots of a polynomial which are inside and outside the unit circle by using quadratic
forms and Sturm sequences for the separation of roots (briefly reviewed in [4, Section2]). All expanding polynomials are classified by this theory. We will show
Theorem 1.1. Let d = 3, 4 and A ∈ Md (Z) be an expanding matrix with | det A| = q
and D = {0, v, · · · , (q − 1)v} with v ∈ Rd \{0}. Then T (A, D) is connected.
Proofs are given separately in Theorem 2.1 and 2.2. Especially for Theorem 2.2,
we have a lot of subcases. This result gives an evidence of a widely believed speculation that all such ‘consecutive integer digit tiles’ may be connected. This gives agood contrast with the second part of this paper.
We do not intend to consider general digit sets but only use digits which correspond to consecutive integers. One reason of this restriction is that this case isessential and widely studied in relation to canonical number systems. For canonicalnumber systems and associated tilings, see Kátai-Kőrnyei [16], Kovács-Pethő [19],Gilbert [11]. Recent progress on topological studies of this tiling can be seen in
Akiyama-Thuswaldner [5, 6].
Now we shall explain the second part of this paper. Let β > 1 be a real number
which is not an integer. A greedy expansion of a positive real x in base β is an
expansion of the form:x=∞X

a−i β −i = a−N0 , a−N0 −1 , · · ·i=N0with ai ∈ [0, β) ∩ Z and a greedy condition
0≤x−

N
X

a−i β −i < β −NN02∀N ≥ N0 .

Let 1 = d−1 β −1 +d−2 β −2 + · · · be an expansion of 1 defined by the algorithm
c−i = βc−i+1 − bβc−i+1 c,

d−i = bβc−i+1 c(2)with c0 = 1, where bxc denotes the maximal integer not exceeding x.
dβ (1) = .d−1 , d−2 , · · · is called β−expansion of 1. (Here we have that d−1 = bβc.)Parry [23] has shown that a sequence x = x1 , x2 , · · · of nonnegative integers is
realized as a β−expansion of some positive real number if and only if it satisfies thefollowing lexicographical condition:∀p ≥ 0,σ p (x) <lex d∗ (1)½

dβ (1),
if dβ (1) is infinite;(d−1 , d−2 , · · · , d−n+1 , d−n − 1),ω if dβ (1) = d−1 , · · · , d−n ,where σ is a shift map which acts on right infinite words defined by σ(x1 , x2 , . . . ) =x2 , x3 . . . . Here aω is a periodic right infinite word generated by a. In this case thissequence x = x1 , x2 , · · · is called admissible. A tail of the word x is a word σ n (x)for some positive integer n.
From now on, let β be a Pisot number which is a real algebraic integer greaterthan 1 whose Galois conjugates other than itself have modulus smaller than 1. LetQ(β)≥0 be the nonnegative elements of the minimum field containing the rational
numbers Q and β. Bertrand [9] and Schmidt [26] showed that any greedy expansionof x ∈ Q(β)≥0 is eventually periodic. Here we call a Pisot unit a Pisot number whichis also a unit of the integer ring of Q(β).
The symbolic dynamical system associated to β−expansion is sofic if and only
if the β−expansion of 1 is eventually periodic. Especially when β is a Pisot number
it gives a sofic system. Thurston [29] introduced an idea to construct a self-affine
tiling generated by a Pisot unit β in connection to this sofic system. Akiyama [1]and Praggastis [24] studied in detail such self-affine tilings. G. Rauzy [25] alreadyconstructed this kind of tiling in a different approach closely related to substitutions.
This tiling has a strong connection to an explicit construction of Markov partitionsof toral automorphisms. See also P. Arnoux and Sh. Ito [7].
Let us recall the definition of this tiling according to [1]. Letwith d∗ (1) =β = β (1) , β (2) , · · · , β (r1 ) and β (r1+1) , β (r1+1) , · · · , β (r1 +r2 ) , β (r1 +r2 )be the real and the complex conjugates of β, respectively. We also denote by x(j)(j = 1, 2, · · · , r1 + 2r2 ) the corresponding conjugates of x ∈ Q(β). Define a map
Φ : Q(β) → R r1 +2 r2 −1 by:³´
Φ(x) = x(2) , · · · , x(r1 ) , <(x(r1 +1) ), =(x(r1 +1) ), · · · , <(x(r1 +r2 ) ), =(x(r1 +r2 ) ) .
Let A = . a−1 , a−2 , · · · be a greedy expansion in base β. Define SA to be the setof elements of Z[β]≥0 whose greedy expansion has its fractional part A. In otherwords we just classify all elements of Z[β]≥0 by their fractional parts and map via
Φ to have a protile TA = Φ(SA ). Akiyama [3] has shown that the Euclidean spaceis covered by these tiles, there are only finitely many tiles up to translation and thenumber of tiles coincides with the number of different tails of the β−expansion of
1. So, unlike in the first part of the paper, the dual tiling of the space has several
tile types (see [3]), and they are obtained by a graph-directed iterated function
system, rather than the standard iterated function system (see [4]). The finitenesscondition in [10] or its weaker version in [3] implies that these TA will give a nonoverlapping tiling of the space Rr1 +2r2 −1 (see also [1]).
The second aim of this paper is to explore connectedness problem of resulting
tiles of Pisot dual tiling of low degree. We use again the classical theory on the3separation of roots of polynomials, and a sufficient condition for connectedness of
Pisot dual tiles which is established in Akiyama-Gjini [4, Theorem 4.1]. We prove
that
Theorem 1.2. Each tile corresponding to a Pisot unit β is arcwise connected if
dβ (1) is finite and terminates with 1.
Our main result is:
Theorem 1.3. Let β be a Pisot unit of degree 3 or 4 defined by the monic polynomial
p(x) ∈ Z[x]. If deg β = 3 or p(0) = 1 then each tile is connected. If deg β = 4 andp(0) = −1 then each tile is connected if and only if
a + c − 2[β] 6= 1
for p(x) = x4 − ax3 − bx2 − cx − 1.
These statements are a combination of Theorem 3.1, 3.2, 3.3 and 3.4. In spiteof a quite simple nature of the statements, the proof is pretty involved having a lotof subcases.
In fact, if deg β = 4, p(0) = −1 and a + c − 2[β] = 1, there exists a disconnected
tile. As far as we know, no example of disconnected Pisot dual tiles was known
before. As these tiles are generated by consecutive integers, it was even expectedthat Pisot dual tiles are always connected. Thus this result gives an unfortunate
surprise that there exists a concrete family of Pisot units one of whose dual tiles is
disconnected. The idea of the proof of disconnectedness is found in Lemma 3.1.

