# Density of the level sets of the metric mean dimension for homeomorphisms

Jeovanny M. Acevedo, Sergio Romaña, Raibel Arias  Jeovanny de Jesus Muentes
Acevedo, Facultad de Ciencias Básicas, Universidad Tecnológica de Bolívar,
Cartagena de Indias - Colombia<EMAIL_ADDRESS>Sergio Augusto Romaña
Ibarra, Instituto de Matemática - Universidade Federal do Rio de Janeiro, Av.
Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C - Cidade
Universitária - Ilha do Fundão, cep: 21941-909, Rio de Janeiro-Brasil
<EMAIL_ADDRESS>Raibel de Jesus Arias Cantillo, Universidade Federal de
Maranhão, Campus Balsas, Brasil<EMAIL_ADDRESS>
###### Abstract.

Let $N$ be an $n$-dimensional compact riemannian manifold, with $n\geq 2$. In
this paper, we prove that for any $\alpha\in[0,n]$, the set consisting of
homeomorphisms on $N$ with lower and upper metric mean dimensions equal to
$\alpha$ is dense in $\text{Hom}(N)$. More generally, given
$\alpha,\beta\in[0,n]$, with $\alpha\leq\beta$, we show the set consisting of
homeomorphisms on $N$ with lower metric mean dimension equal to $\alpha$ and
upper metric mean dimension equal to $\beta$ is dense in $\text{Hom}(N)$.
Furthermore, we also give a proof that the set of homeomorphisms with upper
metric mean dimension equal to $n$ is residual in $\text{Hom}(N)$.

###### Key words and phrases:

mean dimension, metric mean dimension, topological entropy, genericity

###### 2020 Mathematics Subject Classification:

##  1\. Introduction

In the late 1990’s, M. Gromov introduced the notion of mean topological
dimension for a continuous map $\phi:X\rightarrow X$, which is denoted by
$\text{mdim}(X,\phi)$, where $X$ is a compact topological space. The mean
topological dimension is an invariant under conjugacy. Furthermore, this is a
useful tool in order to characterize dynamical systems that can be embedded in
$(([0,1]^{m})^{\mathbb{Z}},\sigma)$, where $\sigma$ is the left shift map on
$([0,1]^{m})^{\mathbb{Z}}$. In [8], Lindenstrauss and Weiss proved that any
homeomorphism $\phi:X\rightarrow X$ that can be embedded in
$(([0,1]^{m})^{\mathbb{Z}},\sigma)$ must satisfy that $\text{mdim}(X,\phi)\leq
m$. In [6], Gutman and Tsukamoto showed that, if $(X,\phi)$ is a minimal
system with $\text{mdim}(X,\phi)<m/2$, then we can embed it in
$(([0,1]^{m})^{\mathbb{Z}},\sigma)$. In [10], Lindenstrauss and Tsukamoto
presented an example of a minimal system with mean topological dimension equal
to $m/2$ that cannot be embedded into $(([0,1]^{m})^{\mathbb{Z}},\sigma)$,
which show the constant $m/2$ is optimal.

The mean topological dimension is difficult to calculate. Therefore,
Lindenstrauss and Weiss in [8] introduced the notion of metric mean dimension,
which is an upper bound for the mean topological dimension. The metric mean
dimension is a metric-dependent quantity (this dependence is not continuous,
as we can see in [3]), therefore, it is not an invariant under topological
entropy.

###  1.1. Metric mean dimension

Let $X$ be a compact metric space endowed with a metric $d$. For any
$n\in\mathbb{N}$, we define $d_{n}:X\times X\to[0,\infty)$ by

$d_{n}(x,y)=\max\\{d(x,y),d(\phi(x),\phi(y)),\dots,d(\phi^{n-1}(x),\phi^{n-1}(y))\\}.$

Fix $\varepsilon>0$. We say that $A\subset X$ is an
$(n,\phi,\varepsilon)$-separated set if $d_{n}(x,y)>\varepsilon$, for any two
distinct points $x,y\in A$. We denote by $\text{sep}(n,\phi,\varepsilon)$ the
maximal cardinality of any $(n,\phi,\varepsilon)$-separated subset of $X$. Set

$\text{sep}(\phi,\varepsilon)=\underset{n\to\infty}{\limsup}\frac{1}{n}\log\text{sep}(n,\phi,\varepsilon).$

We say that $E\subset X$ is an $(n,\phi,\varepsilon)$-spanning set for $X$ if
for any $x\in X$ there exists $y\in E$ such that $d_{n}(x,y)<\varepsilon$. Let
$\text{span}(n,\phi,\varepsilon)$ be the minimum cardinality of any
$(n,\phi,\varepsilon)$-spanning subset of $X$. Set

$\text{span}(\phi,\varepsilon)=\underset{n\to\infty}{\limsup}\frac{1}{n}\log\text{span}(n,\phi,\varepsilon).$

######  Definition 1.1.

The _topological entropy_ of $\phi:X\rightarrow X$ is defined by

$h_{\text{top}}(\phi)=\lim_{\varepsilon\to
0}\text{sep}(\phi,\varepsilon)=\lim_{\varepsilon\to
0}\text{span}(\phi,\varepsilon).$

######  Definition 1.2.

We define the _lower metric mean dimension_ and the _upper metric mean
dimension_ of $(X,d,\phi)$ by

$\displaystyle\underline{\text{mdim}}_{\text{M}}(X,d,\phi)$
$\displaystyle=\liminf_{\varepsilon\to
0}\frac{\text{sep}(\phi,\varepsilon)}{|\log\varepsilon|}=\liminf_{\varepsilon\to
0}\frac{\text{span}(\phi,\varepsilon)}{|\log\varepsilon|}$
$\displaystyle\text{and }\quad\overline{\text{mdim}}_{\text{M}}(X,d,\phi)$
$\displaystyle=\limsup_{\varepsilon\to
0}\frac{\text{sep}(\phi,\varepsilon)}{|\log\varepsilon|}=\limsup_{\varepsilon\to
0}\frac{\text{span}(\phi,\varepsilon)}{|\log\varepsilon|},$

respectively.

######  Remark 1.3.

Throughout the paper, we will omit the underline and the overline on the
notations $\underline{\text{mdim}}_{\text{M}}$ and
$\overline{\text{mdim}}_{\text{M}}$ when the result be valid for both cases,
that is, we will use $\text{mdim}_{\text{M}}$ for the both cases.

In recent years, the metric mean dimension has been the subject of multiple
investigations, which can be verified in the bibliography of the present work.
The purpose of this manuscript is to complete the research started in [5],
[14] and [2], concerning to the topological properties of the level sets of
the metric mean dimension map.

