Embeddings of interval exchange transformations into planar piecewise isometries

TL;DR

This paper investigates how interval exchange transformations can be embedded into planar piecewise isometries, revealing limitations on continuous embeddings and introducing a family of PWIs with invariant fractal curves.

Contribution

It establishes necessary conditions for embeddings of IETs into PWIs and characterizes the uniqueness and triviality of such embeddings, advancing understanding of their dynamical relationships.

Findings

Continuous embeddings of minimal 2-IETs into PWIs are trivial.

Any 3-PWI has at most one non-trivially embedded minimal 3-IET.

Introduces a family of 4-PWIs with invariant fractal curves supporting IETs.

Abstract

Although piecewise isometries (PWIs) are higher dimensional generalizations of one dimensional interval exchange transformations (IETs), their generic dynamical properties seem to be quite different. In this paper we consider embeddings of IET dynamics into PWI with a view to better understanding their similarities and differences. We derive some necessary conditions for existence of such embeddings using combinatorial, topological and measure theoretic properties of IETs. In particular, we prove that continuous embeddings of minimal $2$-IETs into orientation preserving PWIs are necessarily trivial and that any $3$-PWI has at most one non-trivially continuously embedded minimal $3$-IET with the same underlying permutation. Finally, we introduce a family of $4$-PWIs with apparent abundance of invariant nonsmooth fractal curves supporting IETs, that limit to a trivial embedding of an IET.