Extended branching Rauzy induction

TL;DR

This paper introduces an extended form of branching Rauzy induction that applies to all standard interval exchange transformations, including non-minimal cases, and demonstrates clustering of return words in their language.

Contribution

It generalizes branching Rauzy induction to arbitrary IETs, including non-minimal ones, using merging and splitting steps, and analyzes return words clustering.

Findings

All return words in IETs cluster in the Burrows-Wheeler sense.

The extended induction handles equal-length cuts and invariant components.

The method applies to non-minimal IETs.

Abstract

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.