In [5], Theorem C, the authors proved the set consisting of continuous maps
$\phi:[0,1]\rightarrow[0,1]$ such that
$\underline{\text{mdim}}_{\text{M}}([0,1],|\cdot|,\phi)=\overline{\text{mdim}}_{\text{M}}([0,1],|\cdot|,\phi)=\alpha$,
for a fixed $\alpha\in[0,1]$, is dense in $C^{0}([0,1])$ (see also [2],
Theorem 4.1). Furthermore, in Theorem A they showed if $N$ is an
$n$-dimensional compact riemannian manifold, with $n\geq 2$, and riemannian
metric $d$, the set of homeomorphisms $\phi:N\rightarrow N$ such that
$\overline{\text{mdim}}_{\text{M}}(N,d,\phi)=n$ contains a residual set in
$\text{Hom}(N)$ (see a particular case of this fact in [14], Proposition 10).
Next, for any $n\geq 1$, the set consisting of continuous maps
$\phi:N\rightarrow N$ such that

$\underline{\text{mdim}}_{\text{M}}(N,d,\phi)=\overline{\text{mdim}}_{\text{M}}(N,d,\phi)=\alpha$

is dense in $C^{0}(N)$, for a fixed $\alpha\in[0,n]$ (see [2], Theorem 4.5).

We consider the next level sets of the metric mean dimension for
homeomorphisms:

######  Definition 1.4.

For $\alpha,\beta\in[0,n]$, with $\alpha\leq\beta$, we will set

${H}_{\alpha}^{\beta}(N)=\\{\phi\in\text{Hom}(N):\underline{\text{mdim}}_{\text{M}}(N,d,\phi)=\alpha\,\,\text{and}\,\,\overline{\text{mdim}}_{\text{M}}(N,d,\phi)=\beta\\}.$

If $\alpha=\beta$, we denote ${H}_{\alpha}^{\alpha}(N)$ by ${H}_{\alpha}(N)$.

If $n=1$, then ${H}_{\alpha}^{\beta}(N)=\emptyset$ for $0<\alpha\leq\beta\leq
1$, because any homeomorphism on a one dimensional compact riemannian manifold
has zero topological entropy and hence has zero metric mean dimension. Our
first result is:

######  Theorem 1.5.

Let $n\geq 2$. For any $\alpha,\beta\in[0,n]$, with $\alpha\leq\beta$, the set
${H}_{\alpha}^{\beta}(N)$ is dense in $\emph{Hom}(N)$.

Using the techniques employed to prove Theorem 1.5, we will provide a new
proof of Theorem A in [5], that is:

######  Theorem 1.6.

The set
$\overline{H}(N)=\\{\phi\in\emph{Hom}(N):\overline{\emph{mdim}}_{\emph{M}}(N,d,\phi)=n\\}$
contains a residual set in $\emph{Hom}(N)$.

Yano, in [15], defined a kind of horseshoe in order to prove the set
consisting of homeomorphisms $\phi:N\rightarrow N$ with infinite entropy is
generic in $\text{Hom}(N)$, where $N$ is an $n$ dimensional compact manifold,
with $n\geq 2$. If we want to construct a continuous map with infinite entropy
we can consider an infinite sequence of horseshoes such that the number of
legs is unbounded. For the metric mean dimension case, in [5] and [14] the
authors used horseshoe in order to prove the set consisting of homeomorphisms
$\phi:N\rightarrow N$ with upper metric mean dimension equal to $n$ (which is
the maximal value of the metric mean dimension for any map defined on $N$) is
generic in $\text{Hom}(N)$. To get metric mean dimension equal to $n$ we must
construct a sequence of horseshoes such that the number of legs increases very
quickly compared to the decrease in their diameters.

Estimating the precise value of the metric mean dimension for a homeomorphism,
and hence to obtain a homeomorphism with metric mean dimension equal to a
fixed $\alpha\in(0,n)$, is harder and not trivial sake. We need to establish a
precise relation between the sizes of the horseshoes together with the number
of appropriated legs to control the metric mean dimension. This is our main
tool (see Lemma 2.3).

The paper is organized as follows: In Section 2, we will construct
homeomorphisms, defined on an $n$-cube, with metric mean dimension equal to
$\alpha$, for a fixed $\alpha\in(0,n]$. Furthermore, given
$\alpha,\beta\in[0,n]$, with $\alpha\leq\beta$, we will construct examples of
homeomorphisms $\phi:[0,1]^{n}\rightarrow[0,1]^{n}$ such that
$\underline{\text{mdim}}_{\text{M}}([0,1]^{n},d,\phi)=\alpha$ and
$\overline{\text{mdim}}_{\text{M}}([0,1]^{n},d,\phi)=\beta.$ In Section 3, we
will prove Theorem 1.5. Finally, in Section 4, we will show Theorem 1.6.

##  2\. Homeomorphisms on the n-cube with positive metric mean dimension

Let $n\geq 2$. Given $\alpha,\beta\in[0,n]$, $\alpha\leq\beta$, in this
section we will construct a homeomorphism $\phi_{\alpha,\beta}$, defined on an
$n$-cube, with lower metric mean dimension equal to $\alpha$ and upper metric
mean dimension equal to $\beta$ (see Lemma 2.4). This construction will be the
main tool to prove the Theorem 1.5, since if a homeomorphism present a
periodic orbit, then we can glue, in the $C^{0}$-topology, along this orbit
the dynamic of $\phi_{\alpha,\beta}$.

The construction of $\phi_{\alpha,\beta}$ requires a special type of
horseshoe. So, let us present the first definition.

######  Definition 2.1 ($n$-dimensional $(2k+1)^{n-1}$-horseshoe).

Fix $n\geq 2$. Take $E=[a,b]^{n}$ and set $|E|=b-a$. For a fixed natural
number $k>1$, take the sequence $a=t_{0}<t_{1}<\cdots<t_{4k}<t_{4k+1}=b$, with
$|t_{i}-t_{i-1}|=\frac{b-a}{4k+1}$, and consider

$H_{i_{1},i_{2},\dots,i_{n-1}}=[a,b]\times[t_{i_{1}-1},t_{i_{1}}]\times\cdots\times[t_{i_{n-1}-1},t_{i_{n-1}}],\quad\text{for
}i_{j}\in\\{1,\dots,4k+1\\}.$

Take $a=s_{0}<s_{1}<\cdots<s_{2(2k+1)^{n-1}-2}<s_{2(2k+1)^{n-1}-1}=b$, with
$|s_{i}-s_{i-1}|=\frac{b-a}{2(2k+1)^{n-1}-1}$, and consider

$V_{l}=[s_{l-1},s_{l}]\times[a,b]^{n-1},\quad\text{for
}l=1,2,\dots,2(2k+1)^{n-1}-1.$

We say that $E\subseteq A\subseteq\mathbb{R}^{n}$ is an $n$-dimensional
$(2k+1)^{n-1}$-horseshoe for a homeomorphism $\phi:A\rightarrow A$ if:

  * •

$\phi(a,a,\dots,a,b)=(a,a,\dots,a,b)$ and
$\phi(b,b,\dots,b,a)=(b,b,\dots,b,a)$;

  * •

For any $H_{i_{1},i_{2},\dots,i_{n-1}}$, with $i_{j}\in\\{1,3,\dots,4k+1\\}$,
there exists some $l\in\\{1,3,\dots,2(2k+1)^{n-1}-1\\}$ with

$\phi(V_{l})=H_{i_{1},i_{2},\dots,i_{n-1}}\quad\text{and}\quad\phi|_{V_{l}}:V_{l}\rightarrow
H_{i_{1},i_{2},\dots,i_{n-1}}\quad\text{is linear}.$

  * •

For any $l=2,4,\dots,2(2k+1)^{n-1}-2$, $\phi(V_{l})\subseteq A\setminus E$.

