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Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words
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- Journal:
- RAIRO - Theoretical Informatics and Applications / Volume 44 / Issue 3 / July 2010
- Published online by Cambridge University Press:
- 20 July 2010, pp. 313-337
- Print publication:
- July 2010
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- Article
- Export citation
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A word u defined over an alphabet $\mathcal A$
is c-balanced (c∈$\mathbb N$
) if for all pairs of factors v, w of u of the same length
and for all letters a∈$\mathcal A$, 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 $\varphi_p$
defined by $\varphi_p$(L) = LpS, $\varphi_p$(S) = M,
$\varphi_p$(M) = Lp–1S where p> 1.
We prove that the fixed point of $\varphi_p$, formally written as $\varphi_p^\infty$
(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.
