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 2. (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 u 0 = 0 u 1 = 01 u 2 = 010 u 3 = 01001 u 4 = 01001010 u 5 = 0100101001001 u 6 = 010010100100101001010. . .