Inverse problems of symbolic dynamics

Tom 94 / 2011

Alexei Ya. Belov, Grigorii V. Kondakov, Ivan V. Mitrofanov Banach Center Publications 94 (2011), 43-60 MSC: Primary 37B; Secondary 05E. DOI: 10.4064/bc94-0-2


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 a substitutional system, and dynamical properties are considered as criteria for a superword being generated by an interval exchange transformation. As a consequence, one can get a morphic word appearing in an interval exchange transformation such that the frequencies of the letters are algebraic numbers of an arbitrary degree.

Concerning multidimensional systems, our main result is the following. Let $P(n)$ be a polynomial, having an irrational coefficient of the highest degree. A word $w$ $(w=(w_n)$, $\def \nit{\mathbb Z}n\in \nit)$ consists of a sequence of the first binary numbers of $\{P(n)\}$, i.e. $w_n=[2\{P(n)\}]$. Denote the number of different subwords of $w$ of length $k$ by $T(k)$. We prove that there exists a polynomial $Q(k)$, depending only on the power of the polynomial $P$, such that $T(k)=Q(k)$ for sufficiently large $k$.


  • Alexei Ya. BelovMoscow Institute of Open Education
    Aviatsionnyi per.
    125167 Moscow, Russia
    Shanghai University
    99 Shangda Road
    BaoShan District
    200444 Shanghai, China
  • Grigorii V. KondakovMoscow Institute of Physics and Technology
    9, Institutski per.
    141700 Dolgoprudny
    Moscow Region, Russia
  • Ivan V. MitrofanovMoscow State University
    Leninskie Gory
    119992 Moscow, Russia

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