# On abelian versions of critical factorization theorem

@article{Avgustinovich2012OnAV, title={On abelian versions of critical factorization theorem}, author={Sergey V. Avgustinovich and Juhani Karhum{\"a}ki and Svetlana Puzynina}, journal={RAIRO Theor. Informatics Appl.}, year={2012}, volume={46}, pages={3-15} }

In the paper we study abelian versions of the critical factorization theorem. We investigate both similarities and differences between the abelian powers and the usual powers. The results we obtained show that the constraints for abelian powers implying periodicity should be quite strong, but still natural analogies exist.

## 19 Citations

Abelian Properties of Words

- MathematicsWORDS
- 2019

A short overview of some directions of research on abelian properties of words is given, and two new problems are discussed in more detail: small abelia complexity of two-dimensional words, and abelians subshifts.

Weak Abelian Periodicity of Infinite Words

- MathematicsTheory of Computing Systems
- 2015

This paper establishes necessary and sufficient conditions for the weak abelian periodicity of fixed points of uniform binary morphisms, and considers its relation with the notions of balance and letter frequency, and study operations preserving weak abeilic periodicity.

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We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on…

Abelian returns in Sturmian words

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- 2011

It is proved that a word is Sturmian if and only if each of its factors has two or three abelian returns.

Aperiodic Two-Dimensional Words of Small Abelian Complexity

- MathematicsElectron. J. Comb.
- 2019

It is proved that if a two-dimensional recurrent word contains at most two abelian factors for each pair (n,m) of integers, then it has a periodicity vector and it is shown that aTwo-dimensional aperiodic recurrent word must have more than two abelsian factors infinitely often.

Abelian powers and repetitions in Sturmian

- Mathematics
- 2016

Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79–95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent k for every k > 0. We improve on this result by…

ABELIAN POWERS AND IN STURMIAN WORDS

- Mathematics
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. Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79–95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent k for every k > 0. We improve on this result…

Dyck Words, Lattice Paths, and Abelian Borders

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- 2017

A formula for the exact number of binary words of a given length with a given minimal abelian border length is derived, tightening a bound on that number from Christodoulakis et al. (Discrete Applied Mathematics, 2014).

Local Squares, Periodicity and Finite Automata

- MathematicsRainbow of Computer Science
- 2011

Nine variants of the problem when local regularity implies the global one in the setting where the existence of a square of certain length in every position of an infinite word are considered, and an amazing unavoidable result for 2-abelian squares is obtained.

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