Inverse problems of symbolic dynamics

What is it about?

A point $x\in X$, a map $f\:X\to X$, a subset $U\subset X$ and a binary alphabet $\{a,b\}$ define an infinite binary word $W=W(x,f,U)$ whose $n$th term is $a$ if $f^n(x)\in U$ and $b$ if not. This paper is concerned with aspects of the inverse problem, starting with a binary word $W$ and describing properties of possible collections $(x,f,U)$ with $W=W(x,f,U)$. In the setting of unipotent toral maps (and hence of polynomials with an irrational coefficient) the following result is found. Given a polynomial $P$ with irrational lead coefficient, associate to it the sequence $w$ with $w_n=\lfloor 2\{ P(n)\}\rfloor$ (that is, the sequence of $0$s and $1$s given by the first binary digit of the fractional part of $P(n)$). Then there is a polynomial $Q$ such that the number of different subwords of length $k$ in $w$ is given by $Q(k)$ for all $k\ge1$. {For the collection containing this paper see MR2885263.} Reviewed by Thomas Ward

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