10
5
2

4

6

-5
-10
-15
Figure 1: The projection of a disconnected tile generated by the Pisot unit β withthe minimal equation x4 − 3x3 − 7x2 − 6x − 1 = 0

2Connectedness of self-affine tiles generated by
an expanding matrix

First suppose that tiles are generated by an expanding matrix A of degree 3. Our
proof of connectedness is based on the following lemma (c.f. [27], [28]) which givesa characterization of the expanding polynomials of degree 3.

4Lemma 2.1. A polynomial p(x) = x3 + ax2 + bx + c with integer coefficients is
expanding if and only if½| b − ac | < c2 − 1(3)| b + 1 | < | a + c |.
Theorem 2.1. Let A ∈ M3 (Z) be an expanding matrix with | det A| = q and
D = {0, v, · · · , (q − 1)v} with v ∈ R3 \{0} . Then T (A, D) is connected.
Proof. Let p(x) = x3 +ax2 +bx+c with a, b, c ∈ Z be the characteristic polynomial ofA, which is expanding. We use the Kirat-Lau Condition to show the connectedness.
Among a lot of subcases, let us only show the case c ≥ 2, c ≤ b ≤ 2c − 1 and a = 1+c
for example. Then we have b ≥ c+2, −c+1 < b−2c < 0, −2c+2 ≤ 2c− 2b+1 ≤ 0.
♦ If −c+1 ≤ 2c−2b+1 ≤ 0 then the required polynomial h(x) is
x5+(c−1)x4+(b−2c−1)x3+(2c−2b+1)x2+(b−2c)x+c = (x−1)2 p(x)♦ If −2c+2 ≤ 2c−2b+1 ≤ −c then −c+1 < 3c−2b+1 ≤ 0 and −c ≤ 2b−4c−1 < −1.
¦ If 2b−4c−1 > −c then the required polynomial h(x) isx7+(c−1)x6+(b−2c)x5+(3c−2b)x4+(2b−4c−1)x3+(3c−2b+1)x2+(b−2c)x+c = (x2+1)(x−1)2p(x)¦ If 2b−4c−1 = −c then the required polynomial h(x) is
x6+(c−1)x5+(b−2c)x4+ (2c−b)x−c = (x3−2x2+2x−1)p(x)In the following part of this Section, we shall use a necessary and sufficientcondition (c.f. [27], [28]) on coefficients of polynomial p(x) of degree 4 for p(x) tobe an expanding polynomial. Also we claim that the attractor generated by an
expanding integral matrix of degree 4 is connected.
Lemma 2.2. The polynomial p(x) = x4 +ax3 +bx2 +cx+d with integer coefficientsis expanding if and only if

