Decoding Rauzy Induction: An Effective Answer to Bufetov's Question

TL;DR

This paper presents an algorithm to reconstruct initial interval exchange transformations from finite Rauzy induction matrices, improving previous results by Veech and demonstrating the necessity of sequence length for uniqueness.

Contribution

It introduces a method to determine initial transformations from finite Rauzy induction sequences and proves the minimal length needed for uniqueness.

Findings

An algorithm for reconstructing initial transformations from finite matrices.

Proof that finite sequences can correspond to multiple transformations, showing length necessity.

Enhanced understanding of Rauzy induction's informational content.

Abstract

A typical interval exchange transformation has an infinite sequence of matrices associated to it by successive iterations of Rauzy induction. In 2010, W. A. Veech answered a question of A. Bufetov by showing that the interval exchange itself may be recovered from these matrices and must be unique up to topological conjugation. In this work, we will improve upon these results by providing an algorithm to determine the initial transformation from a sufficiently long finite subsequence of these matrices. We also show the defined length to be necessary by constructing finite sequences of Rauzy induction with multiple distinct (even up to conjugacy) initial transformations.