In that case, the sets $H_{i_{1},i_{2},\dots,i_{n-1}}$, with
$i_{j}\in\\{1,3,\dots,4k+1\\}$ are called the legs of the horseshoe.

(a) $V_{i}$, $i=1,\dots,9$

(b) $H_{i}$, $i=1,3,\dots,9$

Figure 2.1. 2-dimensional $5$-horseshoe

(a) $V_{i}$, $i=1,\dots,17$

(b) $H_{i,j}$, $i,j\in\\{1,3,5\\}$

Figure 2.2. 3-dimensional 9-horseshoe

In Figure 2.1 we present an example of a 2-dimensional 5-horseshoe. In that
case, $k=2$, we have $2(2k+1)-1$ divisions $V_{i}$ of $[a,b]^{2}$, and $2k+1$
legs $H_{i}$, for $i=1,3,5,7,9$. In Figure 2.2 we have a 3-dimensional
9-horseshoe (we only show $\phi(E)\cap E$ in that figure). In that case,
$k=1$, we have $2(2k+1)^{2}-1$ divisions $V_{i}$ of $[a,b]^{3}$, and
$(2k+1)^{2}$ legs $H_{i,j}$, for $i,j=1,3,5$.

Note if $E=[a,b]^{n}$ is an $n$-dimensional $(2k+1)^{n-1}$-horseshoe for
$\phi$, then

$\phi^{2}(V_{l_{1}}\cap H_{i_{1}^{(1)},\dots,i_{n-1}^{(1)}})\cap V_{l_{2}}\cap
H_{i_{1}^{(2)},\dots,i_{n-1}^{(2)}}\neq\emptyset,$

for any $l_{1},l_{2}\in\\{1,3\dots,2(2k+1)^{n-1}\\}$ and
$i_{1}^{(1)},\dots,i_{n-1}^{(1)},i_{1}^{(2)},\dots,i_{n-1}^{(2)}\in\\{1,3,\dots,4k+1\\}$
(see Figure 2.3).

(a) $V_{i}$, $H_{i}$

(b) $\phi(E)$

(c) $\phi^{2}(E)$

Figure 2.3. $E$ is the first square, is a 2-dimensional 3-horseshoe for $\phi$

The assumptions $\phi(a,a,\dots,a,b)=(a,a,\dots,a,b)$ and
$\phi(b,b,\dots,b,a)=(b,b,\dots,b,a)$ is just to be able to extend $\phi$ to a
homeomorphism $\tilde{\phi}:E^{\prime}\rightarrow E^{\prime}$, where
$E^{\prime}$ is an $n$-cube with $E\subset E^{\prime}$, such that
$\tilde{\phi}|_{\partial E^{\prime}}\equiv Id$ and
$h_{\text{top}}(\phi^{\prime})=h_{\text{top}}(\phi|_{E\cap\phi(E)})$ (see
Lemma 2.2). Thus, our strategy to prove the Theorem 1.5 will be to use local
charts to glue such horseshoes along of periodic orbits of a homeomorphism.

######  Lemma 2.2.

Let $E,E^{\prime}\subseteq\mathbb{R}^{n}$ be closed $n$-cubes with
$E\subsetneqq(E^{\prime})^{\circ}$ and fix $k\in\mathbb{N}$. There exists a
homeomorphism $\phi:E^{\prime}\longrightarrow E^{\prime}$ such that
$\phi|_{E}:E\longrightarrow E$ is an $n$-dimensional $(2k+1)$-horseshoe,
$\phi|_{\partial E^{\prime}}\equiv Id$ and
$h_{\emph{top}}(\phi)=h_{\emph{top}}(\phi|_{E\cap\phi(E)})$.

Inspired by the results shown in [5] and [14] to obtain a homeomorphism
$\phi:N\rightarrow N$ with upper metric mean dimension equal to
$\text{dim}(N)$, we present the next lemma, which proves for any
$\alpha\in(0,n]$, there exists a homeomorphism
$\phi:\mathcal{\mathcal{C}}:=[0,1]^{n}\rightarrow\mathcal{\mathcal{C}}$, with
(upper and lower) metric mean dimension equal to $\alpha$.

On any $n$-cube $E\subseteq\mathbb{R}^{n}$, we will consider the metric
inherited from $\mathbb{R}^{n}$, $\|\cdot\|$, given by

$\|(x_{1},\dots,x_{n})-(y_{1},\dots,y_{n})\|=\sqrt{(x_{1}-y_{1})^{2}+\cdots+(x_{n}-y_{n})^{2}}.$

######  Lemma 2.3.

Let $\phi:{\mathcal{C}}\rightarrow{\mathcal{C}}$ be a homeomorphism,
$E_{k}=[a_{k},b_{k}]^{n}$ and $E_{k}^{\prime}$ sequences of cubes such that:

  * C1.

$E_{k}\subset(E_{k}^{\prime})^{\circ}$ and
$(E_{k}^{\prime})^{\circ}\cap(E_{s}^{\prime})^{\circ}=\emptyset$ for $k\neq
s$.

  * C2.

$S:=\cup_{k=1}^{\infty}E^{\prime}_{k}\subseteq{\mathcal{C}}$.

  * C3.

each $E_{k}$ is an $n$-dimensional $3^{k(n-1)}$-horseshoe for $\phi$;

  * C4.

For each $k$, $\phi|_{E_{k}^{\prime}}:E_{k}^{\prime}\rightarrow
E_{k}^{\prime}$ satisfies the properties in Lemma 2.2;

  * C5.

$\phi|_{{\mathcal{C}}\setminus S}:{\mathcal{C}}\setminus
S\rightarrow{\mathcal{C}}\setminus S$ is the identity.

