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Coboundaries of 3-IETs

Przemysław Berk, Carlos Ospina

TL;DR

This work analyzes coboundaries and skew-product ergodicity for interval exchanges on 3 intervals with $f\in C^{1+AC}$ and zero mean. By reducing to induced maps that are rotations and employing continued-fraction dynamics, Rokhlin towers, and essential-value criteria, it proves that for typical symmetric 3-IETs, $f$ is a coboundary iff $f(0)=\lim_{x\to1}f(x)=0$, while nonzero endpoint values yield ergodicity for a broad set of 3-IETs and even dense ergodic families; rare non-ergodic examples exist as well. The arguments hinge on solving the cohomological equation on induced rotations and applying the Conze–Fra\c{c}ek essential-value criterion to construct essential values with controlled Birkhoff sums. The results partially answer questions by Chaika and Robertson on typical IETs with cocycles like $f(x)=\cos(2\pi x)$, and elucidate when skew products exhibit ergodicity versus coboundary behavior in the 3-interval setting.

Abstract

In this note, we investigate the coboundaries of interval exchange transformations of 3 intervals (3-IETs). More precisely, we show that a differentiable function with absolutely continuous derivative with bounded variation, whose integral and integral of its derivative is 0, is a coboundary for typical 3-IET if and only if the values at the endpoints of the domain are zero. We also show the existence of rare counterexamples for both cases of possible values at the endpoints of the interval. We obtain our result by studying the properties of associated skew products.

Coboundaries of 3-IETs

TL;DR

This work analyzes coboundaries and skew-product ergodicity for interval exchanges on 3 intervals with and zero mean. By reducing to induced maps that are rotations and employing continued-fraction dynamics, Rokhlin towers, and essential-value criteria, it proves that for typical symmetric 3-IETs, is a coboundary iff , while nonzero endpoint values yield ergodicity for a broad set of 3-IETs and even dense ergodic families; rare non-ergodic examples exist as well. The arguments hinge on solving the cohomological equation on induced rotations and applying the Conze–Fra\c{c}ek essential-value criterion to construct essential values with controlled Birkhoff sums. The results partially answer questions by Chaika and Robertson on typical IETs with cocycles like , and elucidate when skew products exhibit ergodicity versus coboundary behavior in the 3-interval setting.

Abstract

In this note, we investigate the coboundaries of interval exchange transformations of 3 intervals (3-IETs). More precisely, we show that a differentiable function with absolutely continuous derivative with bounded variation, whose integral and integral of its derivative is 0, is a coboundary for typical 3-IET if and only if the values at the endpoints of the domain are zero. We also show the existence of rare counterexamples for both cases of possible values at the endpoints of the interval. We obtain our result by studying the properties of associated skew products.
Paper Structure (17 sections, 16 theorems, 65 equations, 2 figures)

This paper contains 17 sections, 16 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Rotations and interval exchange transformations
  3. Induced maps of 3-IETs are rotations
  4. Continued fractions
  5. Rokhlin towers for rotations
  6. Skew products and ergodicity criterion
  7. Essential values of a cocycle
  8. Proof of
  9. Reducing to
  10. The solution of the cohomological equation
  11. Ergodic examples and the concluding argument
  12. Proof of
  13. Initial reductions.
  14. Bounds of the Birkhoff sums.
  15. -balanced points
  16. ...and 2 more sections

Key Result

Theorem 1

Let $f \in C^{1+AC}([0,1))$ such that $f(0)=\lim_{x\to 1}f(x)=0$ with $\int f=0$, then $f$ is a coboundary for almost every symmetric 3-IET $T$. In particular, the skew product $T_f$ is not ergodic. However, there exists a dense subset of 3-IETs, for which the skew product $T_f$ is ergodic.

Figures (2)

  • Figure 1: Rokhlin towers for a rotation by $\alpha$.
  • Figure 2: Behavior of the Birkhoff sums $S_{k}f$ for different values of $k$. The cocycle function $f$ has discontinuities at $0$, $1-\alpha$, $\beta$. The argument to prove ergodicity in \ref{['thm:main2']} is to estimate the value of the jumps at specific times $k$ and combine this with \ref{['thm: FUcriterion']}.

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Corollary 4
  • Corollary 5
  • Remark 6
  • Remark 7
  • Proposition 8
  • Remark 9
  • Proposition 10: aaronson1997
  • ...and 12 more