8 research outputs found
Languages invariant under more symmetries: overlapping factors versus palindromic richness
Factor complexity and palindromic complexity of
infinite words with language closed under reversal are known to be related by
the inequality for any \,. Word for which
the equality is attained for any is usually called rich in palindromes. In
this article we study words whose languages are invariant under a finite group
of symmetries. For such words we prove a stronger version of the above
inequality. We introduce notion of -palindromic richness and give several
examples of -rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur
Constructions of words rich in palindromes and pseudopalindromes
A narrow connection between infinite binary words rich in classical
palindromes and infinite binary words rich simultaneously in palindromes and
pseudopalindromes (the so-called -rich words) is demonstrated.
The correspondence between rich and -rich words is based on the operation
acting over words over the alphabet and defined by
, where .
The operation enables us to construct a new class of rich words and a new
class of -rich words.
Finally, the operation is considered on the multiliteral alphabet
as well and applied to the generalized Thue--Morse words. As a
byproduct, new binary rich and -rich words are obtained by application of
on the generalized Thue--Morse words over the alphabet .Comment: 26 page
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
Episturmian words: a survey
In this paper, we survey the rich theory of infinite episturmian words which
generalize to any finite alphabet, in a rather resembling way, the well-known
family of Sturmian words on two letters. After recalling definitions and basic
properties, we consider episturmian morphisms that allow for a deeper study of
these words. Some properties of factors are described, including factor
complexity, palindromes, fractional powers, frequencies, and return words. We
also consider lexicographical properties of episturmian words, as well as their
connection to the balance property, and related notions such as finite
episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize
the skew words of Morse and Hedlund.Comment: 36 pages; major revision: improvements + new material + more
reference
On different generalizations of episturmian words
In this paper we study some classes of infinite words generalizing episturmian words, and analyse the relations occurring among such classes. In each case, the reversal operator R is replaced by an arbitrary involutory antimorphism ϑ of the free monoid A ∗. In particular, we define the class of ϑ-words with seed, whose “standard ” elements (ϑ-standard words with seed) are constructed by an iterative ϑ-palindrome closure process, starting from a finite word u0 called the seed. When the seed is empty, one obtains ϑ-words; episturmian words are exactly the R-words. One of the main theorems of the paper characterizes ϑ-words with seed as infinite words closed under ϑ and having at most one left special factor of each length n ≥ N (where N is some nonnegative integer depending on the word). When N = 0 we call such words ϑ-episturmian. Further results on the structure of ϑ-episturmian words are proved. In particular, some relationships between ϑ-words (with or without seed) and ϑ-episturmian words are shown