A class of cubic Rauzy Fractals

# A class of cubic Rauzy Fractals

J. Bastos, A. Messaoudi, D. Smania , T. Rodrigues Departamento de Matem tica, UNESP - Universidade Estadual Paulista, Rua Cristóvão Colombo, 2265, Jardim Nazareth, 15054-000, São José do Rio Preto, SP, Brazil. Departamento de Matemática, ICMC-USP, Avenida do Trabalhador São Carlense, 400, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil. Departamento de Matemática, UNESP - Universidade Estadual Paulista, AV. Eng. Luiz Ed. Carrijo Coube, 14-01, Vargem Limpa, 17033-360, Bauru, SP, Brazil.
August 19, 2019
###### Abstract.

In this paper, we study arithmetical and topological properties for a class of Rauzy fractals given by the polynomial where is an integer. In particular, we prove the number of neighbors of in the periodic tiling is equal to . We also give explicitly an automaton that generates the boundary of . As a consequence, we prove that is homeomorphic to a topological disk.

###### Key words and phrases:
Rauzy fractals, Numeration System, Automaton, Topological Properties
###### 2000 Mathematics Subject Classification:
A.M. was supported by Brasilian CNPq grant 305939/2009-2
D.S. was partially supported by CNPq 303669/2009-8 305537/2012-1 and FAPESP 2008/02841-4, 2010/08654-1

## 1. Introduction

In 1982, G. Rauzy [29] defined a compact subset of called classical Rauzy Fractal as

 E={+∞∑i=0εiαi,εi∈{0,1},εiεi+1εi+2≠111, ∀i≥0},

where is one of the two complex roots of modulus of the polynomial .

The classical Rauzy fractal has many beautiful properties: It is a connected set, with interior simply connected, and boundary fractal. Moreover, it induces a periodic tiling of the plane modulo the group

The Rauzy fractal was studied by many mathematicians and was connected to to many topics as: numeration systems ([23],[25], [27]), geometrical representation of symbolic dynamical system ([4], [5], [6], [8], [17], [22], [28], [32], [31]), multidimensional continued fractions and simultaneous approximations ([7], [10], [9], [18]), auto-similar tilings ([2], [1], [4], [27]) and Markov partitions of Hyperbolic automorphisms of Torus ([19], [22], [27]).

There are many ways of constructing Rauzy fractals, one of them is by -expansions.

Let be a real number and . Using greedy algorithm, we can write in base as where and belong to the set where if or otherwise, where is the integer part of . The sequence is called expansion of and is also denoted by The greedy algorithm can be defined as follows (see [26] and [14]): denote by the fractional party of a number . There exists an integer such . Let and . Then for put and . We get

 x=xkβk+xk−1βk−1+⋯

if , we put If an expansion satisfies for all , it is said to be finite and the ending zeros are omitted. It will be denoted by or .

Now, assume that is a Pisot number of degree , that means that is an algebraic integer of degree whose Galois’ conjugates have modulus less than one. We denote by the real Galois conjugates of and by its complex Galois conjugates. Let and put for all

The Rauzy fractal is by definition the set

 R=Rβ={+∞∑i=0aiψi,(ai)i≥0∈Eβ},

where

 Eβ={(xi)i≥k,k∈Z|∀n≥k,(xi)n≥i≥k is a finite β expansion }.

Observe that is a compact subset of .

For example, if is a root of the polynomial , we obtain the classical Rauzy fractal .

An important class of Pisot numbers are those such that the associated Rauzy fractal has as an interior point. This numbers were characterized by Akiyama in [3]. They are exactly the Pisot numbers that satisfy

 Z[β]∩[0,+∞[⊂Fin(β) (called % property (F)) ,

where is the set of nonnegative real numbers which have a finite -expansion.

In this paper we study properties of the Rauzy fractal associated to a class of cubic unit Pisot numbers that satisfy property (F). These numbers were characterized in [1] as being exactly the set of dominant roots of the polynomial (with integers coefficients)

 Pa,b(x)=x3−ax2−bx−1, a≥0, −1≤b≤a+1.

(If add the restriction ).

In particular, this set divided into three subsets:

• , and in this case

• In this case

• and in this case , where is the Rényi -representation of (see [30]).

Geometrical and arithmetical properties of the Rauzy fractal associated to polynomials were studied in [20]. Here we will study the case . In this case the polynomial , where and , and the Rauzy fractal

 Ra={∞∑i=0aiαi,aiai−1ai−2ai−3

where is the lexicographic order on finite words.

On the other hand, consider the sequence It is known, using greedy algorithm that for all nonnegative integer can can be written as . The sequence is called a greedy R-expansion.

