38 research outputs found
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
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
Snake graphs and continued fractions
This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the continued fractions as well as the Euclidean division algorithm. We apply our findings to establish results on sums of squares, palindromic continued fractions, Markov numbers and other statements in elementary number theory
Periodicity, repetitions, and orbits of an automatic sequence
We revisit a technique of S. Lehr on automata and use it to prove old and new
results in a simple way. We give a very simple proof of the 1986 theorem of
Honkala that it is decidable whether a given k-automatic sequence is ultimately
periodic. We prove that it is decidable whether a given k-automatic sequence is
overlap-free (or squareefree, or cubefree, etc.) We prove that the
lexicographically least sequence in the orbit closure of a k-automatic sequence
is k-automatic, and use this last result to show that several related
quantities, such as the critical exponent, irrationality measure, and
recurrence quotient for Sturmian words with slope alpha, have automatic
continued fraction expansions if alpha does.Comment: preliminary versio