Clustering of return words in languages of interval exchanges
🤖 AI Summary
This paper addresses the clustering property of return words in languages generated by regular interval exchange transformations (IETs), resolving an open question posed by Lapointe (2021): whether all return words are *clustered*, i.e., have contiguous occurrences of identical letters in their Burrows–Wheeler transforms (BWTs). Methodologically, the authors introduce, for the first time, the extended Rauzy induction framework to analyze the combinatorial structure of return words, integrating tools from symbolic dynamics and IET theory. They establish a rigorous proof that, for any regular IET, every return word in its associated language is BWT-clustered. This result partially confirms Lapointe’s conjecture and reveals a deep connection between the combinatorial structure of return words and dynamical invariants of IETs. Moreover, it provides a novel analytical tool and conceptual perspective for studying the combinatorial properties of IET languages.
Application Category
📝 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).
Problem
Research questions and friction points this paper is trying to address.
Study clustering words in interval exchange languages
Link clustering words to interval exchange transformations
Prove return words in regular transformations are clustering
Innovation
Methods, ideas, or system contributions that make the work stand out.
Extended Rauzy induction for clustering analysis
Clustering words via Burrows-Wheeler transform
Link return words to interval exchange transformations
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