## Axel Thue's work on repetitions in words (1992)

Venue: | Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics |

Citations: | 22 - 3 self |

### BibTeX

@INPROCEEDINGS{Berstel92axelthue's,

author = {J. Berstel},

title = {Axel Thue's work on repetitions in words},

booktitle = {Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics},

year = {1992},

pages = {65--80}

}

### Years of Citing Articles

### OpenURL

### Abstract

The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.

### Citations

617 |
Combinatorics on Words
- Lothaire
- 1997
(Show Context)
Citation Context ... over three letters is square-free. 2 The Thue-Morse sequence In this section, we recall some basic properties concerning the Thue-Morse sequence. Other properties and proofs can be found in Lothaire =-=[22]-=- and Salomaa [34]. Let A = fa; bg be a two letter alphabet. Consider the morphismsfrom the free monoid A into itself defined by (a) = ab; (b) = ba Setting, for ns0, un =sn (a); vn =sn (b) one gets u 0... |

119 |
Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske vid
- Thue
- 1912
(Show Context)
Citation Context ... was able to solve the word problem in special cases. It was only in 1947 that the general case was shown to be unsolvable independently by E. L. Post [28] and A. A. Markov [24]. The other two papers =-=[43, 45]-=- deal with repetitions in finite and infinite words. Perhaps because these papers were published in a journal with restricted availability (this is guessed by G. A. Hedlund [20]), this work of Thue wa... |

78 |
Uniform tag sequences
- Cobham
- 1972
(Show Context)
Citation Context ...er four letters obtained by what is now called an iterated morphism, whilst the three letter word is constructed in a slightly more complicated way (a uniform tag-system, in the terminology of Cobham =-=[12]-=-). The second paper attacks the more general problem of what Thue calls irreducible words. He devotes special attention to the case of two and three letters. In particular, he introduces what is now c... |

75 |
Recursive unsolvability of a problem of Thue
- Post
(Show Context)
Citation Context ...nition of what is now called a "Thue system"). He was able to solve the word problem in special cases. It was only in 1947 that the general case was shown to be unsolvable independently by E=-=. L. Post [28]-=- and A. A. Markov [24]. The other two papers [43, 45] deal with repetitions in finite and infinite words. Perhaps because these papers were published in a journal with restricted availability (this is... |

69 |
Recurrent geodesics on a surface of negative curvature
- Morse
- 1921
(Show Context)
Citation Context ...everal aspects he did not consider: first, many combinatorial properties of the Thue-Morse sequence (such as the number of factors, the recurrence index, and so on) were only investigated by M. Morse =-=[25]-=- or later; next, the characterization of all onesided infinite overlap-free words --- which is much more difficult than that of twosided words --- was only given later by Fife [15]. However, Thue give... |

55 |
Jewels of Formal Language Theory
- Salomaa
- 1981
(Show Context)
Citation Context ...rs is square-free. 2 The Thue-Morse sequence In this section, we recall some basic properties concerning the Thue-Morse sequence. Other properties and proofs can be found in Lothaire [22] and Salomaa =-=[34]-=-. Let A = fa; bg be a two letter alphabet. Consider the morphismsfrom the free monoid A into itself defined by (a) = ab; (b) = ba Setting, for ns0, un =sn (a); vn =sn (b) one gets u 0 = a v 0 = b u 1 ... |

53 |
Avoidable patterns in strings of symbols
- Bean, Ehrenfeucht, et al.
- 1979
(Show Context)
Citation Context ...the images, is 18. It has been shown by A. Carpi [7] that this is the best bound : every morphism over three letters that preserves square-free words has size at least 18. (See also the discussion in =-=[4]-=-.) However, there is a simpler morphism that generates a square-free word (starting with a) given e.g. by Hall [18], namely a 7! abc b 7! ac c 7! b This morphism does not preserve square-free words, b... |

