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On the ergodicity of anti-symmetric skew products with singularities and its applications

Przemysław Berk, Krzysztof Frączek, Frank Trujillo

TL;DR

The paper develops a novel ergodicity framework for skew products $T_f$ over interval exchange transformations with piecewise smooth cocycles that have endpoint singularities, extending beyond logarithmic-type singularities by using a Borel–Cantelli–type argument combined with anti-symmetric cocycles. It leverages Rauzy–Veech and Kontsevich–Zorich dynamics to impose Diophantine-type controls and proves ergodicity in two principal settings: (i) flows with no saddle loops, where logarithmic-type symmetry suffices, and (ii) hyperelliptic flows with a single perfect saddle, where an antisymmetric decomposition isolates the ergodic component. The work also constructs ergodic antisymmetric skew extensions for new classes of degenerate saddles and provides applications to the deviation spectrum and equidistribution of Birkhoff error terms for locally Hamiltonian flows, including cases with saddle loops. Overall, the results broaden the range of cocycles and singularities for which ergodicity and equidistribution can be established on translation surfaces, with implications for spectral analysis of locally Hamiltonian systems.

Abstract

We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel-Cantelli-type arguments from Fayad and Lemańczyk (2006). The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and antisymmetric cocycles. Moreover, its most significant advantage is its ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles.

On the ergodicity of anti-symmetric skew products with singularities and its applications

TL;DR

The paper develops a novel ergodicity framework for skew products over interval exchange transformations with piecewise smooth cocycles that have endpoint singularities, extending beyond logarithmic-type singularities by using a Borel–Cantelli–type argument combined with anti-symmetric cocycles. It leverages Rauzy–Veech and Kontsevich–Zorich dynamics to impose Diophantine-type controls and proves ergodicity in two principal settings: (i) flows with no saddle loops, where logarithmic-type symmetry suffices, and (ii) hyperelliptic flows with a single perfect saddle, where an antisymmetric decomposition isolates the ergodic component. The work also constructs ergodic antisymmetric skew extensions for new classes of degenerate saddles and provides applications to the deviation spectrum and equidistribution of Birkhoff error terms for locally Hamiltonian flows, including cases with saddle loops. Overall, the results broaden the range of cocycles and singularities for which ergodicity and equidistribution can be established on translation surfaces, with implications for spectral analysis of locally Hamiltonian systems.

Abstract

We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel-Cantelli-type arguments from Fayad and Lemańczyk (2006). The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and antisymmetric cocycles. Moreover, its most significant advantage is its ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles.
Paper Structure (16 sections, 24 theorems, 294 equations, 6 figures)

This paper contains 16 sections, 24 theorems, 294 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Applications to locally Hamiltonian flows on compact surfaces
  3. Historical overview
  4. Deviation spectrum in full generality
  5. Main results and technical novelties
  6. Interval exchange transformations and translation surfaces
  7. Horizontal spacing
  8. Diophantine conditions
  9. Ergodicity Criterion and Borel-Cantelli Argument
  10. Antisymmetric skew products over symmetric IETs
  11. Applications to skew product with logarithmic singularities
  12. Applications to locally Hamiltonian flows with perfect saddles
  13. Locally Hamiltonian flows with imperfect saddles
  14. Degenerate simple saddles
  15. Imperfect multi-saddles
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\theta:[x_0,+\infty)\to\mathbb{R}_{>0}$, with $x_0>0$, be a slowly varying increasing $C^1$-map such that $\int_{x_0}^{+\infty}\frac{dx}{x\theta(x)}=+\infty$ and the map $x\mapsto x^2\theta'(x)$ is increasing. Then for a.e. symmetric IET $T$ if then the skew product $T_f$ is ergodic.

Figures (6)

  • Figure 1: A chain of adjacent saddle loops.
  • Figure 2: The polygonal representation of the surface $(\pi,\lambda,\tau)$ with indicated zippered rectangles. The picture shows three different orbits, corresponding to three of the cases described in the proof of Lemma \ref{['lem: horizontalspacing']}.
  • Figure 3: The phase portrait of the local Hamiltonian on the rectangle $R$. The map $r_{\pi}$ is the rotation by $\pi$. The symmetry of the picture is due to \ref{['eq: symmetricH']}.
  • Figure 4:
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.2
  • Conjecture 2.3
  • Theorem 2.4
  • Proposition 4.1
  • Lemma 4.2
  • proof
  • proof : Proof of Proposition \ref{['prop: horizontal spacing']}
  • Proposition 5.1
  • ...and 43 more