36 research outputs found
On the number of factors of Sturmian words
AbstractWe prove that for m⩾1, card(Am) = 1+∑mi=1 (m−i+1)ϕ(i) where Am is the set of factors of length m of all the Sturmian words and ϕ is the Euler function. This result was conjectured by Dulucq and Gouyou-Beauchamps (1987) who proved that this result implies that the language (∪m⩾0Am)c is inherently ambiguous. We also give a combinatorial version of the Riemann hypothesis
Complexity and growth for polygonal billiards
We establish a relationship between the word complexity and the number of
generalized diagonals for a polygonal billiard. We conclude that in the
rational case the complexity function has cubic upper and lower bounds. In the
tiling case the complexity has cubic asymptotic growth.Comment: 12 pages, 4 figure
A new geometric approach to Sturmian words
We introduce a new geometric approach to Sturmian words by means of a mapping
that associates certain lines in the n x n -grid and sets of finite Sturmian
words of length n. Using this mapping, we give new proofs of the formulas
enumerating the finite Sturmian words and the palindromic finite Sturmian words
of a given length. We also give a new proof for the well-known result that a
factor of a Sturmian word has precisely two return words.Comment: 12 pages, 7 figures. A preprint of a paper to appear in Theoretical
Computer Scienc
A Characterization of Bispecial Sturmian Words
A finite Sturmian word w over the alphabet {a,b} is left special (resp. right
special) if aw and bw (resp. wa and wb) are both Sturmian words. A bispecial
Sturmian word is a Sturmian word that is both left and right special. We show
as a main result that bispecial Sturmian words are exactly the maximal internal
factors of Christoffel words, that are words coding the digital approximations
of segments in the Euclidean plane. This result is an extension of the known
relation between central words and primitive Christoffel words. Our
characterization allows us to give an enumerative formula for bispecial
Sturmian words. We also investigate the minimal forbidden words for the set of
Sturmian words.Comment: Accepted to MFCS 201
Supercritical holes for the doubling map
For a map and an open connected set ( a hole) we
define to be the set of points in whose -orbit avoids
. We say that a hole is supercritical if (i) for any hole such
that the set is either empty or contains
only fixed points of ; (ii) for any hole such that \barH\subset H_0
the Hausdorff dimension of is positive.
The purpose of this note to completely characterize all supercritical holes
for the doubling map .Comment: This is a new version, where a full characterization of supercritical
holes for the doubling map is obtaine
The number of binary rotation words
We consider binary rotation words generated by partitions of the unit circle
to two intervals and give a precise formula for the number of such words of
length n. We also give the precise asymptotics for it, which happens to be
O(n^4). The result continues the line initiated by the formula for the number
of all Sturmian words obtained by Lipatov in 1982, then independently by
Berenstein, Kanal, Lavine and Olson in 1987, Mignosi in 1991, and then with
another technique by Berstel and Pocchiola in 1993.Comment: Submitted to RAIRO IT
On the Structure of Bispecial Sturmian Words
A balanced word is one in which any two factors of the same length contain
the same number of each letter of the alphabet up to one. Finite binary
balanced words are called Sturmian words. A Sturmian word is bispecial if it
can be extended to the left and to the right with both letters remaining a
Sturmian word. There is a deep relation between bispecial Sturmian words and
Christoffel words, that are the digital approximations of Euclidean segments in
the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic}
bispecial Sturmian words are precisely the maximal internal factors of
\emph{primitive} Christoffel words. We extend this result by showing that
bispecial Sturmian words are precisely the maximal internal factors of
\emph{all} Christoffel words. Our characterization allows us to give an
enumerative formula for bispecial Sturmian words. We also investigate the
minimal forbidden words for the language of Sturmian words.Comment: arXiv admin note: substantial text overlap with arXiv:1204.167
The complexity of tangent words
In a previous paper, we described the set of words that appear in the coding
of smooth (resp. analytic) curves at arbitrary small scale. The aim of this
paper is to compute the complexity of those languages.Comment: In Proceedings WORDS 2011, arXiv:1108.341