Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On asymptotic expansions of ergodic integrals for $\Z^d$-extensions of translation flows

Henk Bruin, Charles Fougeron, Davide Ravotti, Dalia Terhesiu

TL;DR

The paper develops a rigorous framework for asymptotic expansions of ergodic integrals on $\mathbb{Z}^d$-covers of compact translation surfaces, using a renormalization by homogeneous pseudo-Anosov lifts and a sophisticated operator Local Limit Theorem. The authors prove higher-order asymptotics for $d=1,2$ with optimal weak rational ergodicity rates, and provide explicit expansions involving Gaussian terms and polynomial corrections in a renormalization parameter $K$. The approach hinges on anisotropic Banach spaces for twisted transfer operators, enabling precise spectral analysis of $\mathcal{L}_u$ and its leading eigenvalue, which feeds into detailed ergodic-sum expansions. They also illustrate the theory on self-similar staircases and Ehrenfest wind-tree models, expanding ergodic results to $\mathbb{Z}^d$-covers and providing new insights into the ergodic structure and fluctuations of such infinite covers. Overall, the work advances quantitative ergodic theory for higher-rank abelian covers of translation surfaces and offers tools with potential applications to related billiard models and self-similar flows.

Abstract

We obtain expansions of ergodic integrals for $\Z^d$-covers of compact self-similar translation flows, and as a consequence we obtain a form of weak rational ergodicity with optimal rates. As examples, we consider the so-called self-similar $(s,1)$-staircase flows ($\Z$-extensions of self-similar translations flows of genus-$2$ surfaces), and particular cases of the Ehrenfest wind-tree model.

On asymptotic expansions of ergodic integrals for $\Z^d$-extensions of translation flows

TL;DR

The paper develops a rigorous framework for asymptotic expansions of ergodic integrals on -covers of compact translation surfaces, using a renormalization by homogeneous pseudo-Anosov lifts and a sophisticated operator Local Limit Theorem. The authors prove higher-order asymptotics for with optimal weak rational ergodicity rates, and provide explicit expansions involving Gaussian terms and polynomial corrections in a renormalization parameter . The approach hinges on anisotropic Banach spaces for twisted transfer operators, enabling precise spectral analysis of and its leading eigenvalue, which feeds into detailed ergodic-sum expansions. They also illustrate the theory on self-similar staircases and Ehrenfest wind-tree models, expanding ergodic results to -covers and providing new insights into the ergodic structure and fluctuations of such infinite covers. Overall, the work advances quantitative ergodic theory for higher-rank abelian covers of translation surfaces and offers tools with potential applications to related billiard models and self-similar flows.

Abstract

We obtain expansions of ergodic integrals for -covers of compact self-similar translation flows, and as a consequence we obtain a form of weak rational ergodicity with optimal rates. As examples, we consider the so-called self-similar -staircase flows (-extensions of self-similar translations flows of genus- surfaces), and particular cases of the Ehrenfest wind-tree model.
Paper Structure (24 sections, 26 theorems, 116 equations, 7 figures)

This paper contains 24 sections, 26 theorems, 116 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Abelian covers and homogeneous automorphisms
  3. Homological properties of $\mathbb Z^d$-covers
  4. Homogeneous pseudo-Anosov lifts
  5. Staircases
  6. Wind-tree billiards
  7. The wind-tree model with plus-shaped obstacles
  8. The classical wind-tree model
  9. Ergodic properties
  10. Ergodic integrals and (weak) rational ergodicity via Local Limit Laws
  11. Main results
  12. Strategy of the proof
  13. An (operator) local limit theorem (LLT) for $F_{K}$
  14. Proof of the main results
  15. Banach spaces and Proof of Proposition \ref{['prop:op']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $G \in C^1(X_{\Gamma})$ be compactly supported. Then, there exist real bounded functions $g_{k,j}$ so that for all $N \ge 1$ and $\boldsymbol{m}$-a.e. $x \in X_{\Gamma}$, as $T \to \infty$ and $K = \log^* T = \lceil -\frac{\log T}{\log \lambda} \rceil$.

Figures (7)

  • Figure 1: The $(3,1)$-rectangle (left) and the $(3,1)$-staircase (right). On the left, $\Sigma$ consists of a single point with cone angle $6\pi$; it splits into countably many singularities on the staircase, all with cone angle $6\pi$.
  • Figure 2: The $0$th step and its image under $\psi_\Gamma$ with matrix $\binom{1 \ 3}{2 \ 7}$.
  • Figure 3: Dehn twist on an horizontal cylinder
  • Figure 4: Left: Wind-tree model with plus-shapes wind-tree model (and finite horizon in horizontal and vertical directions). A fundamental domain in dotted lines. Right: Four copies forming a fundamental domain of the unfolded translation surface of the plus-shaped wind-tree model.
  • Figure 5: A fundamental domain for the surface $X$. The four colored dots represents the four singularities in $\Sigma$, each of angle $6\pi$. The elements $\gamma_1, \dots, \gamma_{13} \in H(X,\Sigma,\mathbb Z)$ form a basis of the relative homology: the homology class of any oriented path connecting two elements of $\Sigma$ can be written as a linear combination with integer coefficients of $\gamma_1, \dots, \gamma_{13}$ (for example, the path marked as $\eta_1$ corresponds to the homology class $\gamma_2+\gamma_4+\gamma_5$, since the concatenation of the paths $\gamma_2, \gamma_4,\gamma_5$ and $-\eta_1$ bounds a disk and hence is trivial in homology).
  • ...and 2 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 43 more