Symbolic dynamics of piecewise contractions

TL;DR

This paper explores the symbolic dynamics of piecewise contractions on the interval, showing their codings relate closely to interval exchange transformations, revealing structural similarities in their long-term behavior.

Contribution

It establishes a correspondence between natural codings of injective piecewise contractions and topologically transitive interval exchange transformations, extending understanding of their symbolic dynamics.

Findings

Natural codings of injective n-PCs contain periodic orbits or areomorphic segments to n-IET codings.

Every natural coding of a topologically transitive n-IET can be realized by an injective n-PC.

The results connect the dynamics of contractions with interval exchange transformations, enriching symbolic dynamics theory.

Abstract

A map $f{:}\,[0,1)\to [0,1)$ is a {\it piecewise contraction of $n$ intervals} ($n$-PC) if there exist $0<\lambda<1$ and a partition of $[0,1)$ into intervals $I_1,\ldots,I_n$ such that $f\vert_{I_i}$ is $\lambda$-Lipschitz for every $1\le i\le n$. An infinite word $\theta=\theta_0\theta_1\ldots$ over the alphabet $\mathcal{A}=\{1,\ldots,n\}$ is a {\it natural coding of} $f$ if there exists $x\in I$ such that $\theta_k=i$ if and only if $f^k(x)\in I_i$. We prove that if $\theta$ is a natural coding of an injective $n$-PC, then some infinite subword of $\theta$ is either periodic or isomorphic to a natural coding of a topologically transitive $m$-interval exchange transformation ($m$-IET), where $m\le n$. Conversely, every natural coding of a topologically transitive $n$-IET is also a natural coding of some injective $n$-PC.