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.

翻译：虽然片形缩略图(PWIs)是一维间距交换变异(IETs)的更高维度的常规化,但其通用动态特性似乎大不相同。 在本文中,我们考虑将IET动态植入PWI, 以便更好地了解它们的相似性和差异。 我们为这种嵌入的存在创造了一些必要的条件, 使用IET的组合、 地貌学和度量性理论性能。 特别是, 我们证明, 将最少2美元- IET 嵌入方向保护 PWis 必然微不足道, 任何 $3 美元- PWI 都最多有一个非三维的嵌入最小的 3美元- IET 和 相同的基本变异性。 最后, 我们引入了一个有4美元- PWIWI 的家族, 其明显富含支持 IET 的无变异非移动形的分形曲线, 仅限于一小块嵌入 IET 。