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arXiv · 2605.00186

Topological Prevalence of Finite Type Interval Translation Maps

Kostiantyn Drach·Leon Staresinic·Sebastian van Strien
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Abstract

An interval translation map (ITM) is a map $T \colon I \to I$ defined as a piecewise translation on a finite partition of an interval $I$ into $r \ge 2$ subintervals. Unlike classical interval exchange transformations (IETs), the images of these subintervals are allowed to overlap, making ITMs a natural generalisation of IETs. An ITM $T$ is said to be \textit{of finite type} if its attractor $\bigcap_{n\ge 0} T^n(I)$ is a finite union of intervals; in this case, restricted to this invariant set, $T$ is bijective and hence behaves like an IET. Otherwise, $T$ is of infinite type. In this paper, for every $r \ge 2$, we prove that the set of finite type ITMs contains an open and dense subset in the space of all possible ITMs on $r$ subintervals. This confirms a topological version of a long-standing conjecture due to Boshernitzan and Kornfeld.

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Kostiantyn Drach, Leon Staresinic, Sebastian van Strien. 2026-04-30. Topological Prevalence of Finite Type Interval Translation Maps. https://arxiv.org/abs/2605.00186

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