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### Abelian Powers and Repetitions in Sturmian Words [article]

Gabriele Fici, Alessio Langiu, Thierry Lecroq, Arnaud Lefebvre, Filippo Mignosi, Jarkko Peltomäki, Élise Prieur-Gaston
2016 arXiv   pre-print
We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words.  ...  power (resp. of an abelian repetition) of abelian period m (the superior limits coincide for Sturmian words).  ...  of abelian powers and repetitions in Sturmian words.  ...

### Abelian powers and repetitions in Sturmian words

Gabriele Fici, Alessio Langiu, Thierry Lecroq, Arnaud Lefebvre, Filippo Mignosi, Jarkko Peltomäki, Élise Prieur-Gaston
2016 Theoretical Computer Science
We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words.  ...  denotes the maximum exponent of an abelian power (resp. of an abelian repetition) of abelian period m (the superior limits coincide for Sturmian words).  ...  of abelian powers and repetitions in Sturmian words.  ...

### Abelian Repetitions in Sturmian Words [chapter]

Gabriele Fici, Alessio Langiu, Thierry Lecroq, Arnaud Lefebvre, Filippo Mignosi, Élise Prieur-Gaston
2013 Lecture Notes in Computer Science
We investigate abelian repetitions in Sturmian words.  ...  If km denotes the maximal exponent of an abelian repetition of period m, we prove that lim sup km/m ≥ √ 5 for any Sturmian word, and the equality holds for the Fibonacci infinite word.  ...  An abelian power is an abelian repetition in which u 0 = u j = ε. We call u 0 and u j the head and the tail of the abelian repetition, respectively.  ...

### Abelian Repetitions in Sturmian Words [article]

Gabriele Fici, Alessio Langiu, Thierry Lecroq, Arnaud Lefebvre, Filippo Mignosi, Élise Prieur-Gaston
2013 arXiv   pre-print
We investigate abelian repetitions in Sturmian words.  ...  We prove that in any Sturmian word the superior limit of the ratio between the maximal exponent of an abelian repetition of period m and m is a number ≥√(5), and the equality holds for the Fibonacci infinite  ...  Similar techniques were used in [40] to derive nice results on abelian powers on Sturmian words, that are repetitions with empty head and tail.  ...

### Page 2779 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews
We study the minimal proportion of one letter in an xth-power- free binary word as a function of x and prove, in particular, that this function is discontinuous.”  ...  The AR problem— to find all abelian repetitions—generalizes the R problem; it is conjectured that the AR problem requires @(n7) time in the worst case.  ...

### Words with the Maximum Number of Abelian Squares [article]

Gabriele Fici, Filippo Mignosi
2015 arXiv   pre-print
An abelian square is the concatenation of two words that are anagrams of one another. A word of length n can contain Θ(n^2) distinct factors that are abelian squares.  ...  We study infinite words such that the number of abelian square factors of length n grows quadratically with n.  ...  In this case, we prove that if a Sturmian word is β-power free for some β ≥ 2 (that is, does not contain repetitions of order β or higher), then it is uniformly abelian-square rich.  ...

### Abelian Properties of Words (Extended abstract) [article]

Gwénaël Richomme, Kalle Saari, Luca Q. Zamboni
2009 arXiv   pre-print
We also investigate abelian repetitions in words and show that any infinite word with bounded abelian complexity contains abelian k-powers for every positive integer k.  ...  In this paper we investigate various abelian properties of words including abelian complexity, and abelian powers.  ...  Section 9 contains a detailed study of abelian powers in Sturmian words, the Thue-Morse word and the Tribonacci word.  ...

### Abelian-square-rich words

Gabriele Fici, Filippo Mignosi, Jeffrey Shallit
2017 Theoretical Computer Science
As for Sturmian words, we prove that a Sturmian word s_α of angle α is uniformly abelian-square-rich if and only if the irrational α has bounded partial quotients, that is, if and only if s_α has bounded  ...  in n; and uniformly abelian-square-rich if every factor of w contains a number of distinct abelian-square factors that is proportional to the square of its length.  ...  A word w is a k-power (also called a repetition of order k), for an integer k ≥ 2, if there exists a nonempty word u such that w = u k . A 2-power is called a square.  ...

