54 research outputs found
Balancedness of Arnoux-Rauzy and Brun words
We study balancedness properties of words given by the Arnoux-Rauzy and Brun
multi-dimensional continued fraction algorithms. We show that almost all Brun
words on 3 letters and Arnoux-Rauzy words over arbitrary alphabets are finitely
balanced; in particular, boundedness of the strong partial quotients implies
balancedness. On the other hand, we provide examples of unbalanced Brun words
on 3 letters
Factor Complexity of S-adic sequences generated by the Arnoux-Rauzy-Poincar\'e Algorithm
The Arnoux-Rauzy-Poincar\'e multidimensional continued fraction algorithm is
obtained by combining the Arnoux-Rauzy and Poincar\'e algorithms. It is a
generalized Euclidean algorithm. Its three-dimensional linear version consists
in subtracting the sum of the two smallest entries to the largest if possible
(Arnoux-Rauzy step), and otherwise, in subtracting the smallest entry to the
median and the median to the largest (the Poincar\'e step), and by performing
when possible Arnoux-Rauzy steps in priority. After renormalization it provides
a piecewise fractional map of the standard -simplex. We study here the
factor complexity of its associated symbolic dynamical system, defined as an
-adic system. It is made of infinite words generated by the composition of
sequences of finitely many substitutions, together with some restrictions
concerning the allowed sequences of substitutions expressed in terms of a
regular language. Here, the substitutions are provided by the matrices of the
linear version of the algorithm. We give an upper bound for the linear growth
of the factor complexity. We then deduce the convergence of the associated
algorithm by unique ergodicity.Comment: 36 pages, 16 figure
Uniformly balanced words with linear complexity and prescribed letter frequencies
We consider the following problem. Let us fix a finite alphabet A; for any
given d-uple of letter frequencies, how to construct an infinite word u over
the alphabet A satisfying the following conditions: u has linear complexity
function, u is uniformly balanced, the letter frequencies in u are given by the
given d-uple. This paper investigates a construction method for such words
based on the use of mixed multidimensional continued fraction algorithms.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Random product of substitutions with the same incidence matrix
Any infinite sequence of substitutions with the same matrix of the Pisot type
defines a symbolic dynamical system which is minimal. We prove that, to any
such sequence, we can associate a compact set (Rauzy fractal) by projection of
the stepped line associated with an element of the symbolic system on the
contracting space of the matrix. We show that this Rauzy fractal depends
continuously on the sequence of substitutions, and investigate some of its
properties; in some cases, this construction gives a geometric model for the
symbolic dynamical system
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
Extremal properties of (epi)Sturmian sequences and distribution modulo 1
Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in
distribution of real numbers modulo 1 via combinatorics on words, we survey
some combinatorial properties of (epi)Sturmian sequences and distribution
modulo 1 in connection to their work. In particular we focus on extremal
properties of (epi)Sturmian sequences, some of which have been rediscovered
several times
Palindromes in infinite ternary words
We study infinite words u over an alphabet A satisfying the property
P : P(n)+ P(n+1) = 1+ #A for any n in N, where P(n) denotes the number of
palindromic factors of length n occurring in the language of u. We study also
infinite words satisfying a stronger property PE: every palindrome of u has
exactly one palindromic extension in u. For binary words, the properties P and
PE coincide and these properties characterize Sturmian words, i.e., words with
the complexity C(n)=n+1 for any n in N. In this paper, we focus on ternary
infinite words with the language closed under reversal. For such words u, we
prove that if C(n)=2n+1 for any n in N, then u satisfies the property P and
moreover u is rich in palindromes. Also a sufficient condition for the property
PE is given. We construct a word demonstrating that P on a ternary alphabet
does not imply PE.Comment: 12 page
Directive words of episturmian words: equivalences and normalization
Episturmian morphisms constitute a powerful tool to study episturmian words.
Indeed, any episturmian word can be infinitely decomposed over the set of pure
episturmian morphisms. Thus, an episturmian word can be defined by one of its
morphic decompositions or, equivalently, by a certain directive word. Here we
characterize pairs of words directing a common episturmian word. We also
propose a way to uniquely define any episturmian word through a normalization
of its directive words. As a consequence of these results, we characterize
episturmian words having a unique directive word.Comment: 15 page
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