A class of cubic Rauzy Fractals
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 Properties2000 Mathematics Subject Classification:
1. Introduction
In 1982, G. Rauzy [29] defined a compact subset of called classical Rauzy Fractal as
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]), autosimilar 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
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
where
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
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)
(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
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 Rexpansion.
The Rauzy fractal is equal
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
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:
and the initial conditions
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
and
3. Properties of and its boundary
Theorem 3.1.
The set induces a periodic tiling of the complex plane, that is,

;

implies que .
Remark 3.2.
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
and for all . Let where and are defined in item (v) in Proposition 3.3. We have .
On the other hand,
where e . Then, .
On the other hand, for all
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) 
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) 
Therefore, from item (ii) of Proposition 3.3, we have for all Then,
According to item (v) of Proposition 3.3, we have
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
Since is a interior point of (item (iii) of Proposition 3.3), then there exists a nonnegative integer such that
Since then where and . Therefore,
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:

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,
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:
Hence, . Therefore, and belong to . We also show that
Then, . Therefore, belongs to .

Let then
(3) where and . We have to consider the following cases:

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

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) 
In the following we prove that all the , , belong to . Clearly, for all ,
(5) 
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