Density of Stable Interval Translation Maps

Assume that the interval $I=[0,1)$ is partitioned into finitely many intervals $I_1,\dots ,I_r$ and consider a map $T\text{\hspace{0.17em}⁣}:I\to I$ so that $T_{\mathrm{\mid }{I}_s}$ 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.