37 research outputs found
Integers in number systems with positive and negative quadratic Pisot base
We consider numeration systems with base and , for quadratic
Pisot numbers and focus on comparing the combinatorial structure of the
sets and of numbers with integer expansion in base
, resp. . Our main result is the comparison of languages of
infinite words and coding the ordering of distances
between consecutive - and -integers. It turns out that for a
class of roots of , the languages coincide, while for other
quadratic Pisot numbers the language of can be identified only with
the language of a morphic image of . We also study the group
structure of -integers.Comment: 19 pages, 5 figure
On the number of return words in infinite words with complexity 2n+1
In this article, we count the number of return words in some infinite words
with complexity 2n+1. We also consider some infinite words given by codings of
rotation and interval exchange transformations on k intervals. We prove that
the number of return words over a given word w for these infinite words is
exactly k.Comment: see also http://liafa.jussieu.fr/~vuillon/articles.htm
Occurrences of palindromes in characteristic Sturmian words
This paper is concerned with palindromes occurring in characteristic Sturmian
words of slope , where is an irrational.
As is a uniformly recurrent infinite word, any (palindromic) factor
of occurs infinitely many times in with bounded gaps. Our
aim is to completely describe where palindromes occur in . In
particular, given any palindromic factor of , we shall establish
a decomposition of with respect to the occurrences of . Such a
decomposition shows precisely where occurs in , and this is
directly related to the continued fraction expansion of .Comment: 17 page
On a generalization of Abelian equivalence and complexity of infinite words
In this paper we introduce and study a family of complexity functions of
infinite words indexed by k \in \ints ^+ \cup {+\infty}. Let k \in \ints ^+
\cup {+\infty} and be a finite non-empty set. Two finite words and
in are said to be -Abelian equivalent if for all of length
less than or equal to the number of occurrences of in is equal to
the number of occurrences of in This defines a family of equivalence
relations on bridging the gap between the usual notion of
Abelian equivalence (when ) and equality (when We show that
the number of -Abelian equivalence classes of words of length grows
polynomially, although the degree is exponential in Given an infinite word
\omega \in A^\nats, we consider the associated complexity function \mathcal
{P}^{(k)}_\omega :\nats \rightarrow \nats which counts the number of
-Abelian equivalence classes of factors of of length We show
that the complexity function is intimately linked with
periodicity. More precisely we define an auxiliary function q^k: \nats
\rightarrow \nats and show that if for
some k \in \ints ^+ \cup {+\infty} and the is ultimately
periodic. Moreover if is aperiodic, then if and only if is Sturmian. We also
study -Abelian complexity in connection with repetitions in words. Using
Szemer\'edi's theorem, we show that if has bounded -Abelian
complexity, then for every D\subset \nats with positive upper density and for
every positive integer there exists a -Abelian power occurring in
at some position $j\in D.
Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers
We study arithmetical and combinatorial properties of -integers for
being the root of the equation . We determine with the accuracy of the maximal number of
-fractional positions, which may arise as a result of addition of two
-integers. For the infinite word coding distances between
consecutive -integers, we determine precisely also the balance. The word
is the fixed point of the morphism and . In the case the corresponding infinite word is
sturmian and therefore 1-balanced. On the simplest non-sturmian example with
, we illustrate how closely the balance and arithmetical properties of
-integers are related.Comment: 15 page
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
Factor-balanced -adic languages
A set of words, also called a language, is letter-balanced if the number of
occurrences of each letter only depends on the length of the word, up to a
constant. Similarly, a language is factor-balanced if the difference of the
number of occurrences of any given factor in words of the same length is
bounded. The most prominent example of a letter-balanced but not
factor-balanced language is given by the Thue-Morse sequence. We establish
connections between the two notions, in particular for languages given by
substitutions and, more generally, by sequences of substitutions. We show that
the two notions essentially coincide when the sequence of substitutions is
proper. For the example of Thue-Morse-Sturmian languages, we give a full
characterisation of factor-balancedness
Generalizations of Sturmian sequences associated with -continued fraction algorithms
Given a positive integer and irrational between zero and one, an
-continued fraction expansion of is defined analogously to the classical
continued fraction expansion, but with the numerators being all equal to .
Inspired by Sturmian sequences, we introduce the -continued fraction
sequences and , which are related to the
-continued fraction expansion of . They are infinite words over a two
letter alphabet obtained as the limit of a directive sequence of certain
substitutions, hence they are -adic sequences. When , we are in the
case of the classical continued fraction algorithm, and obtain the well-known
Sturmian sequences. We show that and are
-balanced for some explicit values of and compute their factor
complexity function. We also obtain uniform word frequencies and deduce unique
ergodicity of the associated subshifts. Finally, we provide a Farey-like map
for -continued fraction expansions, which provides an additive version of
-continued fractions, for which we prove ergodicity and give the invariant
measure explicitly.Comment: 23 pages, 2 figure
Relations on words
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.
In the second part, we mainly focus on abelian equivalence, -abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and -equivalence. In particular, some new refinements of abelian equivalence are introduced