You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We prove a variational principle for the metric mean dimension analog to the one in [@LT]. Instead of using the rate distortion function we use the function $h_\mu(\e,T,\delta)$ that is closely related to the entropy $h_\mu(T)$ of $\mu$. Our formulation has the advantage of being, in the authors opinion, more natural when doing computations. As a corollary we obtain a proof of the standard variational principle. We also obtain some relations between the rate distortion function with our function $\widetilde{h}_\mu(\e,T,\delta)$, a modification of $h_\mu(\e,T,\delta)$ when replacing the dynamical metrics with the average dynamical metrics. Using our methods we also reprove the main result in [@LT]. We will explain how to construct homeomorphisms on closed manifolds with maximal metric mean dimension. We end this paper with some questions that naturally arise from this work.'
address:
- 'Princeton University, Princeton NJ 08544-1000, USA.'
- 'Facultad de Matemáticas, Pontificia Universidad Católica de Chile (PUC), Avenida Vicuña Mackenna 4860, Santiago, Chile'
author:
- Anibal Velozo
- Renato Velozo
title: 'Rate distortion theory, metric mean dimension and measure theoretic entropy'
---

Introduction
============

The *topological entropy* is a fundamental quantity that allows us to quantify the chaoticity of a dynamical system. If the ambient space is compact and the dynamics is Lipschitz, then the topological entropy is finite. On the other hand, if the dynamics is just continuous, the topological entropy might be infinite. In fact, K. Yano proved in [@yan] that on a closed manifold of dimension at least two the topological entropy is infinite for generic homeomorphisms. It is then natural to consider a dynamical quantity that distinguishes systems with infinite topological entropy. The *mean dimension* is a meaningful quantity when the topological entropy is infinite. This invariant was introduced by Gromov in [@gro], and further studied by E. Lindenstrauss and B. Weiss in [@LW]. This invariant has found many applications to embedding problems, in other words to the problem of when a dynamical system can be embedded into another or not, see for instance [@l],[@glt] and references therein. In this paper we will mainly focus on the *metric mean dimension*. This is an invariant of the dynamical system defined in [@LW], which in contrast with the topological entropy might depend on the metric on the ambient space. For completeness we proceed to define the relevant quantities. Let $(\X,d)$ be a compact metric space and $T:\X\to \X$ a continuous map. Define $N_d(n,\e)$ as the maximal cardinality of a $(n,\e)$-separated subset of $\X$ and $$S(\X,d,\e)=\limsup_{n\to \infty}\dfrac{1}{n}\log N_d(n,\e).$$ For precise definitions see Section \[prelim\]. The topological entropy is defined as $$h_{top}(\X,T)=\lim_{\e\to 0} S(\mathcal{X},d,\e).$$ The topological entropy is known to be independient of the compatible metric $d$ since $\X$ is compact. The upper metric mean dimension is defined as $$\overline{mdim}(\X,d,T)=\limsup_{\e\to 0}\dfrac{S(\mathcal{X},d,\e)}{|\log\e|}.$$

Recently E. Lindenstrauss and M. Tsukamoto in [@LT] established a variational principle for the metric mean dimension. In this formula the role of the measure theoretic entropy is replaced by the rate distortion function $R_\mu(\e)$ (for its definition see Section \[prelim\]). Let ${\mathcal{M}}_T(\X)$ be the space of $T$-invariant probability measures on $\X$. They proved the following result.

\[teolin\]Suppose $(\X,d)$ satisfy Condition 1.2 in [@LT]. Then $$\overline{mdim}(\X,d,T)=\limsup_{\e\to 0}\dfrac{\sup_{\mu\in{\mathcal{M}}_T(\X)} R_\mu(\e)}{|\log\e|}.$$

For the meaning of Condition 1.2 see Definition \[cond\]. In this paper we will prove an analog formula where instead of the rate distortion function we use some terms that appear in Katok’s entropy formula. This has the advantage to simplify some computations and to make them, in the authors opinion, more intuitive. For $\mu\in {\mathcal{M}}_T(\X)$ define $N_\mu(n,\e,\delta)$ as the minimum number of $(n,\e)$-dynamical balls needed to cover a set of measure strictly bigger than $1-\delta$. Then define $h_\mu(\e,T,\delta)=\limsup_{n\to\infty} \frac{1}{n}\log N_\mu(n,\e,\delta)$. It was proven by A. Katok in [@ka] that $h_\mu(T)=\lim_{\e\to 0} h_\mu(\e,T,\delta)$ for every ergodic measure $\mu$ and any $\delta\in (0,1)$, where $h_\mu(T)$ is the measure theoretic entropy of $\mu$. One of the main results of this paper is the following theorem.

\[theo1\] Let $(\X,d)$ be a compact metric space and $T:\X\to\X$ continuous. Then $$\overline{mdim}(\X,d,T)=\lim_{\delta\to 0}\limsup_{\e\to 0}\dfrac{\sup_{\mu\in{\mathcal{M}}_T(\X)}h_\mu(\e,T,\delta)}{|\log\e|}.$$

From the proof of Theorem \[theo1\] we also recover a proof of the standard variational principle. An easy application of Theorem \[theo1\] is the computation $$mdim(([0,1]^n)^{\mathbb{Z}},d_T,T)=n,$$ where the transformation $T$ is the shift map and $d_T(x,y)=\sum_{k\in{\mathbb{Z}}} \frac{1}{2^{|k|}}d(x_k,y_k)$, where $x=(...,x_{-1},x_0,x_1...)$, $y=(...,y_{-1},y_0,y_1...)$ and $d$ is the standard metric on $[0,1]^n$. We also investigate the connection between the rate distortion function $R_\mu(\e)$ and our replacement $h_\mu(\e,T,\delta)$. For reasons that will be clear to the reader the function $R_\mu(\e)$ is closely related to $\widetilde{h}_\mu(\e,T,\delta)$, where instead of using $(n,\e)$-dynamical balls we use $(n,\e)$-average dynamical balls. We are in particular able to reprove Theorem \[teolin\]. We also prove that the rate distortion function recovers the measure theoretic entropy in the ergodic case.

\[theo2\] Let $(\X,d)$ be a compact metric space and $\mu\in {\mathcal{M}}_T(\X)$ an ergodic measure. Then $$h_\mu(T)=\lim_{\e\to 0} R_\mu(\e)=\widetilde{h}_\mu(T,\delta).$$

We then obtain a result analog to the one in Theorem \[theo1\] when we replace the $(n,\e)$-dynamical balls with the $(n,\e)$-average dynamical balls, i.e. when replacing $h_\mu(\e,T,\delta)$ by $\widetilde{h}_\mu(\e,T,\delta)$. We use this formula to reprove Theorem \[teolin\].

The paper is organized as follows. In Section \[prelim\] we recall some basic definitions from ergodic and information theory. In Section \[sec3\] we prove Theorem \[theo1\] and we compute the metric mean dimension of the shift over $([0,1]^n)^{\mathbb{Z}}$. In Section \[sec3\] we also obtain a proof of the standard variational principle. In Section \[sec4\] we discuss the connections between the rate distortion function $R_\mu(\e)$ and the function $\widetilde{h}_\mu(\e,T,\delta)$. We use the results in Section \[sec4\] to reprove Theorem \[teolin\]. In Section \[sec5\] we discuss how generic is for a continuous map or homeomorphism on a manifold to have positive or maximal metric mean dimension. In Section \[sec6\] we make some final remarks and state some natural questions, we also suggest the definition of what should be the metric mean dimension of a measure.

The authors would like to thank G. Iommi and M. Tsukamoto for their interest in this work. The first author would like to thanks to his advisor G. Tian for his constant support and encouragements. The last part of this paper was written when the first author was visiting Pontificia Universidad Católica de Chile, he is very grateful to G. Iommi for the invitation. Finally, the second author would like to thanks to his advisor J. Bochi for his continued guidance and encouragements.

Preliminaries {#prelim}
=============

Let $(\X,d)$ be a metric space and $T:\X\to\X$ a continuous map. We refer to the triple $(\X,d,T)$ a *dynamical system*. We will not always assume that $\X$ is compact, we will specify when that assumption is required. The metric $d$ induces a topology on $\X$ and this topology endows $\X$ with the borelian $\sigma$-algebra. Any measure on $\X$ is assumed to be defined on the borelian $\sigma$-algebra of $(\X,d)$. We will use standard concepts in ergodic theory, for completeness we briefly define the notions more relevants to this work. A probability measure $\mu$ on $\X$ is said to be invariant under $T$ or *$T$-invariant* if for any measurable set $A$ we have $\mu(A)=\mu(T^{-1}A)$. A probability $T$-invariant measure $\mu$ is said to be *ergodic* if any measurable set $A$ satisfying $T^{-1}A=A$ has measure zero or one. To a partition $\p=\{\p_1,...,\p_m\}$ of $\X$ we can assign the value $$H_\mu(\p)=-\sum_{i=1}^m \mu(\p_i)\log\mu(\p_i).$$ Observe that everytime we have a measurable map $Z:\X\to W$ with finite image we can associate a partition on $\X$, the *preimage partition* of $Z$ (which is a finite partition since the map has finite image). In this case we denote by $H_\mu(Z)$ to the entropy of the preimage partition of $Z$. Let $\p\vee \mathcal{Q}$ be the refinement of the partitions $\p$ and $\mathcal{Q}$. Given two random variables with finite image $Z_1:\X\to W_1$ and $Z_2:\X\to W_2$ we can define the *mutual information of $Z_1$ and $Z_2$* as $$I_\mu(Z_1,Z_2)=H_\mu(Z_1)+H_\mu(Z_2)-H_\mu(Z_1,Z_2),$$ where $H_\mu(Z_1,Z_2)$ is the entropy of the refinement of the preimage partitions of $Z_1$ and $Z_2$. We define the partitions $\p^n:=\p\vee T^{-1}\p\vee ...\vee T^{-(n-1)}\p$ and the quantity $$h_\mu(\p,T)=\inf_{n}\dfrac{1}{n}H_\mu(\p^n).$$ We define the *entropy of the measure* $\mu$ as $$h_\mu(T)=\sup_\p h_\mu(\p,T),$$ where the supremum runs over finite partitions of $\X$. For more detailed information about the entropy of a measure and its relevance in ergodic theory we refer the reader to [@W]. Using the map $T$ and the metric $d$ we can define two new families of metrics on $\X$ by the following expressions $$d_n(x,y)=\max_{k\in \{0,...,n-1\}}d(T^kx,T^ky),$$ $$\widetilde{d}_n(x,y)=\dfrac{1}{n}\sum_{k=0}^{n-1} d(T^kx,T^ky).$$ We denote a ball of radius $\e$ in the metric $d_n$ as a *$(n,\e)$-dynamical ball* and a ball of radius $\e$ in the metric $\widetilde{d}_n$ as a *$(n,\e)$-average dynamical ball*. The following definition allows us to state Theorem \[kat\] in a cleaner way and also motivates Definition \[ent2\].

