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Rigidity for piece-wise smooth circle maps and certain GIETs

Przemysław Berk, Frank Trujillo

Abstract

The goal of this article is to show a rigidity property of conjugacies of generalized interval exchange transformations (GIETs). More precisely, we show that if two piecewise $C^3$ GIETs $f$ and $g$ of generic rotation number with mean-non-linearity 0 are homeomorphic, boundary-equivalent and their renormalizations approach in an appropriate way the set of affine interval exchange transformations, then their respective renormalizations converge to each other and the conjugating map is $C^1$. Moreover, if $f$ and $g$ are GIETs with rotation type combinatorial data, generic rotation number and they are break-equivalent as piecewise circle diffeomorphisms, they are actually $C^1$-conjugated as circle diffeomorphisms. These results generalize the work of K. Cunha and D. Smania \cite{cunha_rigidity_2014} in the case of piecewise $C^3$ circle maps, where the authors prove an analogous result for GIETs with rotation type combinatorial data and bounded rotation number.

Rigidity for piece-wise smooth circle maps and certain GIETs

Abstract

The goal of this article is to show a rigidity property of conjugacies of generalized interval exchange transformations (GIETs). More precisely, we show that if two piecewise GIETs and of generic rotation number with mean-non-linearity 0 are homeomorphic, boundary-equivalent and their renormalizations approach in an appropriate way the set of affine interval exchange transformations, then their respective renormalizations converge to each other and the conjugating map is . Moreover, if and are GIETs with rotation type combinatorial data, generic rotation number and they are break-equivalent as piecewise circle diffeomorphisms, they are actually -conjugated as circle diffeomorphisms. These results generalize the work of K. Cunha and D. Smania \cite{cunha_rigidity_2014} in the case of piecewise circle maps, where the authors prove an analogous result for GIETs with rotation type combinatorial data and bounded rotation number.
Paper Structure (23 sections, 25 theorems, 129 equations)

This paper contains 23 sections, 25 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. A notion of typicality for infinitely renormalizable GIETs.
  3. Description of results
  4. Outline of the article
  5. Preliminaries
  6. Notations
  7. Generalized interval exchange transformations
  8. Piecewise smooth circle homeomorphisms
  9. An appropriate subsequence of Zorich times
  10. Exponential convergence of renormalizations
  11. Statements of the results
  12. Proof of the main results
  13. Proof of Theorem \ref{['thm: rigidity']}
  14. Proof of Theorem \ref{['thm: affine_shadow']}
  15. Proof of Theorem \ref{['thm: linearization']}
  16. ...and 8 more sections

Key Result

Proposition 2.1

Let $f$ be a $C^1$ GIET with genus $\kappa$ satisfying Keane's condition. Then the following holds.

Theorems & Definitions (50)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4: Cohomological equation
  • Definition 2.5: EC Condition
  • Corollary 2.6
  • Theorem 3.1: Rigidity
  • Theorem 3.2: Affine shadow
  • Theorem 3.3: Linearization
  • ...and 40 more