Abelian properties of Parry words

Abstract

Abelian complexity of a word $\mathbf{u}$ is a function that counts the number of pairwise non-abelian-equivalent factors of $\mathbf{u}$ of length $n$. We prove that for any $c$-balanced Parry word $\mathbf{u}$, the values of the abelian complexity function can be computed by a finite-state automaton. The proof is based on the notion of relative Parikh vectors. The approach works for any function $F(n)$ that can be expressed in terms of the set of relative Parikh vectors corresponding to the length $n$. For example, we show that the balance function of a $c$-balanced Parry word is computable by a finite-state automaton as well.

arXiv comment19 pages

References in corpus (8)

• Abelian Complexity in Minimal Subshifts
• Balance and Abelian complexity of the Tribonacci word
• Abelian Complexity of Infinite Words Associated with Quadratic Parry Numbers
• Balance properties of Arnoux-Rauzy words
• Abelian complexity and Abelian co-decomposition
• Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words
• Abelian complexity function of the Tribonacci word
• Balances of $m$-bonacci words

Cited by in corpus (1)

• Abelian complexity function of the Tribonacci word