# Existence of non-trivial embeddings of Interval Exchange Transformations into Piecewise Isometries

## TL;DR

This paper demonstrates that most interval exchange transformations with higher genus surfaces can be embedded into piecewise isometries, revealing invariant curves similar to KAM curves that are not simple unions of circles or line segments.

## Contribution

It proves the existence of non-trivial, isometric embeddings of interval exchange transformations into piecewise isometries for higher genus surfaces, a novel connection in dynamical systems.

## Key findings

- Almost every interval exchange transformation can be embedded into piecewise isometries.
- Embedded invariant curves are not unions of simple geometric shapes.
- The result extends understanding of invariant structures in dynamical systems.

## Abstract

We prove that almost every interval exchange transformation, with an associated translation surface of genus $g\geq 2$, can be non-trivially and isometrically embedded in a family of piecewise isometries. In particular this proves the existence of invariant curves for piecewise isometries, reminiscent of KAM curves for area preserving maps, which are not unions of circle arcs or line segments.