The Rauzy fractal is equal

 Ra={∞∑i=0aiαi, ∀N≥0 (ai)0≤i≤N is a greedy R-expansion}.

We will also study properties of another set very closed to the Rauzy Fractal. We call this set the -Rauzy fractal and define it by

 Ga={∞∑i=0aiαi,∀N≥0, (ai)0≤i≤N is a greedy G-expansion },

where where

The set was defined in [18] by Hubert and Messaoudi. They used it to prove that is the sequence of best approximations of the vector (for a certain norm on called the Rauzy norm

In the case where and it is known (see [18]) that the set of -expansions is equal to the set of that satisfy the following conditions:

 εiεi−1εi−2εi−3

and the initial conditions

 ε0

Observe that the above initial conditions from the fact that:

Many topological properties of are known (see [1, 16, 22, 25, 29]): It’s a connected compact subset of , with interior simply connected and fractal boundary, moreover it induces a periodic tiling of the plane modulo . It can be also seen as geometrical realization of the dynamical system associated to the substitution defined by: .

To our knowledge, geometrical and topological properties of the set were not yet studied. In this paper, we show that induces a periodic tiling of the complex plane. We also construct an explicit finite state automaton that generates both boundaries of and . With this we prove that for all has neighbors while has neighbors (in the periodic tiling). The interest of giving explicitly the automaton remains in the fact that the study of properties of give topological and metrical information about the boundary and .

Here, we prove that the boundary of is homeomorphic to a topological circle. This study can be done for all integer .

The paper is divided by the following manner. In the second section, we give some notations. In the third section, we study some properties of the boundary of , in the fourth section, we construct an explicit finite state automaton that recognizes the boundaries of and for all . The fifth section is devoted to the study topological properties of the boundary of . In particular, using the automaton, we prove that the boundary of is homeomorphic to a circle.

## 2. Notations and definitions

Denote by (resp. ) the set of sequences belonging to such that, there exists an integer satisfying and for all , moreover for all , the sequence is a -expansion (resp. -expansion ). That is

and Observe that

We will identify a sequence belonging to such that for all with the sequence

Let be an element of . Assume that there exists such that for all This sequence will be denoted by .

For technical reasons, we will consider

 Ra={∞∑i=2aiαi,aiai−1ai−2ai−3

and

 Ga={∞∑i=2aiαi, ∀ N≥2 ,(ai)2≤i≤N is a Greedy G-expansion}.

## 3. Properties of Ga and its boundary

###### Theorem 3.1.

The set induces a periodic tiling of the complex plane, that is,

1. ;

2. implies que .

###### Remark 3.2.

The proof can be deduced from [29] (done in case of Rauzy fractal , see also [11]). For clarity, we will give the proof here.

Consider the sequence by . Then for all integer .

###### Proposition 3.3.

The following properties are valid:

• All natural integer can be written by unique way as where .

• Let and be two elements of (resp. ) such that and If then and for all and for all

• Let (resp. ) then (resp. ). In particular, (resp. ).

• Let then there exist a sequence such that .

• For all we have . In particular if then where and

• Let and be elements of (resp. ). Then if, only if .

• Let such that then and are -Linearly independent.

###### Remark 3.4.

The results given in Proposition 3.3 are classical. For i) and vi), see [21]. For ii) see [18]. The results iii) and vii) can be found in [1].

For iv), see [13]. v) is left to the reader and can bem done by induction.

Proof of Theorem 2.1.

Let and . Using item (vii) of Proposition 3.3 and Kronecker’s Theorem, we deduce that the set is dense in . Then there exists a sequence such that

 zk=nkα2+pkα+rk, nk∈N, pk, qk∈Z

and for all . Let where and are defined in item (v) in Proposition 3.3. We have .

On the other hand,

 Ak=nkα2+pkα+rk+(r(nk)−pk)α+(s(nk)−rk)=zk+tkα+mk,

where e . Then, .

On the other hand, for all

 ∣z+tkα+mk∣≤∣z−zk∣+∣zk+tkα+mk∣<ϵ+d,

where is the diameter of . Since is a lattice, there exists an increasing sequence of integer numbers such that for all   Then, there exist such that and for all . As and is a closed set, we have that

To prove, item b), it is sufficient to establish that if where then .

Assume that there exist and an element such that . Thus there is an integer such that for all

 (1) n∑i=2εiαi+p+qα∈Ga.

Case 1: The set is infinite.