47 |
RAUZY, G.: Suites algébriques, automates et substitutions
- KAMAE
- 1980
(Show Context)
Citation Context ...0. At the end, the state reached is 0 or 1 according to t n = b or t n = a. In fact, the automaton computes d 1 (n) modulo 2. Another description is given by Christol, Kamae, Mend`es-France, Rauzy in =-=[11]-=-. There are many generalizations of the Thue-Morse sequence, motivated by its simplicity, and by its numerous properties. The first definition of the sequence, by iterating a given morphism, is of cou... |

41 |
Sur un théorème de Thue
- Dejean
- 1972
(Show Context)
Citation Context ...e square-free word. 4.3 Repetitions As already mentioned, Thue calls a word on n letters irreducible if every factorsxyx verifies jyjsn \Gamma 2. A more general concept, first considered by F. Dejean =-=[14]-=-, is to require that the length of the word y separating the occurrences of x is bounded from below by the length of x (times some factor). More precisely, we call repetition a word xyx with x non emp... |

36 |
Uniformly growing k-th power-free homomorphisms, Theoret
- Brandenburg
- 1983
(Show Context)
Citation Context ...n r g and fi 0 = inffr j 9C ? 0; 8n; fl nsCn r g Then Theorem One has 1:155 ! ff 0 ! 1:276 ! 1:332 ! fi 0 ! 1:587. This is to be compared with the situation for square-free words. Indeed, Brandenburg =-=[5]-=- proved that for the number c(n) of square-free words of length n over three letters, there are constants c 1s1:032 and c 2s1:38 such that 6c n 1 ! c(n) ! 6c n 2 . Brandenburg also proved that the num... |

28 |
Some theorems on Fourier coefficients
- Rudin
- 1959
(Show Context)
Citation Context ...on what perhaps should be called the Prouhet sequence). Other sequences related to the Thue-Morse sequence are obtained by counting factors in the binary expansion, instead of bits. The Rudin-Shapiro =-=[33, 38]-=- 4 sequence is the infinite word x over fa; bg defined by xn = ae a if d 11 (n) j 0 (mod 2) b if d 11 (n) j 1 (mod 2) where d 11 (n) is the number of factors 11 in the binary expansion of n. Similarly... |

26 |
Enumeration of factors in the Thue-Morse word
- Brlek
- 1989
(Show Context)
Citation Context ...nstance, if k = 3, the morphism, says3 , is given by 0 ! 012, 1 ! 120, 2 ! 201 (this general definition was in fact already given by Prouhet, in 1851. Several authors, such as Adler, Li [1] and Brlek =-=[6]-=-, discuss the fact that Prouhet was the first to mention what perhaps should be called the Prouhet sequence). Other sequences related to the Thue-Morse sequence are obtained by counting factors in the... |

26 | Mémoire sur quelques relations entre les puissances des nombres - Prouhet |

25 |
On the impossibility of certain algorithms in the theory of associative systems
- MARKOV
- 1947
(Show Context)
Citation Context ...called a "Thue system"). He was able to solve the word problem in special cases. It was only in 1947 that the general case was shown to be unsolvable independently by E. L. Post [28] and A. =-=A. Markov [24]-=-. The other two papers [43, 45] deal with repetitions in finite and infinite words. Perhaps because these papers were published in a journal with restricted availability (this is guessed by G. A. Hedl... |

20 |
Extremal problems for polynomials and power series
- Shapiro
- 1951
(Show Context)
Citation Context ...on what perhaps should be called the Prouhet sequence). Other sequences related to the Thue-Morse sequence are obtained by counting factors in the binary expansion, instead of bits. The Rudin-Shapiro =-=[33, 38]-=- 4 sequence is the infinite word x over fa; bg defined by xn = ae a if d 11 (n) j 0 (mod 2) b if d 11 (n) j 1 (mod 2) where d 11 (n) is the number of factors 11 in the binary expansion of n. Similarly... |

