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A surprising discrepancy in the regularity of conjugacies between generalized interval exchange transformations and their inverses at freezing

Krzysztof Frączek, Łukasz Kotlewski

Abstract

Generalized interval exchange transformations (GIETs) are semi-conjugate to interval exchange transformations (IETs) when the Rauzy-Veech combinatorics is $\infty$-complete. When this semi-conjugacy is a homeomorphism, a fundamental problem is to understand the regularity of the conjugacy and its inverse. Contrary to the usual expectation that their Hölder regularities degenerate simultaneously, we exhibit a strongly asymmetric behavior. For self-similar IETs of hyperbolic periodic type and a natural one-parameter central family of affine IET deformations obtained via a freezing (zero-temperature) limit, the conjugacy becomes arbitrarily irregular while its inverse remains uniformly Hölder. Using thermodynamic formalism for renormalization and zero-temperature limits, we obtain sharp asymptotics for Hausdorff dimensions of invariant and conformal measures and for the supremal Hölder exponents of the conjugacy and its inverse.

A surprising discrepancy in the regularity of conjugacies between generalized interval exchange transformations and their inverses at freezing

Abstract

Generalized interval exchange transformations (GIETs) are semi-conjugate to interval exchange transformations (IETs) when the Rauzy-Veech combinatorics is -complete. When this semi-conjugacy is a homeomorphism, a fundamental problem is to understand the regularity of the conjugacy and its inverse. Contrary to the usual expectation that their Hölder regularities degenerate simultaneously, we exhibit a strongly asymmetric behavior. For self-similar IETs of hyperbolic periodic type and a natural one-parameter central family of affine IET deformations obtained via a freezing (zero-temperature) limit, the conjugacy becomes arbitrarily irregular while its inverse remains uniformly Hölder. Using thermodynamic formalism for renormalization and zero-temperature limits, we obtain sharp asymptotics for Hausdorff dimensions of invariant and conformal measures and for the supremal Hölder exponents of the conjugacy and its inverse.
Paper Structure (6 sections, 12 theorems, 70 equations, 1 figure, 3 tables)

This paper contains 6 sections, 12 theorems, 70 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. GIETs, AIETs and their conjugacy with IETs
  3. Self-similar IETs and corresponding AIETs
  4. Thermodynamical formalism and zero-temperature regime
  5. Asymptotic behavior
  6. Example

Key Result

Theorem 1.1

Suppose that $T$ is an IET of hyperbolic periodic type and $\omega\in\mathbb{R}^{\mathcal{A}}$ is a log-slope vector of central type. Let us consider the family of AIETs $f_{t\omega}\in\operatorname{Aff}(T,t\omega)$, for $t\in\mathbb{R}$. Then,

Figures (1)

  • Figure 1: Plots showing the behavior of Hausdorff dimensions and Hölder exponents as the parameter $t$ varies

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 3.1: Theorems 1.3 and 1.6 in BFKT
  • Theorem 3.2: Theorem 5.2 in BFKT
  • Theorem 3.3: Propositions 14.3 and 15.3 in BFKT
  • Theorem 4.1: Theorem 1.1 in ChGU
  • Remark 4.2
  • Proposition 4.3: Proposition 3 in ChGU
  • Proposition 4.4
  • proof
  • Proposition 4.5
  • ...and 11 more