We have:

  * (i)

For a fixed $r\in(0,\infty)$, if $|E_{k}|=\frac{B}{3^{kr}}$ for each
$k\in\mathbb{N}$, where $B>0$ is a constant, then

$\emph{mdim}_{\emph{M}}({\mathcal{C}},\|\cdot\|,\phi^{2})=\frac{n}{r+1}.$

  * (ii)

If $|E_{k}|=\frac{B}{k^{2}}$ for each $k\in\mathbb{N}$, where $B>0$ is a
constant, then

$\emph{mdim}_{\emph{M}}({\mathcal{C}},\|\cdot\|,\phi^{2})=n.$

###### Proof.

We will prove (i), since (ii) can be proved analogously. Set
$\varphi=\phi^{2}$. Note that

${{\text{mdim}}_{\text{M}}}({\mathcal{C}},\|\cdot\|,\varphi)={{\text{mdim}}_{\text{M}}}(S,\|\cdot\|,\varphi|_{S}).$

Take any $\varepsilon\in(0,1)$. For any $k\geq 1$, set
$\varepsilon_{k}=\frac{|E_{k}|}{2(3^{k})-1}=\frac{B}{(2(3^{k})-1)3^{kr}}$.
There exists $k\geq 1$ such that
$\varepsilon\in[\varepsilon_{k+1},\varepsilon_{k}]$. We have

$\text{sep}(n,\varphi,\varepsilon)\geq\text{sep}(n,\varphi,\varepsilon_{k})\geq\text{sep}(n,\varphi|_{E_{k}\cap\phi(E_{k})},\varepsilon_{k})\quad\text{for
any }n\geq 1.$

Since $E_{k}$ is an $n$-dimensional $3^{k(n-1)}$-horseshoe for $\phi$, for
each $k\geq 1$, consider

${H}^{k}_{i_{1},i_{2},\dots,i_{n-1}}=[a_{k},b_{k}]\times[t_{i_{1}-1},t_{i_{1}}]\times\cdots\times[t_{i_{n-1}-1},t_{i_{n-1}}],\quad\text{for
}i_{j}=1,\dots,2(3^{k})^{n-1}-1,$

and

$V_{l}^{k}=[s_{l-1},s_{l}]\times[a_{k},b_{k}]^{n-1},\quad\text{for
}l=1,\dots,2(3^{k})^{n-1}-1,$

as in Definition 2.1. For each $j=1,\dots,n-1$, let
$i_{j}\in\\{1,\dots,2(3^{k})^{n-1}-1\\}$ and take

$C^{k}_{l,i_{1},\dots,i_{n-1}}=H^{k}_{i_{1},i_{2},\dots,i_{n-1}}\cap
V^{k}_{l}\quad\text{for each }l\in\\{1,\dots,2(3^{k})^{n-1}-1\\}.$

For $t=1,\dots,m$, let $i_{1}^{(t)},$
$\dots,i_{n-1}^{(t)}\in\\{1,3,\dots,2(3^{k})-1\\}$,
$l_{j_{t}}\in\\{1,3,\dots,2(3^{k})^{n-1}-1\\}$ and set

$\displaystyle
C^{k}_{l_{j_{2}},i_{1}^{(2)},\dots,i_{n-1}^{(2)},l_{j_{1}},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}$
$\displaystyle=\varphi^{-1}\left[\varphi\left(C^{k}_{l_{j_{2}},i_{1}^{(2)},\dots,i_{n-1}^{(2)}}\right)\cap
C^{k}_{l_{j_{1}},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}\right]$
$\displaystyle\vdots$ $\displaystyle
C^{k}_{l_{j_{m}},i_{1}^{(m)},\dots,i_{n-1}^{(m)},\dots,l_{j_{1}},i_{1}^{(1)},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}$
$\displaystyle=\varphi^{-(m-1)}\left[\varphi^{m-1}\left(C^{k}_{l_{j_{m}},i_{1}^{(m)},\dots,i_{n-1}^{(m)},\dots,l_{j_{2}},i_{1}^{(2)},\dots,i_{n-1}^{(2)}}\right)\cap
C^{k}_{l_{j_{1}},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}\right]$

From the definition of $\varphi$, these sets are non-empty. Furthermore, we
can choose a sequence
$s_{1},\dots,s_{3^{k}}\in\\{1,3,\dots,2(3^{k})^{n-1}-1\\}$ such that, if $x$
and $y$ belong to different sets
$C^{k}_{s^{(m)},i_{1}^{(m)},\dots,i_{n-1}^{(m)},\dots,s_{(m)},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}$,
where $s^{(j)}\in\\{s_{1},\dots,s_{3^{k}}\\}$, then
$d_{m}(x,y)>\varepsilon_{k}$. We have $3^{km}3^{k(n-1)m}=3^{knm}$ sets of this
form. Hence,

$\text{sep}(m,\varphi,\varepsilon_{k})\geq 3^{knm}\quad\text{and
thus}\quad\text{sep}(\varphi,\varepsilon)\geq\log{3^{kn}}.$

Therefore,

$\displaystyle{\underline{\text{mdim}}_{\text{M}}}(S,\|\cdot\|,\varphi|_{S})$
$\displaystyle=\liminf_{\varepsilon\rightarrow
0}\frac{\text{sep}(\varphi,\varepsilon)}{|\log{\varepsilon}|}\geq\lim_{k\rightarrow\infty}\frac{\log
3^{kn}}{|\log\varepsilon_{k+1}|}$
$\displaystyle=\lim_{k\rightarrow\infty}\frac{\log
3^{kn}}{\log[(2(3^{k+1})-1)3^{(k+1)r}B^{-1}]}=\lim_{k\rightarrow\infty}\frac{\log
3^{kn}}{\log[3^{(k+1)+(k+1)r}]}$ $\displaystyle=\frac{n}{r+1}.$

On the other hand, note that $\frac{\log
3^{kn}}{\log[4(2(3^{k})-1)3^{kr}B^{-1}]}\rightarrow\frac{n}{1+r}$ as
$k\rightarrow\infty$. Hence, for any $\delta>0$, there exists $k_{0}\geq 1$,
such that, for any $k>k_{0}$, we have $\frac{\log
3^{kn}}{\log[4(2(3^{k})-1)3^{kr}B^{-1}]}<\frac{n}{1+r}+\delta$. Hence, suppose
that $\varepsilon>0$ is small enough such that $\varepsilon<\epsilon_{k_{0}}$.
Set $\tilde{\Omega}_{k}=\underset{n\in\mathbb{Z}}{\bigcap}\varphi^{n}(E_{k})$
and take $\tilde{\Omega}=\underset{k\in\mathbb{N}}{\bigcup}\Omega_{k}$. We
have

${{\text{mdim}}_{\text{M}}}({\mathcal{C}},\|\cdot\|,\varphi)={{\text{mdim}}_{\text{M}}}(\tilde{\Omega},\|\cdot\|,\varphi|_{\tilde{\Omega}}).$

If $x\in\tilde{\Omega}$, then $x$ belongs to some
$C^{k}_{l_{j_{m}},i_{1}^{(m)},\dots,i_{n-1}^{(m)},\dots,l_{j_{1}},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}$.
Take $s^{(j_{1})},\dots s^{(j_{m})}\in\\{s_{1},\dots,s_{3^{k}}\\}$ with

$|s^{(j_{i})}-l_{j_{i}}|=\underset{t=1,\dots,m}{\min}|s^{(t)}-l_{j_{i}}|,\quad\text{fot
each }i=1,\dots,m,$

and furthermore $d_{m}(x,y)\leq 4\varepsilon_{k}$ for any $y\in
C^{k}_{s^{(j_{m})},i_{1}^{(m)},\dots,i_{n-1}^{(m)},\dots,s^{(j_{1})},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}$.
Hence, if $Y_{k}=\cup_{j=1}^{k}E_{j}$, for every $m\geq 1$, we have

