Giuseppe Pirillo

Episturmian words and some constructions of de Luca and Rauzy

Abstract In this paper we study infinite episturmian words which are a natural generalization of Sturmian words to an arbitrary alphabet. A characteristic property is: they are closed under reversal and have at most one right special factor of each length. They are first obtained by a construction due to de LUCA which utilizes the palindrome closure. They can also be obtained by the way of extended RAUZY rules.

Palindromes and Sturmian words

Abstract An infinite word x over the alphabet A is Sturmian if and only if g x ( n ) = n + 1 for any integer n , where g x ( n ) is the number of distinct words of length n occurring in x . A palindrome is a word that can be read indistinctly from left to right or from right to left. We prove that x is Sturmian if and only if h x ( n ) = 1 + ( n mod 2) for any integer n , where h x ( n ) is the number of palindromes of length n occurring in x . An infinite word x over the alphabet A is generated by a morphism f if there exists a letter c ϵ A such that lim n →∞ f n ( c ) = x . We prove the existence of a morphism that generates the palindromes of any infinite Sturmian word generated by a morphism.

Episturmian words and episturmian morphisms

Infinite episturmian words are a generalization of Sturmian words which includes the Arnoux-Rauzy sequences. We continue their study and that of episturmian morphisms, begun previously, in relation with the action of the shift operator. Palindromic and periodic factors of these words are described. We consider, in particular, the case where these words are generated by morphisms and introduce then a notion of intercept generalizing that of Sturmian words. Finally, we prove that the frequencies of the factors in a strong sense do exist for all episturmian words.

A relation between trinucleotide comma-free codes and trinucleotide circular codes

The comma-free codes and circular codes are two important classes of codes in code theory and in genetics. Fifty years ago before the discovery of the genetic code, a trinucleotide (triletter) comma-free code was proposed for associating the codons of genes with the amino acids of proteins. More recently, in the last ten years, trinucleotide circular codes have been identified statistically in different genomes. Here, we identify a relation between these two classes of trinucleotide codes by constructing a hierarchy of comma-free and circular codes.

Fractional powers in Sturmian words

Abstract Given an infinite Sturmian word s , we calculate the function L(m) which gives the length of the longest factor of s having period m . The expression of L(m) makes use of the continued fraction of the irrational α associated with s .

Varieties of comma-free codes

New varieties of comma-free codes CFC of length 3 on the 4-letter alphabet are defined and analysed: self-complementary comma-free codes (CCFC), C^3 comma-free codes (C^3CFC), C^3 self-complementary comma-free codes (C^3CCFC), self-complementary maximal comma-free codes (CMCFC), C^3 maximal comma-free codes (C^3MCFC) and C^3 self-complementary maximal comma-free codes (C^3CMCFC). New properties with words of length 3, 4, 5 and 6 in comma-free codes are used for the determination of growth functions in the studied code varieties.

Brief communication: Identification of all trinucleotide circular codes

A new trinucleotide proposition is proved here and allows all the trinucleotide circular codes on the genetic alphabet to be identified (their numbers and their sets of words). This new class of genetic motifs, i.e. circular codes (or synchronizing genetic motifs), may be involved in the structure and the origin of the genetic code, and in reading frames of genes.

A Characterization for a Set of Trinucleotides to be a Circular Code

The Arques-Michel autocomplementary circular codes are 216 sets of 20 trinucleotides. They have a lot of very interesting properties. Studying them, we found the following result: a set X of trinucleotides is a circular code if and only if it has no 5-necklaces.

EPISTURMIAN WORDS: SHIFTS, MORPHISMS AND NUMERATION SYSTEMS

Episturmian words, which include the Arnoux-Rauzy sequences, are infinite words on a finite alphabet generalizing the Sturmian words and sharing many of their same properties. This was studied in previous papers. Here we gain a deeper insight into these properties. This leads in particular to consider numerations systems similar to the Ostrowski ones and to give a matrix formula for computing the number of representations of an integer in such a system. We also obtain a complete answer to the question: if an episturmian word is morphic, which shifts of it, if any, also are morphic ?

On uniformly repetitive semigroups

Letk be an integer greater than 1 andS be a finitely generated semigroup. The following propositions are equivalent: 1) the semigroup of non negative integers is not uniformlyk-repetitive; 2) any finitely generated and uniformlyk-repetitive semigroup is finite. As a consequence we prove that any finitely generated and uniformly 4-repetitive semigroup is finite.