Michelangelo Bucci

A connection between palindromic and factor complexity using return words

In this paper we prove that for any infinite word w whose set of factors is closed under reversal, the following conditions are equivalent:(I)all complete returns to palindromes are palindromes; (II)P(n)+P(n+1)=C(n+1)-C(n)+2 for all n, where P (resp. C) denotes the palindromic complexity (resp. factor complexity) function of w, which counts the number of distinct palindromic factors (resp. factors) of each length in w.

Rich and Periodic-Like Words

In this paper we investigate the periodic structure of rich words (i.e., words having the highest possible number of palindromic factors), giving new results relating them with periodic-like words. In particular, some new characterizations of rich words and rich palindromes are given. We also prove that a periodic-like word is rich if and only if the square of its fractional root is also rich.

Enumeration and structure of trapezoidal words

Trapezoidal words are words having at most n+1 distinct factors of length n for every n>=0. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmian palindrome if and only if its longest repeated prefix is a palindrome. We also define a new class of words, semicentral words, and show that they are characterized by the property that they can be written as uxyu, for a central word u and two different letters x,y. Finally, we investigate the prefixes of the Fibonacci word with respect to the property of being open or closed trapezoidal words, and show that the sequence of open and closed prefixes of the Fibonacci word follows the Fibonacci sequence.

Characteristic morphisms of generalized episturmian words

In a recent paper with L.Q. Zamboni, the authors introduced the class of @q-episturmian words. An infinite word over A is standard @q-episturmian, where @q is an involutory antimorphism of A^*, if its set of factors is closed under @q and its left special factors are prefixes. When @q is the reversal operator, one obtains the usual standard episturmian words. In this paper, we introduce and study @q-characteristic morphisms, that is, morphisms which map standard episturmian words into standard @q-episturmian words. They are a natural extension of standard episturmian morphisms. The main result of the paper is a characterization of these morphisms when they are injective. In order to prove this result, we also introduce and study a class of biprefix codes which are overlap-free, i.e., any two code words do not overlap properly, and normal, i.e., no proper suffix (prefix) of any code-word is left (right) special in the code. A further result is that any standard @q-episturmian word is a morphic image, by an injective @q-characteristic morphism, of a standard episturmian word.

On a Family of Morphic Images of Arnoux-Rauzy Words

In this paper we prove the following result. Let s be an infinite word on a finite alphabet, and N *** 0 be an integer. Suppose that all left special factors of s longer than N are prefixes of s , and that s has at most one right special factor of each length greater than N . Then s is a morphic image, under an injective morphism, of a suitable standard Arnoux-Rauzy word.

Some Characterizations of Sturmian Words in Terms of the Lexicographic Order

In this paper we present three new characterizations of Sturmian words based on the lexicographic ordering of their factors.

Aperiodic pseudorandom number generators based on infinite words

In this paper we study how certain families of aperiodic infinite words can be used to produce aperiodic pseudorandom number generators (PRNGs) with good statistical behavior. We introduce the well distributed occurrences (WELLDOC) combinatorial property for infinite words, which guarantees absence of the lattice structure defect in related pseudorandom number generators. An infinite word u on a d-ary alphabet has the WELLDOC property if, for each factor w of u, positive integer m, and vector v ź Z m d , there is an occurrence of w such that the Parikh vector of the prefix of u preceding such occurrence is congruent to v modulo m. (The Parikh vector of a finite word v over an alphabet A has its i-th component equal to the number of occurrences of the i-th letter of A in v.) We prove that Sturmian words, and more generally Arnoux-Rauzy words and some morphic images of them, have the WELLDOC property. Using the TestU01 11 and PractRand 5 statistical tests, we moreover show that not only the lattice structure is absent, but also other important properties of PRNGs are improved when linear congruential generators are combined using infinite words having the WELLDOC property.

Infinite Words with Well Distributed Occurrences

In this paper we introduce the well distributed occurrences WDO combinatorial property for infinite words, which guarantees good behavior no lattice structure in some related pseudorandom number generators. An infinite word u on a d-ary alphabet has the WDO property if, for each factor w of u, positive integer m, and vector v

Reversible Christoffel factorizations

\in{\mathbb Z}_{m}^{d}

On additive properties of sets defined by the Thue-Morse word

, there is an occurrence of w such that the Parikh vector of the prefix of u preceding such occurrence is congruent to v modulo m. We prove that Sturmian words, and more generally Arnoux-Rauzy words and some morphic images of them, have the WDO property.