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A characterization of zero entropy loosely Bernoulli flows via FK-pseudometric

Alexandre Trilles

TL;DR

This work defines the Feldman-Katok pseudometric for flows $\tilde{\rho}_{FK}$ and uses it to extend the discrete-time theory of loosely Bernoulli/loosely Kronecker systems to continuous-time dynamics. It establishes a measure-theoretic characterization: a measure-preserving flow $(X,\Phi,\mu)$ is loosely Kronecker if and only if there exists a full-measure set $H$ with $\tilde{\rho}_{FK}(x,y)=0$ for all $x,y\in H$, and a purely topological characterization: a flow is topologically loosely Kronecker exactly when it is uniquely ergodic and its unique invariant measure is loosely Kronecker. The results connect measure-theoretic and topological notions via Ratner-like criteria in the flow setting and show that suspension flows and time-changes preserve these properties under appropriate conditions. Overall, the paper unifies continuous-time and discrete-time viewpoints on zero-entropy loosely Kronecker dynamics, providing practical criteria and illuminating the relationship between LK behavior and Kakutani equivalence.

Abstract

We introduce the Feldman-Katok pseudometric (FK-pseudometric for short) for flows. We then provide a characterization of zero entropy loosely Bernoulli measures for continuous flows via the FK-pseudometric extending the result known for discrete-time dynamical systems. We also provide a purely topological characterization of uniquely ergodic continuous flows whose unique invariant measure is zero entropy loosely Bernoulli.

A characterization of zero entropy loosely Bernoulli flows via FK-pseudometric

TL;DR

This work defines the Feldman-Katok pseudometric for flows and uses it to extend the discrete-time theory of loosely Bernoulli/loosely Kronecker systems to continuous-time dynamics. It establishes a measure-theoretic characterization: a measure-preserving flow is loosely Kronecker if and only if there exists a full-measure set with for all , and a purely topological characterization: a flow is topologically loosely Kronecker exactly when it is uniquely ergodic and its unique invariant measure is loosely Kronecker. The results connect measure-theoretic and topological notions via Ratner-like criteria in the flow setting and show that suspension flows and time-changes preserve these properties under appropriate conditions. Overall, the paper unifies continuous-time and discrete-time viewpoints on zero-entropy loosely Kronecker dynamics, providing practical criteria and illuminating the relationship between LK behavior and Kakutani equivalence.

Abstract

We introduce the Feldman-Katok pseudometric (FK-pseudometric for short) for flows. We then provide a characterization of zero entropy loosely Bernoulli measures for continuous flows via the FK-pseudometric extending the result known for discrete-time dynamical systems. We also provide a purely topological characterization of uniquely ergodic continuous flows whose unique invariant measure is zero entropy loosely Bernoulli.
Paper Structure (10 sections, 23 theorems, 33 equations)

This paper contains 10 sections, 23 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Flows
  4. Ergodic Theory
  5. Feldman-Katok pseudometric
  6. Discrete Feldman-Katok pseudometric
  7. Feldman-Katok pseudometric for flows
  8. Loosely Kronecker flows
  9. Topologically loosely Kronecker flows
  10. Examples and applications

Key Result

Theorem 1

Let $\Phi$ be a continuous flow on $X$ and $\mu$ be an ergodic $\Phi$-invariant measure. The measure-preserving flow $(X, \Phi, \mu)$ is loosely Kronecker if and only if there exists a Borel set $H \subseteq X$ such that $\mu(H) = 1$ and $\operatorname{\tilde{\rho}_{FK}}(x,y) = 0$ for every $x,y \in

Theorems & Definitions (48)

  • Theorem 1
  • Theorem 2
  • Corollary
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Definition 3.2
  • ...and 38 more