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Dynamical metric order

Maria Carvalho, Fagner B. Rodrigues

Abstract

We introduce the notion of dynamical metric order of a continuous map on a compact metric space, %study its basic properties, and compute it for several classes of maps. This concept which is a counterpart of the metric mean dimension with the role of the box-counting dimension being played by the metric order. It is devised for maps acting on spaces with infinite box-counting dimension but finite metric order. For example, it brings forward new information about full shifts whose alphabets have infinite box-counting dimension; and provides a sharper estimate of complexity for the induced map determined by a continuous transformation on a compact metric space, whose upper metric mean dimension is known to admit only two values (zero or infinity). We also show that it satisfies a variational principle where maximization is taken over the space of invariant probability measures and whose equilibrium states always exist.

Dynamical metric order

Abstract

We introduce the notion of dynamical metric order of a continuous map on a compact metric space, %study its basic properties, and compute it for several classes of maps. This concept which is a counterpart of the metric mean dimension with the role of the box-counting dimension being played by the metric order. It is devised for maps acting on spaces with infinite box-counting dimension but finite metric order. For example, it brings forward new information about full shifts whose alphabets have infinite box-counting dimension; and provides a sharper estimate of complexity for the induced map determined by a continuous transformation on a compact metric space, whose upper metric mean dimension is known to admit only two values (zero or infinity). We also show that it satisfies a variational principle where maximization is taken over the space of invariant probability measures and whose equilibrium states always exist.

Paper Structure

This paper contains 25 sections, 16 theorems, 223 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions
  3. Bowen dynamical metrics
  4. Separated sets, spanning sets and open covers
  5. Box-counting dimension
  6. Metric mean dimension
  7. Metric order
  8. Dynamical metric order
  9. $W_p$ and $LP$ metrics on $\mathcal{P}(X)$
  10. Quantization order
  11. Shifts on infinite-dimensional alphabets
  12. Dynamical quantization number
  13. Complexity of the induced map
  14. A particular case: Lipschitz maps
  15. Growth rate of the metric order
  16. ...and 10 more sections

Key Result

Lemma 2.2

BB The quantization number $Q_{\mu, LP}(\varepsilon)$ for the $LP$ metric is the least number of closed balls with radius $\varepsilon$ that cover any subset of $X$ with $\mu$-measure at least $1-\varepsilon$. Therefore, for every $\varepsilon > 0$ and $\mu \in \mathcal{P}(X)$

Figures (1)

  • Figure 1: Graph of the map $T$.

Theorems & Definitions (39)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Theorem A
  • Example 3.1
  • Lemma 3.2
  • proof
  • Example 3.3
  • Lemma 3.4
  • ...and 29 more