2014/08/23 by Huang, Yuke, Zhang, Hanxiong
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1408.5495
Let ω be a factor of Doubling sequence D_∞=x1x2⋯, then it occurs in the sequence infinitely many times. Let ωp be the p-th occurrence of ω and Gp(ω) be the gap between ωp and ωp+1. In this paper, we discuss the structure of the gap sequence \Gp(ω)\p≥1. We prove that all factors can be divided into two types, one type has exactly two distinct gaps G1(ω) and G2(ω), the other type has exactly three distinct gaps G1(ω), G2(ω) and G4(ω). We determine the expressions of gaps completely. And also give the substitution of each gap sequence. The main tool in this paper is "envelope word", which is a new notion, denoted by Em,i. As an application, we determine the positions of all ωp, discuss some combinatorial properties of factors, and count the distinct squares beginning in D_∞[1,N] for N≥1.