Characterisation of Stability for Interval Translation Maps

An interval translation map (ITM) is a piece-wise translation $T\text{\hspace{0.17em}⁣}:I\to I$ defined on a finite partition $I_1,\dots ,I_r$ of an interval $I$ into $r\ge 2$ subintervals. In contrast to classical interval exchange transformations (IETs), we do not require that the images of these subintervals are disjoint; in particular, ITMs are not assumed to be bijective. Thus, ITMs provide a natural non-invertible generalisation of IETs. In this paper, we formulate an appropriate notion of stability for general interval translation mappings and prove a characterisation of stability in terms of two dynamically natural properties called the Absence of Critical Connections and Matching. This result can be viewed as the foundational step towards the stability theory of general ITMs.