Gap Sequence of Cutting Sequence with Slope $θ=[0;\dot{d}]$

Abstract

In this paper, we consider the factor properties and gap sequence of a special type of cutting sequence with slope $θ=[0;\dot{d}]$, denoted by $F_{d,\infty}$. Let $ω$ be a factor of $F_{d,\infty}$, then it occurs in the sequence infinitely many times. Let $ω_p$ be the $p$-th occurrence of $ω$ and $G_p(ω)$ be the gap between $ω_p$ and $ω_{p+1}$. We define the $d$ types of kernel words and envelope words, give two versions of "uniqueness of kernel decomposition property". Using them, we prove the gap sequence $\{G_p(ω)\}_{p\geq1}$ has exactly two distinct elements for each $ω$, and determine the expressions of gaps completely. Furthermore, we prove that the gap sequence is $σ_i(F_{d,\infty})$, where $σ_i$ is a substitution depending only on the type of $Ker(ω)$, i.e. the kernel word of $ω$. We also determine the position of $ω_p$ for all $(ω,p)$. As applications, we study some combinatorial properties, such as the power, overlap and separate property between $ω_p$ and $ω_{p+1}$ for all $(ω,p)$, and find all palindromes in $F_{d,\infty}$.