250 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
Occurrences of palindromes in characteristic Sturmian words
This paper is concerned with palindromes occurring in characteristic Sturmian
words of slope , where is an irrational.
As is a uniformly recurrent infinite word, any (palindromic) factor
of occurs infinitely many times in with bounded gaps. Our
aim is to completely describe where palindromes occur in . In
particular, given any palindromic factor of , we shall establish
a decomposition of with respect to the occurrences of . Such a
decomposition shows precisely where occurs in , and this is
directly related to the continued fraction expansion of .Comment: 17 page
A characterization of fine words over a finite alphabet
To any infinite word w over a finite alphabet A we can associate two infinite
words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the
lexicographically smallest (resp. greatest) amongst the factors of w of the
same length. We say that an infinite word w over A is "fine" if there exists an
infinite word u such that, for any lexicographic order, min(w) = au where a =
min(A). In this paper, we characterize fine words; specifically, we prove that
an infinite word w is fine if and only if w is either a "strict episturmian
word" or a strict "skew episturmian word''. This characterization generalizes a
recent result of G. Pirillo, who proved that a fine word over a 2-letter
alphabet is either an (aperiodic) Sturmian word, or an ultimately periodic (but
not periodic) infinite word, all of whose factors are (finite) Sturmian.Comment: 16 pages; presented at the conference on "Combinatorics, Automata and
Number Theory", Liege, Belgium, May 8-19, 2006 (to appear in a special issue
of Theoretical Computer Science
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
Extremal properties of (epi)Sturmian sequences and distribution modulo 1
Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in
distribution of real numbers modulo 1 via combinatorics on words, we survey
some combinatorial properties of (epi)Sturmian sequences and distribution
modulo 1 in connection to their work. In particular we focus on extremal
properties of (epi)Sturmian sequences, some of which have been rediscovered
several times
Characterization of infinite LSP words and endomorphisms preserving the LSP property
Answering a question of G. Fici, we give an -adic characterization of
thefamily of infinite LSP words, that is, the family of infinite words having
all their left special factors as prefixes.More precisely we provide a finite
set of morphisms and an automaton such that an infinite word is
LSP if and only if it is -adic and one of its directive words is
recognizable by .Then we characterize the endomorphisms that preserve
the property of being LSP for infinite words.This allows us to prove that there
exists no set of endomorphisms for which the set of infinite LSP words
corresponds to the set of -adic words. This implies that an automaton is
required no matter which set of morphisms is used.Comment: arXiv admin note: text overlap with arXiv:1705.0578
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