Density of Stable Interval Translation Maps

TL;DR

This paper investigates the density of stable interval translation maps, showing that a large subset of these maps have finite type with dynamics akin to circle rotations, and that their structure varies continuously.

Contribution

It establishes that stable maps form an open dense subset in the space of all interval translation maps, with stable maps exhibiting finite type behavior and continuous dependence on parameters.

Findings

Stable maps are dense and form an open subset in the space of all interval translation maps.

Stable maps of finite type have first return maps equivalent to circle rotations.

The map from stable maps to their attractor sets is continuous in the Hausdorff topology.

Abstract

Assume that the interval $I=[0,1)$ is partitioned into finitely many intervals $I_1,\dots,I_r$ and consider a map $T\colon I\to I$ so that $T_{\vert I_s}$ is a translation for each $1 \le s \le r$. We do not assume that the images of these intervals are disjoint. Such maps are called Interval Translation Maps. Let $ITM(r)$ be the space of all such transformations, where we fix $r$ but not the intervals $I_1,\dots,I_r$, nor the translations. The set $X(T):=\bigcap_{n\ge 0} T^n[0,1)$ can be a finite union of intervals (in which case the map is called of finite type), or is a disjoint union of finitely many intervals and a Cantor set (in which case the map is called of infinite type). In this paper we show that there exists an open and dense subset $\mathcal{S}(r)$ of $ITM(r)$ consisting of stable maps, i.e. each $T\in \mathcal{S}(r)$ is of finite type, the first return map to any component of $X(T)$ corresponds to a circle rotation and $\mathcal{S}(r) \ni T \mapsto X(T)$ is continuous in the Hausdorff topology.