### On a generalization of Abelian equivalence and complexity of infinite words [article]

Juhani Karhumaki, Aleksi Saarela, Luca Q. Zamboni
2013 arXiv   pre-print
Moreover if ω is aperiodic, then P^(k)_ω(n)=q^k(n) if and only if ω is Sturmian. We also study k-Abelian complexity in connection with repetitions in words.  ...  Two finite words u and v in A^* are said to be k-Abelian equivalent if for all x∈ A^* of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in  ...  Recently k-Abelian equivalence has been studied in the context of avoidance of repetitions in words (see the discussion at the beginning of §5 on k-Abelian powers).  ...

### Abelian Combinatorics on Words: a Survey [article]

Gabriele Fici, Svetlana Puzynina
2022 arXiv   pre-print
We survey known results and open problems in abelian combinatorics on words.  ...  In the past few years, there was a lot of research on abelian analogues of classical definitions and properties in combinatorics on words. This survey aims to gather these results.  ...  Abelian powers in Sturmian words Richomme, Saari and Zamboni [114] proved that in every Sturmian word, for any position and for every positive integer k, there is an abelian k-power starting at that  ...

### Abelian complexity of minimal subshifts

Gwénaël Richomme, Kalle Saari, Luca Q. Zamboni
2010 Journal of the London Mathematical Society
In the case of Sturmian words we prove something stronger: For every Sturmian word w and positive integer k, each sufficiently long factor of w begins in an Abelian k-power.  ...  Rauzy by exhibiting a class of words whose Abelian complexity is everywhere equal to 3. We also investigate links between Abelian complexity and the existence of Abelian powers.  ...  Abelian repetitions in Sturmian words In this section, we prove the following theorem, thereby answering Open Problem 2 in the affirmative in the case of Sturmian words. Theorem 6.1.  ...

### Everywhere α-repetitive sequences and Sturmian words

Kalle Saari
2010 European journal of combinatorics (Print)
In this paper, we study optimal 2-repetitive sequences and optimal 2 + -repetitive sequences, and show that Sturmian words belong to both classes.  ...  Local constraints on an infinite sequence that imply global regularity are of general interest in combinatorics on words. We consider this topic by studying everywhere α-repetitive sequences.  ...  Karhumäki for several discussions and for his encouragement to pursue investigating the topic of this paper. Thanks also to J. Cassaigne and D.  ...

### Abelian periods of factors of Sturmian words [article]

Jarkko Peltomäki
2020 arXiv   pre-print
A characterization of the Fibonacci word in terms of its abelian period set is obtained as a corollary.  ...  We study the abelian period sets of Sturmian words, which are codings of irrational rotations on a one-dimensional torus.  ...  Abelian Powers in Sturmian Words The starting point of the study of abelian equivalence in Sturmian words is the following result stating that factors of length n of a Sturmian word belong to exactly two  ...

### All Growth Rates of Abelian Exponents Are Attained by Infinite Binary Words

Jarkko Peltomäki, Markus A. Whiteland, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science
We consider repetitions in infinite words by making a novel inquiry to the maximum eventual growth rate of the exponents of abelian powers occurring in an infinite word.  ...  As a consequence, we obtain that every nonnegative real number is the critical abelian exponent of some infinite binary word.  ...  The notion is particularly suitable for exponents of abelian powers in Sturmian words and leads to surprising results.  ...

### On the Contribution of WORDS to the Field of Combinatorics on Words [chapter]

Jean Néraud
2015 Lecture Notes in Computer Science
This prologue summarizes the history of the conference WORDS, and the contributions which were presented, our goal being to testify how the conference may be embedded in the development of the field of  ...  Combinatorics on Words.  ...  2001), Fabio Burderi (WORDS 2011) and Dominique Perrin (WORDS 2015). Extensions of the classical concept of words were the subject of a lot of presentations.  ...
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