Extended branching Rauzy induction

2025-11-27

AI Summary

This work addresses the limitation of classical Rauzy induction—which applies only to minimal interval exchange transformations (IETs)—by introducing an extended *branched* Rauzy induction for arbitrary standard IETs, including non-minimal cases. To handle equal-length cuts and invariant components, we propose a two-step induction mechanism: *merging* and *splitting*, thereby generalizing branched Rauzy induction to the full class of IETs for the first time. Integrated with stepwise morphisms, this framework unifies the dynamical behavior and combinatorial language structure of IETs. Our main result establishes that, for any IET, all return words in its associated language are *strictly clustered* under the Burrows–Wheeler transform (BWT). This breakthrough extends the scope of Rauzy-type methods beyond minimality assumptions and reveals, for the first time, an intrinsic combinatorial regularity of IET languages under BWT—offering a novel tool and perspective at the intersection of ergodic theory and symbolic dynamics.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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.

Problem

• Extends branching Rauzy induction to arbitrary standard IETs
• Generalizes merging and splitting steps for equal-length cuts
• Analyzes return words clustering in arbitrary IET languages

Innovation

• Extended branching Rauzy induction for arbitrary IETs
• Generalizes merging and splitting steps for equal-length cuts
• Stepwise morphic argument for return word clustering