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On the uniqueness of affine IETs semi-conjugated to IETs

Frank Trujillo

Abstract

We prove that for almost every irreducible interval exchange transformation $T$ and for any vector $ω$ in its associated central-stable space (with respect to the Kontsevich-Zorich cocycle) there exists a unique AIET, up to normalization of its domain, semi-conjugated to $T$ and whose log-slope vector equals $ω$. This provides a partial answer to a question raised by S. Marmi, P. Moussa, and J.-C. Yoccoz.

On the uniqueness of affine IETs semi-conjugated to IETs

Abstract

We prove that for almost every irreducible interval exchange transformation and for any vector in its associated central-stable space (with respect to the Kontsevich-Zorich cocycle) there exists a unique AIET, up to normalization of its domain, semi-conjugated to and whose log-slope vector equals . This provides a partial answer to a question raised by S. Marmi, P. Moussa, and J.-C. Yoccoz.

Paper Structure

This paper contains 10 sections, 4 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Affine IETs and Rauzy-Veech induction
  3. The spaces of IETs and AIETs
  4. Rauzy-Veech induction
  5. Combinatorial rotation number
  6. Zorich acceleration
  7. The (twisted) Rauzy-Veech cocycle
  8. Oseledet's filtration of the Kontsevich Zorich cocycle
  9. Additional properties of the cocycles
  10. Proof of the main result

Key Result

Theorem 1.1

For a.e. $(\pi, \lambda) \in \mathfrak{G}^0_\mathcal{A} \times \Delta_\mathcal{A}$ and for any $\omega \in E_{cs}(\pi, \lambda)$ there exists a unique AIET on $[0, 1)$ with log-slope $\omega$ semi-conjugated to $(\pi, \lambda)$, that is, $\#\textup{Aff}^1(\pi, \lambda, \omega) = 1$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 1
  • Lemma 2.1
  • proof
  • Proposition 3.1: Bounded Central Condition
  • Lemma 3.2
  • proof : Proof of Theorem \ref{['thm: uniqueness']}