The Tribonacci Substitution

We study the discretised segments generated by the iterated Tribonacci substitution and the projections of the integer points on them to some plane. After suitable transformations we get a sequence of finite two-dimensional words which tends to a doubly rotational word on Z. (Without scaling we would get the Rauzy fractal.) As an introduction we start with the corresponding case of the Fibonacci substitution. 1. The Fibonacci Word If there would exist Miss Word elections, the Fibonacci word would be an excellent candidate to win. In this section we give an overview of the properties of the Fibonacci word. For background information for this and other sections we refer to [L] and [B]. The Fibonacci substitution is the substitution φ over the 2-letter alphabet A := {0, 1} defined by φ(0) = 01, φ(1) = 0. If we start with 0 and repeatedly apply φ we get successively u0 = 0 u1 = 01 u2 = 010 u3 = 01001 u4 = 01001010 u5 = 0100101001001 u6 = 010010100100101001010 . . . INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(3) (2005), #A13 2 Note that un = un−1un−2 for every integer n > 1. This sequence of words converges to the so-called Fibonacci word f = (fm) ∞ m=1. If we define F0 = 0, F1 = 1 and Fn = Fn−1 +Fn−2 for any integer n > 1, then the number of symbols of un, denoted by |un|, equals Fn+2 and the number of 0’s and 1’s in un, denoted by |un|0 and |un|1, equals Fn+1 and Fn, respectively. The Fibonacci word has the following properties: • The frequency of 0 equals limn→∞ Fn+1 Fn+2 = −2 + 12 √ 5 =: γ, the frequency of 1 equals limn→∞ Fn Fn+2 = 3 2 − 1 2 √ 5 = γ, hence γ+γ = 1, [L] sect.2.1.1. Since the frequencies are irrational, the Fibonacci sequence is non-periodic. • The Fibonacci word is balanced, which means that for all subwords u, v of f of equal lengths we have ||u|1 − |v|1| ≤ 1, [L] sect.2.1.1. Note that ||u|0 − |v|0| ≤ 1 is an equivalent requirement. • The Fibonacci word is sturmian, that is, P (n) = n + 1 for every n, where P (n) equals the number of different subwords of f of length n, [L] sect.2.1.1. Because a word is (ultimately) periodic if there exists an n for which P (n) ≤ n, [CH] sect.2, a sturmian word is in this sense the most regular non-periodic word. • The Fibonacci word is a rotation word, [L] sect.2.1.2. In fact ∀m ≥ 1 : fm = { 0 if {(m + 1)γ} ∈ (0, γ] 1 if {(m + 1)γ} ∈ {0} ∪ (γ, 1) where {·} denotes the fractional part. • The Fibonacci word is a Beatty sequence, [L] sect.2.1.2. In fact ∀m ≥ 1 : fm = (m+ 1)γ − mγ . • The Fibonacci word is a cutting sequence, [S]. In fact, in the x-y-plane the broken Fibonacci halfline that you get by starting in (0, 0) and going 1 in the direction of the x-axis when fm = 0 and 1 in the direction of the y-axis when fm = 1 is an ideal discrete approximation to the halfline given by y = γx, x ≥ 0. See Figure 1. Up to now we have considered one-sided words and halflines. The definition of the broken Fibonacci word as a Beatty sequence (or as a rotation sequence) allows a straightforward extension to a biinfinite word (fm)m∈Z. Actually there is another natural extension, viz. ∀m ∈ Z : f ∗ m = (m+ 1)γ − mγ = { 0 if {(m+ 1)γ} ∈ [0, γ) 1 if {(m+ 1)γ} ∈ [γ, 1). The sequences (fm)m∈Z and (f ∗ m)m∈Z coincide except that f ∗ −1 = f0 = 0, f−1 = f ∗ 0 = 1. Furthermore fm = f−m−1 for all m > 0 (cf. Theorem 2.4). Starting at the origin and going in both directions according to (fm)m∈Z we obtain an ideal discretisation, called INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(3) (2005), #A13 3