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

Abstract

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

16 pages, 6 figures