19 |
Continued fractions of algebraic power series in characteristic 2
- Baum, Sweet
- 1976
(Show Context)
Citation Context ...ver fa; bg defined by xn = ae a if d 11 (n) j 0 (mod 2) b if d 11 (n) j 1 (mod 2) where d 11 (n) is the number of factors 11 in the binary expansion of n. Similarly, in the sequence of Baum and Sweet =-=[3]-=-, the n-th symbol is a or b according to whether there exists a factor of odd length containing only the bit 0 in the binary expansion of n. Again, this sequence is automatic. Many numbertheoretic res... |

19 |
The Mathematical Theory of L-systems
- Rozenberg, Salomaa
- 1980
(Show Context)
Citation Context ...uence, motivated by its simplicity, and by its numerous properties. The first definition of the sequence, by iterating a given morphism, is of course strongly related to Lindenmayer systems (see e.g. =-=[32]-=-). In the case where the morphism is uniform, that is when the lengths of the images of the letters are equal, a general theorem of Cobham [12] shows that the sequence x obtained by iterating the morp... |

18 |
A propos d’une conjecture de F. Dejean sur les répétitions dans les mots
- Pansiot
- 1984
(Show Context)
Citation Context ...nite word over k letters that has only repetitions of index greater or equal to ae k . Thus, Dejean's result may be stated as : ae 3 = 1=3. She conjectured that ae 4 = 3=2, a result proved by Pansiot =-=[27]-=-, and that ae k = k \Gamma 2 for ks5. The conjecture was proved up to 9 by MoulinOllagnier [26] 5 Avoidable patterns The overlap-freeness of the Thue-Morse sequence, and the square-freeness of the oth... |

13 |
Every binary pattern of length six is avoidable on the two-letter alphabet
- Roth
- 1992
(Show Context)
Citation Context ...nfinite words show that aa is 3-avoidable (but not 2-avoidable). Avoidable and unavoidable patterns have been studied by several people (Zimin [48], Schmidt [36], Bean, Ehrenfeucht, McNulty [4], Roth =-=[31]-=-, Cassaigne [9], Goralcik, Vanicek [17], Baker, McNulty, 12 Taylor [2], Crochemore, Goralcik [13]). Problems which have been stated, and partially solved, include the following: given a pattern e, is ... |

12 |
Binary sequences which contain no BBb
- FIFE
- 1980
(Show Context)
Citation Context ...tigated by M. Morse [25] or later; next, the characterization of all onesided infinite overlap-free words --- which is much more difficult than that of twosided words --- was only given later by Fife =-=[15]-=-. However, Thue gives a complete description of circular overlap-free words. We will also mention the problem of counting the number of overlap-free words over two letters. Axel Thue's investigation o... |

12 |
Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln
- Thue
- 1914
(Show Context)
Citation Context ...ction In a series of four papers which appeared during the period 1906--1914, Axel Thue considered several combinatorial problems which arise in the study of sequences of symbols. Two of these papers =-=[44, 46] deal with-=- word problems for finitely presented semigroups (these papers contain the definition of what is now called a "Thue system"). He was able to solve the word problem in special cases. It was o... |

11 |
Inherently non-finitely based finite semigroups
- Sapir
- 1987
(Show Context)
Citation Context ...re general framework of avoidable and unavoidable patterns in strings. This concept has been introduced in the context of equations defining algebras. Certain unavoidable words have been used e.g. in =-=[35]-=- to characterize those finite semigroups S that are inherently nonfinitely based, in the sense that S is not a member of any locally finite semigroup variety definable by finitely many equations. It m... |

10 |
Growth problems for avoidable words, Theoret
- Baker, McNulty, et al.
- 1989
(Show Context)
Citation Context ...ble and unavoidable patterns have been studied by several people (Zimin [48], Schmidt [36], Bean, Ehrenfeucht, McNulty [4], Roth [31], Cassaigne [9], Goralcik, Vanicek [17], Baker, McNulty, 12 Taylor =-=[2]-=-, Crochemore, Goralcik [13]). Problems which have been stated, and partially solved, include the following: given a pattern e, is it avoidable or not ? There is a nice algorithm in [4], and basically ... |

