Forcing Minimal Interval Patterns as Interval Exchange Transformations

TL;DR

This paper establishes a mathematical connection between over-twist patterns and interval exchange transformations, showing that such patterns can be represented with bounded segments, independent of certain parameters.

Contribution

It proves that over-twist patterns are conjugate to interval exchange transformations with a bounded number of isometric segments, depending only on the pattern's modality.

Findings

Over-twist patterns are conjugate to interval exchange transformations.

The number of segments is bounded independently of period and over-rotation number.

Bound depends solely on the pattern's modality.

Abstract

We prove that any over-twist pattern is conjugate to an interval exchange transformation with bounded number of segments of isometry, restricted on one of its cycles. The bound is independent of the period and over-rotation number of the over-twist pattern and depends only on its modality.