## Toeplitz Words, Generalized Periodicity and Periodically Iterated Morphisms (1997)

Venue: | European J. of Combinatorics |

Citations: | 7 - 2 self |

### BibTeX

@ARTICLE{Cassaigne97toeplitzwords,,

author = {Julien Cassaigne and Juhani Karhumäki},

title = {Toeplitz Words, Generalized Periodicity and Periodically Iterated Morphisms},

journal = {European J. of Combinatorics},

year = {1997},

volume = {18},

pages = {497--510}

}

### OpenURL

### Abstract

We consider so-called Toeplitz words which can be viewed as generalizations of one-way infinite periodic words. We compute their subword complexity, and show that they can always be generated by iterating periodically a finite number of morphisms. Moreover, we define a structural classification of Toeplitz words which is reflected in the way how they can be generated by iterated morphisms.

### Citations

649 |
Algebraic Combinatorics on Words
- Lothaire
- 2002
(Show Context)
Citation Context ...s is exactly the same as what happened to the well known periodicity lemma of Fine and Wilf, cf. [6]: it was introduced for real functions, but became really fundamental in connection with words, cf. =-=[11]-=-! 2 Basic Definitions For a finite alphabet \Sigma, let \Sigma , \Sigma + and \Sigma ! denote the sets of all finite, finite non-empty, and one-way infinite words over \Sigma, respectively. We call an... |

250 |
Recurrence in Ergodic Theory and Combinatorial Number Theory
- Furstenberg
- 1981
(Show Context)
Citation Context ...e, if p and q are relative primes then the period we computed for T i (w) is its smallest period. It follows from above that Toeplitz words are uniformly recurrent in the sense of ergodic theory, cf. =-=[7]-=-, i.e. for each number k there exists a constant n(k) such that whenever u with length k occurs as a factor in T (w), then it occurs in any factor of T (w) of length n(k). 3 Toeplitz Words and Iterati... |

106 |
Uniqueness theorems for periodic functions
- Fine, Wilf
- 1965
(Show Context)
Citation Context ...ironment, however, is that of words, and we believe they provide most elegant results on this area. This is exactly the same as what happened to the well known periodicity lemma of Fine and Wilf, cf. =-=[6]-=-: it was introduced for real functions, but became really fundamental in connection with words, cf. [11]! 2 Basic Definitions For a finite alphabet \Sigma, let \Sigma , \Sigma + and \Sigma ! denote th... |

32 |
la complexitè des suites infinies
- Allouche, Sur
- 1994
(Show Context)
Citation Context ...ty, or briefly complexity, of an infinite word u is the function pu : N! N such that pu (n) = the number of factors of u of length n: Recently quite a lot of research has been done on this field, cf. =-=[1]-=-. It follows from our Theorems 1 and 2, by a result in [5], that for our first two cases of Toeplitz words, i.e. when the number of holes divides the length of the pattern, the subword complexity is l... |

16 |
Subword complexities of various classes of deterministic developmental languages without interaction, Theoret. Comput. Sci. 1
- Lee, Rozenberg
- 1975
(Show Context)
Citation Context ...nction pu : N! N such that pu (n) = the number of factors of u of length n: Recently quite a lot of research has been done on this field, cf. [1]. It follows from our Theorems 1 and 2, by a result in =-=[5]-=-, that for our first two cases of Toeplitz words, i.e. when the number of holes divides the length of the pattern, the subword complexity is linear. In this section we show that arbitrary Toeplitz wor... |

13 |
Toeplitz sequences, paperfolding, towers of Hanoi and progression-free sequences of integers, Ens
- Allouche, Bacher
- 1992
(Show Context)
Citation Context ...] these words, now referred to as Toeplitz words, have been considered from the combinatorial point of view. Surprising connections of these words to different combinatorial problems are discussed in =-=[2]-=-. The combinatorial research of Toeplitz words seems to have concentrated on a special case of our later classification, corresponding to so-called paperfolding words, cf. [2, 4, 12]. The Toeplitz wor... |

