RAIRO-Theor. Inf. Appl.
Volume 53, Number 3-4, July–December 2019
|Page(s)||125 - 151|
|Published online||01 November 2019|
Derived sequences of complementary symmetric Rote sequences
Czech Technical Univ. in Prague,
120 00 Prague 2,
Trojanova 13, Czech Republic.
2 Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 7 October 2019
Complementary symmetric Rote sequences are binary sequences which have factor complexity C(n) = 2n for all integers n ≥ 1 and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derived sequences to the prefixes of any complementary symmetric Rote sequence v which is associated with a standard Sturmian sequence u. We show that any non-empty prefix of v has three return words. We prove that any derived sequence of v is coding of three interval exchange transformation and we determine the parameters of this transformation. We also prove that v is primitive substitutive if and only if u is primitive substitutive. Moreover, if the sequence u is a fixed point of a primitive morphism, then all derived sequences of v are also fixed by primitive morphisms. In that case we provide an algorithm for finding these fixing morphisms.
Mathematics Subject Classification: 68R15
Key words: Derived sequence / return word / Rote sequence / Sturmian sequence
© EDP Sciences, 2020
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