Inverse problems of symbolic dynamics

• Autorzy: Alexei Ya. Belov , Grigorii V. Kondakov , Ivan V. Mitrofanov
• Języki publikacji: EN

Abstrakty

EN

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 = (wₙ), n ∈ ℤ) consists of a sequence of the first binary numbers of {P(n)}, i.e. wₙ = [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.

Kategorie tematyczne

• 05Exx: Algebraic combinatorics
• 37Bxx: Topological dynamics

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2011
• Tom: 94
• Numer: 1
• Strony: 43-60

Daty

wydano

2011

Twórcy

autor

Alexei Ya. Belov

• Moscow Institute of Open Education, Aviatsionnyi per., 125167 Moscow, Russia
• Shanghai University, 99 Shangda Road, BaoShan District, 200444 Shanghai, China

autor

Grigorii V. Kondakov

• Moscow Institute of Physics and Technology, 9, Institutski per., 141700 Dolgoprudny, Moscow Region, Russia

autor

Ivan V. Mitrofanov

• Moscow State University, Leninskie Gory, 119992 Moscow, Russia

Identyfikatory

DOI

10.4064/bc94-0-2