arXiv subjects

Leon Staresinic

Publications and source records attributed to Leon Staresinic.

4 recordsLinked to original sources

arXiv2026

Transversality for Interval Translation Maps

An interval translation map (ITM) is a piece-wise translation $T \colon I \to I$ defined on a finite partition $I_1, \ldots, I_r$ 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 prove a transversality theorem for a family of dynamically defined vector subspaces that encode the dynamics of a given ITM. As a consequence, we establish a perturbation result that gives a precise control of the first return dynamics to subintervals in $I$, while preserving the remaining global dynamics of the system. Beyond their independent interest, these results are a key technical ingredient in the proof of the Characterisation of Stability of ITMs in arXiv:2605.00190, and in the establishment of the topological version of the Boshernitzan--Kornfeld Conjecture in arXiv:2605.00186.

math.DS↗

arXiv2026

Topological Prevalence of Finite Type Interval Translation Maps

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.

math.DS↗

arXiv2026

Characterisation of Stability for Interval Translation Maps

An interval translation map (ITM) is a piece-wise translation $T \colon I \to I$ defined on a finite partition $I_1, \ldots, I_r$ 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.

math.DS↗

arXiv2025

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 $T\colon I\to I$ 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.

math.DS↗