Inverse problems of symbolic dynamics

TL;DR

This paper explores inverse problems in symbolic dynamics, linking combinatorial properties of sequences to dynamical systems like interval exchange transformations, and proves polynomial bounds on subword complexity for certain sequences.

Contribution

It establishes a connection between combinatorial properties and dynamical systems, and proves that subword complexity for sequences derived from polynomial functions eventually follows a polynomial growth.

Findings

Subword complexity T(k) is eventually polynomial in k.

Sequences from polynomial functions have algebraic letter frequencies.

Interval exchange transformations generate morphic words with algebraic frequencies.

Abstract

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. $w_n=[2\{P(n)\}]$. 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 $k$.}