d≥2



|c − ad| ≤ d2 − 2|a + c| < 1 + b + d−1+b−ac+c2 +d+a2 d−2bd−acd+d2+bd2−d3 < 0
or
(4)

d
≤
−2



|c − ad| ≤ d2 − 2|a + c| < −1 − b − d−1+b−ac+c2 +d+a2 d−2bd−acd+d2+bd2−d3 > 0.
Theorem 2.2. Let A ∈ M4 (Z) be an expanding matrix with | det A| = q and
D = {0, v, · · · , (q − 1)v} with v ∈ R4 \{0} . Then T (A, D) is connected.
Remark 1. The characteristic polynomial of the matrix A is not necessary irreducible.

53

Connectedness of self-affine tiles of the tilinggenerated by a Pisot unit of low degree

Let β > 1 be a Pisot unit. Let us recall the definition of graph directed attractors andgraph directed iterated function systems. Let G = G(V, E) be a strongly connected
graph where V = {1, . . . , q} is the set of vertices and E is the set of directed edges.
Let Ei,j be the set of edges from i to j. Now for each e ∈ E define a uniformly
contractive map Fe : Rd → Rd . Then by [22, Theorem 1] there exists a uniquefamily K1 , . . . , Kq of compact non-empty sets satisfying
Ki =

q
[
[

Fe (Kj ).

(5)

j=1 e∈Ei,j

The set of contractions {Fe | e ∈ E} is called a graph directed iterated function
system and the sets Ki are called graph directed attractors. Connectedness andarcwise connectedness of these graph directed attractors are studied in [21] as well.
For words a, b, we denote by a ⊕ b the concatenation of words. Let G−1 be thenatural map defined by the following commutative diagram:×βQ(β) −−−−→ Q(β)




Φy
yΦ

(6)

Rd−1 −−−−→ Rd−1 .
G−1Then G−1 is contractive since β is a Pisot number. The set equations are given inthis form:[T.A =G−1 (T.i⊕A ),
(7)
.i⊕A

where the summation is taken over all possible i ∈ [0, β) ∩ Z such that i ⊕ A
is admissible (see [3]). Note that we identify .i ⊕ A with the corresponding β−
expansion to realize it as a non negative real number. Since there are finitely many
tiles up to translation, it is easy to show that they form graph directed self-affine
attractors by using Parry’s result mentioned in Section 1.
Define η by
η := max{µ : µ is a tail of the β − expansion of 1}
The maximum exists because, since β is a Pisot unit, β−expansion of 1 has only
a finite number of different tails. Let us call T.η the smallest tile (the name isjustified because η gives the strongest constraint on its integer parts). AkiyamaGjini [4] proved that every tile of the dual tiling generated by a Pisot unit β is
arcwise connected if
T.η ∩ (T.η − Φ(β −1 )) 6= ∅(8)where T.η is the smallest
tile. Especially if there exist ai ∈ Z (i = 1, 2, · · · ) suchP∞
that |ai | < bβc and 1 + i=1 ai Φ(β i ) = 0 then
∞
X
i=1

i−1
a+
) + Φ(η) =
i Φ(β

∞
X

i−1
a−
) + Φ(η) − Φ(1/β)
i Φ(β

i=1

−
where a+i = max{ai , 0} and ai = − min{ai , 0}. Since |ai | < bβc, the words
+ + +
− − −
· · · , a3 a2 a1 .η and · · · , a3 a2 a1 .η are admissible. So the above condition (8) issatisfied. Usually this trick works but there are some cases to be treated separately.

6In course of proofs of Theorem 3.1, Theorem 3.2 and Theorem 3.3, we are
required to check that each formal expansion is admissible.
Theorem 3.1. Let β be a Pisot unit of degree 3. Then each tile is arcwise connected.
Proof. To prove this theorem we use the characterization of Pisot units of degree 3
given by Akiyama [2]. Also we use the β− expansion of 1 (see Gjini [12]) to findthe common point of tiles which appear in (8).
To study the connectedness of each tile of the dual tiling of degree 4 first we
give the characterization of Pisot units of degree 4:
Proposition 3.1. Let β > 1 be an algebraic unit and let
p(x) = x4 − ax3 − bx2 − cx − dwith d = ±1 be its minimal polynomial. Then β is a Pisot unit if and only if(|b − 2| < a + c
a−c>0

