Topological Prevalence of Finite Type Interval Translation Maps

TL;DR

This paper proves that for any number of subintervals, the set of finite type interval translation maps (ITMs) is topologically large, confirming a long-standing conjecture about their prevalence.

Contribution

It establishes that finite type ITMs form an open and dense subset in the space of all ITMs for any number of subintervals, confirming a topological conjecture.

Findings

Finite type ITMs are topologically prevalent for all r ≥ 2.

The set of finite type ITMs contains an open and dense subset.

This confirms a long-standing conjecture by Boshernitzan and Kornfeld.

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.