\[ent\] Let $\mu$ be a $T$-invariant probability measure. For $\delta\in (0,1)$, $n\in {\mathbb{N}}$ and $\epsilon>0$ we define $N_\mu(n,\epsilon,\delta)$ to be the minimum number of $(n,\epsilon)$-dynamical balls needed to cover a set of $\mu$-measure strictly bigger than $1-\delta$. Set $$h_\mu(\epsilon,T,\delta)=\limsup_{n\to \infty}\dfrac{1}{n}\log N_\mu(n,\epsilon,\delta).$$

The following theorem was proven in [@ka].

\[kat\] Let $(\X,d)$ be a compact metric space, $T:\X\to \X$ a continuous transformation and $\mu$ an ergodic $T$-invariant probability measure. Then $$h_\mu(T)=\lim_{\epsilon\to 0} h_\mu(\epsilon,T,\delta),$$ where $h_\mu(T)$ is the measure theoretic entropy of $\mu$. In particular the limit above does not depend on $\delta\in (0,1)$.

\[rem2\] Theorem \[kat\] is proven by establishing inequalities between LHS and RHS. The inequality LHS $\ge$ RHS can be stated more precisely as $$h_\mu(\mathcal{P},T)\ge \limsup_{n\to\infty}\dfrac{1}{n}\log N_\mu(n, \epsilon,\delta),$$ where $\p$ is any partition of diameter less than $\epsilon$. This fact will be used in the proof of Proposition \[prop1\]. We emphasize this inequality is independient of $\delta$.

The following definition mimics Definition \[ent\] when we use $\widetilde{d}_n$ instead of $d_n$. We use Katok’s formula to motivate the definition of the *average measure theoretic entropy* of an ergodic measure $\mu$.

\[ent2\] Let $\mu$ be a $T$-invariant probability measure on $\X$. We define $$\widetilde{h}_\mu(\epsilon, T,\delta)=\limsup_{n\to \infty}\dfrac{1}{n} \log \widetilde{N}_\mu(n,\epsilon,\delta),$$ where $\delta\in (0,1)$ and $\widetilde{N}_\mu(n,\epsilon,\delta)$ is the minimum number of $(n,\epsilon)$-average dynamical balls needed to cover a set of $\mu$-measure strictly bigger than $1-\delta$. We define the *average measure theoretic entropy* of $\mu$ as the limit $$\widetilde{h}_\mu(T,\delta)=\lim_{\epsilon\to 0} \widetilde{h}_\mu(\epsilon, T,\delta).$$

One of the goals of this paper is to establish relations between the *rate distortion function* $R_\mu(\e)$ recently used by E. Lindenstrauss and M. Tsukamoto [@LT] in the study of the metric mean dimension, and the quantities defined above. We start by recalling the definition of $R_\mu(\e)$.

\[cond\] We say that the pair $(X,Y)$ satisfies condition $(*)_{n,\e}$ if the following properties are satisfied.

1.  There exists a probability space $(\Omega,{\mathbb{P}})$ such that $X:\Omega\to \X$ and $Y=(Y_0,...,Y_{n-1}):\Omega\to \X^n$ are measurable functions.

2.  The measure $X_*{\mathbb{P}}$ is a $T$-invariant probability measure on $\X$.

3.  ${\mathbb{E}}(\frac{1}{n}\sum_{k=0}^{n-1}d(T^kX,Y_k))\le \epsilon$.

We say that the pair $(X,Y)$ satisfies condition $(*)_{n,\e,\mu}$ if moreover $X_*{\mathbb{P}}=\mu$.

Given a $T$-invariant probability measure $\mu$ on $\X$, define the *rate distortion function* as $$R_\mu(\e)=\inf \dfrac{1}{n}I(X,Y),$$ where the infimum runs over pairs $(X,Y)$ satisfying condition $(*)_{n,\e,\mu}$.

In the definition above $I(X,Y)$ is the mutual information of the random variables $X$ and $Y$. If $X$ and $Y$ have finite image this was defined in the second paragraph of the introduction. For the general case (assuming $\X$ is a standard probability space) we refer the reader to Section 5.5 in [@G].

The variational principle {#sec3}
=========================

In this section we prove Theorem \[theo1\] and the standard variational principle. Define $N_d(n,\e)$ as the maximal cardinality of a $(n,\e)$-separated set in $(\X,d)$ and $$S(\X,d,\e)=\limsup_{n\to \infty}\dfrac{1}{n}\log N_d(n,\e).$$ In case the dynamical system has been specified, we will frequently use the simplified notations $S(\e)=S(\X,d,\e)$. We start with the following lemma which is important for our results.

\[1111\] Assume $(\X,d)$ is compact and $T:\X\to \X$ a continuous map. Given $\e>0$, there exists a $T$-invariant measure $\mu_\e$ such that $$h_{\mu_\e}(\e,T,\delta)\ge  S(\X,d,2\e)-3\delta S(\X,d,\e).$$

Let $E_n=\{x_1,...,x_{N_d(n,\e)}\}$ be a maximal collection of $(n,\e)$-separated points in $\X$. Define $$\sigma_n=\dfrac{1}{|E_n|}\sum_{x\in E_n} \delta_x,$$ where $\delta_x$ is the probability measure supported at $x$. Then define $$\overline{\sigma}_n=\dfrac{1}{n}\sum_{k=0}^{n-1} T^k_*\sigma_n.$$ Consider a subsequence $\{n_k\}_{k\in {\mathbb{N}}}$ such that $$S(\X,d,\e)=\lim_{k\to \infty}\dfrac{1}{n_k}\log N_d(n_k,\e).$$ By standard arguments we can find a subsequence of $\{n_k\}$, which we still denote as $\{n_k\}$, such that $\{\overline{\sigma}_{n_k}\}_{k\in{\mathbb{N}}}$ converges to a $T$-invariant probability measure $\mu$. We can arrange the sequence such that $\lim_{k\to \infty} \frac{n_k}{k}=\infty$. Let $K$ be a subset of $\X$ with $\mu(K)> 1-\delta$, and $N(K,n,\e)=N_\mu(n,\e,\delta)$, where $N(K,n,\e)$ is defined to be the minimum number of $(n,\e)$-dynamical balls needed to cover $K$. We can assume that $K$ is open in $\X$. There exists $k_0$ such that for every $k\ge k_0$ we have $\overline{\sigma}_{n_k}(K)> 1-\delta$. Let $$L_n=\{(i,j)\in{\mathbb{N}}^2: 0\le i\le n-1, 1\le j\le N_d(n,\e)\}.$$ We assign to each point in $L_n$ either a $0$ or $1$ in the following way. If $T^ix_j\in K$ then assign $1$ to the point $(i,j)$, assign $0$ to $(i,j)$ otherwise. By definition of the measure $\overline{\sigma}_{n_k}$ we know that the number of ones in $L_{n_k}$ is bigger or equal than $n_k N_d(n_k,\e)(1-\delta)$. For $s>1$ we define $L_{n_k}(s)$ as the set of points in $L_{n_k}$ with first coordinate in the interval $[[n_k/k],n_k-[n_k/s]-1]$. The number of ones in $L_{n_k}(s)$ is at least $n_k N_d(n_k,\e)(1-\delta-\frac{1}{n_k}[\frac{n_k}{s}]-\frac{1}{n_k}[\frac{n_k}{k}])\ge n_k N_d(n_k,\e)(1-\delta-\frac{1}{s}-\frac{1}{k})$. From now on we assume $s>\frac{1}{1-2\delta-\frac{1}{k}}$, in particular $1-\delta-\frac{1}{s}-\frac{1}{k}>\delta$. We will moreover assume $s<\frac{1}{1-3\delta}$, this can be done if we assume $k$ is sufficiently large so that $\delta>1/k$. We conclude that the number of ones in $L_{n_k}(s)$ is at least $n_k N_d(n_k,\e)\delta$. Since $L_{n_k}$ has $(n_k-[n_k/s]-[n_k/k])$ columns, there exists an index $m_k$ such that the $m_k$th column has at least $n_k N_d(n_k,\e)\delta/(n_k-[n_k/s]-[n_k/k])$ ones and $[n_k/k]\le m_k< n_k-[n_k/s]$. Observe that $\X$ can be covered by at most $N_d(m_k,\e/2)$ $(m_k,\e/2)$-dynamical balls, in particular with $N_d(m_k,\e/2)$ subsets of $d_{m_k}$-diameter smaller than $\e$. Also observe that if $i\ne j$ and $d_{m_k}(x_i,x_j)\le \e$, then $d_{n_k-m_k}(T^{m_k}x_i,T^{m_k} x_j)>\e$, because $d_{n_k}(x_i,x_j)>\e$ by the definition of $E_{n_k}$. We can conclude that there exists an subset $I\subset \{1,....,N_d(n_k,\e)\}$ such that for $i\in I$ we have $T^{m_k}x_i\in K$, and $|I|\ge n_k N_d(n_k,\e)\delta/(n_k-[n_k/s]-[n_k/k])>N_d(n_k,\e)\delta$. In other words there exists a subset $A$ of $\{T^{m_k} x_i\}_{i\in I}$ such that the diameter of $A$ with respect to $d_{m_k}$ is at most $\e$ and $$|A|\ge \dfrac{ N_d(n_k,\e)\delta}{N_d(m_k,\e/2)  }.$$ This implies that if $a,b\in A$ and $a\ne b$, then $d_{n_k}(a,b)\ge d_{n_k-m_k}(a,b)\ge \e$. Then $$N_\mu(n_k,\e/2,\delta)=N(K,n_k,\e/2)\ge \dfrac{ N_d(n_k,\e)\delta}{N_d(m_k,\e/2)  }.$$ Therefore $$h_\mu(\e/2,T,\delta)=\limsup_{n\to \infty}\dfrac{1}{n}\log N_\mu(n,\e/2,\delta)\ge \limsup_{k\to\infty} \dfrac{1}{n_k}\log N_d(n_k,\e)-\dfrac{m_k}{n_k}\dfrac{1}{m_k}\log N_d(m_k,\e/2).$$ Recall that by construction $m_k<n_k-[n_k/s]$, then $\frac{m_k}{n_k}\le 1-\frac{1}{s}+\frac{1}{n_k}$. Therefore $$\begin{aligned}
 h_\mu(\e/2,T,\delta)&\ge \limsup_{k\to\infty} \dfrac{1}{n_k}\log N_d(n_k,\e)-(1-\dfrac{1}{s}+\dfrac{1}{n_k})\dfrac{1}{m_k}\log N_d(m_k,\e/2).\\
&\ge \limsup_{k\to\infty} \dfrac{1}{n_k}\log N_d(n_k,\e)-(3\delta+\dfrac{1}{n_k})\dfrac{1}{m_k}\log N_d(m_k,\e/2).\end{aligned}$$ Also observe that the inequality $[n_k/k]\le m_k$ implies that $m_k\to \infty$ as $k\to \infty$. By construction $S(\X,d,\e)=\lim_{k\to \infty}\dfrac{1}{n_k}\log N_d(n_k,\e)$, therefore $$h_\mu(\e/2,T,\delta)\ge  S(\e)-3\delta S(\e/2).$$