$\displaystyle\text{span}(m,\varphi|_{Y_{k}},4\varepsilon)$
$\displaystyle\leq\sum_{j=1}^{k}\frac{3^{jnm}}{\varepsilon}\leq
k\frac{3^{knm}}{\varepsilon}.$

Therefore,

$\displaystyle\frac{\text{span}(\varphi|_{Y_{k}},4\varepsilon)}{|\log
4\varepsilon|}$
$\displaystyle\leq\limsup_{m\rightarrow\infty}\frac{\log\left[k\frac{3^{knm}}{\varepsilon}\right]}{m|\log
4\varepsilon_{k}|}=\frac{\log
3^{kn}}{\log[4(2(3^{k})-1)3^{kr}B^{-1}]}<\frac{n}{1+r}+\delta.$

This fact implies that for any $\delta>0$ we have

$\displaystyle{\overline{\text{mdim}}_{\text{M}}}(S,\|\cdot\|,\varphi|_{S})<\frac{n}{r+1}+\delta\quad\text{and
hence
}\quad{\overline{\text{mdim}}_{\text{M}}}(S,\|\cdot\|,\varphi|_{S})\leq\frac{n}{r+1}.$

The above facts prove that
${{\text{mdim}}_{\text{M}}}(\mathcal{C},\|\cdot\|,\varphi)=\frac{n}{r+1}.$ ∎

Given $\alpha,\beta\in[0,n]$, with $\alpha\leq\beta$, in the next lemma we
construct an homeomorphism $\phi:[0,1]^{n}\rightarrow[0,1]^{n}$ such that

$\underline{\text{mdim}}_{\text{M}}([0,1]^{n},\|\cdot\|,\phi)=\alpha\quad\text{and}\quad\overline{\text{mdim}}_{\text{M}}([0,1]^{n},\|\cdot\|,\phi)=\beta.$

######  Lemma 2.4.

For every $\alpha,\beta\in[0,n]$, with $\alpha<\beta$, there exists a
homeomorphism $\phi_{\alpha,\beta}:[0,1]^{n}\rightarrow[0,1]^{n}$ such that

$\alpha=\underline{\emph{mdim}}_{\emph{M}}([0,1]^{n},\|\cdot\|,\phi_{\alpha,\beta})<\overline{\emph{mdim}}_{\emph{M}}([0,1]^{n},\|\cdot\|,\phi_{\alpha,\beta})=\beta.$

###### Proof.

We will prove the case $\beta<n$. First, we will show there exists a
homeomorphism $\phi_{0,\beta}:[0,1]^{n}\rightarrow[0,1]^{n}$ such that

$0=\underline{\text{mdim}}_{\text{M}}([0,1]^{n},\|\cdot\|,\phi_{0,\beta})<\overline{\text{mdim}}_{\text{M}}([0,1]^{n},\|\cdot\|,\phi_{0,\beta})=\beta.$

Fix $r>0$ and set $b=\frac{n}{r+1}$. Set $a_{0}=0$ and
$a_{n}=\sum_{i=0}^{n-1}\frac{C}{3^{ir}}$ for $n\geq 1$, where
$C=\frac{1}{\sum_{i=0}^{\infty}\frac{1}{3^{ir}}}=\frac{3^{r}-1}{3^{r}}$. Let
$E_{k}=[a_{k},b_{k}]^{n}$ and $E_{k}^{\prime}$ satisfying the conditions C1
and C2, C4 and C5 in Lemma 2.3, with $|E_{k}|=\frac{B}{3^{kr}}$, and such that
each $E_{k^{k}}$ is an $n$-dimensional $3^{nk^{k}}$-horseshoe for
$\phi_{0,\beta}$ and otherwise $\phi_{0,\beta}|_{E_{k}}$ is the identity on
$E_{k}$. We can prove
$\underline{\text{mdim}}_{\text{M}}([0,1]^{n},\|\cdot\|,\phi_{0,\beta})=0$ and
$\overline{\text{mdim}}_{\text{M}}([0,1]^{n},\|\cdot\|,\phi_{0,\beta})=\beta.$

Now consider $\alpha<\beta$. From Lemma 2.3(i), we have there exists
$\phi_{\alpha}\in\text{Hom}([0,1]^{n})$ such that

${\underline{\text{mdim}}_{\text{M}}}([0,1]^{n},\|\cdot\|,\phi_{\alpha})={\overline{\text{mdim}}_{\text{M}}}([0,1]^{n},\|\cdot\|,\phi_{\alpha})=\alpha.$

Set ${\phi}_{\alpha,\beta}\in\text{Hom}([0,1]^{n})$ be defined by

${\phi}_{\alpha,\beta}(x)=\begin{cases}T_{1}^{-1}\circ\phi_{0,\beta}\circ
T_{1}(x)&\text{if }x\in[0,\frac{1}{2}]^{n},\\\
T_{2}^{-1}\circ\phi_{\alpha}\circ T_{2}(x)&\text{if
}x\in[\frac{1}{2},1]^{n},\\\ x&\text{otherwise},\end{cases}$

where $T_{1}:[0,\frac{1}{2}]^{n}\rightarrow[0,1]^{n}$ and
$T_{2}:[\frac{1}{2},1]^{n}\rightarrow[0,1]^{n}$ are bi-lipchitsz maps. We have

$\displaystyle{\underline{\text{mdim}}_{\text{M}}}([0,1]^{n}$
$\displaystyle,\|\cdot\|,{\phi}_{\alpha,\beta})$
$\displaystyle=\max\\{{\underline{\text{mdim}}_{\text{M}}}([0,{1}/{2}]^{n},\|\cdot\|,{\phi}_{\alpha,\beta}|_{[0,\frac{1}{2}]^{n}}),{\underline{\text{mdim}}_{\text{M}}}([{1}/{2},1]^{n},\|\cdot\|,{\phi}_{\alpha,\beta}|_{[\frac{1}{2},1]^{n}})\\}$
$\displaystyle=\max\\{{\underline{\text{mdim}}_{\text{M}}}([0,1]^{n},\|\cdot\|,\phi_{0,\beta}),{\underline{\text{mdim}}_{\text{M}}}([0,1]^{n},\|\cdot\|,\phi_{\alpha})\\}$
$\displaystyle={\underline{\text{mdim}}_{\text{M}}}([0,1]^{n},\|\cdot\|,\phi_{\alpha})=\alpha.$

Analogously, we prove
${\overline{\text{mdim}}_{\text{M}}}([0,1]^{n},\|\cdot\|,\phi_{\alpha,\beta})={\overline{\text{mdim}}_{\text{M}}}([0,1]^{n},\|\cdot\|,\phi_{0,\beta})=\beta.$
∎

##  3\. Homeomorphisms on manifolds with positive metric mean dimension

Throughout this section, $N$ will denote an $n$ dimensional compact riemannian
manifold with $n\geq 2$ and $d$ a riemannian metric on $N$. On $\text{Hom}(N)$
we will consider the metric

$\hat{d}(\phi,\varphi)=\max_{p\in
N}\\{d(\phi(p),\varphi(p)),d(\phi^{-1}(p),\varphi^{-1}(p))\\}\quad\quad\text{
for any }\phi,\varphi\in\text{Hom}(N).$

It is well-known $(\text{Hom}(N),\hat{d})$ is a complete metric space. The
main goal of this section is to prove Theorem 1.5.

