527 research outputs found
Sturmian numeration systems and decompositions to palindromes
We extend the classical Ostrowski numeration systems, closely related to
Sturmian words, by allowing a wider range of coefficients, so that possible
representations of a number better reflect the structure of the associated
Sturmian word. In particular, this extended numeration system helps to catch
occurrences of palindromes in a characteristic Sturmian word and thus to prove
for Sturmian words the following conjecture stated in 2013 by Puzynina, Zamboni
and the author: If a word is not periodic, then for every it has a prefix
which cannot be decomposed to a concatenation of at most palindromes.Comment: Submitted to European Journal of Combinatoric
Nested quasicrystalline discretisations of the line
One-dimensional cut-and-project point sets obtained from the square lattice
in the plane are considered from a unifying point of view and in the
perspective of aperiodic wavelet constructions. We successively examine their
geometrical aspects, combinatorial properties from the point of view of the
theory of languages, and self-similarity with algebraic scaling factor
. We explain the relation of the cut-and-project sets to non-standard
numeration systems based on . We finally examine the substitutivity, a
weakened version of substitution invariance, which provides us with an
algorithm for symbolic generation of cut-and-project sequences
The recurrence function of a random Sturmian word
This paper describes the probabilistic behaviour of a random Sturmian word.
It performs the probabilistic analysis of the recurrence function which can be
viewed as a waiting time to discover all the factors of length of the
Sturmian word. This parameter is central to combinatorics of words. Having
fixed a possible length for the factors, we let to be drawn
uniformly from the unit interval , thus defining a random Sturmian word
of slope . Thus the waiting time for these factors becomes a random
variable, for which we study the limit distribution and the limit density.Comment: Submitted to ANALCO 201
Permutation Complexity Related to the Letter Doubling Map
Given a countable set X (usually taken to be the natural numbers or
integers), an infinite permutation, \pi, of X is a linear ordering of X. This
paper investigates the combinatorial complexity of infinite permutations on the
natural numbers associated with the image of uniformly recurrent aperiodic
binary words under the letter doubling map. An upper bound for the complexity
is found for general words, and a formula for the complexity is established for
the Sturmian words and the Thue-Morse word.Comment: In Proceedings WORDS 2011, arXiv:1108.341
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