In particular, last lemma proves that $$\sup_\mu h_\mu(\e,T,\delta)\ge S(2\e)-3\delta S(\e),$$ where the supremum runs over the set of $T$-invariant probability measures. This observation will be useful in the next corollary. A straightforward application of Lemma \[1111\] is a proof of the standard variational principle and the proof of Theorem \[theo1\].

Let $(\X,d)$ be a compact metric space and $T:\X\to\X$ a continuous map. Then $$h_{top}(\X,T)=\sup_{\mu\in{\mathcal{M}}_T(\X)} h_\mu(T).$$

From Proposition \[1111\] we know $$\sup_\mu h_\mu(\e,T,\delta)\ge S(2\e)-3\delta S(\e).$$ Let $\mu=\int \mu_x d\mu(x)$ be the ergodic decomposition of $\mu$. The proof of the inequality in Remark \[rem2\] can be adapted to obtain $$h_\mu(\e,T,\delta)\le \sup_{x\in \X} h_{\mu_x}(\Q)\le \sup_{\mu\text{ ergodic}}h_\mu(T),$$ for any partition $\Q$ of diameter smaller than $\e$. Also observe that Theorem \[kat\] implies $$h_\mu(T)\le h_{top}(\X,T),$$ for every ergodic measure $\mu$. Finally $$S(2\e)-3\delta S(\e)\le \sup_\mu h_\mu(\e,T,\delta)\le \sup_{\mu\text{ ergodic}}h_\mu(T)\le h_{top}(T).$$ Finally taking $\e\to 0$ and $\delta\to 0$ we get that $$\sup_{\mu\text{ ergodic}}h_\mu(T)=h_{top}(T).$$

Let $(\X,d)$ be a compact metric space and $T:\X\to\X$ a continous map. Then $$\overline{mdim}(\X,d,T)=\lim_{\delta\to 0}\limsup_{\e\to 0}\dfrac{\sup_{\mu\in{\mathcal{M}}_T(\X)}h_\mu(\e,T,\delta)}{|\log\e|}.$$

The inequality $LHS\ge RHS$ follows directly from Lemma \[1111\]. The other inequality follows trivially, because by definition $h_\mu(\e,T,\delta)\le S(\epsilon)$.

As an application of this formula we will compute the metric mean dimension of $([0,1]^{\mathbb{Z}},d_T,T)$. It is well known that $\overline{mdim}( [0,1]^{{\mathbb{Z}}},d_T,T)=1$, see for instance [@LW] and [@LT]. We will proceed to verify this fact. We recall that the metric $d_T$ is given by $d_T(x,y)=\sum_{k\in{\mathbb{Z}}} \frac{1}{2^{|k|}}d(x_k,y_k)$, where $x=(...,x_{-1},x_0,x_1...)$, $y=(...,y_{-1},y_0,y_1...)$ and $d$ is the standard metric on $[0,1]$.

The metric mean dimension of the shift map on $[0,1]^{\mathbb{Z}}$ with the metric $d_T$ is given by $$\overline{mdim}( [0,1]^{{\mathbb{Z}}},d_T,T)=1.$$

For $k\ge 1$ consider the set $P_k=\{p_1, p_2, \cdots, p_k\}$, where $p_i=\frac{2i-1}{2k}$. Define $\lambda_k$ as the probability measure on $[0,1]$ that equidistribute the points in $P_k$ and let $\mu_k= (\lambda_k)^{\otimes {\mathbb{Z}}}$, the product measure on $[0,1]^{\mathbb{Z}}$. Define $$A_{i_0,i_2,\dots, i_{n-1}}=\{x\in [0,1]^{{\mathbb{Z}}}: x_0=p_{i_0}, \dots , x_{n-1}=p_{i_{n-1}}\}.$$ We will need the following fact.

Let $r<1/2k$ and $q \in [0,1]^{{\mathbb{Z}}}$. Then there is a unique set $A_{i_0,i_2,\dots, i_{n-1}}$ such that $$supp (\mu_k) \cap B_n(q,r) \subset A_{i_0,i_2,\dots, i_{n-1}}.$$

Let $x \in supp(\mu_k) \cap B_n(q,r)$. By definition $$d(x_j,q_j)\le d(T^jx,T^jq)\le d_n(x,q)\le r<\frac{1}{2k},\quad \forall j \in\{0,\cdots,n-1\}.$$ Since $x \in supp(\mu_k)$ we conclude $x_j\in P_k$. Otherwise the neighbourhood $\mathcal{N}_j=\cdots\times [0,1] \times U_j\times [0,1] \times \cdots$ of $x$ has zero $\mu_k$-measure, where $U_j\subset [0,1]$ is an open set containing $x_j$ with $U_j \cap P_k=\emptyset$. From the choice of $r$ we conclude $x_j$ can take only one value in $P_k$, say $p_{i_j}$.

This lemma in particular implies that $$\mu_k(B_n(q,r))\le  \mu_k(A_{i_0,i_2,\dots, i_{n-1}})=\frac{1}{k^n},$$ for every $q\in [0,1]^{\mathbb{Z}}$. Then if $A$ satisfy $\mu_k(A)>1-\delta$ and $A\subset \bigcup_{i=1}^L B_n(z_i,r)$, then $$1-\delta<\mu_k(A)\le \mu_k(\bigcup_{i=1}^L B_n(z_i,r))\le \frac{L}{k^n}.$$ This implies that $N_{\mu_k}(n,r,\delta)\ge(1-\delta)k^n$, for every $n\ge 1$ and $\delta\in (0,1)$. Finally

$$\lim_{r \to 0}\frac{\sup_\mu h_{\mu}(r,T,\delta)}{|\log r|}\ge \lim_{k \to \infty}\frac{h_{\mu_k}(\frac{1}{3k},T,\delta)}{\log 3k}\ge \lim_{k \to \infty} \frac{\log k}{\log 3k}=1.$$ The opposite inequality is easier, we refer the reader to [@LT] for the details.