#### Good Charts

For each $p\in N$, consider the exponential map

$\text{exp}_{p}:B_{\delta^{\prime}}(0_{p})\subseteq T_{p}N\rightarrow
B_{\delta^{\prime}}(p)\subseteq N,$

where $0_{p}$ is the origin in the tangent space $T_{p}N$, $\delta^{\prime}$
is the injectivity radius of $N$ and $B_{\epsilon}(x)$ denote the open ball of
radius $\epsilon>0$ with center $x$. We will take
${\delta}_{N}=\frac{\delta^{\prime}}{2}$. The exponential map has the
following properties:

  * •

Since $N$ is compact, $\delta^{\prime}$ does not depends on $p$.

  * •

$\text{exp}_{p}(0_{p})=p$ and
$\text{exp}_{p}[B_{\delta_{N}}(0_{p})]=B_{\delta_{N}}({p})$;

  * •

$\text{exp}_{p}:B_{\delta_{N}}(0_{p})\rightarrow B_{\delta_{N}}({p})$ is a
diffeomorphism;

  * •

If $v\in B_{\delta_{N}}(0_{p})$, taking $q=\text{exp}_{p}(v)$ we have
$d(p,q)=\|v\|$.

  * •

The derivative of $\text{exp}_{p}$ at the origin is the identity map:

$D(\text{exp}_{p})(0)=\text{id}:T_{p}N\rightarrow T_{p}N.$

Since $\text{exp}_{p}:B_{\delta_{N}}(0_{p})\rightarrow B_{\delta_{N}}({p})$ is
a diffeomorphism and $D(\text{exp}_{p})(0)=\text{id}:T_{p}N\rightarrow
T_{p}N,$ we have $\text{exp}_{p}:B_{\delta_{N}}(0_{p})\rightarrow
B_{\delta_{N}}({p})$ is a bi-Lipschitz map with Lipschitz constant close to 1.
Therefore, we can assume that if $v_{1},v_{2}\in B_{\delta_{N}}(0_{p})$,
taking $q_{1}=\text{exp}_{p}(v_{1})$ and $q_{2}=\text{exp}_{p}(v_{2})$, we
have $d(q_{1},q_{2})=\|v_{1}-v_{2}\|$. Furthermore, we will identify
$B_{\delta_{N}}(0_{p})\subset T_{p}N$ with
$B_{\delta_{N}}(0)=\\{x\in\mathbb{R}^{n}:\|x\|<\delta_{N}\\}\subseteq\mathbb{R}^{n}$.

Recall that if $\alpha=\beta\in[0,n]$, then

${H_{\alpha}^{\alpha}}(N):={H}_{\alpha}(N)=\\{\phi\in\text{Hom}(N):\underline{\text{mdim}}_{\text{M}}(N,d,\phi)=\overline{\text{mdim}}_{\text{M}}(N,d,\phi)=\alpha\\}.$

######  Theorem 3.1.

For any $\alpha\in[0,n]$, the set $H_{\alpha}(N)$ is dense in $\emph{Hom}(N)$.

###### Proof.

Let $P^{r}(N)$ be the set of $C^{r}$-diffeomorphisms on $N$ with a periodic
point. This set is $C^{0}$-dense in $\text{Hom}(N)$ (see [4], [7]). Hence, in
order to prove the theorem, it is sufficient to show that $H_{\alpha}(N)$ is
dense in $P^{r}(N)$. Fix $\psi\in P^{r}(N)$ and take any
$\varepsilon\in(0,\delta_{N})$. Suppose that $p\in N$ is a periodic point of
$\psi$, with period $k$. Let $\beta>0$, small enough, such that
$[-\beta,\beta]^{n}\subset B_{\frac{\varepsilon}{4}}(0)$. For each
$i=0,\dots,k-1$,
$N\setminus\psi(\exp_{\psi^{i}(p)}\partial[-\beta,\beta]^{n})$ has two
connected components. We denote by $C_{i}$ the connected component that
contains $\psi^{i}(p)$. Consider the positive number

$\gamma=\min_{i=0,\dots,k-1}d(\psi^{i+1}(p),\psi(\exp_{\psi^{i}(p)}\partial[-\beta,\beta]^{n}))>0.$

Let $\lambda\in(0,\min\\{\gamma/2,\beta\\}),$ such that

$\exp_{\psi^{i+1}(p)}[-\lambda,\lambda]^{n}\subset C_{i+1}\quad\text{for each
}i=0,\dots,k-1.$

The regions $[-\beta,\beta]^{n}\setminus(-\lambda,\lambda)^{n}$ and
$C_{i+1}\setminus\exp_{\psi^{i}(p)}(-\lambda,\lambda)^{n}$ are homeomorphic.
For each $i=0,\dots,k-1,$ take

$\displaystyle h_{i}:\partial[-\beta,\beta]^{n}$
$\displaystyle\to\exp^{-1}_{\psi^{i+1}(p)}\left(\partial{C}_{i+1}\right)$
$\displaystyle u$
$\displaystyle\mapsto\exp^{-1}_{\psi^{i+1}(p)}\circ\psi\circ\exp_{\psi^{i}(p)}(u).$

Then, there exists a homeomorphism

$H_{i}:[-\beta,\beta]^{n}\setminus(-\lambda,\lambda)^{n}\to\exp^{-1}_{\psi^{i+1}(p)}[C_{{i+1}}\setminus\exp_{\psi^{i+1}(p)}(-\lambda,\lambda)^{n}],$

such that

$H_{i}|_{\partial[-\lambda,\lambda]^{n}}=Id\ \ \text{and}\ \
H_{i}|_{\partial[-\beta,\beta]^{n}}=h_{i}.$

If $q\in
R_{i}=\exp_{\psi^{i}(p)}\left([-\beta,\beta]^{n}\setminus(-\lambda,\lambda)^{n}\right)$,
we can write

$q=\exp_{\psi^{i}(p)}(z),\quad\text{for some
}z\in[-\beta,\beta]^{n}\setminus(-\lambda,\lambda)^{n}.$

Set $\varphi_{\alpha}:N\rightarrow N$, given by

$\varphi_{\alpha}(q)=\begin{cases}\exp_{\psi^{i+1}(p)}H_{i}(z),&\text{if }q\in
R_{i}=\exp_{\psi^{i}(p)}\left([-\beta,\beta]^{n}\setminus(-\lambda,\lambda)^{n}\right)\\\
\exp_{\psi^{i+1}(p)}\phi_{\alpha}(\exp_{\psi^{i}(p)}^{-1}(q)),&\text{if }q\in
D_{i}:=\exp_{\psi^{i}(p)}\left([-\lambda,\lambda]^{n}\right)\\\
\psi(q),&\text{otherwise},\end{cases}$

where $\phi_{\alpha}\colon[-\lambda,\lambda]^{n}\to[-\lambda,\lambda]^{n}$ is
a homeomorphism which satisfies
$\phi_{\alpha}|_{\partial[-\lambda,\lambda]^{n}}=Id$ and