In this case, as then there exists a integer such that By item (iv) of Proposition 3.3 we deduce that

 (2) N∑i=2εiβi+p+qβ=M∑i=ldiβi, where(di)l≤i≤M∈E(R), l, M∈Z.

From (1) and (2) we have that .

Therefore, from item (ii) of Proposition 3.3, we have for all Then,

 ∑Ni=2εiβi+p+qβ=∑Mi=2eiβi=∑Mi=ldiβi.

According to item (v) of Proposition 3.3, we have

 ˜nβ2+(r(˜n)+q)β+(s(˜n)+p)=˜lβ2+r(˜l)β+s(˜l).

where and . Therefore, and for all (by (i) of Proposition 3.3). Thus, .

Case 2: The set is finite.

Let . If then we use the same argument than in case 1.

Assume that We have

 N∑i=2εiαi+p+qα=∞∑i=2diαi∈Ga⊂Ra, where (di)i≥2∈E(R).

Since is a interior point of (item (iii) of Proposition 3.3), then there exists a non-negative integer such that

 −p−qα+M∑i=2diαi=∞∑i=2eiαi∈Ra.

Since then where and . Therefore,

 −p−qα+M∑i=2diαi=K∑i=lfiαi=∞∑i=2eiαi.

By item (ii) of Proposition 3.3, we deduce that for all and by the same argument used in case 1, we have that

###### Proposition 3.5.

The boundary of satisfies the following properties:

• where is a finite set belonging to , whose cardinality is even and greater than or equal to and .

• Let then there exist and such that and .

Proof:

1. Let , then there exists a sequence such that

By Theorem 3.1 (a), there exists a sequence of elements such that for all Then is bounded. Since is a discrete group then there exists a subsequence such that for all Since , we have . Therefore,

 ∂Ga⊂⋃p∈Z+ZαGa∩(Ga+p).

On the other hand, if Then by Theorem 3.1 (b), . Therefore, . Hence, where . Since , we deduce that is finite set. Finally, the cardinality of is even because if then .

Now, we prove that .

In fact, it’s easy to see that can be written in the following ways:

 −α3=(a−1)∑∞i=1(α4i+1+α4i+2)=α+(a−2)α3+(a−1)∑∞i=1(α4i+α4i+1)=1+(a−1)α2+(a−2)α3+(a−1)∑∞i=1(α4i+α4i+1).

Hence, . Therefore, and belong to . We also show that

 z=α−1+(a−1)∑∞i=1(α4i+α4i+1)=(a−1)∑∞i=1(α4i−2+α4i+1)=α+(a−2)α2+∑∞i=1(α4i−1+α4i+2).

Then, . Therefore, belongs to .

2. Let then

 (3) z=n+pα++∞∑i=2εiαi=∞∑i=2ε′iαi,

where and . We have to consider the following cases:

1. If the set is finite, then , which contradicts item b) of Theorem 3.1.

2. Assume that the set is infinite. Let be an integer and . We have that . On the other hand, there exists an integer such that for all , . Hence and . Moreover, because otherwise . On the other hand, there exists such that for all integer (because is bounded). Then, for all integer where . Since compact, we have . Therefore, .

## 4. Definition of the automaton recognizing the points with at least two expansions

In this section we proceed to the construction of the automaton that characterize the boundary of and . The set of states of the automaton (see Theorem 4.2) is the set

Let and be two states. The set of edges is the set of satisfying . The set of initial states is .

Let us explain the behaviour of this automaton. Let and belonging (resp. E(G)), and . Suppose . For all we put

 (4) Ak(ε,ε′)=α−k+2k∑i=l(εi−ε′i)αi.

In the following we prove that all the , , belong to . Clearly, for all ,

 (5) Ak+1(ε,ε′)=Ak(ε,ε′)α+(εk+1−ε′k+1)α2.

Let be the smallest integer such that . Hence for . Suppose . Then, . From (5) we deduce which should belong to . Hence if or if , where . Continuing by the same way and using the fact that the set of states is finite, we obtain a finite state automaton.

###### Remark 4.1.

The idea of using finite state automaton to recognize points that have at least 2 - expansions is old. It was done in the case of where is a Pisot number and the digits belong to a finite set of integer numbers by Frougny in [14]. In [34] Thurston proved the same result in the case where is a Pisot complex numbers and the digits are in a finite subset of algebraic integers in (see also [17], [24]). The difficulty remains in the fact that it is not easy to find exactly the set of states. The classical method uses the modulus of . In this work, we give a method which does not use the modulus of , with this we could find all the states for the automata associated to a class of cubic Pisot unit numbers.

### 4.1. Characterization of the points with two expansions

Let and in (resp. ) where and