This computation trivially extends to higher dimensions. We recover the formula $$\overline{mdim}(([0,1]^d)^{\mathbb{Z}},d_T,T)=d.$$ This can also be generalized to more general metric spaces. Let $Y$ be a compact metric space with metric $d$. Given $\e>0$, define $N(\e)$ as the maximal cardinality of an $\e$-separated set in $(Y,d)$. The upper Minkowski dimension or upper box dimension of $Y$ is defined as $$\overline{dim}_{B}(Y,d)=\limsup_{\e\to 0}\frac{N(\e)}{|\log\e|}.$$ Consider a decreasing sequence $\{\e_k\}_{k\in{\mathbb{N}}}$ converging to zero such that $\lim_{k\to \infty} \frac{N(\e_k)}{|\log\e_k|}= \overline{dim}_{B}(Y)$. The role of the points $P_k$ is replaced by a maximal collection of $\e_k$-separated points in $Y$. The measures $\lambda_k$ is the probability measure that equidistributes a maximal collection of $\e_k$-separated points. As before we get the inequality $\mu_k(B_n(q,r))<N(\e_k)^{-n}$, for every $q\in Y^{\mathbb{Z}}$ and $r<\frac{\e_k}{2}$. We finally get the inequality $$\limsup_{\e \to 0}\frac{\sup_\mu h_{\mu}(r,T,\delta)}{|\log r|}\ge \limsup_{k \to \infty}\frac{h_{\mu_k}(\e_k/3,T,\delta)}{|\log(\e_k/3)|}\ge \lim_{k \to \infty} \frac{\log N(\e_k)}{|\log \e_k|}=\overline{dim}_{B}(Y).$$ We conclude $\overline{mdim}(Y^{\mathbb{Z}},d_T,T)\ge \overline{dim}_{B}(Y).$ Let $\epsilon \in (0,1)$ and consider a natural number $l$ such that $\sum_{|j|\ge l} 2^{-|j|}<\epsilon /(2\cdot \diam (Y))$. Let $M$ the maximum cardinality of an $\e$-separated subset $B=\{x_i\}_{i=1}^M$ of $\mathcal{X}$. Consider the function defined by $f(x)=x_i$, where $x_i$ is the closest point to $x$ in the subset $B$, whenever $x_i$ is uniquely defined. We can extend $f$ to a measurable function on $\X$. Define the sets $A_i=f^{-1}(x_i)$, and $$S_{i_{-l},\dots, i_{n+l}}=\{y\in Y^{\mathbb{Z}}: y_j\in A_{i_j}\textrm{ for all } -l\le j \le n+l\},$$ where $i_j\in \{1,...,M\}$ for each $j$. Observe that $z,y \in S_{i_{-l},\dots, i_{n+l}}$ implies $$d(T^iz, T^iy)\le\diam(Y)\sum_{|j|\ge l} 2^{-|j|}+\sum_{|j|\le l}2^{-|j|}d(z_{i+j},y_{i+j})<\frac{\epsilon}{2}+2\epsilon \sum_{|j|\le l}2^{-|j|}<7\epsilon,$$ for every $0\le i\le n$. This inequality proves that every set $S_{i_{-l},\dots, i_{n+l}}$ has diameter less than $7\epsilon$ with respect to the metric $d_n$. Since the collection of sets $S_{i_{-l},\dots, i_{n+l}}$ is a covering of $Y^{\mathbb{Z}}$ we conclude $N_{d_T}(n,7\e)\le M^{n+2l+1}=N(\e)^{n+2l+1}.$ Since for any measure $\mu$ we have $N_{\mu}(n,\e,\delta)\le N_{d_T}(n,\e)$, we conclude $$\begin{aligned}
\limsup_{\e \to 0}\frac{\sup_\mu h_{\mu}(\e,T,\delta)}{|\log \e|} \le & \lim_{\epsilon \to 0}\frac{\limsup_{n\to \infty} \frac{1}{n}\log N_{d_T}(n,\e)}{|\log \epsilon|}\\ \le&  \limsup_{\epsilon \to 0}\frac{\log N(\epsilon)}{|\log \epsilon/7|}=\overline{dim}_B(Y,d).\end{aligned}$$

We summarize this in the following result.

\[calculo\] With the notation above we have $$\overline{mdim}(Y^{\mathbb{Z}},d_T,T)=\overline{dim}_B(Y,d).$$

Rate distortion function and $\widetilde{h}_\mu(\e,T,\delta)$ {#sec4}
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\[prop1\] Let $\mu$ be an ergodic $T$-invariant probability measure. Assume $(\X,d)$ is compact and denote its diameter by $D$. Then $$R_\mu(\epsilon)\le \widetilde{h}_\mu(\e,T, \e/2D).$$ Let $\p$ be any partition of diameter less than $\e$. Then $$R_\mu(\epsilon)\le h(\p,T).$$

Recall that by definition $R_\mu(\epsilon)$ is the infimum of the quantities $\frac{1}{n}I(X,Y)$, where $(X,Y)$ satisfies condition $(*)_{n,\epsilon,\mu}$ (here $n$ is also allowed to vary). In particular if we want to give an upper bound to $R_\mu(\epsilon)$ we can just exhibit a good choice of pair $(X,Y)$. Since $(\X,d)$ is a compact metric space it has finite diameter, call $D$ to the diameter of $(\X,d)$. Choose $K\subset \X$ such that $\widetilde{N}(K,n,\epsilon)=\widetilde{N}_\mu(n,\epsilon, \frac{\epsilon}{2D})= N$, i.e. there exist points $\{x_i\}_{i=1}^N$ in $\X$ such that $K\subset \bigcup_{i=1}^N \widetilde{B}_n(x_i, \e/2D)$ and $\mu(K)>1-\frac{\epsilon}{2D}$, where last union is disjoint. Pick a point $p\in \X\setminus \{x_i\}_{i=1}^N$ and define a measurable function $f:\X\to\X$ in such way that for $x\in K$ we have that $f(x)=x_i$ for some $x_i$ with $x\in \widetilde{B}_n(x_i,\e/2)$, if $x\in \X\setminus K$, then $f(x)=p$. By construction $|f(\X)|=N+1$. Define $Y_k=T^k\circ f\circ X$. Observe that $$\begin{aligned}
& {\mathbb{E}}\Big(\dfrac{1}{n}\sum_{k=0}^{n-1}d(T^kX,Y_k)\Big)=\int_\X \dfrac{1}{n}\sum_{k=0}^{n-1}d(T^k(x),T^kf(x))d\mu(x) \\
 &= \int_K \dfrac{1}{n}\sum_{k=0}^{n-1}d(T^k(x),T^kf(x))d\mu(x)+\int_{\X\setminus K} \dfrac{1}{n}\sum_{k=0}^{n-1}d(T^k(x),T^kf(x))d\mu(x)\\
&\le \int_K \dfrac{1}{n}\sum_{k=0}^{n-1}d(T^k(x),T^kf(x))d\mu(x)+\mu(\X\setminus K) D .\end{aligned}$$ Finally observe that by definition of $f$ and if $x\in K$ then we have $\widetilde{d}_n(x,f(x))\le \epsilon/2$. We immediately conclude that $${\mathbb{E}}\Big(\dfrac{1}{n}\sum_{k=0}^{n-1}d(T^kX,Y_k)\Big)\le \e.$$ Since by definition of $I(X,Y)$ we have $I(X,Y)\le \log|f(\X)|=\log(N+1)$, then $$R_\mu(\e)=\inf \dfrac{1}{n}I(X,Y)\le \liminf_{n\to\infty}\dfrac{1}{n}\log \widetilde{N}(n,\e,\e/(2D)).$$ If we construct $f$ using sets well approximated by $(n,\e)$-dynamical balls, i.e. in the construction above we take $K$ to satisfy $N(K,n,\frac{\e}{2D})=N_\mu(n,\e,\frac{\e}{2D})$, we obtain the analogous bound $$R_\mu(\e)\le \liminf_{n\to\infty}\dfrac{1}{n}\log N(n,\e,\e/(2D)).$$ Finally use Remark \[rem2\] to conclude.

The following theorem is an equivalent version of the Source Coding Theorem, which is a fundamental result in Information theory. We will use Theorem \[rd\] in the proof of Proposition \[prop2\]. For a proof and history about this theorem we refer the reader to [@G].

\[rd\] Let $X:(\Omega,{\mathbb{P}})\to (\X,\mu)$ be a measurable function with $X_*{\mathbb{P}}=\mu$, an ergodic $T$-invariant probability measure on $\X$. Given $\e_0>0$, there exists $n(\e_0)\in {\mathbb{N}}$ such that the following holds. For each $n\ge n(\e_0)$, there exists a measurable function $f_n:\X\to \X^n$ such that the pair $(X,f_n\circ X)$ satisfies condition $(*)_{n,\e+\e_0}$ and $|f_n(\X)|\le e^{n(R_\mu(\e)+\e_0)}$.

The following definition has similar content than Definition \[cond\] but will help to simplify the language used in the proof of Proposition \[prop2\].

A map $Z$ is called a $(n,\epsilon)$-approximation of $(\X,T)$ if $Z:\X\to\X^n$ is a measurable map with finite image and it satisfies $$\int_\X \frac{1}{n}\sum_{k=0}^{n-1} d(T^k(x), Z_k(x))d\mu(x)\le \epsilon.$$

\[lem\] Given $Z$ a $(n,\epsilon)$-approximation of $(\X,T)$, we can find a $(n,2\epsilon)$-approximation of $(\X,T)$, say $Z'=(Z'_0,...,Z'_{n-1})$, such that $Z'_k=T^kZ'_0$. Moreover the preimage partitions of $Z$ and $Z'$ coincide.

Denote by $\Q=\{Q_1,...,Q_T\}$ the preimage partition of $Z$, this means that $Z_{|Q_j}\equiv z_j=(z_{j,0},...,z_{j,n-1})\in \X^n$. Then we have $$\sum_{k=1}^T\int_{Q_k}\frac{1}{n}\sum_{i=0}^{n-1}d(T^i(x), z_{k,i})d\mu(x)\le \epsilon.$$ To simplify notation define $g_k:Q_k\to {\mathbb{R}}$ as $g_k(x)=\frac{1}{n}\sum_{i=0}^{n-1}d(T^i(x), z_{k,i})$. Therefore the above inequality can be writen as $$\sum_{k=1}^T \int_{Q_k}g_k(x)d\mu(x)\le \epsilon.$$ Let $a_k:= \int_{Q_k}g_k(x)d\mu(x)$. Assume $a_k\ne 0$ for some $k$, we claim that there exists $x_k\in Q_k$ such that $g_k(x_k)\le a_k/\mu(Q_k)$. If $g_k(x)> a_k/\mu(Q_k)$ for all $x\in Q_k$, then integrating over $Q_k$ will lead to a contradiction by the definition of $a_k$. If $a_k=0$, we pick any element in $Q_k$ and call it $x_k$. We define $Z'_i(x)=T^i(x_k)$ whenever $x\in Q_k$. By construction the preimage partions of $Z$ and $Z'$ coincide and $$\begin{aligned}
\int_{Q_k}\frac{1}{n}\sum_{i=0}^{n-1}d(T^i(x), T^i(x_k))d\mu(x)&\le \int_{Q_k}\frac{1}{n}\sum_{i=0}^{n-1}d(T^i(x), z_{k,i})d\mu(x)+\mu(Q_k)g_k(x_k)\\
& \le 2\epsilon.\end{aligned}$$

\[prop2\] Let $\mu$ be an ergodic $T$-invariant probability measure. The following inequality holds

$$\widetilde{h}_\mu(4L\epsilon, T,1/L)\le  R_\mu(\epsilon).$$

Fix $\e_0>0$ arbitrarily small. For $n\ge n(\e)$ consider the map $f_{n}=(f_{n,0},...,f_{n,n-1}):\X\to \X^n$ provided by Theorem \[rd\]. Since $(X,f_n\circ X)$ satisfies condition $(*)_{n,\epsilon+\e_0}$ we have the inequality $$\int_\X \frac{1}{n}\sum_{k=0}^{n-1}d(T^k(x),f_{n,k}(x))d\mu(x)\le \epsilon+\e_0.$$ In particular each map $f_n$ is a $(n,\epsilon+\e_0)$-approximation of $(\X,T)$. Moreover $$\begin{aligned}
 \frac{1}{n}\log|f_n(\X)|\le & R_\mu(\epsilon)+\e_0.
$$