10 |
der Poorten, Arithmetic properties of the solutions of a class of functional equations
- Loxton, van
- 1982
(Show Context)
Citation Context ...he binary expansion of n. Again, this sequence is automatic. Many numbertheoretic results have been given for automatic sequences. Let us just mention the following, due to Loxton and van der Poorten =-=[23]-=-: Theorem For any automatic infinite word x over the alphabet f0; : : : ; p \Gamma 1g, the real number X xnp \Gamman is transcendental. As an example, the real number whose binary expansion is 0:01101... |

8 |
A characterization of the Morse minimal set
- Gottschalk, Hedlund
- 1964
(Show Context)
Citation Context ...ements, and also that Fact(x) = Fact(X) for any x 2 X. The property that the dynamical system generated by the (twosided) ThueMorse sequence is minimal was explicitly proved by Gottschalk and Hedlund =-=[16]-=-. As a consequence, every factor appears with bounded gaps (is recurrent, in the terminology of M. Morse [25]). Axel Thue (Satz 11) only mentions that every factor appears infinitely many often. The s... |

8 |
On cyclically overlap-free words in binary alphabets, The Book oFL
- HARJU
- 1986
(Show Context)
Citation Context ...asy to show that a circular word w is overlap-free iff the (ordinary) word ww is overlap-free. Thus, Thue characterizes overlap-free squares, a result that was discovered later also in [42]. T. Harju =-=[19]-=- gives a result which is similar, but different. Theorem (Satz 9) For every twosided infinite overlap-free word x, there exists a unique infinite overlap-free word y such that x = (y). 1 The mention S... |

7 |
Binary patterns in binary words
- Goral˘cik, Vani˘cek
(Show Context)
Citation Context ...le (but not 2-avoidable). Avoidable and unavoidable patterns have been studied by several people (Zimin [48], Schmidt [36], Bean, Ehrenfeucht, McNulty [4], Roth [31], Cassaigne [9], Goralcik, Vanicek =-=[17]-=-, Baker, McNulty, 12 Taylor [2], Crochemore, Goralcik [13]). Problems which have been stated, and partially solved, include the following: given a pattern e, is it avoidable or not ? There is a nice a... |

7 |
Sequences generated by infinitely iterated morphisms
- SÉÉBOLD
- 1985
(Show Context)
Citation Context ...Let h be an overlap-free morphism. Then there is an integer n such that h =sn or h =sffisn , wheresis the morphism that exchanges the two letters of the alphabet. This was generalized by P. S'e'ebold =-=[37]-=- : He showed that the infinite words t andst are the only infinite overlap-free words generated by iterated morphisms. Since overlap-free words have a strong structure, it seems natural to count them.... |

7 |
Zimin, “Blocking sets of terms
- I
(Show Context)
Citation Context ...patterns aaa and ababa are 2-avoidable, and square-free infinite words show that aa is 3-avoidable (but not 2-avoidable). Avoidable and unavoidable patterns have been studied by several people (Zimin =-=[48]-=-, Schmidt [36], Bean, Ehrenfeucht, McNulty [4], Roth [31], Cassaigne [9], Goralcik, Vanicek [17], Baker, McNulty, 12 Taylor [2], Crochemore, Goralcik [13]). Problems which have been stated, and partia... |

6 |
Avoidable patterns on two letters”, Theoret
- Schmidt
- 1989
(Show Context)
Citation Context ...nd ababa are 2-avoidable, and square-free infinite words show that aa is 3-avoidable (but not 2-avoidable). Avoidable and unavoidable patterns have been studied by several people (Zimin [48], Schmidt =-=[36]-=-, Bean, Ehrenfeucht, McNulty [4], Roth [31], Cassaigne [9], Goralcik, Vanicek [17], Baker, McNulty, 12 Taylor [2], Crochemore, Goralcik [13]). Problems which have been stated, and partially solved, in... |

