1,137 research outputs found

    Towards a statement of the S-adic conjecture through examples

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    The SS-adic conjecture claims that there exists a condition CC such that a sequence has a sub-linear complexity if and only if it is an SS-adic sequence satisfying Condition CC for some finite set SS of morphisms. We present an overview of the factor complexity of SS-adic sequences and we give some examples that either illustrate some interesting properties or that are counter-examples to what could be believed to be "a good Condition CC".Comment: 2

    Morphic words and equidistributed sequences

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    The problem we consider is the following: Given an infinite word ww on an ordered alphabet, construct the sequence νw=(ν[n])n\nu_w=(\nu[n])_n, equidistributed on [0,1][0,1] and such that ν[m]<ν[n]\nu[m]<\nu[n] if and only if σm(w)<σn(w)\sigma^m(w)<\sigma^n(w), where σ\sigma is the shift operation, erasing the first symbol of ww. The sequence νw\nu_w exists and is unique for every word with well-defined positive uniform frequencies of every factor, or, in dynamical terms, for every element of a uniquely ergodic subshift. In this paper we describe the construction of νw\nu_w for the case when the subshift of ww is generated by a morphism of a special kind; then we overcome some technical difficulties to extend the result to all binary morphisms. The sequence νw\nu_w in this case is also constructed with a morphism. At last, we introduce a software tool which, given a binary morphism φ\varphi, computes the morphism on extended intervals and first elements of the equidistributed sequences associated with fixed points of φ\varphi

    Canonical Representatives of Morphic Permutations

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    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 {0,…,q−1}\{0,\ldots,q-1\} 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

    Inverse problems of symbolic dynamics

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    This paper reviews some results regarding symbolic dynamics, correspondence between languages of dynamical systems and combinatorics. Sturmian sequences provide a pattern for investigation of one-dimensional systems, in particular interval exchange transformation. Rauzy graphs language can express many important combinatorial and some dynamical properties. In this case combinatorial properties are considered as being generated by substitutional system, and dynamical properties are considered as criteria of superword being generated by interval exchange transformation. As a consequence, one can get a morphic word appearing in interval exchange transformation such that frequencies of letters are algebraic numbers of an arbitrary degree. Concerning multydimensional systems, our main result is the following. Let P(n) be a polynomial, having an irrational coefficient of the highest degree. A word ww (w=(w_n), n\in \nit) consists of a sequence of first binary numbers of {P(n)}\{P(n)\} i.e. wn=[2{P(n)}]w_n=[2\{P(n)\}]. Denote the number of different subwords of ww of length kk by T(k)T(k) . \medskip {\bf Theorem.} {\it There exists a polynomial Q(k)Q(k), depending only on the power of the polynomial PP, such that T(k)=Q(k)T(k)=Q(k) for sufficiently great kk.

    Quasicrystals, model sets, and automatic sequences

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    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
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