20 research outputs found
On the critical exponent of generalized Thue-Morse words
For certain generalized Thue-Morse words t, we compute the "critical
exponent", i.e., the supremum of the set of rational numbers that are exponents
of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer
Science (accepted October 15, 2007
Relation between powers of factors and recurrence function characterizing Sturmian words
In this paper we use the relation of the index of an infinite aperiodic word
and its recurrence function to give another characterization of Sturmian words.
As a byproduct, we give a new proof of theorem describing the index of a
Sturmian word in terms of the continued fraction expansion of its slope. This
theorem was independently proved by Carpi and de Luca, and Damanik and Lenz.Comment: 11 page
Repetitions in beta-integers
Classical crystals are solid materials containing arbitrarily long periodic
repetitions of a single motif. In this paper, we study the maximal possible
repetition of the same motif occurring in beta-integers -- one dimensional
models of quasicrystals. We are interested in beta-integers realizing only a
finite number of distinct distances between neighboring elements. In such a
case, the problem may be reformulated in terms of combinatorics on words as a
study of the index of infinite words coding beta-integers. We will solve a
particular case for beta being a quadratic non-simple Parry number.Comment: 11 page
Characterization of repetitions in Sturmian words: A new proof
We present a new, dynamical way to study powers (that is, repetitions) in
Sturmian words based on results from Diophantine approximation theory. As a
result, we provide an alternative and shorter proof of a result by Damanik and
Lenz characterizing powers in Sturmian words [Powers in Sturmian sequences,
Eur. J. Combin. 24 (2003), 377--390]. Further, as a consequence, we obtain a
previously known formula for the fractional index of a Sturmian word based on
the continued fraction expansion of its slope.Comment: 9 pages, 1 figur
Powers in Sturmian sequences
AbstractWe consider Sturmian sequences and explicitly determine all the integer powers occurring in them. Our approach is purely combinatorial and is based on canonical decompositions of Sturmian sequences and properties of their building blocks
The repetition threshold for binary rich words
A word of length is rich if it contains nonempty palindromic factors.
An infinite word is rich if all of its finite factors are rich. Baranwal and
Shallit produced an infinite binary rich word with critical exponent
() and conjectured that this was the least
possible critical exponent for infinite binary rich words (i.e., that the
repetition threshold for binary rich words is ). In this article,
we give a structure theorem for infinite binary rich words that avoid
-powers (i.e., repetitions with exponent at least 2.8). As a consequence,
we deduce that the repetition threshold for binary rich words is
, as conjectured by Baranwal and Shallit. This resolves an open
problem of Vesti for the binary alphabet; the problem remains open for larger
alphabets.Comment: 16 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