Using Lemma \[lem\] we find maps $\widetilde{f}_n:\X\to\X^n$, each being a $(n,2\epsilon+2\e_0)$-approximation of $(\X,T)$ with the additional property that $\widetilde{f}_{n,k}=T^k\widetilde{f}_{n,0}$. If we define $g_n:=\widetilde{f}_{n,0}$, then $\widetilde{f}_n=(g_n,Tg_n,...,T^{n-1}g_n)$. Observe that $H_\mu(f_n)=H_\mu(\widetilde{f}_n)$ and $|f_n(\X)|=|\widetilde{f}_n(\X)|=|g_n(\X)|$, in particular $$\frac{1}{n}\log|g_n(\X)|\le  R_\mu(\epsilon)+\e_0.$$ We have now all the ingredients to relate $R_\mu(\epsilon)$ with the quantity $\widetilde{h}_\mu(2L\e,T,1/L)$. First notice that $$\int_\X \widetilde{d}_n(x,g_n(x) ) d\mu(x) = \int_\X \dfrac{1}{n}\sum_{k=0}^{n-1} d(T^k(x),T^kg_n(x))d\mu(x)\le 2\epsilon+2\e_0.$$ Define $\c=g_n(\X)\subset \X$ and set $A=\bigcup_{p\in \c} \widetilde{B}_n(p,2L(\epsilon+\e_0))$. Observe that if $x\in \X\setminus A$ then $\widetilde{d}_n(x,g_n(x))\ge 2L(\epsilon+\e_0)$, this immediately implies $\mu(\X\setminus A)\le 1/L$ or equivalently $\mu(A)\ge 1-\frac{1}{L}$. By construction the set $A$ can be covered by $|\c|=|g_n(\X)|$ $(n,2L(\epsilon+\e_0))$-average dynamical balls. This implies $$\dfrac{1}{n}\log \widetilde{N}_\mu(n,2L(\epsilon+\e_0),1/L)\le \dfrac{1}{n}\log |g_n(\X)|\le R_\mu(\epsilon)+\e_0,$$ for each $n>n(\e_0)$. This implies $$\limsup_{n\to \infty} \dfrac{1}{n}\log \widetilde{N}_\mu(n,2L(\epsilon+\e_0),1/L)\le  R_\mu(\epsilon)+\e_0,$$ for each $\e_0>0$. Taking $\e_0\to 0$, $$\limsup_{n\to \infty} \dfrac{1}{n}\log \widetilde{N}_\mu(n,4L\e,1/L)\le \limsup_{n\to \infty} \dfrac{1}{n}\log \widetilde{N}_\mu(n,2L(\epsilon+\e_0),1/L)\le   R_\mu(\epsilon)+\e_0,$$ we get $$\widetilde{h}_\mu(4L\e,T,1/L)=\limsup_{n\to \infty} \dfrac{1}{n}\log \widetilde{N}_\mu(n,4L\e,1/L)\le R_\mu(\epsilon).$$

We can summarize Proposition \[prop1\] and Proposition \[prop2\] in the following result.

\[teo1\] Let $(\X,d)$ be a compact metric space of diameter $D$ and $T:\X\to \X$ a continous transformation. Let $\mu$ be an ergodic $T$-invariant probability measure, then we have $$\widetilde{h}_\mu(4L\epsilon,T,1/L)\le R_\mu(\epsilon)\le \widetilde{h}_\mu(\epsilon,T,\e/2D).$$

Theorem \[teo1\] will be used in the proof of Theorem \[theo2\].

Given $x\in\X$, $n\ge0$ and $r\in (0,1)$ we define $$B'_n(x,\e,r)=\{y\in X: \dfrac{1}{n}\#\{0\le i\le n-1:d(T^ix,T^iy)\le \e\}\ge 1-r\}.$$ Define $N_\mu(n,\e,\delta,r)$ as the minimum number of balls $B'_n(x,\e,r)$ needed to cover a set of measure bigger or equal than $\delta$.

The following proposition follows from the main result of [@ZZC].

\[prop3\] Let $\mu$ be an ergodic $T$-invariant probability measure. Then for each $\delta\in (0,1)$ we have the inequality $$h_\mu(T)\le \lim_{r\to 0}\lim_{\e\to 0}\limsup_{n\to \infty} \dfrac{1}{n}\log N_\mu(n,\e,\delta,r).$$ If we moreover assume that $\X$ is compact, then the equality holds.

An easy application of Proposition \[prop3\] is the following result.

\[prop4\] Let $\mu$ be an ergodic $T$-invariant probability measure. Then for every $\delta\in (0,1)$ we have $$h_\mu(T)\le \widetilde{h}_\mu(T,\delta).$$

Observe that if $y\in \widetilde{B}_n(x,\e)$, then $$\#\{0\le i\le n-1: d(T^ix,T^iy)>L\e\}\le n/L,$$ or equivalently $$\dfrac{1}{n}\#\{0\le i\le n-1: d(T^ix,T^iy)\le L\e\}\ge 1-\dfrac{1}{L} .$$ This implies that $ \widetilde{B}_n(x,\e)\subset B_n'(x,L\e,1/L)$, and therefore $$N_\mu(n,L\e,\delta,1/L)\le \widetilde{N}_\mu(n,\e,\delta).$$ By taking limits we get $$\begin{aligned}
\lim_{\e\to 0}\limsup_{n\to\infty} \dfrac{1}{n}\log N_\mu(n,\e,\delta,r) \le& \lim_{\e\to 0}\limsup_{n\to\infty}\dfrac{1}{n}\log \widetilde{N}_\mu(n,\e,\delta)=\widetilde{h}_\mu(T,\delta).\end{aligned}$$ Finally using Proposition \[prop3\] we get $$h_\mu(T)\le \widetilde{h}_\mu(T,\delta).$$

We can finally prove Theorem \[theo2\].

Under the hypothesis of Theorem \[teo1\] we have $$\widetilde{h}_\mu(T,\delta)= \lim_{\epsilon\to 0} R_\mu(\epsilon)= h_\mu(T)$$

Combining Theorem \[teo1\] and Proposition \[prop1\] we get $$\widetilde{h}_\mu(T,\delta)\le \lim_{\epsilon\to 0} R_\mu(\epsilon)\le  h_\mu(T).$$ Then Proposition \[prop4\] gives the equality of the three quantities.

We will now prove a version of the variational principle for the average dynamical distances. The proof follows closely the proof of Lemma \[1111\] with minor modifications. For completeness we explain it in detail. Define $\widetilde{N}_d(n,\e)$ as the maximal cardinality of a $(n,\e)$-average separated set in $(\X,d)$ and $$\widetilde{S}(\X,d,\e)=\limsup_{n\to \infty}\dfrac{1}{n}\log \widetilde{N}_d(n,\e).$$ In case the dynamical system has been specified, we will frequently use the simplified notation $\widetilde{S}(\e)=\widetilde{S}(\X,d,\e)$.

\[2222\] Assume $(\X,d)$ is compact and $T$ is injective. Given $\e>0$ and $\delta\in (0,1/4)$, there exists a $T$-invariant probability measure $\mu_\e$ such that $$\widetilde{h}_{\mu_\e}(\e,T,\delta)\ge  \widetilde{S}\Big(\dfrac{2\e}{1-4\delta}\Big)-3\delta  \widetilde{S}\Big(\dfrac{\e}{1-4\delta}\Big).$$

