Abstract
A D0L word on an alphabet Σ =\0,1,\ldots,q-1\ is called symmetric if it is a fixed point w=\varphi(w) of a morphism \varphi:Σ ^* → Σ ^* defined by \varphi(i)=øverlinet_1 + i øverlinet_2 + i\ldots øverlinet_m + i for some word t_1t_2\ldots t_m (equal to \varphi(0)) and every i ∈ Σ ; here øverlinea means a \bmod q. We prove a result conjectured by J. Shallit: if all the symbols in \varphi(0) are distinct (i.e., if t_i ≠q t_j for i ≠q j), then the symmetric D0L word w is overlap-free, i.e., contains no factor of the form axaxa for any x ∈ Σ ^* and a ∈ Σ .
Highlights
In his classical 1912 paper [15], A
Thue gave the first example of an overlap-free infinite word, i. e., of a word which contains no subword of the form axaxa for any symbol a and word x
Theorem 2 If φ : Σ∗q → Σ∗q is a growing symmetric morphism, and if all symbols occurring in φ(0) are distinct, the fixed point w = w(φ) is overlap-free
Summary
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Shallit: if all the symbols in φ(0) are distinct (i.e., if ti = t j for i = j), the symmetric D0L word w is overlap-free, i.e., contains no factor of the form axaxa for any x ∈ Σ∗ and a ∈ Σ.
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