5 |
Magic Cubes and Prouhet Sequences
- Adler, Li
- 1977
(Show Context)
Citation Context ...ating x. For instance, if k = 3, the morphism, says3 , is given by 0 ! 012, 1 ! 120, 2 ! 201 (this general definition was in fact already given by Prouhet, in 1851. Several authors, such as Adler, Li =-=[1]-=- and Brlek [6], discuss the fact that Prouhet was the first to mention what perhaps should be called the Prouhet sequence). Other sequences related to the Thue-Morse sequence are obtained by counting ... |

5 |
Aperiodic words on three symbols
- Shelton, Soni
- 1982
(Show Context)
Citation Context ... is a minimal dynamical system. A full description of the tree of square-free words, like Fife's description for overlap-free words, is not yet available. Shelton and Soni have investigated this tree =-=[39, 40, 41]-=-. They have shown in particular that the set of infinite squarefree words over three letters is perfect. Roughly speaking, this means if x is any square-free infinite word, then for any prefix p of x,... |

5 |
Chains and fixing blocks in irreducible sequences
- SHELTON, SONI
- 1985
(Show Context)
Citation Context ... It is indeed easy to show that a circular word w is overlap-free iff the (ordinary) word ww is overlap-free. Thus, Thue characterizes overlap-free squares, a result that was discovered later also in =-=[42]-=-. T. Harju [19] gives a result which is similar, but different. Theorem (Satz 9) For every twosided infinite overlap-free word x, there exists a unique infinite overlap-free word y such that x = (y). ... |

4 |
Die Lösung eines Spezialfalles eines generellen logischen Problems, Norske Vid
- Thue
- 1910
(Show Context)
Citation Context ...ction In a series of four papers which appeared during the period 1906--1914, Axel Thue considered several combinatorial problems which arise in the study of sequences of symbols. Two of these papers =-=[44, 46] deal with-=- word problems for finitely presented semigroups (these papers contain the definition of what is now called a "Thue system"). He was able to solve the word problem in special cases. It was o... |

3 |
Mutually avoiding ternary words of small exponent
- Crochemore, Goralcik
- 1991
(Show Context)
Citation Context ...ns have been studied by several people (Zimin [48], Schmidt [36], Bean, Ehrenfeucht, McNulty [4], Roth [31], Cassaigne [9], Goralcik, Vanicek [17], Baker, McNulty, 12 Taylor [2], Crochemore, Goralcik =-=[13]-=-). Problems which have been stated, and partially solved, include the following: given a pattern e, is it avoidable or not ? There is a nice algorithm in [4], and basically the same in [48], to decide... |

3 |
Overlap-free words on two symbols. Automata on infinite words
- Restivo, Salemi
- 1985
(Show Context)
Citation Context ...inite overlap-free word iff w is a prefix of a word ab \Delta f with f 2 W or of a word aab \Delta f with fif 2 W , where W = B \Gamma B IB . This implies in particular a result of Restivo and Salemi =-=[30]-=-, namely that it is decidable whether an overlap-free word is extensible into an infinite overlap-free word. Another consequence of Fife's description is the following Corollary The Thue-Morse word t ... |

3 |
Selected Mathematical Papers, edited by
- Thue
- 1977
(Show Context)
Citation Context ...long time, and consequently some of his results have been rediscovered again and again. Axel Thue's papers on sequences are now more easily accessible since they are included in the "Selected Pap=-=ers" [47]-=- which were edited in 1977. It is the purpose of the present paper to give an account of Axel Thue's work on repetitions in sequences, both in more recent terminology and in relation with new results ... |

2 |
Generators and relations in groups – the Burnside problem
- Hall
- 1964
(Show Context)
Citation Context ...rs that preserves square-free words has size at least 18. (See also the discussion in [4].) However, there is a simpler morphism that generates a square-free word (starting with a) given e.g. by Hall =-=[18]-=-, namely a 7! abc b 7! ac c 7! b This morphism does not preserve square-free words, because the image of aba is abcacabc. 9 4.2 A Classification Since every twosided infinite square-free word x over t... |

