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Naive mean dimension

Bingbing Liang, Kesong Yan

Abstract

We investigate the dynamical property of the naive mean dimension for continuous actions of any countable group on compact metrizable spaces. It is shown that naive mean dimension serves as an upper bound of sofic mean dimension for actions of nonamenable groups. For algebraic actions we obtain more satisfactory results by looking at the naive version of mean rank for modules over integral group rings. We also consider the naive metric mean dimension and investigate its relations with sofic metric mean dimension.

Naive mean dimension

Abstract

We investigate the dynamical property of the naive mean dimension for continuous actions of any countable group on compact metrizable spaces. It is shown that naive mean dimension serves as an upper bound of sofic mean dimension for actions of nonamenable groups. For algebraic actions we obtain more satisfactory results by looking at the naive version of mean rank for modules over integral group rings. We also consider the naive metric mean dimension and investigate its relations with sofic metric mean dimension.

Paper Structure

This paper contains 8 sections, 26 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Amenable and sofic groups
  4. Group rings and algebraic actions
  5. Naive mean dimension
  6. Naive mean rank
  7. Naive metric mean dimension
  8. Naive small boundary property

Key Result

Theorem 1.1

Let $\Gamma \curvearrowright X$ be any continuous action of a sofic group $\Gamma$ on a compact metrizable space. If the naive mean dimension of $\Gamma \curvearrowright X$ is zero, then the sofic mean dimension of $\Gamma \curvearrowright X$ is nonpositive.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • ...and 45 more