5 research outputs found
A generalized palindromization map in free monoids
The palindromization map in a free monoid was introduced in 1997
by the first author in the case of a binary alphabet , and later extended by
other authors to arbitrary alphabets. Acting on infinite words,
generates the class of standard episturmian words, including standard
Arnoux-Rauzy words. In this paper we generalize the palindromization map,
starting with a given code over . The new map maps to the
set of palindromes of . In this way some properties of are
lost and some are saved in a weak form. When has a finite deciphering delay
one can extend to , generating a class of infinite words
much wider than standard episturmian words. For a finite and maximal code
over , we give a suitable generalization of standard Arnoux-Rauzy words,
called -AR words. We prove that any -AR word is a morphic image of a
standard Arnoux-Rauzy word and we determine some suitable linear lower and
upper bounds to its factor complexity.
For any code we say that is conservative when
. We study conservative maps and
conditions on assuring that is conservative. We also investigate
the special case of morphic-conservative maps , i.e., maps such that
for an injective morphism . Finally,
we generalize by replacing palindromic closure with
-palindromic closure, where is any involutory antimorphism of
. This yields an extension of the class of -standard words
introduced by the authors in 2006.Comment: Final version, accepted for publication on Theoret. Comput. Sc
Combinatorics on Words. New Aspects on Avoidability, Defect Effect, Equations and Palindromes
In this thesis we examine four well-known and traditional concepts of combinatorics on words. However the contexts in which these topics are treated are not the traditional ones. More precisely, the question of avoidability is asked, for example, in terms of k-abelian squares. Two words are said to be k-abelian equivalent if they have the same number of occurrences of each factor up to length k. Consequently, k-abelian equivalence can be seen as a sharpening of abelian equivalence. This fairly new concept is discussed broader than the other topics of this thesis.
The second main subject concerns the defect property. The defect theorem is a well-known result for words. We will analyze the property, for example, among the sets of 2-dimensional words, i.e., polyominoes composed of labelled unit squares.
From the defect effect we move to equations. We will use a special way to define a product operation for words and then solve a few basic equations over constructed partial semigroup. We will also consider the satisfiability question and the compactness property with respect to this kind of equations.
The final topic of the thesis deals with palindromes. Some finite words, including all binary words, are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. The famous Thue-Morse word has the property that for each positive integer n, there exists a factor which cannot be generated by fewer than n palindromes. We prove that in general, every non ultimately periodic word contains a factor which cannot be generated by fewer than 3 palindromes, and we obtain a classification of those binary words each of whose factors are generated by at most 3 palindromes. Surprisingly these words are related to another much studied set of words, Sturmian words.Siirretty Doriast