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The equivalence of precompactness, zero maximal pattern entropy and bounded mean complexity for finite partitions

Jian Li, Tao Yu, Xianliang Zhong

Abstract

In this paper, we investigate several types of low complexity of finite partitions, including precompactness, zero maximal pattern entropy, bounded mean complexity and mean equicontinuity. We first show that a collection of finite partitions in a standard probability space is precompact in the Rokhlin metric if and only if it has zero maximal pattern entropy if and only if the collection of the characteristic functions of atoms in those partitions is precompact in $L^2$ if and only if it has bounded mean complexity with respect the Hamming distance. Next, we show that for a countably infinite discrete amenable group acting on a standard probability space, a finite partition has zero maximal pattern entropy if and only if each characteristic function of atom in the partition is almost periodic if and only if it has bounded mean complexity with respect to some (and hence any) Følner sequence if and only if it is mean equicontinuous with respect to some (and hence any) tempered Følner sequence.

The equivalence of precompactness, zero maximal pattern entropy and bounded mean complexity for finite partitions

Abstract

In this paper, we investigate several types of low complexity of finite partitions, including precompactness, zero maximal pattern entropy, bounded mean complexity and mean equicontinuity. We first show that a collection of finite partitions in a standard probability space is precompact in the Rokhlin metric if and only if it has zero maximal pattern entropy if and only if the collection of the characteristic functions of atoms in those partitions is precompact in if and only if it has bounded mean complexity with respect the Hamming distance. Next, we show that for a countably infinite discrete amenable group acting on a standard probability space, a finite partition has zero maximal pattern entropy if and only if each characteristic function of atom in the partition is almost periodic if and only if it has bounded mean complexity with respect to some (and hence any) Følner sequence if and only if it is mean equicontinuous with respect to some (and hence any) tempered Følner sequence.
Paper Structure (11 sections, 20 theorems, 86 equations)

This paper contains 11 sections, 20 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the complexity of finite partitions
  3. Finite measurable partition and entropy
  4. Sequence entropy of partitions
  5. Bounded mean complexity of partitions
  6. Proof of Theorem \ref{['thm:main1']}
  7. The complexity of finite partitions in measurable-preserving systems
  8. The complexity of measurable actions
  9. The complexity of measure-preserving systems
  10. The complexity of amenable group actions
  11. The complexity of invariant measures of a topological dynamical system

Key Result

Theorem 1.1

Let $(X, \mathcal{B}, \mu)$ be a standard probability space. For $r\geq 2$, let $\mathfrak{P}_r$ be the collection of all partitions of $X$ with at most $r$ atoms. Then for a subset $K$ of $\mathfrak{P}_r$, the following assertions are equivalent:

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 28 more