## Overlap-Free Symmetric D0L words (2001)

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Citations: | 5 - 0 self |

### BibTeX

@MISC{Frid01overlap-freesymmetric,

author = {Anna E. Frid},

title = {Overlap-Free Symmetric D0L words},

year = {2001}

}

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### Abstract

Introduction In his classical 1912 paper [15] (see also [3]), 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. Thue's example is known now as the Thue-Morse word w TM = 01101001100101101001011001101001 : : :: It was rediscovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey [1]). The set of all overlap-free words was studied e. g. by Fife [8] who described all binary overlap-free infinite words and Seebold [13] who proved that the Thue-Morse word is essentially the only binary overlap-free word which is a fixed point of a morphism. Nowadays the theory of overlap-free words is a part of a more general theory of pattern avoidance [5]. J.-P. Allouche and J. Shallit [2] asked if the initial Thue's construction of an overlap-free wo

### Citations

126 |
Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihern. Norske vid
- Thue
- 1912
(Show Context)
Citation Context ... D0L word w is overlap-free, i.e., contains no factor of the form axaxa for any x ∈ Σ ∗ and a ∈ Σ. Keywords: overlap-free word, D0L word, symmetric morphism 1 Introduction In his classical 1912 paper =-=[15]-=- (see also [3]), 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. Thue’s example is known now ... |

118 | Combinatorics of words
- Chorut, Karhumäki
- 1997
(Show Context)
Citation Context ...i are symbols, is called (locally) p-periodic if ui = ui+p for all i ∈ {1,...,n − p} such that ui �= ⋄ and ui+p �= ⋄. The following result is a generalization of the classical Fine and Wilf’s theorem =-=[9, 6]-=-: Theorem 3 ([4]) Let u be a partial word of length n which is p-periodic and q-periodic. If u contains only one hole, and if n ≥ p + q, then u is gcd(p,q)-periodic. Now let us start the proof of Theo... |

107 |
Uniqueness theorems for periodic functions
- Fine, Wilf
- 1965
(Show Context)
Citation Context ...i are symbols, is called (locally) p-periodic if ui = ui+p for all i ∈ {1,...,n − p} such that ui �= ⋄ and ui+p �= ⋄. The following result is a generalization of the classical Fine and Wilf’s theorem =-=[9, 6]-=-: Theorem 3 ([4]) Let u be a partial word of length n which is p-periodic and q-periodic. If u contains only one hole, and if n ≥ p + q, then u is gcd(p,q)-periodic. Now let us start the proof of Theo... |

59 | The Ubiquitous Prouhet-Thue-Morse Sequence
- Allouche, Shallit
- 1999
(Show Context)
Citation Context ... Thue-Morse word wT M = 01101001100101101001011001101001.... It was rediscovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey =-=[1]-=-). The set of all overlap-free words was studied e. g. by E. D. Fife [8] who described all binary overlapfree infinite words and P. Séébold [13] who proved that the Thue-Morse word is essentially the ... |

41 |
Sur un théorème de Thue
- Dejean
- 1972
(Show Context)
Citation Context ...ord. For example, the Thue-Morse word wT M is a fixed point of a symmetric morphism ϕT M: � ϕT M(0) = 01, ϕT M(1) = 10. Symmetric D0L words include also other useful examples, such as the Dejean word =-=[7]-=-, the Keränen word [11] and others (see Section 10.5 in [12], where in particular the term “symmetric” is introduced). Note that the class of symmetric D0L words is included in a wider class of unifor... |

25 | Axel Thue’s work on repetitions in words
- Berstel
- 1992
(Show Context)
Citation Context ...overlap-free, i.e., contains no factor of the form axaxa for any x ∈ Σ ∗ and a ∈ Σ. Keywords: overlap-free word, D0L word, symmetric morphism 1 Introduction In his classical 1912 paper [15] (see also =-=[3]-=-), 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. Thue’s example is known now as the Thue-Mo... |