We will fix $\e>0$ and $\delta\in (0,1/4)$. Let $E_n=\{x_1,...,x_{\widetilde{N}_d(n,\e)}\}$ be a maximal collection of $(n,\e)$-average separated points in $\X$. Define $$\sigma_n=\dfrac{1}{|E_n|}\sum_{x\in E_n} \delta_x,$$ where $\delta_x$ is the probability measure supported at $x$. Then define $$\overline{\sigma}_n=\dfrac{1}{n}\sum_{k=0}^{n-1} T^k_*\sigma_n.$$ Consider a subsequence $\{n_k\}$ such that $$\widetilde{S}(\X,d,\e)=\lim_{k\to \infty}\dfrac{1}{n_k}\log \widetilde{N}_d(n_k,\e).$$ By standard arguments we can find a subsequence of $\{n_k\}$, that we still denote by $\{n_k\}$, such that $\{\overline{\sigma}_{n_k}\}_{k\in{\mathbb{N}}}$ converges to a $T$-invariant probability measure $\mu$. We arrange the sequence such that $\lim_{k\to \infty} \frac{n_k}{k}=\infty$. Let $K$ be a subset of $\X$ with $\mu(K)> 1-\delta$ and $\widetilde{N}_d(K,n,(1-4\delta)\e/2)=\widetilde{N}_\mu(n,(1-4\delta)\e/2,\delta)$ where $\widetilde{N}(K,m,r)$ is defined to be the minimum number of $(m,r)$-average dynamical balls needed to cover $K$. We can assume that $K$ is open in $\X$. There exists $k_0$ such that for every $k\ge k_0$ we have $\overline{\sigma}_{n_k}(K)>  1-\delta$. Let $$L_n=\{(i,j)\in{\mathbb{N}}^2: 0\le i\le n-1, 1\le j\le \widetilde{N}(n,\e)\}.$$ We assign to each point in $L_n$ either a $0$ or $1$ in the following way. If $T^ix_j\in K$ then assign $1$ to the point $(i,j)$, assign $0$ to $(i,j)$ otherwise. By definition of the measure $\overline{\sigma}_{n_k}$ we know that the number of ones in $L_{n_k}$ is bigger or equal than $n_k \widetilde{N}_d(n_k,\e)(1-\delta)$. For $s>1$ we define $L_{n_k}(s)$ as the set of points in $L_{n_k}$ with first coordinate in the interval $[[n_k/k],n_k-[n_k/s]-1]$. The number of ones in $L_{n_k}(s)$ is at least $n_k \widetilde{N}_d(n_k,\e)(1-\delta-\frac{1}{n_k}[\frac{n_k}{s}]-\frac{1}{n_k}[\frac{n_k}{k}])\ge n_k \widetilde{N}_d(n_k,\e)(1-\delta-\frac{1}{s}-\frac{1}{k})$. From now on we assume $s>\frac{1}{1-2\delta-\frac{1}{k}}$, in particular $1-\delta-\frac{1}{s}-\frac{1}{k}>\delta$. We will moreover assume $s<\frac{1}{1-3\delta}$. This can be done if we assume $k$ is sufficiently large so that $\delta>1/k$. We conclude that the number of ones in $L_{n_k}(s)$ is at least $n_k \widetilde{N}_d(n_k,\e)\delta$. Since $L_{n_k}$ has $(n_k-[n_k/s]-[n_k/k])$ columns, then by the pigeonhole principle there exists an index $m_k$ such that the $m_k$th column has at least $\frac{n_k \widetilde{N}_d(n_k,\e)\delta}{(n_k-[n_k/s]-[n_k/k])}$ ones and $[n_k/k]\le m_k< n_k-[n_k/s]$. Observe that $\X$ can be covered by at most $\widetilde{N}_d(m,\e/2)$ $(m,\e/2)$-average dynamical balls, in particular with $\widetilde{N}_d(m_k,\e/2)$ subsets of $d_{m_k}$-diameter smaller than $\e$. Also observe that if $i\ne j$ and $\widetilde{d}_{m_k}(x_i,x_j)\le \e$, then $$\begin{aligned}
m_k\e+\sum_{p=0}^{n_k-m_k-1} d(T^{m_k+p}x_i,T^{m_k+p}x_j)\ge& \sum_{p=0}^{m_k-1}d(T^{p} x_i,T^{p}x_j)+\sum_{p=m_k}^{n_k-1} d(T^px_i,T^px_j)\\ &> n_k\e. \end{aligned}$$ In the last inequality, we are using the fact that $\widetilde{d}_{n_k}(x_i,x_j)>\e$. This implies that $\widetilde{d}_{n_k-m_k}(T^{m_k}x_i,T^{m_k}x_j)> \e$. We can conclude that there exists an subset $I\subset \{1,....,\widetilde{N}_d(n_k,\e)\}$ such that for $i\in I$ we have $T^{m_k}x_i\in K$, and $|I|\ge n_k \widetilde{N}_d(n_k,\e)\delta/(n_k-[n_k/s]-[n_k/k])>\widetilde{N}_d(n_k,\e)\delta$. By the pigeonhole principle there exists a subset $A$ of $\{T^{m_k} x_i\}_{i\in I}$ such that the diameter of $A$ with respect to $\widetilde{d}_{m_k}$ is at most $\e$ and $$|A|\ge \dfrac{ \widetilde{N}_d(n_k,\e)\delta}{\widetilde{N}_d(m_k,\e/2)  }.$$ As mentioned above this implies that if $a,b\in A$ and $a\ne b$, then $\widetilde{d}_{n_k-m_k}(a,b)\ge \e$. Therefore $$\widetilde{d}_{n_k}(a,b)>\frac{n_k-m_k}{n_k}\e>\frac{\frac{n_k}{s}-1}{n_k}\e>\Big(\dfrac{1}{s}-\dfrac{1}{n_k}\Big)\e>\Big(1-3\delta-\frac{1}{n_k}\Big)\e>(1-4\delta)\e.$$ We have assumed $k$ is sufficiently large so that $\delta>\frac{1}{n_k}$. Finally we obtain the bound $$\widetilde{N}_\mu(n_k,(1-4\delta)\e/2,\delta)=\widetilde{N}_d(K,n_k,(1-4\delta)\e/2)\ge  P \ge \dfrac{ \widetilde{N}_d(n_k,\e)\delta}{\widetilde{N}_d(m_k,\e/2)  },$$ where $P$ is the maximum number of $(n,(1-4\delta)\e)$-average separated points in $K$. Finally $$\begin{aligned}
\widetilde{h}_\mu((1-4\delta)\e/2,T,\delta)=&\limsup_{n\to \infty}\dfrac{1}{n}\log \widetilde{N}_\mu(n,(1-4\delta)\e/2,\delta)\\ \ge & \limsup_{k\to\infty} \dfrac{1}{n_k}\log \widetilde{N}_d(n_k,\e)-\dfrac{m_k}{n_k}\dfrac{1}{m_k}\log \widetilde{N}_d(m_k,\e/2).\end{aligned}$$ Recall that by construction $m_k<n_k-[n_k/s]$, then $\frac{m_k}{n_k}\le 1-\frac{1}{s}+\frac{1}{n_k}$. Therefore $$\begin{aligned}
 \widetilde{h}_\mu((1-4\delta)\e/2,T,\delta)&\ge \limsup_{k\to\infty} \dfrac{1}{n_k}\log \widetilde{N}_d(n_k,\e)-(1-\dfrac{1}{s}+\dfrac{1}{n_k})\dfrac{1}{m_k}\log \widetilde{N}_d(m_k,\e/2).\\
&\ge \limsup_{k\to\infty} \dfrac{1}{n_k}\log \widetilde{N}_d(n_k,\e)-(3\delta+\dfrac{1}{n_k})\dfrac{1}{m_k}\log \widetilde{N}_d(m_k,\e/2).\end{aligned}$$ Also observe that the inequality $[n_k/k]\le m_k$ implies that $m_k\to \infty$ as $k\to \infty$, and by construction $\widetilde{S}(\X,d,\e)=\lim_{k\to \infty}\dfrac{1}{n_k}\log \widetilde{N}_d(n_k,\e)$, therefore $$\widetilde{h}_\mu((1-4\delta)\e/2,T,\delta)\ge  \widetilde{S}(\e)-3\delta \widetilde{S}(\e/2).$$ This implies in particular that $$\sup_{\mu\in{\mathcal{M}}_T(\X)} \widetilde{h}_{\mu}(\e,T,\delta)\ge  \widetilde{S}\Big(\dfrac{2\e}{1-4\delta}\Big)-3\delta  \widetilde{S}\Big(\dfrac{\e}{1-4\delta}\Big).$$

As corollary of Lemma \[2222\] we will obtain a proof of Theorem \[teolin\]. For this we need a version of Theorem \[rd\] for non-ergodic measures. To state this result we need to introduce some notation. On $\X^n$ we consider the metric $\rho_n$ given by $\rho_n(x,y)=\frac{1}{n}\sum_{i=1}^n d(x_i,y_i)$, where $x=(x_1,...,x_n)$ and $y=(y_1,...,y_n)$. We define $\mu^n$ as the image of $\mu$ under the map $\X\to \X^n$ given by $x\mapsto (x,Tx,...,T^{n-1}x)$. For a finite subset $C_n\subset \X^n$ we define ${\mathbb{E}}_\mu(C_n)$ as the integral $\int _{\X^n}\min_{c\in C_n} \rho_n(x,c)d\mu^n(x)$.

Given $R>0$ define $$\delta_\mu(R)=\inf\{{\mathbb{E}}_\mu(C_n)|  \text{ }C_n\subset \X^n \text{ and }|C_n|\le e^{nR}\},$$ and $$D_\mu(R)=\inf\{{\mathbb{E}}_\mu\Big(\frac{1}{n}\sum_{k=0}^{n-1}d(T^kX,Y_k\circ X)\Big)|  \text{ }\frac{1}{n}I_\mu(X,(Y_0,...,Y_{n-1}))\le R\}.$$

In this language the Source Coding Theorem (see Theorem \[rd\]) can be stated as

Let $\mu$ be an ergodic $T$-invariant probability measure on $\X$. Then $$\delta_\mu(R)=D_\mu(R).$$

The following properties of $\delta_\mu(R)$ are proven in [@G].

The function $\mu\mapsto \delta_\mu(R)$ is affine and upper semicontinuous. The function $R\mapsto \delta_\mu(R)$ is convex and decreasing.

As a corollary we obtain the formula $$\delta_\mu(R)=\int_\X \delta_{\mu_x}(R)d\mu(x),$$ where $\mu=\int \mu_x d\mu(x)$ is the ergodic decomposition of $\mu$. It is easy to see that $\inf_{R\ge 0} \delta_\mu(R)=0$. This together with the convexity of $\delta_\mu(R)$ implies that $\delta_\mu(R)$ is strictly decreasing at $R$ if $\delta_\mu(R)>0$. Define $$D_\infty(R)=\sup_x D_{\mu_x}(R)=\sup_x \delta_{\mu_x}(R).$$ Since the supremum of convex function is still convex we conclude $D_\infty(R)$ is convex, in particular continuous. It follows easily from the definition that $\inf_{R\ge 0} D_\infty(R)=0$. As before this implies that $D_\infty(R)$ is strictly decreasing at $R$ if $D_\infty(R)>0$. We remark that $R_\mu(\e)$ is the ‘formal’ inverse of $D_\mu(R)$, with this we mean it is the inverse in the region where the functions are strictly decreasing. For the next remark we will use the following fact.

\[lemconv\] Let $D_{\mu_x}: (0,\infty) \to {\mathbb{R}}$ be a convex decreasing function for each $x\in\X$ and define $$D_{\infty}(R)=\sup_x D_{\mu_x}(R).$$ Let $I=\{t\in (0,\infty): D_\infty(t)>0\}$ and $J=Im(D_\infty)$. Then $D_\infty^{-1}:J\to I$ is well defined and for every $\e\in J$ we have $$D_{\infty}^{-1}(\epsilon)=\sup_{x} D_{\mu_x}^{-1}(\epsilon).$$

