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Skew-product systems over infinite interval exchange transformations

Henk Bruin, Olga Lukina

Abstract

We study the ergodic properties (recurrence, discrepancy, diffusion coefficients and ergodicity itself) of a class of $\mathbb Z$-extensions over infinite interval exchange transformations called rotated odometers. The choice of a skew-function is motivated by the use in the study of parallel flows on a particular staircase manifold of infinite genus.

Skew-product systems over infinite interval exchange transformations

Abstract

We study the ergodic properties (recurrence, discrepancy, diffusion coefficients and ergodicity itself) of a class of -extensions over infinite interval exchange transformations called rotated odometers. The choice of a skew-function is motivated by the use in the study of parallel flows on a particular staircase manifold of infinite genus.
Paper Structure (28 sections, 18 theorems, 89 equations, 1 figure)

This paper contains 28 sections, 18 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Rotated odometers
  3. Non-ergodicity of $\mathbb{Z}$-extensions of rotated odometers
  4. Diffusion coefficient
  5. Discrepancy
  6. Skew-products of rotated odometers as first return maps
  7. Flows on a finite area translation surface
  8. The staircase
  9. Preliminaries of rotated odometers and essential values
  10. Substitutions
  11. Renormalization and symbolic dynamics
  12. The associated matrix in Frobenius form
  13. Doubling the middle symbol
  14. Essential values
  15. Coboundaries for rotated odometers
  16. ...and 13 more sections

Key Result

Theorem 1.9

Let $(I,F_\pi,\hbox{Leb})$ be a stationary rotated odometer. Let $T_\pi: I \times \mathbb{Z} \to I \times \mathbb{Z}$ be its $\mathbb{Z}$-extension with skew-function $\psi$ as in skew-function, and $\boldsymbol{d} := \gcd\{ \psi(\chi(a)) : a \in \mathcal{A}\}$. If $\boldsymbol{d} \geq 2$, then the

Figures (1)

  • Figure 1: Left: The unit square with identifications given by \ref{['eq-simrel']} for $\tau = {\rm id}$. Dashed lines represent a straight-line flows line at an angle $\theta = \tan^{-1}\frac{q}{p}$ and its decomposition as the horizontal translation by $\frac{p}{q}$ and the vertical translation by $1$. Right: Three steps of the staircase with identifications.

Theorems & Definitions (59)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.6
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Theorem 1.11
  • Theorem 1.12
  • Theorem 2.1
  • ...and 49 more