Transversality for Interval Translation Maps

TL;DR

This paper develops a transversality theorem for interval translation maps (ITMs), a non-invertible generalization of interval exchange transformations, enabling precise control of their local dynamics while maintaining global behavior.

Contribution

It introduces a transversality result for ITMs and a perturbation technique to control return dynamics, advancing the understanding of their stability and topological properties.

Findings

Proved a transversality theorem for ITMs.

Established a perturbation method for local dynamics.

Contributed to characterizing stability and topological conjectures for ITMs.

Abstract

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.