As mentioned above the convexity of $D_{\mu_x}$ implies the convexity and therefore the continuity of $D_\infty$. We also know that $D_\infty$ is strictly decreasing at $R$ if $D_\infty(R)>0$. This implies that $D_\infty^{-1}:J\to I$ is well defined and $J$ is a connected open interval. The inequality $D_{\infty}^{-1}(\epsilon)\ge \sup_{x} D_{\mu_x}^{-1}(\epsilon)$ follows from the decreasing assumption. For the other inequality we argue by contradiction, i.e. we assume there exists $\e\in J$ and $\epsilon_0>0$ such that for all $x\in\X$ we have $D_{\infty}^{-1}(\epsilon)-\epsilon_0 >D_{\mu_x}^{-1}(\epsilon)$, whenever $D_{\mu_x}^{-1}(\epsilon)$ is well defined. Define $b=D_{\infty}^{-1}(\epsilon)$ and $a=D_{\infty}^{-1}(\epsilon)-\epsilon_0/2$. By the definition of $D_\infty$ there exists $x$ such that $D_{\infty}(a)\ge D_{\mu_x}(a) >\epsilon=D_{\infty}(b)\ge D_{\mu_x}(b)$. By the intermediate value theorem exists $c\in [a,b)$ such that $D_{\mu_x}(c)=\epsilon.$ This gives a contradiction since $c> D_{\infty}^{-1}(\epsilon)-\epsilon_0$.

\[remnoerg\] Given $R>0$ and $\e_0>0$, there exists $n(\e_0,R)\in {\mathbb{N}}$ such that the following holds. For $n\ge n(\e_0,R)$ there exists $C_n\subset \X^n$ such that $|C_n|\le e^{n(R+\e_0)}$ and ${\mathbb{E}}_\mu(C_n)\le \int D_{\mu_x}(R)d\mu(x) +\e_0$. In particular ${\mathbb{E}}_\mu(C_n)\le D_\infty(R) +\e_0$. If $R_\infty(\e)$ is the inverse of $D_\infty(R)$, then by Lemma \[lemconv\] we know $R_\infty(\e)=\sup_{\mu_x} R_{\mu_x}(\e)$. In particular for $R=R_\infty(\e)$ and $n\ge n(\e_0,R)$ we get a code $C_n\subset \X^n$ such that $|C_n|\le e^{n(R_\infty(\e)+\e_0)}$ and ${\mathbb{E}}_\mu(C_n)\le \e+\e_0$. Following the proof of Proposition \[prop2\] we get $$\widetilde{h}_\mu(4L\e,T,1/L)\le \sup_x R_{\mu_x}(\e)\le\sup_{\mu\text{ ergodic}} R_{\mu}(\e).$$

Before reproving Theorem \[teolin\] we are going to state Condition 1.2 introduced in [@LT] and mentioned in the statement of Theorem \[teolin\].

\[cond\] Let $(\mathcal{X},d)$ be a compact metric space. It satisfy the Condition 1.2, if for every $\delta>0$, we have $$\lim_{\epsilon\to 0} \epsilon^{\delta} \log \#(\mathcal{X},d,\epsilon)=0,$$ where $\#(\mathcal{X},d,\epsilon)$ is the minimal number of $\e$-balls needed to cover $\X$.

\[remfin\] It is important to mention that every compact metrizable space admits a distance satisfying the Condition 1.2. This remark corresponds to Lemma 1.3 in [@LT]. It is also important to mention that Condition 1.2 implies that $$\limsup_{\e\to 0} \dfrac{\widetilde{S}(\X,d,\e)}{|\log\e|}=\overline{mdim}(\X,d,T).$$

\[proprate\] Let $(\X,d)$ be a compact metric space and $T:\X\to \X$ a continuous map. Then $$\limsup_{\e\to 0} \dfrac{\sup_\mu R_\mu(\e)}{|\log \e|}= \limsup_{\e\to 0} \dfrac{\widetilde{S}(\X,d,\e)}{|\log\e|}.$$ If $(\X,d)$ satisfy Condition 1.2 in [@LT], then $$\overline{mdim}(\X,d,T)=\limsup_{\e\to 0} \dfrac{\sup_\mu R_\mu(\e)}{|\log \e|}.$$

The inequality $$\limsup_{\e\to 0} \dfrac{\sup_\mu R_\mu(\e)}{|\log \e|}\le  \limsup_{\e\to 0} \dfrac{\widetilde{S}(\X,d,\e)}{|\log\e|},$$ is the easy part of the statement. We refer the reader to [@LT] for a proof. We will prove here the reversed inequality. Observe that by Remark \[remnoerg\] we have $$\widetilde{h}_\mu(4L\epsilon,T,1/L)\le \sup_{\mu\text{ ergodic} }R_\mu(\epsilon).$$ In particular $$\widetilde{h}_\mu(4\e/\delta,T,\delta)\le \sup_{\mu\text{ ergodic} }R_\mu(\epsilon).$$ This implies $$\sup_{\mu \text{ ergodic}} R_\mu(\e)\ge \sup_\mu \widetilde{h}_\mu(4\e/\delta,T,\delta)\ge \widetilde{S}\Big(\frac{8\e}{\delta(1-4\delta)}\Big)-3\delta\widetilde{S}\Big(\frac{4\e}{\delta(1-4\delta)}\Big).$$ Finally dividing by $|\log \e|$, taking limsup in $\e$ and then $\delta \to 0$ we obtain the lower bound $$\limsup_{\e\to 0} \dfrac{\sup_{\mu \text{ ergodic}}R_\mu(\e)}{|\log \e|}\ge \limsup_{\e\to 0} \dfrac{\widetilde{S}(\e)}{|\log\e|}.$$

Combining Proposition \[proprate\] and Remark \[remfin\] we obtain a proof of Theorem \[teolin\].

Dynamics on manifolds {#sec5}
=====================

We start this section with the following observation.

\[remmm\] Let $F:\X\to \X$ be a dynamical system. The map $x\mapsto (x,Fx,F^2x,...)$ embedds $(\X,F)$ into $(\X^{\mathbb{N}},T)$, where $T$ is the shift map. In other words we have a $T$-invariant subset $Y\subset \X^{\mathbb{N}}$ where $(Y,T)$ is conjugate to $(\X,F)$. The metric $d_T$ on $\X^{\mathbb{N}}$ restricted to $Y$ induces a metric on $\X$. We still denote this metric by $d_T$. More explicitely we have $d_T(x,y)=\sum_{k\ge 0} \frac{1}{2^k}d(T^kx,T^ky)$. Since $d(x,y)\le d_T(x,y)$ for all $x,y\in \X$ we conclude that $$\begin{aligned}
 \overline{mdim}(\X,d,F)\le & \overline{mdim}(\X,d_T,F)=\overline{mdim}(Y,d_T,T)\\ \le & \overline{mdim}(\X^{\mathbb{N}},d_T,T) =\overline{dim}_B(\X,d).\end{aligned}$$ In particular the dimension of $\X$ is always an upper bound for the mean dimension.

A map $F$ that realize the equality in the above inequality is said to have maximal metric mean dimension. We will construct an example of a continuous map on the interval with maximal metric mean dimension. From now on $d$ will always stand for the euclidean distance in $[0,1]$. We start with the following Lemma.

\[exam\] Let $J_k=[a_{k-1},a_k]\subset [0,1]$ and $b_k=a_k-a_{k-1}$. Decompose $J_k$ into $2l_k$ equal intervals $\{J^s_k\}_{s=1}^{2l_k}$ and define $\e_k=\frac{b_k}{2l_k}$. Suppose that $f:[0,1]\to [0,1]$ is a continuous map such that $J_k\subset f(J^s_k)$. Then $$S([0,1],d,\e_k)\ge \log(l_k).$$

Define the cylinders $$A_{i_0,...,i_{m-1}}=\{x\in J_k: f^{h}x\in J^{i_h}_s, \quad \textrm{for every } h\in\{0,...,m-1\}\}.$$ We will only consider those sets with all $i_h$’s odd. If $x$ and $y$ are in distinct cylinders, then $d_n(x,y)\ge e_k$. Moreover by the assumption $J_k\subset f(J^s_k)$ we know each cylinder is nonempty. This implies that the maximal number of $(m,\e_k)$-separated points in $J_k$ is at least $(l_k)^m$ for each $m\ge 1$. In particular $$S([0,1],d,\e_k)\ge S(J_k,d,\e_k)\ge \log l_k.$$

\[existence\] There exists $f:[0,1]\to [0,1]$ continuous such that $$\overline{mdim}([0,1],d,f)=1.$$

Let $b_k=C/k^2$ such that $\sum_{k\ge1} b_k=1$ and $l_k=k^k$. Define $a_k=\sum_{1\le s\le k} b_s$, $a_0=0$ and $J_k=[a_{k-1},a_k]$. It is easy to construct a continuous function $f$ such that the requirements in Lemma \[exam\] are satisfied. We can conclude that $$mdim([0,1],d,f)=\limsup_{\e\to 0} \frac{S([0,1],d,\e)}{|\log\e|}\ge \limsup_{k\to \infty} \frac{S([0,1],d,\e_k)}{|\log\e_k|}\ge  \limsup_{k\to \infty} \frac{\log l_k}{|\log\e_k|},$$ where $\e_k=\frac{b_k}{2l_k}=\frac{C}{2k^{k+2}}$. Then $$\overline{mdim}([0,1],d,f)\ge  \limsup_{k\to \infty} \frac{\log l_k}{|\log\e_k|} = \limsup_{k\to \infty} \frac{\log k^k}{\log (2k^{k+2}/C)}=1.$$ The opposite inequality holds for any continuous map by Remark \[remmm\] since $$dim_B([0,1],d)=1.$$

A similar construction allows us to prove that if $f(x)=x$ has infinitely many solutions, then $f$ can be approximated in the $C^0$ topology by continuous functions with metric mean dimension equal to one (slightly perturb the function $f$ nearby the fixed points by a map as in Lemma \[exam\] using $l_k$ sufficiently large in comparison with $b_k$). This argument allows us to prove that the space $$A=\{F\in C([0,1]): \overline{mdim}([0,1],d,F)=1\},$$ where $C([0,1])$ is the space of continuous self maps on the interval $[0,1]$ with the uniform topology, is dense in $C([0,1])$. To see this start with $G\in C([0,1])$. By the mean value theorem there exists $x_G\in [0,1]$ such that $G(x_G)=x_G$. Nearby $x_G$ perturb $G$ to a function $G'$ with infinitely many fixed points. Then mimic the construction done in Proposition \[existence\] as mentioned above. We summarize this in the following result.

