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Quantitative orbit equivalence for $\mathbb{Z}$-odometers

Petr Naryshkin, Spyridon Petrakos

TL;DR

The paper addresses the quantitative classification of orbit equivalences for $\mathbb{Z}$-odometers under sub-$L^1$ conditions. It introduces a back-and-forth inductive construction on finite factors to realize a limiting orbit equivalence with cocycles whose $\omega$-norm is sublinear, thereby achieving sub-$L^1$-orbit equivalence. The main result shows that any two $\mathbb{Z}$-odometers are sub-$L^1$-orbit equivalent, implying that $L^1$-orbit equivalence collapses to conjugacy (flip-conjugacy) within this class. The framework extends prior work and suggests a viable strategy for systems that can be approximated by finite actions.

Abstract

We prove that any two $\mathbb{Z}$-odometers are sub-$L^1$-orbit equivalent, greatly strengthening previous results and giving a definitive picture of quantitative orbit equivalence for these systems.

Quantitative orbit equivalence for $\mathbb{Z}$-odometers

TL;DR

The paper addresses the quantitative classification of orbit equivalences for -odometers under sub- conditions. It introduces a back-and-forth inductive construction on finite factors to realize a limiting orbit equivalence with cocycles whose -norm is sublinear, thereby achieving sub--orbit equivalence. The main result shows that any two -odometers are sub--orbit equivalent, implying that -orbit equivalence collapses to conjugacy (flip-conjugacy) within this class. The framework extends prior work and suggests a viable strategy for systems that can be approximated by finite actions.

Abstract

We prove that any two -odometers are sub--orbit equivalent, greatly strengthening previous results and giving a definitive picture of quantitative orbit equivalence for these systems.
Paper Structure (5 sections, 7 theorems, 21 equations, 3 figures)

This paper contains 5 sections, 7 theorems, 21 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantitative orbit equivalence
  4. Z-odometers
  5. The main result

Key Result

Theorem 1

If two ergodic p.m.p. transformations $T$ and $S$ on a standard probability space are integrably orbit equivalent, then $T$ and $S$ are flip-conjugate (i.e., either $T\cong S$ or $T\cong S^{-1}$).

Figures (3)

  • Figure 1: Inverse system defining an odometer.
  • Figure 3: Definition of $\psi_{n+1}$.
  • Figure 4: Almost commutative diagram defining $\varphi_o$ and $\varphi_e$.

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem A: Theorem \ref{['thm:main with omega']}
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: see DKLMT22
  • Definition 2.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 5 more