Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On quantization and the classical variational principle for the metric mean dimension

Maria Carvalho, Gustavo Pessil

Abstract

For a homeomorphism $T\colon X \to X$ on a compact metric space $(X,d)$, one can find in the literature several distinct notions of measure-theoretic $\varepsilon$-entropy maps $F(μ,\varepsilon)$, defined on the space of Borel $T$-invariant probability measures of $X$, satisfying $$ \mathrm{\overline{mdim}_M}(X,d,T) \, =\, \limsup_{\varepsilon \, \to \, 0^+}\, \sup_{μ\,\in\, P_T(X)} \, \frac{F(μ,\varepsilon)}{\log\,(1/\varepsilon)} $$ where $\mathrm{\overline{mdim}_M}(X,d,T)$ stands for the upper metric mean dimension. A major question regards the change in the order in which $\limsup_{\varepsilon}$ and $\sup_{μ\,\in\, P_T(X)}$ appear in the previous equality. We introduce the notion of mean quantization dimension of a measure and prove that $$ \mathrm{\overline{mdim}_M}(X,d,T) \,\, =\, \max_{μ\,\in\, P_T(X)} \, \mathrm{\overline{mdim}_Q}(μ). $$ This concept exhibits a fundamental property that prompted our search for sufficient conditions on $F$ under which the aforementioned change in order is valid. We show that some of the well-known maps $F$ do not satisfy this property and fail to comply with a classical variational principle.

On quantization and the classical variational principle for the metric mean dimension

Abstract

For a homeomorphism on a compact metric space , one can find in the literature several distinct notions of measure-theoretic -entropy maps , defined on the space of Borel -invariant probability measures of , satisfying where stands for the upper metric mean dimension. A major question regards the change in the order in which and appear in the previous equality. We introduce the notion of mean quantization dimension of a measure and prove that This concept exhibits a fundamental property that prompted our search for sufficient conditions on under which the aforementioned change in order is valid. We show that some of the well-known maps do not satisfy this property and fail to comply with a classical variational principle.
Paper Structure (27 sections, 15 theorems, 150 equations)

This paper contains 27 sections, 15 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Quantization
  4. Mean quantization dimension
  5. Main property
  6. Variational principles for maps in ${\mathcal{D}}$
  7. Ergodic equilibrium states
  8. Organization of the paper
  9. Basic definitions
  10. Bowen dynamical metrics
  11. Separated sets, spanning sets, open covers, closed covers
  12. Box-counting dimension
  13. Metric mean dimension
  14. $W_p$ and $LP$ metrics on $\mathcal{P}(X)$
  15. Measure-theoretic $\varepsilon$-entropies
  16. ...and 12 more sections

Key Result

Theorem 2.3

BB Let $(X,d)$ be a compact metric space and $T\colon X\to X$ be a continuous map. Given $D \in \{W_p\colon p \in [1,+\infty)\}\cup \{LP\}$ and $\mu \in {\mathcal{E}}_T(X)$, then

Theorems & Definitions (36)

  • Example 1.1
  • Example 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem A
  • Definition 2.5
  • Theorem B
  • Theorem C
  • ...and 26 more