Abelian Complexity of Infinite Words Associated with Quadratic Parry Numbers
We derive an explicit formula for the Abelian complexity of infinite words associated with quadratic Parry numbers.
math.CO↗
arXiv subjects
Ondřej Turek
Publications and source records attributed to Ondřej Turek.
Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words
A word $u$ defined over an alphabet $\mathcal{A}$ is $c$-balanced ($c\in\mathbb{N}$) if for all pairs of factors $v$, $w$ of $u$ of the same length and for all letters $a\in\mathcal{A}$, the difference between the number of letters $a$ in $v$ and $w$ is less or equal to $c$. In this paper we consider a ternary alphabet $\mathcal{A}=\{L,S,M\}$ and a class of substitutions $ϕ_p$ defined by $ϕ_p(L)=L^pS$, $ϕ_p(S)=M$, $ϕ_p(M)=L^{p-1}S$ where $p>1$. We prove that the fixed point of $ϕ_p$, formally written as $ϕ_p^\infty(L)$, is 3-balanced and that its Abelian complexity is bounded above by the value 7, regardless of the value of $p$. We also show that both these bounds are optimal, i.e. they cannot be improved.
math.CO↗
Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers
We study arithmetical and combinatorial properties of $β$-integers for $β$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq n+2\geq 3$. We determine with the accuracy of $\pm 1$ the maximal number of $β$-fractional positions, which may arise as a result of addition of two $β$-integers. For the infinite word $u_β$ coding distances between consecutive $β$-integers, we determine precisely also the balance. The word $u_β$ is the fixed point of the morphism $A \to A^{m-1}B$ and $B\to A^{m-n-1}B$. In the case $n=1$ the corresponding infinite word $u_β$ is sturmian and therefore 1-balanced. On the simplest non-sturmian example with $n\geq 2$, we illustrate how closely the balance and arithmetical properties of $β$-integers are related.
cs.DM↗