(

for d = −1;|b| < a + c
a2 + 4b − c2 > 0

for d = 1

Theorem 3.2. Let β be a Pisot unit of degree 4 having its minimal polynomial
p(x) = x4 − ax3 − bx2 − cx + 1. Then each tile is arcwise connected.
Proof. We write βi instead of Φ(β i ) for simplicity. Let us prove only the case a ≥ 1,
a+1 ≤ b ≤ 2a and b − a − 1 ≤ c ≤ a − 1.
∗ If c−b+a ≥ 0 then bβc = a+1 and dβ (1) = .a+1, (b−a−1, c+a−b, b−c−1, c, a)ω .
P
8i
Since every conjugate of β is also a root of p(x)(x−1)(x4−1) ∞i=0 x = 0 then1−(c+1)β1 +((c−b)β2+(b−a)β3+aβ4+cβ5 ) (1−x4 )P∞

i=0 β8i = 0

and all the coefficients have absolute value less than bβc.
∗ If c − b + a = −1 and b ≥ a + 2 then dβ (1) = .a + 1, (b − a − 2, a + 1, b − a −2, 0, a−1, b−a,P
a−1, b−a−1, a)ω . Since every conjugate of β is also a root of∞2p(x)(x −x+1) i=0 x5i = 0 thenP∞
1 −(c+1)β1 −(aβ2 −β3 +cβ4 +aβ5 +cβ6 ) i=0 β5i = 0
and all the coefficients have absolute value less than bβc = a + 1.
ω∗ If c−b+a = −1 and b = a+1 then c = 0, bβc = a and dβ (1) =P.∞a, a, (a, a−1) .
2
Since every conjugate of β is also a root of the p(x)(x +1) i=0 β4i = 0 then.η = ω (a − 1, a), a, 0.η− 0.1

is a common point of the smallest tile T.η and T.η − Φ(β −1 ).

When β is a Pisot unit of degree 4 with minimal polynomial p(x) = x4 − ax3 −bx − cx − 1, we have the following Lemma:
2Lemma 3.1. If the negative root γ of the polynomial x2 − bβcx − 1 has the propertyp(γ) > 0then at least one of the tiles is not connected.
7

The rough geometric idea of this lemma is to project the tiles along the negative
conjugate direction as in Figure 1 and to show that the restriction on digits whichcomes from the sofic system, gives an obstacle to connect subdivided pieces. Using
this Lemma we get that
Theorem 3.3. Let β be a Pisot unit of degree 4 with its minimal polynomial p(x) =x4 − ax3 − bx2 − cx − 1. Then each tile is arcwise connected except for the following
cases








a ≥ 3a ≥ 3a ≥ 1a ≥ 5
c=a−1
c=a+1
c=a+3
c=a−3
1−a
1+a
5+3a
5−3a





≤ b ≤ −a 
≤ b ≤ −1 ≤ b ≤ a−1 ≤ b ≤ 2a+222
22
Proof. Let us prove only the case: c = a + 1 then a+1
2 ≤ b ≤ 2a andω
if b ≤ a−1
.a , b+1, (0 , a−b, b , b , a−b+1, 0 , b)ω
if b = adβ (1) = .a+1, 0 , 0 , (0 , a , 0 , 0 , a , a , 1).a+1, b−a−1, 2a−b+1, b−a, a , 1
if b ≥ a+1∗ For b ≤ a − 1, to show that one of the tiles is not connected, according toLemma 3.1, it is enough to prove that p(γ) > 0. Since γ 2 − aγ − 1 = 0 wehave thatp(γ) ≥ γ 4 − aγ 3 − (a − 1)γ 2 − (a + 1)γ − 1 = γ 2 (1 − γ) > 0.
P∞
∗ For b = a, since every conjugate of β is also a root of p(x)(x−1) i=0 x3i = 0,thenP∞
1+aβ1 −β2+ (β3−β4) i=0 β3i = 0
and all the coefficients have absolute value less than bβc = a + 1.
∗ For b ≥ a + 1, we use Theorem 1.2.

From the proof of this Theorem we can easily see that a + c − 2bβc = 1 for thecases when at least one of the tiles is disconnected and a + c − 2bβc ≤ 0 for the cases
when each tile is connected. So, the above theorem can be written in the following
equivalent way:
Theorem 3.4. Let β be a Pisot unit of degree 4 with its minimal polynomial p(x) =x4 − ax3 − bx2 − cx − 1. Then
• a + c − 2bβc ≤ 1,
• each tile is arcwise connected if and only if a + c − 2bβc ≤ 0.

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Shigeki Akiyama
Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050,
NIIGATA 950-2181, JAPANe-mail address: akiyama@math.sc.niigata-u.ac.jp
Nertila Gjini
Department of Mathematics, Faculty of Science, Tirana University, TIRANA, ALBANIA
e-mail address: nkokona@hotmail.com

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