Branching Rauzy induction is a two-sided form of Rauzy induction that acts on regular interval exchange transformations (IETs). We introduce an extended form of branching Rauzy induction that applies to arbitrary standard IETs, including non-minimal ones. The procedure generalizes the branching Rauzy method with two induction steps, merging and splitting, to handle equal-length cuts and invariant components respectively. As an application, we show, via a stepwise morphic argument, that all return words in the language of an arbitrary IET cluster in the Burrows-Wheeler sense.

翻译：分支Rauzy归纳法是作用于正则区间交换变换（IETs）的双向Rauzy归纳形式。本文引入一种适用于任意标准IETs（包括非极小情形）的扩展分支Rauzy归纳法。该方法通过合并与分裂两个归纳步骤，将分支Rauzy方法推广至处理等长分割与不变分量。作为应用，我们通过逐步态射论证表明：在Burrows-Wheeler意义下，任意IET语言中的所有回归词均呈现聚类特性。