${{\text{mdim}}_{\text{M}}}([-\lambda,\lambda]^{n},\|\cdot\|,\phi_{\alpha})=\alpha,$

(see Lemma 2.3). Set $K:=\cup_{i=0}^{k-1}D_{i}$. Note that $N\setminus K$ is
$\varphi_{\alpha}$ invariant and

$\text{mdim}_{\text{M}}(N\setminus K,d,\varphi_{\alpha}|_{N\setminus K})=0.$

Hence,

$\text{mdim}_{\text{M}}(N,d,\varphi_{\alpha})=\text{mdim}_{\text{M}}(K,d,\varphi_{\alpha}|_{K}).$

Note if $q\in D_{i}$, we have

$(\varphi_{\alpha})^{s}(q)=\text{exp}_{\psi^{(i+s)\text{ mod
}k}(p)}\circ(\phi_{\alpha})^{s}\circ\text{exp}^{-1}_{\psi^{i}(p)}(q)\quad\text{and}\quad(\varphi_{\alpha})^{k}(q)=\text{exp}_{\psi^{i}(p)}\circ(\phi_{\alpha})^{k}\circ\text{exp}^{-1}_{\psi^{i}(p)}(q).$

Hence, $D\subseteq D_{i}$ is an $(s,\phi_{\alpha},\epsilon)$-separated set if
and only if $\text{exp}_{\psi^{i}(p)}(D)\subseteq N$ is an
$(s,\varphi_{\alpha},\epsilon)$-separated set for any $\epsilon>0$. Therefore,
$\text{sep}(s,\varphi_{\alpha}|_{K},\epsilon)=k\,\text{sep}(s,\phi_{\alpha},\epsilon).$
Therefore,

$\text{mdim}_{\text{M}}(K,d,\varphi_{\alpha}|_{K})=\text{mdim}_{\text{M}}([-\lambda,\lambda]^{n},\|\cdot\|,\phi_{\alpha}),$

which proves the theorem. ∎

The last theorem proved Theorem 1.5 in the case $\alpha=\beta=n$. The proof of
the general case will be a consequence of the above arguments, in fact:

######  _Proof of the Theorem 1.5_.

The proof follows similarly to the proof of Theorem 3.1, changing
$\phi_{\alpha}$ by a continuous map
$\phi_{\alpha,\beta}\colon[-\lambda,\lambda]^{n}\to[-\lambda,\lambda]^{n}$
such that

${\underline{\text{mdim}}_{\text{M}}}([-\lambda,\lambda]^{n},\|\cdot\|,{\phi}_{\alpha,\beta})=\alpha\quad\text{and}\quad{\overline{\text{mdim}}_{\text{M}}}([-\lambda,\lambda]^{n},\|\cdot\|,\phi_{\alpha,\beta})=\beta,$

as in the Lemma 2.4. ∎

##  4\. Tipical homeomorphism has maximal metric mean dimension

To complete this work, in this section we show Theorem 1.6, which was proved
in Theorem A in [5], however, we will present an alternative proof of this
fact using the techniques of Section 2 and Section 3.

######  Definition 4.1 ($n$-dimensional strong horseshoe).

Let $E=[a,b]^{n}$ and set $|E|=b-a$. For a fixed natural number $k>1$, set
$\delta_{k}=\frac{b-a}{4k+1}$ and $\epsilon_{k}=\frac{b-a}{2(2k+1)^{n-1}-1}$.
For $i=0,1,2,\dots,4k+1$, set $t_{i}=a+i\delta_{k}$ and consider

$H_{i_{1},i_{2},\dots,i_{n-1}}=[a,b]\times[t_{i_{1}-1},t_{i_{1}}]\times\cdots\times[t_{i_{n-1}-1},t_{i_{n-1}}],$

for $i_{j}\in\\{1,\dots,4k+1\\}.$ For $l=0,1,2,\dots,2(2k+1)^{n-1}-2$, set
$s_{l}=a+l\epsilon_{k}$ and consider

$V_{l+1}=[s_{l},s_{l+1}]\times[a,b]^{n-1}.$

We say that $E\subseteq A\subseteq\mathbb{R}^{n}$ is an $n$-dimensional strong
$(\epsilon,(2k+1)^{n-1})$-horseshoe for a homeomorphism $\phi:A\rightarrow A$
if $|E|>\epsilon$ and furthermore:

  * •

For any $H_{i_{1},i_{2},\dots,i_{n-1}}$, with $i_{j}\in\\{1,3,\dots,4k+1\\}$,
there exists $l\in\\{1,3,\dots,2(2k+1)^{n-1}-1\\}$ with

$H_{i_{1},i_{2},\dots,i_{n-1}}\subseteq\phi(V_{l})^{\circ}.$

  * •

For any $l=2,4,\dots,2(2k+1)^{n-1}-2$, $\phi(V_{l})\subseteq A\setminus E$.

######  Remark 4.2.

Note if $\epsilon^{\prime}>\epsilon$, then any $n$-dimensional strong
$(\epsilon^{\prime},(2k+1)^{n-1})$-horseshoe for $\phi$ is an $n$-dimensional
strong $(\epsilon,(2k+1)^{n-1})$-horseshoe for $\phi$.

Using local charts, the last definition definition can be done on the manifold
$N$.

######  Definition 4.3.

Let $N$ be an $n$-dimensional riemannian manifold and fix $k\geq 1$. We say
that $\phi\in\text{Hom}(N)$ has an $n$-dimensional strong
$(\epsilon,(2k+1)^{n-1})$-horseshoe $E=[a,b]^{n}$, if there is $s$ and an
exponential charts $\text{exp}_{i}:B_{\delta_{N}}(0)\rightarrow N$, for
$i=1,\dots,s$, such that:

  * •

$\phi_{i}=\text{exp}_{(i+1)\text{mod}\,s}\circ\phi\circ\text{exp}^{-1}_{i}:B_{\delta}(0)\rightarrow
B_{\delta_{N}}(0)$ is well defined for some $\delta\leq\delta_{N}$;

  * •

$E$ is an $n$-dimensional strong $(\epsilon,(2k+1)^{n-1})$-horseshoe for
$\phi_{i}$.

To simplify the notation, we will set $\phi_{i}=\phi$ for each $i=1,\dots,s$.

For $\epsilon>0$ and $k\in\mathbb{N}$, we consider the sets

$\displaystyle H(\epsilon,k)=\\{\phi^{2}\in\text{Hom}(N):\phi\text{ has a
strong }(\epsilon,k)\text{-horseshoe}\\}$ $\displaystyle
H(k)=\bigcup_{i\in\mathbb{N}}H\left(\frac{1}{i^{2}},3^{ki(n-1)}\right)$
$\displaystyle\mathcal{H}=\overset{\infty}{\underset{k=1}{\bigcap}}H(k).$

######  Lemma 4.4.

The set $\mathcal{H}$ is residual.

