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Unbalancedness of the conjugacy relation of ergodic measure-preserving transformations

Bo Peng

Abstract

We show that the isomorphism of ergodic measure-preserving transformations is not Borel reducible to the relation induced by the conjugacy action of the full group of an ergodic measure-preserving transformation on itself. This answers a question of Le Maître in the negative and gives a positive indication towards a conjecture of Sabok. In fact, we prove that the isomorphism of ergodic measure-preserving transformation is unbalanced, which answers another question of Le Maître in the positive.

Unbalancedness of the conjugacy relation of ergodic measure-preserving transformations

Abstract

We show that the isomorphism of ergodic measure-preserving transformations is not Borel reducible to the relation induced by the conjugacy action of the full group of an ergodic measure-preserving transformation on itself. This answers a question of Le Maître in the negative and gives a positive indication towards a conjecture of Sabok. In fact, we prove that the isomorphism of ergodic measure-preserving transformation is unbalanced, which answers another question of Le Maître in the positive.

Paper Structure

This paper contains 4 sections, 11 theorems, 18 equations.

Table of Contents

  1. introduction
  2. A direct proof of the anti-classification
  3. Global unbalancedness
  4. Acknowledgments

Key Result

Theorem 1.6

(Allison and Panagiotopoulos Unblanced) If the Polish $G$-space is generically unbalanced, then the orbit equivalence relation is not classifiable by ${\rm TSI}$ Polish group actions.

Theorems & Definitions (21)

  • Conjecture 1.1
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 2.1
  • proof : Proof of Corollary \ref{['cor']}
  • Lemma 2.2
  • ...and 11 more