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Typical Weak Mixing and Exceptional Spectral Properties for Interval Translation Mappings

Mauro Artigiani, Artur Avila, Sébastien Ferenczi, Pascal Hubert, Alexandra Skripchenko

Abstract

We investigate weak mixing for some classes of interval translation mappings. We give two distinct proofs that a typical Bruin-Troubetzkoy interval translation mapping is weakly mixing. Moreover, we show that the second approach extends to other classes of interval translation mappings. In particular, we show that Bruin interval translation mappings on any number of intervals are typically weak mixing. Finally, we construct the first examples of non weak mixing Bruin-Troubetzkoy ITM of infinite type.

Typical Weak Mixing and Exceptional Spectral Properties for Interval Translation Mappings

Abstract

We investigate weak mixing for some classes of interval translation mappings. We give two distinct proofs that a typical Bruin-Troubetzkoy interval translation mapping is weakly mixing. Moreover, we show that the second approach extends to other classes of interval translation mappings. In particular, we show that Bruin interval translation mappings on any number of intervals are typically weak mixing. Finally, we construct the first examples of non weak mixing Bruin-Troubetzkoy ITM of infinite type.
Paper Structure (25 sections, 23 theorems, 106 equations, 9 figures)

This paper contains 25 sections, 23 theorems, 106 equations, 9 figures.

Table of Contents

  1. Introduction
  2. ITMs
  3. Main results
  4. Organization of the paper
  5. Bruin-Troubetzkoy ITMs
  6. The geometrical interpretation of the cocycles
  7. Band complexes for BT ITMs
  8. The acceleration of the induction and the geometric interpretation
  9. A criterion for positivity of the second Lyapunov exponent
  10. The setup
  11. Bruin $4$-ITMs
  12. The $\mathcal{R}$ induction for Bruin $4$-ITMs
  13. Suspending Bruin $4$-ITMs
  14. A conjugation
  15. From positivity of the second Lyapunov exponent to weak mixing
  16. ...and 10 more sections

Key Result

Theorem 2

Almost all Bruin-Troubetzkoy ITMs of infinite type are weakly mixing.

Figures (9)

  • Figure 1: An example of a Bruin-Troubetzkoy ITM. The intervals below are images of the ones above, color coded.
  • Figure 2: A topological picture of the band complex for the BT ITM in \ref{['fig:BT_ITM']}. We also show, roughly in the middle of each band, one vertical leaf for each band.
  • Figure 3: The first case of the induction $\widetilde{\mathcal{R}}$ for the band complex. Remark that the pointed leftmost horizontal segment in the right hand drawing is not part of the domain of the induced transformation.
  • Figure 4: The second case of the induction $\widetilde{\mathcal{R}}$ for the band complex. Remark that the pointed leftmost horizontal segment in the right hand drawing is not part of the domain of the induced transformation.
  • Figure 5: An example of a Bruin ITM on $4$ intervals. The intervals below are images of the ones above, color coded.
  • ...and 4 more figures

Theorems & Definitions (47)

  • Conjecture 1: Boshernitzan-Kornfeld, BK
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark
  • Proposition 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 37 more