Kernel words and gap sequence of the tribonacci sequence

In this paper, we investigate the factor properties and gap sequence of the Tribonacci sequence, the fixed point of the substitution σ(a, b, c) = (ab, ac, a). Let ωp be the p-th occurrence of ω and Gp(ω) be the gap between ωp and ωp+1. We introduce a notion of kernel for each factor ω, and then give the decomposition of the factor ω with respect to its kernel. Using the kernel and the decomposition, we prove the main result of this paper: for each factor ω, the gap sequence {Gp(ω)}p≤1 is the Tribonacci sequence over the alphabet {G1(ω), G2(ω), G4(ω)}, and the expressions of gaps are determined completely. As an application, for each factor ω and p∈ℕ, we determine the position of ωp. Finally we introduce a notion of spectrum for studying some typical combinatorial properties, such as power, overlap and separate of factors.