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Extended branching Rauzy induction

Francesco Dolce, Christian B. Hughes

TL;DR

The work broadens Rauzy-type inductions to general standard IETs by integrating merging and splitting with the classic right/left Rauzy steps, enabling finite induction chains to obtain first-return maps on cylinders even in irregular or non-minimal cases. It builds a robust language-theoretic framework using extension graphs and alsinic/dendric notions to connect IET dynamics with Burrows-Wheeler clustering, proving that return words are clustering with respect to the induced permutation. The results yield constructive descriptions of return word sets and provide a path to perfect clustering in symmetric IETs, with broader implications for discrete dynamical systems and symbolic coding. The paper also outlines directions for refining the induction via local commutations and extending Bratteli–Vershik models to multi-component Cantor systems derived from 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.

Extended branching Rauzy induction

TL;DR

The work broadens Rauzy-type inductions to general standard IETs by integrating merging and splitting with the classic right/left Rauzy steps, enabling finite induction chains to obtain first-return maps on cylinders even in irregular or non-minimal cases. It builds a robust language-theoretic framework using extension graphs and alsinic/dendric notions to connect IET dynamics with Burrows-Wheeler clustering, proving that return words are clustering with respect to the induced permutation. The results yield constructive descriptions of return word sets and provide a path to perfect clustering in symmetric IETs, with broader implications for discrete dynamical systems and symbolic coding. The paper also outlines directions for refining the induction via local commutations and extending Bratteli–Vershik models to multi-component Cantor systems derived from 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.

Paper Structure

This paper contains 17 sections, 22 theorems, 56 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Combinatorics on Words
  3. Words
  4. Languages
  5. Morphisms
  6. Permutations
  7. Burrows-Wheeler transform
  8. Interval Exchange Transformations
  9. Induced Interval Exchange Transformations
  10. Right Rauzy step
  11. Left Rauzy step
  12. Merging
  13. Splitting
  14. First-return maps to cylinders
  15. Extension Graphs and Alsinicity
  16. ...and 2 more sections

Key Result

Proposition 2.3

Two words $u,v$ over the same ordered alphabet $\mathcal{A}$ are conjugate if and only if ${\rm bwt}_{\mathcal{A}} \left( u \right) = {\rm bwt}_{\mathcal{A}} \left( v \right)$.

Figures (7)

  • Figure 1: An IET with three components and four regions (for simplicity we denote with the letter $a$ both by $I_a$ and $T(I_a)$.
  • Figure 2: A DIET (on the left) and its associated IET (on the right).
  • Figure 3: The IET in Example \ref{['ex:non-minimal-IET']}.
  • Figure 4: A circularly reordered IET of Figure \ref{['fig:non-minimal-IET']}.
  • Figure 5: The IET of Example \ref{['ex:merging']} after the application of $\sigma_\mathcal{B}$.
  • ...and 2 more figures

Theorems & Definitions (53)

  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Proposition 2.9
  • Example 3.1
  • ...and 43 more