Tilings Generated by Ito-Sadahiro and Balanced ( − β )-numeration Systems

Let β > 1 be a cubic Pisot unit. We study forms of Thurston tilings arising from the classical β-numeration system and from the (−β)-numeration system for both the Ito-Sadahiro and balanced definition of the (−β)-transformation.


Introduction
Representations of real numbers in a positional numeration system with an arbitrary base β > 1, so-called β-expansions, were introduced by Rényi [10].During the fifty years since the publication of this seminal paper, β-expansions have been extensively studied from various points of view.
This paper considers tilings generated by β-expansions in the case when β is a Pisot unit.A general method for constructing the tiling of a Euclidean space by a Pisot unit was proposed by Thurston [11], although an example of such a tiling had already appeared in the work of Rauzy [9].Fundamental properties of these tilings were later studied by Praggastis [8] and Akiyama [1,2].
In 2009, Ito and Sadahiro introduced a new numeration system [6], using a non-integer negative base −β < −1.Their approach is very similar to the approach by Rényi.Another definition of a system using a non-integer negative base −β < −1, obtained as a slight modification of the system by Ito and Sadahiro, was considered by Dombek [4].
The main subject of this paper is to transfer the construction by Thurston into the framework of (−β)numeration (both cases) and to provide examples of how tilings (for fixed β) in the positive and negative case can resemble and/or differ from each other.The paper is intended as an entry point into a study of the properties of these tilings.

Rényi β-expansions
Let β > 1 be a real number and let the transformation T β : [0, 1) → [0, 1) be defined by the prescription T β (x) := βx − βx .The representation of a number x ∈ [0, 1) of the form The β-expansion of an arbitrary real number x ≥ 1 can be naturally defined in the following way: Find an exponent k ∈ N such that x β k ∈ [0, 1).Using the transformation T β derive the β-expansion of x β k of the form , and as usual we write where the limit is taken over the usual product topology on {0, 1, . . ., β − 1} N .It can be shown that The characterization of admissible stings is given by the following theorem due to Parry.Theorem 1 ([7]) A string x 1 x 2 x 3 . . .over the alphabet {0, 1, . . ., β − 1} is β-admissible, if and only if for all i = 1, 2, 3, . .., , where lex is the lexicographical order.
Using β-admissible digit strings, one can define the set of non-negative β-integers, denoted Z β , and the set Fin(β) of those x ∈ R + whose β-expansions have only finitely many non-zero coefficients to the right from the fractional point The distances between consecutive β-integers are described in [11].It is shown that they take values in the set {Δ i | i = 0, 1, . ..},where Δ i = ∞ j=1 t i+j β j and d β (1) = t 1 t 2 . ...Moreover, the sequence coding the distances in Z β is known to be invariant under a substitution provided d β (1) is eventually periodic [5].The form of this substitution also depends on d β (1).
If we consider β an algebraic integer, then obviously Fin(β) ⊂ Z[β −1 ] + .The converse inclusion, which is very important for the construction of the tiling and also for the arithmetical properties of the system, does not hold in general.An algebraic integer β for which by the prescription where The representation of x in such a form is called the (−β)-expansion of x and is denoted

By analogy to the case of Rényi β-expansions, we use for the (
. It is shown easily that the digits x i of a (−β)-expansion belong to the set {0, 1, . . ., β }.
In order to describe strings that arise as (−β)-expansions of some , so-called (−β)admissible digit strings, we will use the notation introduced in [6].We denote l β = −β β + 1 and r β = 1 β + 1 the left and right end-point of the definition interval I β of the transformation T −β , respectively.That is and alt is the alternate order.
Recall that the alternate order is defined as follows: We say that Similarly to the Rényi case, one can define the set of (−β)-integers, denoted Z −β , using the admissible digit strings.
The set of distances between consecutive (−β)-integers has been described only for a particular class of β, cf.[3].

Balanced (−β)-numeration system
The last numeration system used in this paper is a slight modification of (−β)-numeration defined by Ito and Sadahiro.Let −β < −1 be the base and consider the transformation given by The balanced where Also in this case we use for the It is shown easily that the digits x i of a balanced (−β)-expansion belong to the set Note that sometimes d is used instead of −d.
. The two following theorems by Dombek [4] prove that also in this case the admissible strings are characterized by the balanced (−β)-expansions of the endpoints of the interval − The set of balanced (−β)-integers, denoted Z B,−β , is defined by analogy to the two previous cases.
The map Φ is used to construct the tiling in the following way.Let w = w 1 . . .w l ∈ {0, 1, . . ., β − 1} * be a finite word such that w0 ω is an admissible digit string.We define the tile T w as The properties of the tiling of the Euclidean space using tiles T w were described by Akiyama; the results are summarized in the following theorems.

Theorem 7 ([1]
) Let β be a Pisot unit of degree d with Property (F).Then • for each x ∈ Z β we have Φ(x) ∈ Inn(T ), where is the empty word and Inn(X) denotes the set of inner points of X; especially, the origin 0 is an inner point of the so-called central tile T , • for each tile T w we have Inn(T w ) = T w , • ∂(T w ) is closed and nowhere dense in R d−1 , where ∂(T w ) is the set of boundary elements of T w , ) ω with m, p the smallest possible.Then there are exactly m + p different tiles up to translation.
Thus the construction of the tiling associated to (−β)numeration follows the same lines, the corresponding mapping Φ − being defined using isomorphisms of the extension fields Q(−β) and Q(−β (j) ), and the following variant of Proposition 6 holds; its proof follows the same lines as in the proof of the original proposition.

Examples of tilings
In the rest of the paper we provide several examples of tilings associated with β cubic Pisot units, i.e., the minimal polynomial of β is of the form x 3 − ax 2 − bx ± 1. Every time all the tiles T w with w of length 0, 1, 2 are plotted.
So far no properties of tilings in the negative case similar to those in Theorem 7 and Theorem 8 have been proved.However, the following examples demonstrate that it is reasonable to anticipate that most of the properties remain valid.On the other hand, one can also observe that for a fixed β when we change the β-numeration into the (−β)-numeration (either Ito-Sadahiro or balanced) the shape and form of the tiles can be either preserved or changed slightly or completely.

Minimal polynomial
and again all three sets of integers have the same possible distances between consecutive elements, Δ i ∈ {1, β − 2, β 2 − 2β − 2, β 2 − 3β + 1}.However, in this case the associated substitutions are not conjugated (the condition is not fulfilled on exactly one of four letters) and even though the tilings do look similar, they are composed of different tiles.See Figure 2.
6.4 Minimal polynomial x 3 − 2x 2 − 1 This β is an example of a base for which two tilings (and the corresponding properties of the sets of integers) are very similar, but the third tiling differs substantially.We have

6. 2 2 = ( 1 (
Minimal polynomial x 3 − 2x 2 − 2x − 1 This β is an example of a base for which the three considered tilings almost do not change.We have d β (1) = 211 , d −β (l β ) = 201 ω , d B,−β − 1 −1)1) ω .All three sets Z β , Z −β and Z B,−β have the same set of three possible distances between consecutive elements, namely {1, β − 2, β 2 − 2β − 2}.The codings of the distances in these sets are generated by substitutions which are pairwise conjugated.Recall that substitutions ϕ and ψ over an alphabet A are said to be conjugated if there exists a word w ∈ A * such that ϕ(a) = wψ(a)w −1 for all a ∈ A. The tilings are composed of the same tiles (up to rotation).See Figure 1.