Combinatorical and Algebraic Structures Seminar

Session details

Date: 11.11.2008
Speaker: Karel Klouda, FJFI, České vysoké učení technické
Title: Index of infinite words associated with quadratic Parry numbers
Abstract: (spoluautoři: Ľ. Balková, E. Pelantová)
We are interested in the maximal powers of factors of $u_\beta$ and in the index of $u_\beta$, where $u_\beta$ is a~right-sided infinite word coding distinct distances between neighboring nonnegative $\beta$-integers. $\beta$ is called Parry number if these distances take only finite values and so if $u_\beta$ is an infinite word over a finite alphabet. For a~finite word $w$, the concatenation $w^n$ of $n$ words $w$ is said to be the $n$-{\em th power} of $w$, $n \in \mathbb N$. Even rational powers may be defined. If we limit our consideration to factors of an infinite word $u$, the ratio of the length of the maximal power of $w$ contained in $u$ and the length of $w$ gives us the {\em index} of $w$, denoted $\text{ind}(w)$. The {\em index of the infinite word} $u$ is then naturally defined as $$\text{ind}(u)=\sup\{\text{ind}(w)\bigm | w \ \text{factor of} \ u\}.$$ It is well-known that $u_\beta$ can be obtained as a fixed point of a substitution. For quadratic non-simple Parry numbers this substitution is given by $\varphi(0)=0^p1$ and $\varphi(1)=0^q1$, where $p \geq q+1$. Employing this fact we will prove that $$ \text{ind}(u_\beta)= \begin{cases} p+1+\frac{2q+1-p}{\beta-1} & \text{ if } p < 3q + 1 \\ \text{ind}(w_0) \geq p+1+\frac{2q+1-p}{\beta-1}& \text{ otherwise,} \end{cases} $$ where $\text{ind}(w_0)$ is the maximal (rational) power of a certain factor $w_0$.

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