21 |
Partial words and a theorem of Fine and
- Berstel, Boasson
- 1999
(Show Context)
Citation Context ...lap-free. The remaining part of the paper is devoted to the proof of this result. 2 Proof of Theorem 2 Let us start with introducing some more notions and citing a result by J. Berstel and L. Boasson =-=[4]-=- which we shall need later. A partial word is a word on the alphabet Σ∪{⋄}, where the symbol ⋄ /∈ Σ is called the hole ‡ . Each hole means an unknown symbol of Σ. A (partial) word u = u1 ...un, where ... |

16 |
Abelian squares are avoidable on 4 letters
- Keränen
- 1992
(Show Context)
Citation Context ...Thue-Morse word wT M is a fixed point of a symmetric morphism ϕT M: � ϕT M(0) = 01, ϕT M(1) = 10. Symmetric D0L words include also other useful examples, such as the Dejean word [7], the Keränen word =-=[11]-=- and others (see Section 10.5 in [12], where in particular the term “symmetric” is introduced). Note that the class of symmetric D0L words is included in a wider class of uniform marked D0L words whos... |

12 |
Binary sequences which contain no BBb
- FIFE
- 1980
(Show Context)
Citation Context ...scovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey [1]). The set of all overlap-free words was studied e. g. by E. D. Fife =-=[8]-=- who described all binary overlapfree infinite words and P. Séébold [13] who proved that the Thue-Morse word is essentially the only binary overlap-free word which is a fixed point of a morphism. Nowa... |

10 | Sum of digits, overlaps, and palindromes
- Allouche, Shallit
- 2000
(Show Context)
Citation Context ...nly binary overlap-free word which is a fixed point of a morphism. Nowadays the theory of overlap-free words is a part of a more general theory of pattern avoidance [5]. J.-P. Allouche and J. Shallit =-=[2]-=- asked if the initial Thue’s construction of an overlap-free word could be generalized and found a whole family of overlap-free infinite words built by a similar principle. This paper contains a furth... |

9 |
personal communication
- Shallit
- 2004
(Show Context)
Citation Context ...lized and found a whole family of overlap-free infinite words built by a similar principle. This paper contains a further generalization of that result; its main theorem was conjectured by J. Shallit =-=[14]-=-. Let us give all the necessary definitions and state the main theorem. Consider a finite alphabet Σ = Σq = {0,1,...,q − 1}. For an integer i, let i denote the residue of i modulo q. A morphism ϕ : Σ ... |

7 |
Sequences generated by infinitely iterated morphisms
- SÉÉBOLD
- 1985
(Show Context)
Citation Context ... occurs in various fields of mathematics (see the survey [1]). The set of all overlap-free words was studied e. g. by E. D. Fife [8] who described all binary overlapfree infinite words and P. Séébold =-=[13]-=- who proved that the Thue-Morse word is essentially the only binary overlap-free word which is a fixed point of a morphism. Nowadays the theory of overlap-free words is a part of a more general theory... |

3 |
Unavoidable patterns
- Cassaigne
- 2002
(Show Context)
Citation Context ...hue-Morse word is essentially the only binary overlap-free word which is a fixed point of a morphism. Nowadays the theory of overlap-free words is a part of a more general theory of pattern avoidance =-=[5]-=-. J.-P. Allouche and J. Shallit [2] asked if the initial Thue’s construction of an overlap-free word could be generalized and found a whole family of overlap-free infinite words built by a similar pri... |

2 | Applying a Uniform Marked Morphism to a Word
- Frid
- 1999
(Show Context)
Citation Context ...2], where in particular the term “symmetric” is introduced). Note that the class of symmetric D0L words is included in a wider class of uniform marked D0L words whose properties were studied e. g. in =-=[10]-=-. Note that an infinite word w = w1w2 ...wn ..., where wi ∈ Σ, is the fixed point of the symmetric morphism ϕ if and only if ∀k ≥ 0 ∀i ∈ {1,...,m} wkm+i = wk+1 +ti. (1) Indeed, this equality means tha... |