195 research outputs found
Quasicrystals, model sets, and automatic sequences
We survey mathematical properties of quasicrystals, first from the point of
view of harmonic analysis, then from the point of view of morphic and automatic
sequences.
Nous proposons un tour d'horizon de propri\'et\'es math\'ematiques des
quasicristaux, d'abord du point de vue de l'analyse harmonique, ensuite du
point de vue des suites morphiques et automatiques
Canonical Representatives of Morphic Permutations
An infinite permutation can be defined as a linear ordering of the set of
natural numbers. In particular, an infinite permutation can be constructed with
an aperiodic infinite word over as the lexicographic order
of the shifts of the word. In this paper, we discuss the question if an
infinite permutation defined this way admits a canonical representative, that
is, can be defined by a sequence of numbers from [0, 1], such that the
frequency of its elements in any interval is equal to the length of that
interval. We show that a canonical representative exists if and only if the
word is uniquely ergodic, and that is why we use the term ergodic permutations.
We also discuss ways to construct the canonical representative of a permutation
defined by a morphic word and generalize the construction of Makarov, 2009, for
the Thue-Morse permutation to a wider class of infinite words.Comment: Springer. WORDS 2015, Sep 2015, Kiel, Germany. Combinatorics on
Words: 10th International Conference. arXiv admin note: text overlap with
arXiv:1503.0618
Directive words of episturmian words: equivalences and normalization
Episturmian morphisms constitute a powerful tool to study episturmian words.
Indeed, any episturmian word can be infinitely decomposed over the set of pure
episturmian morphisms. Thus, an episturmian word can be defined by one of its
morphic decompositions or, equivalently, by a certain directive word. Here we
characterize pairs of words directing a common episturmian word. We also
propose a way to uniquely define any episturmian word through a normalization
of its directive words. As a consequence of these results, we characterize
episturmian words having a unique directive word.Comment: 15 page
A remarkable sequence related to and
We prove that five ways to define entry A086377 in the On-Line Encyclopedia
of Integer Sequences do lead to the same integer sequence
On the number of return words in infinite words with complexity 2n+1
In this article, we count the number of return words in some infinite words
with complexity 2n+1. We also consider some infinite words given by codings of
rotation and interval exchange transformations on k intervals. We prove that
the number of return words over a given word w for these infinite words is
exactly k.Comment: see also http://liafa.jussieu.fr/~vuillon/articles.htm
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