Let Ο be a factor of Doubling sequence Dββ=x1βx2ββ―, then
it occurs in the sequence infinitely many times. Let Οpβ be the p-th
occurrence of Ο and Gpβ(Ο) be the gap between Οpβ and
Οp+1β. In this paper, we discuss the structure of the gap sequence
{Gpβ(Ο)}pβ₯1β. We prove that all factors can be divided into two
types, one type has exactly two distinct gaps G1β(Ο) and G2β(Ο),
the other type has exactly three distinct gaps G1β(Ο), G2β(Ο) and
G4β(Ο). 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,iβ. As an application, we
determine the positions of all Οpβ, discuss some combinatorial
properties of factors, and count the distinct squares beginning in
Dββ[1,N] for Nβ₯1.Comment: 14 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1408.372