On the Invertibility and Structural Properties of Interval Translation Maps - Tajbakhsh, Khosro; Yaghmaeian, Reza (Lobachevskii journal of mathematics, 2025)
In this paper, we present a significant advancement by proving that each Interval Translation Map (ITM) is invertible almost everywhere on the support of its invariant measure. This result has far-reaching consequences, as it implies that ITMs reduce to Interval Exchange Maps (IEM) in some cases. While previous work has noted that the support of the invariant measure may sometimes form a Cantor set, we are able to conclude that every ITM is indeed invertible almost everywhere on its support. This discovery extends the understanding of ITMs and their structural properties, providing a deeper insight into the behavior of these dynamical systems. Additionally, we investigate several properties of ITMs, contributing to a deeper understanding of their structural properties.