Almost every Interval Translation Map of three intervals is finite type

TL;DR

This paper proves that almost all interval translation maps of three intervals are of finite type, showing they can be reduced to simple rotations, which advances understanding of their dynamical behavior.

Contribution

The paper establishes the finiteness conjecture for three-interval ITMs, demonstrating that finite type ITMs form a large, dense subset of all such maps.

Findings

Finite type ITMs of three intervals are prevalent.

Any three-interval ITM reduces to a rotation or double rotation.

The set of finite type ITMs has full Lebesgue measure.

Abstract

Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In this paper, we prove the finiteness conjecture for the ITMs of three intervals. Namely, the subset of ITMs of finite type contains an open, dense, and full Lebesgue measure subset of the space of ITMs of three intervals. For this, we show that any ITM of three intervals can be reduced either to a rotation or to a double rotation.