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Interval Rearrangement Ensembles

Alexey Teplinsky

Abstract

We introduce a new concept of interval rearrangement ensembles (IRE), which is a generalization of interval exchange transformations (IET). This construction expands the space of IETs in accordance with the natural duality that we pinpoint. Induction of Rauzy\,--\,Veech kind is applicable to IREs. It is conjugate to the reverse operation by the duality mentioned above. A natural extension of an IRE is associated with two transversal flows on a flat translation surface with branching points.

Interval Rearrangement Ensembles

Abstract

We introduce a new concept of interval rearrangement ensembles (IRE), which is a generalization of interval exchange transformations (IET). This construction expands the space of IETs in accordance with the natural duality that we pinpoint. Induction of Rauzy\,--\,Veech kind is applicable to IREs. It is conjugate to the reverse operation by the duality mentioned above. A natural extension of an IRE is associated with two transversal flows on a flat translation surface with branching points.

Paper Structure

This paper contains 13 sections, 9 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Classical IETs
  3. Multi-segment IETs
  4. Definition of an IRE
  5. IREs as dynamical systems
  6. Transforming an IRE into an IET on a tree.
  7. Dimensions of real data spaces
  8. Induction for IREs
  9. Duality and time reversing symmetry
  10. Natural extensions of IREs
  11. Surfaces and flows associated with IREs
  12. Example
  13. Conclusions

Key Result

Proposition 1

$\dim(X_\sigma) = d + P.$

Theorems & Definitions (26)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 16 more