147 research outputs found
On prefixal factorizations of words
We consider the class of all infinite words over
a finite alphabet admitting a prefixal factorization, i.e., a factorization
where each is a non-empty prefix of With
each one naturally associates a "derived" infinite word
which may or may not admit a prefixal factorization. We are
interested in the class of all words of
such that for all . Our primary
motivation for studying the class stems from its connection
to a coloring problem on infinite words independently posed by T. Brown in
\cite{BTC} and by the second author in \cite{LQZ}. More precisely, let be the class of all words such that for every finite
coloring there exist and a factorization
with for each In \cite{DPZ}
we conjectured that a word if and only if is purely
periodic. In this paper we show that so
in other words, potential candidates to a counter-example to our conjecture are
amongst the non-periodic elements of We establish several
results on the class . In particular, we show that a
Sturmian word belongs to if and only if is
nonsingular, i.e., no proper suffix of is a standard Sturmian word
Clustering words
We characterize words which cluster under the Burrows-Wheeler transform as
those words such that occurs in a trajectory of an interval exchange
transformation, and build examples of clustering words
Central sets and substitutive dynamical systems
In this paper we establish a new connection between central sets and the
strong coincidence conjecture for fixed points of irreducible primitive
substitutions of Pisot type. Central sets, first introduced by Furstenberg
using notions from topological dynamics, constitute a special class of subsets
of \nats possessing strong combinatorial properties: Each central set
contains arbitrarily long arithmetic progressions, and solutions to all
partition regular systems of homogeneous linear equations. We give an
equivalent reformulation of the strong coincidence condition in terms of
central sets and minimal idempotent ultrafilters in the Stone-\v{C}ech
compactification \beta \nats . This provides a new arithmetical approach to
an outstanding conjecture in tiling theory, the Pisot substitution conjecture.
The results in this paper rely on interactions between different areas of
mathematics, some of which had not previously been directly linked: They
include the general theory of combinatorics on words, abstract numeration
systems, tilings, topological dynamics and the algebraic/topological properties
of Stone-\v{C}ech compactification of \nats.Comment: arXiv admin note: substantial text overlap with arXiv:1110.4225,
arXiv:1301.511
On the least number of palindromes contained in an infinite word
We investigate the least number of palindromic factors in an infinite word.
We first consider general alphabets, and give answers to this problem for
periodic and non-periodic words, closed or not under reversal of factors. We
then investigate the same problem when the alphabet has size two.Comment: Accepted for publication in Theoretical Computer Scienc
The sequence of open and closed prefixes of a Sturmian word
A finite word is closed if it contains a factor that occurs both as a prefix
and as a suffix but does not have internal occurrences, otherwise it is open.
We are interested in the {\it oc-sequence} of a word, which is the binary
sequence whose -th element is if the prefix of length of the word is
open, or if it is closed. We exhibit results showing that this sequence is
deeply related to the combinatorial and periodic structure of a word. In the
case of Sturmian words, we show that these are uniquely determined (up to
renaming letters) by their oc-sequence. Moreover, we prove that the class of
finite Sturmian words is a maximal element with this property in the class of
binary factorial languages. We then discuss several aspects of Sturmian words
that can be expressed through this sequence. Finally, we provide a linear-time
algorithm that computes the oc-sequence of a finite word, and a linear-time
algorithm that reconstructs a finite Sturmian word from its oc-sequence.Comment: Published in Advances in Applied Mathematics. Journal version of
arXiv:1306.225
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
A Coloring Problem for Infinite Words
In this paper we consider the following question in the spirit of Ramsey
theory: Given where is a finite non-empty set, does there
exist a finite coloring of the non-empty factors of with the property that
no factorization of is monochromatic? We prove that this question has a
positive answer using two colors for almost all words relative to the standard
Bernoulli measure on We also show that it has a positive answer for
various classes of uniformly recurrent words, including all aperiodic balanced
words, and all words satisfying
for all sufficiently large, where denotes the number of
distinct factors of of length Comment: arXiv admin note: incorporates 1301.526
Abelian maximal pattern complexity of words
In this paper we study the maximal pattern complexity of infinite words up to
Abelian equivalence. We compute a lower bound for the Abelian maximal pattern
complexity of infinite words which are both recurrent and aperiodic by
projection. We show that in the case of binary words, the bound is actually
achieved and gives a characterization of recurrent aperiodic words
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