8 |
Number Representations and Dragon Curves
- Davis, Knuth
- 1970
(Show Context)
Citation Context ...roblems are discussed in [2]. The combinatorial research of Toeplitz words seems to have concentrated on a special case of our later classification, corresponding to so-called paperfolding words, cf. =-=[2, 4, 12]-=-. The Toeplitz words we are considering here are defined as follows: take an infinite periodic word w ! on the alphabet \Sigma [ f?g, where ? corresponds to a hole. Fill the holes iteratively by subst... |

8 |
M.: 0−1-sequences of Toeplitz type
- Jacobs, Keane
- 1969
(Show Context)
Citation Context ... is reflected in the way how they can be generated by iterated morphisms. 1 Introduction Toeplitz introduced in [14] an iterative construction to define almost periodic functions on the real line. In =-=[8]-=- Jacobs and Keane modified this construction to define infinite words. Their motivation, however, was on topological aspects of words, in particular, on ergodic theory. Starting from [13] these words,... |

7 |
Infinite 0-1-Sequences without Long Adjacent Identical Blocks
- Prodinger, Urbanek
- 1979
(Show Context)
Citation Context ...real line. In [8] Jacobs and Keane modified this construction to define infinite words. Their motivation, however, was on topological aspects of words, in particular, on ergodic theory. Starting from =-=[13]-=- these words, now referred to as Toeplitz words, have been considered from the combinatorial point of view. Surprising connections of these words to different combinatorial problems are discussed in [... |

6 |
der Poorten, Arithmetic and analytic properties of paperfolding sequences
- France, van
- 1981
(Show Context)
Citation Context ...roblems are discussed in [2]. The combinatorial research of Toeplitz words seems to have concentrated on a special case of our later classification, corresponding to so-called paperfolding words, cf. =-=[2, 4, 12]-=-. The Toeplitz words we are considering here are defined as follows: take an infinite periodic word w ! on the alphabet \Sigma [ f?g, where ? corresponds to a hole. Fill the holes iteratively by subst... |

1 |
Lepist o, Alternating iteration of morphisms and the Kolakoski sequence
- aki, A
- 1992
(Show Context)
Citation Context ...art of that we classify different types of Toeplitz words in terms of their subword complexity. We also point out connections between Toeplitz words and different types of iteration of morphisms, cf. =-=[3, 10]-=-. More precisely, let us call the above Toeplitz word (p; q)-Toeplitz word, if the length of the pattern, that is jwj, is p, and if it contains q holes. We show that (i) if q = 1, then the word can be... |

1 |
Self-generating runs, prob
- Kolakoski
- 1966
(Show Context)
Citation Context ...that I(w) = T (w) : Note that in the process of defining I(w) holes are filled one by one by reading the prefix of I(w). Consequently, I(w) is selfreading in the same sense as the Kolakoski word, cf. =-=[9, 3]-=-. We proceed with a few examples. Example 1. The periodic word (112) ! is obtained as a Toeplitz word in a number of different ways : (112) ! = T (112??????) = T (112?12?1?) = T (1?21?211?) : Example ... |

1 |
o, On the power of periodic iteration of morphisms
- Lepist
- 1993
(Show Context)
Citation Context ...art of that we classify different types of Toeplitz words in terms of their subword complexity. We also point out connections between Toeplitz words and different types of iteration of morphisms, cf. =-=[3, 10]-=-. More precisely, let us call the above Toeplitz word (p; q)-Toeplitz word, if the length of the pattern, that is jwj, is p, and if it contains q holes. We show that (i) if q = 1, then the word can be... |

1 |
Beispiele zur Theorie der fastperiodischen Funktionen
- Toeplitz
- 1928
(Show Context)
Citation Context ...er of morphisms. Moreover, we define a structural classification of Toeplitz words which is reflected in the way how they can be generated by iterated morphisms. 1 Introduction Toeplitz introduced in =-=[14]-=- an iterative construction to define almost periodic functions on the real line. In [8] Jacobs and Keane modified this construction to define infinite words. Their motivation, however, was on topologi... |