Clustering of return words in languages of interval exchanges

TL;DR

The paper addresses whether return words in languages generated by regular interval exchange transformations are clustering, affirmatively answering a question of Lapointe. It develops an extended Rauzy induction framework and a family of morphisms that preserve clustering across induction steps, enabling the main result to hold for all return words of regular IETs. The authors also connect clustering words to discrete interval exchanges (DIETs) and introduce alsinicity to extend the theory to multisets of words. This work strengthens the bridge between symbolic dynamics, combinatorics on words, and interval exchange dynamics, with implications for understanding the structure of return words and their Burrows-Wheeler profiles.

Abstract

A word over an ordered alphabet is said to be clustering if identical letters appear adjacently in its Burrows-Wheeler transform. Such words are strictly related to (discrete) interval exchange transformations. We use an extended version of the well-known Rauzy induction to show that every return word in the language generated by a regular interval exchange transformation is clustering, partially answering a question of Lapointe (2021).

Figures (4)

• Figure 1: A DIET (on the left) and its associated IET (on the right).
• Figure 2: Series of Rauzy steps and their associated morphisms.
• Figure 3: Extension graphs of $\varepsilon$ and ${\tt a}$ in $\mathcal{L}(T)$, with $T$ as in Example \ref{['ex:extensiongraph']}.
• Figure 4: Extension graphs of $\varepsilon$, ${\tt a}$ and ${\tt aba}$ as in Example \ref{['ex:multinonclustering']}.

Theorems & Definitions (17)

• theorem 1
• proposition 1
• proposition 2
• proposition 3
• lemma 1
• proof
• lemma 2
• proposition 4: branching
• lemma 3
• proof