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Metric mean dimension via subshifts of compact type

Gustavo Pessil

TL;DR

This paper develops a comprehensive framework for metric mean dimension in the setting of subshifts of compact type and extends it to discontinuous maps. It proves that the metric mean dimension is invariant between a map and its inverse limit, generalizing Bowen’s entropy relation, and shows that mdim_M for discontinuous maps can be defined via the closure of the associated transition subshift. The work then builds a variational principle connecting Friedland’s entropy for free semigroup actions with the CRV random-walk approach to mdim_M, and demonstrates that zero-complexity maps can generate positive mdim_M under semigroup dynamics. Applications include showing that the Gauss map and induced Manneville–Pomeau maps have mdim_M equal to the box dimension of their discontinuity sets, tying mdim_M to the pressure’s critical exponent s_ty and to geometric features of the discontinuities. Collectively, the results provide a robust bridge between symbolic dynamics, thermodynamic formalism, and geometric notions of dimension for a broad class of (possibly discontinuous) dynamical systems.

Abstract

We investigate the metric mean dimension of subshifts of compact type. We prove that the metric mean dimensions of a continuous map and its inverse limit coincide, generalizing Bowen's entropy formula. Building upon this result, we extend the notion of metric mean dimension to discontinuous maps in terms of suitable subshifts. As an application, we show that the metric mean dimension of the Gauss map and that of induced maps of the Manneville-Pomeau family is equal to the box dimension of the corresponding set of discontinuity points, which also coincides with a critical parameter of the pressure operator associated to the geometric potential.

Metric mean dimension via subshifts of compact type

TL;DR

This paper develops a comprehensive framework for metric mean dimension in the setting of subshifts of compact type and extends it to discontinuous maps. It proves that the metric mean dimension is invariant between a map and its inverse limit, generalizing Bowen’s entropy relation, and shows that mdim_M for discontinuous maps can be defined via the closure of the associated transition subshift. The work then builds a variational principle connecting Friedland’s entropy for free semigroup actions with the CRV random-walk approach to mdim_M, and demonstrates that zero-complexity maps can generate positive mdim_M under semigroup dynamics. Applications include showing that the Gauss map and induced Manneville–Pomeau maps have mdim_M equal to the box dimension of their discontinuity sets, tying mdim_M to the pressure’s critical exponent s_ty and to geometric features of the discontinuities. Collectively, the results provide a robust bridge between symbolic dynamics, thermodynamic formalism, and geometric notions of dimension for a broad class of (possibly discontinuous) dynamical systems.

Abstract

We investigate the metric mean dimension of subshifts of compact type. We prove that the metric mean dimensions of a continuous map and its inverse limit coincide, generalizing Bowen's entropy formula. Building upon this result, we extend the notion of metric mean dimension to discontinuous maps in terms of suitable subshifts. As an application, we show that the metric mean dimension of the Gauss map and that of induced maps of the Manneville-Pomeau family is equal to the box dimension of the corresponding set of discontinuity points, which also coincides with a critical parameter of the pressure operator associated to the geometric potential.
Paper Structure (20 sections, 19 theorems, 157 equations, 3 figures)

This paper contains 20 sections, 19 theorems, 157 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Subshifts of compact type
  3. Discontinuous maps
  4. Preliminaries
  5. Metric mean dimension
  6. Box dimension
  7. Katok entropy
  8. Proof of Theorem
  9. Proof of Theorem
  10. Free semigroup actions
  11. Friedland's approach
  12. Carvalho, Rodrigues and Varandas' approach
  13. Variational principle
  14. Zero complexity maps may generate positive metric mean dimension
  15. Proof of Corollary
  16. ...and 5 more sections

Key Result

Theorem A

Let $(X,d)$ be a compact metric space and $\Gamma\subset X\times X$ be a closed subset. Then,

Figures (3)

  • Figure 1: Graph of the map $T$.
  • Figure 2: $\Gamma\,=\, \color{ForestGreen}\operatorname{{graph}}(g_1)\,\color{black}\cup\,\color{red}\operatorname{{graph}}(g_2)\color{black}$.
  • Figure 3: Graphs of $g_1\circ g_2$ and $g_2\circ g_1$, respectively.

Theorems & Definitions (38)

  • Theorem A
  • Theorem B
  • Corollary 1.1
  • Corollary 1.2
  • Example 1.3
  • Example 1.4
  • Definition 1.5
  • Theorem C
  • Example 1.6
  • Example 1.7
  • ...and 28 more