TY - JOUR

T1 - Inverse problems of symbolic dynamics

AU - Belov, A.

AU - Kondakov, G.V.

AU - Mitrofanov, I.

PY - 2011

Y1 - 2011

N2 - 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 w $(w=(w_n), n\in \nit)$ consists of a sequence of 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) .
\medskip {\bf Theorem.} {\it There exists a polynomial Q(k), depending only on the power of the polynomial P, such that T(k)=Q(k) for sufficiently great k.}

AB - 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 w $(w=(w_n), n\in \nit)$ consists of a sequence of 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) .
\medskip {\bf Theorem.} {\it There exists a polynomial Q(k), depending only on the power of the polynomial P, such that T(k)=Q(k) for sufficiently great k.}

UR - http://arxiv.org/abs/1104.5605

M3 - Article

SN - 1735-8787

VL - 94

SP - 43

EP - 60

JO - Banach Journal of Mathematical Analysis

JF - Banach Journal of Mathematical Analysis

ER -