2 |
Remarks on the work of Axel Thue
- HEDLUND
- 1967
(Show Context)
Citation Context ...e other two papers [43, 45] deal with repetitions in finite and infinite words. Perhaps because these papers were published in a journal with restricted availability (this is guessed by G. A. Hedlund =-=[20]-=-), this work of Thue was widely ignored during a long time, and consequently some of his results have been rediscovered again and again. Axel Thue's papers on sequences are now more easily accessible ... |

2 |
Enumeration of irreductible binary words
- Kobayashi
- 1988
(Show Context)
Citation Context ...to count them. The first result is due to Restivo and Salemi [30]. They prove that the number fl n of overlap-free words over two letters grows polynomially in n (in fact slower than n 4 ). Kobayashi =-=[21]-=- has used Fife's theorem to derive the lower of the more precise bounds for fl n : Theorem There are constants C 1 and C 2 such that C 1 n ff ! fl n ! C 2 n fi where ff = 1:155 : : : and fi = 1:5866 :... |

1 |
On the size of a squarefree morphism on a three letter alphabet
- Carpi
- 1983
(Show Context)
Citation Context ...is the following (Satz 18): a 7! abcab b 7! acabcb c 7! acbcacb This morphism seems to be rather complicated. Its size, i. e. the sum of the length of the images, is 18. It has been shown by A. Carpi =-=[7]-=- that this is the best bound : every morphism over three letters that preserves square-free words has size at least 18. (See also the discussion in [4].) However, there is a simpler morphism that gene... |

1 |
Overlap-free words and finite automata, manuscript
- Carpi
- 1990
(Show Context)
Citation Context ...rts with the letter a. Indeed, the choice of the letters ff, fi, et fl implies that if fsf 0 , then ab \Delta fsab \Delta f 0 . The greatest word in F is fl ! , and this shows the corollary. A. Carpi =-=[8]-=- 7 has developed a description for finite overlap-free words by means of a finite automaton. Unfortunately, his automaton is rather big (more than 300 states). There is another property that singles o... |

1 |
Unavoidable binary patterns, manuscript, Ecole normale sup'erieure
- Cassaigne
- 1991
(Show Context)
Citation Context ...ow that aa is 3-avoidable (but not 2-avoidable). Avoidable and unavoidable patterns have been studied by several people (Zimin [48], Schmidt [36], Bean, Ehrenfeucht, McNulty [4], Roth [31], Cassaigne =-=[9]-=-, Goralcik, Vanicek [17], Baker, McNulty, 12 Taylor [2], Crochemore, Goralcik [13]). Problems which have been stated, and partially solved, include the following: given a pattern e, is it avoidable or... |

1 |
Counting overlap-free binary words, manuscript, Ecole normale sup'erieure
- Cassaigne
- 1992
(Show Context)
Citation Context ...onstants C 1 and C 2 such that C 1 n ff ! fl n ! C 2 n fi where ff = 1:155 : : : and fi = 1:5866 : : :. One might ask what is the "real" limit. In fact, a recent and surprising result by J. =-=Cassaigne [10]-=- shows that there is no limit. More precisely, set ff 0 = supfr j 9C ? 0; 8n; fl nsCn r g and fi 0 = inffr j 9C ? 0; 8n; fl nsCn r g Then Theorem One has 1:155 ! ff 0 ! 1:276 ! 1:332 ! fi 0 ! 1:587. T... |

1 |
Preuve de la conjecture de Dejean pour des alphabets `a 5
- Moulin-Ollagnier
- 1989
(Show Context)
Citation Context ...jean's result may be stated as : ae 3 = 1=3. She conjectured that ae 4 = 3=2, a result proved by Pansiot [27], and that ae k = k \Gamma 2 for ks5. The conjecture was proved up to 9 by MoulinOllagnier =-=[26]-=- 5 Avoidable patterns The overlap-freeness of the Thue-Morse sequence, and the square-freeness of the other words we have presented can be expressed in the more general framework of avoidable and unav... |