Let $$A=\{F\in C([0,1]): \overline{mdim}([0,1],d,F)=1\}.$$ Then $A$ is a dense subset of $C([0,1])$.

A similar procedure can be done in a compact manifold $\X$ of dimension bigger or equal than two with a metric $d$ induced from a Riemannian metric $g$. We will prove that any homeomorphism $F$ on $\X$ can be approximated by a homeomorphism with maximal metric mean dimension under the assumption that $F$ has a fixed point. For simplicity in the notation we will explain the procedure for a homeomorphism in a two manifold although the same argument holds in any dimension. Let $p$ be a fixed point of $F$. We make an arbitrary small perturbation $F_1$ of $F$ in a neighborhood of $p$ to create countably many isolated fixed points. We will perturb $F_1$ in a sufficiently small neighborhood of each fixed point (all neighborhood will be disjoint). As done in [@yan] we can perturb $F_1$ in a box of length $b_k$ (for arbitrarily small $b_k$) around the fixed point $p_k$ such that the perturbation contains a $2l_k$-horseshoe. Since our box is sufficiently small we can compare the metric $d$ with the euclidean metric up to multiplication by a uniform positive constant (here we are strongly using the compactness of $\X$). We will explain the local picture required in the perturbation. We start with ${\mathbb{R}}^2$ endowed with the euclidean metric $d^0$ and a square $R$ of lenght $b_k$. We define a map $G_k$ as shown in Figure 1. Let $\{U_i\}_{i=1}^{2l_k}$ be vertical strips of horizontal lenght $b_k/(2l_k)$ and the regions $\{A_i\}$ contain in $G_k(R)$ as in Figure 1. Let $B_{ij}=G_k^{-1}(A_i\cap U_j)$ for $(i,j)$ a pair of odd numbers in $[1,2l_k]$.

(0,0) – (5,0) – (5,5) – (0,5) – (0,0); (-1,4.7) – (-1,4.4); plot coordinates [(-1,4.4) (6,4.4) (6,4.1) (-1.5,4.1) (-1.5,3.2) (6,3.2) (6, 2.9)]{}; plot coordinates [(-1,4.7) (6.5,4.7) (6.5,3.8) (-1,3.8) (-1,3.5) (6.5,3.5) (6.5,2.9)]{}; at (2.3,-0.58) [$\textmd{Figure 1\label{fig1}: Horseshoe}$]{}; (2.4,2.8) – (3.4,2);

plot coordinates [(-1.5,1.9) (-1.5,1.3) (6,1.3) (6, 1) (-1, 1) ]{}; plot coordinates [(-1,1.9) (-1,1.6) (6.5,1.6) (6.5,0.7) (-1, 0.7)]{}; (-1, 1) – (-1, 0.7);

plot \[domain=0:5\] (,1) – plot \[smooth,domain=5:0\] (,0.7) – cycle;

plot \[domain=0:5\] (,1.6) – plot \[smooth,domain=5:0\] (,1.3) – cycle; plot \[domain=0:5\] (,4.7) – plot \[smooth,domain=5:0\] (,4.4) – cycle; plot \[domain=0:5\] (,4.1) – plot \[smooth,domain=5:0\] (,3.8) – cycle; plot \[domain=0:5\] (,3.2) – plot \[smooth,domain=5:0\] (,3.5) – cycle;

(0.5,0) – (0.5,5); (1,0) – (1,5); (1.5,0) – (1.5,5); (2,0) – (2,5); (0,0) – (0,5);

(0,-0.2) – (1.6,-0.2); (3.6,-0.2) – (5,-0.2); (1.7,-0.2) – (3.5,-0.2);

(0,-0.1) – (0,-0.3); (0.5,-0.1) – (0.5,-0.3); (1,-0.1) – (1,-0.3); (1.5,-0.1) – (1.5,-0.3);

at (0.25,-0.2) [$U_1$]{}; at (0.75,-0.2) [$U_2$]{}; at (1.25,-0.2) [$U_3$]{};

at (2.7,4.3) [$A_1$]{}; at (2.7,3.7) [$A_2$]{}; at (2.7,3.1) [$A_3$]{}; at (2.7,1.15) [$A_{2l_k-1}$]{}; at (2.7,0.55) [$A_{2l_k}$]{};

Define the cylinders $$A_{(i_0j_0),...,(i_{m-1}j_{m-1})}=\{x\in R: G_k^s x\in B_{i_sj_s} \quad \textrm{for every } s\in\{1,...,m-1\}\}.$$ It follows from the construction that each cylinder is not empty. Moreover if $x$ and $y$ are in different cylinders then $d^0_m(x,y)>\e_k$, where $\e_k=b_k/(4l_k)$. We can conclude that $S(R,d^0, \e_k)\ge \log l_k^2$. We can perturb $F_1$ nearby the points in $\{p_k\}_{}$ by a map $F_2$ that nearby $p_k$ looks like $G_k$. The same reasoning as in Lemma \[exam\] gives us that $$\overline{mdim}(\X,d,F_2)\ge \limsup_{k\to \infty} \dfrac{\log l_k^2}{\log (4l_k/b_k)}=\limsup_{k\to \infty}\dfrac{2}{1+\frac{\log 4}{\log l_k}+\frac{\log b_k^{-1}}{\log l_k}}.$$ Observe we can assume $b_k^{-1}\ll \log _k$. Therefore we found a map $F_2$ arbitrarily close to $F$ such that $\overline{mdim}(\X,d,F_2)=2$. For a general homeomorphism we can still approximate by homeomorphisms with positive upper metric mean dimension. To see this take $p\in \X$ such that $F^k(p)$ is sufficiently close to $p$, then perturb to create a fixed point using the standard closing lemma for homeomorphism. After this approximation is done we proceed as above or as done in [@yan]. The same argument can be applied to prove that any homeomorphism $F$ with a $k$-periodic point can be approximated by homeomorphisms with upper metric mean dimension at least $\frac{1}{k}\dim(\X)$. We denote by $H(\X)$ to the space of homeomorphisms of $\X$. We remark that under the assumptions above we know $\dim_B(\X,d)=dim(\X)$. We summarize this discussion in the following result.

Let $(\X,g)$ be a closed Riemannian manifold with induced metric $d$. The set $$A=\{F\in H(\X): \overline{mdim}(\X,d,F)=\dim(\X)\},$$ is dense in the subspace of homeomorphisms with a fixed point. The set $$B=\{F\in H(\X): \overline{mdim}(\X,d,F)>0\},$$ is dense in $H(\X)$.

Final Remarks {#sec6}
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For simplicity in previous sections we worked with the upper metric dimension. It also follows from Lemma \[1111\] that if $\overline{mdim}(\X,d,T)<\infty$, then $$\underline{mdim}(\X,d,T)=\lim_{\delta\to 0}\liminf_{\e\to 0} \dfrac{\sup_\mu h_\mu(\e,T,\delta)}{|\log \e|}.$$ A more symmetric formulation of the variational principle would be $$\overline{mdim}(\X,d,T)=\limsup_{\e\to 0} \dfrac{\sup_\delta \sup_\mu h_\mu(\e,T,\delta)}{|\log \e|},$$ and $$\underline{mdim}(\X,d,T)=\liminf_{\e\to 0} \dfrac{\sup_\delta \sup_\mu h_\mu(\e,T,\delta)}{|\log \e|}.$$ This also follows directly from Lemma \[1111\]. Analog statements are true for the average dynamical metric and the corresponding average metric mean dimension.

The computation done in Theorem \[calculo\] also shows that the metric mean dimension depends strongly on the metric. If we have two compatible metrics $d_1$ and $d_2$ on a topological space $Y$ with different box dimension, then the metric mean dimension of the shift map $T$ on $Y^{\mathbb{Z}}$ will differ when considering the metric $(d_1)_T$ and $(d_2)_T$.

It is an interesting question to describe hypothesis on the dynamics that ensure the existence of a measure of ‘maximal metric mean dimension’. In our context this would mean that $$\overline{mdim}(X,d,T)=\lim_{\delta\to 0} \limsup_{\e\to 0} \dfrac{h_\mu(\e,T,\delta)}{|\log\e|}.$$ In the context of [@LT], this would mean $$\overline{mdim}(X,d,T)=\limsup_{\e\to 0} \dfrac{R_\mu(\e)}{|\log\e|}.$$ As our bounds show, there is not major difference between this two perspectives (at least under Condition 1.2 we could link this two sides by using $\widetilde{h}_\mu(\e,T,\delta)$). The authors believe that if $\X$ is a manifold and $T$ preserves a OU measure $\mu$ , i.e. $\mu$ has not atoms and it is positive on every open set, then $\mu$ would have this property (at least when choosing $T$ generic in the space of homeomorphisms that preserve $\mu$). To motivate the importance of this situation see [@ou].

We define the *upper metric mean dimension of a measure* $\mu$ as $$\overline{mdim}_\mu(\X,d,T)=\lim_{\delta\to 0}\limsup_{\e\to 0}\frac{h_\mu(\e,T,\delta)}{|\log\e|}.$$ As before, we could also use the analog formulation in the context of [@LT]. We remark that this quantity depends on the metric, it is not a purely measure theoretic invariant. As remarked in the introduction of [@LT], it is not possible to define a meaningful invariant that only depends on measure theoretic information. Another natural question would be if the variational principle holds for this definition, in other words, if it is true that $\overline{mdim}(\X,d,T)=\sup_\mu \overline{mdim}_\mu(\X,d,T)$.

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