###### Proof.

Clearly, for any $\epsilon\in(0,\delta_{N})$ and $k\in\mathbb{N}$, the set
$H(\epsilon,k)$ is open and nonempty.
We claim that the set $H(k)$ is dense in $\text{Hom}(N)$. In fact: fix
$\psi\in P^{r}(N)$ with a $s$-periodic point. In the same way as the proof of
the Theorem 3.1, every small neighborhood of the orbit of this point can be
perturbed in order to obtain a strong $\left(\frac{1}{i^{2}},3^{nki}\right)$
horseshoe for a $\phi$ such that $\phi^{2}$ be close to $\psi$ for a large
enough $i$. Thus $H(k)$ is a dense set, and _a fortiori_
$\mathcal{H}=\overset{\infty}{\underset{k=1}{\bigcap}}H(k)$ is residual in
$\text{Hom}(N)$. ∎

Finally, we prove Theorem 1.6.

######  _Proof of the Theorem 1.6_.

It is sufficient to prove that $\overline{\text{mdim}}_{\text{M}}(N,d,\phi)=n$
for any $\phi\in\mathcal{H}$. For this sake, take
$\varphi=\phi^{2}\in\mathcal{H}$. We have $\varphi\in H(k)$ for any $k\geq 1$.
Therefore, for any $k\in\mathbb{N}$, there exists $i_{k}$, with
$i_{k}<i_{k+1}$, such that $\phi$ has a strong
$\left(\frac{1}{i_{k}^{2}},3^{n\,k\,i_{k}}\right)$-horseshoe
$E_{{k}}=[a_{k},b_{k}]^{n}$, such that $|E_{{k}}|>\frac{1}{i_{k}^{2}}$,
consisting of rectangles

$H^{i_{k}}_{j_{1},j_{2},\dots,j_{n-1}}=[a_{k},b_{k}]\times[t^{{k}}_{j_{1}-1},t^{{k}}_{j_{1}}]\times\cdots\times[t^{{k}}_{j_{n-1}-1},t^{{k}}_{j_{n-1}}],\quad\text{with}\quad
j_{t}\in\\{1,3,\dots,2(3^{k\,i_{k}})-1\\},$

and

$V^{{k}}_{l}=[s^{k}_{l-1},s^{k}_{l}]\times[a_{k},b_{k}]^{n-1},\quad\text{ for
}l\in\\{1,3,\dots,2({3^{ki_{k}(n-1)}})-1\\},$

with

$H^{{k}}_{j_{1},j_{2},\dots,j_{n-1}}\subseteq\phi(V^{{k}}_{l})^{\circ}\quad\text{for
some }l.$

For $i_{j}\in\\{1,\dots,2(3^{ki_{k}})-1\\}$ and
$l=1,\dots,2(3^{ki_{k}})^{n-1}-1,$ take

$E^{k}_{l,i_{1},\dots,i_{n-1}}=[s^{k}_{l-1},s^{k}_{l}]\times[t^{k}_{i_{1}-1},t^{k}_{i_{1}}]\times\cdots\times[t^{k}_{i_{n-1}-1},t^{k}_{i_{n-1}}].$

For $t=1,\dots,m$, let $i_{1}^{(t)},$
$\dots,i_{n-1}^{(t)}\in\\{1,3,\dots,2(3^{ki_{k}})-1\\}$ and for
$l_{t}\in\\{1,\dots,2(3^{ki_{k}})^{n-1}-1\\},$ set

$\displaystyle C^{k}_{l_{1},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}$
$\displaystyle=E^{k}_{l_{1},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}$ $\displaystyle
C^{k}_{l_{2},i_{1}^{(2)},\dots,i_{n-1}^{(2)},l_{1},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}$
$\displaystyle=\varphi^{-1}\left[\varphi\left(C^{k}_{l_{2},i_{1}^{2},\dots,i_{n-1}^{2}}\right)\cap
E^{k}_{l_{1},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}\right]$ $\displaystyle\vdots$
$\displaystyle
C^{k}_{l_{m},i_{1}^{(m)},\dots,i_{n-1}^{(m)},\dots,l_{1},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}$
$\displaystyle=\varphi^{-(m-1)}\left[\varphi^{m-1}\left(C^{k}_{l_{m},i_{1}^{(m)},\dots,i_{n-1}^{(m)},\dots,l_{2},i_{1}^{(2)},\dots,i_{n-1}^{(2)}}\right)\cap
E^{k}_{l_{1},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}\right]$

Furthermore, set

$\tilde{C}^{k}_{l_{m},i_{1}^{(m)},\dots,i_{n-1}^{(m)},\dots,l_{1},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}:=\text{exp}\left[C^{k}_{l_{m},i_{1}^{(m)},\dots,i_{n-1}^{(m)},\dots,l_{1},i_{1}^{(1)},\dots,i_{n-1}^{(1)}}\right].$

For any $k\geq 1$, take
$\varepsilon_{k}=\frac{1}{4i_{k}^{2}(2(3^{k\,i_{k}})-1)}$. We can choose at
least $3^{ki_{k}}$ sub-indices $p_{1},\dots,p_{3^{ki_{k}}}$ in
$\\{1,\dots,2(3^{ki_{k}})^{n-1}-1\\}$ such that if
$x\in\tilde{C}_{p_{s},i_{1},\dots,i_{n-1}}$ and
$y\in\tilde{C}_{p_{t},i_{1},\dots,i_{n-1}}$, with $s\neq t$, then
$d_{1}(x,y)>\varepsilon_{k}$. Hence, if $x$ and $y$ belong to different sets
$\tilde{C}^{k}_{p^{(m)},i_{1}^{(m)},\dots,i_{n-1}^{(m)},\dots,p^{(1)},i_{1}^{(1)},\dots,i_{n-1}^{(1)}},$
with $p^{(t)}\in\\{p_{1},\dots,p_{3^{ki_{k}}}\\}$ and $i_{1}^{(t)},$
$\dots,i_{n-1}^{(t)}\in\\{1,3,\dots,2(3^{ki_{k}})-1\\}$, we have that
$d_{m}(x,y)>\varepsilon_{k}$. Hence,

$\text{sep}(m,\varphi,\varepsilon_{k})\geq 3^{ki_{k}nm}\quad\text{and
thus}\quad\text{sep}(\varphi,\varepsilon_{k})\geq\log{3^{nki_{k}}}.$

Therefore,

$\displaystyle{\overline{\text{mdim}}_{\text{M}}}(N,d,\varphi)$
$\displaystyle=\limsup_{k\rightarrow\infty}\frac{\text{sep}(\varphi,\varepsilon_{k})}{|\log{\varepsilon_{k}}|}\geq\lim_{k\rightarrow\infty}\frac{\log
3^{nki_{k}}}{|\log\varepsilon_{k}|}=\limsup_{k\rightarrow\infty}\frac{\log
3^{nki_{k}}}{\log[4i_{k}^{2}(2(3^{ki_{k}})-1)]}=n.$

This fact proves the theorem, since for any $\psi\in\text{Hom}(N)$, the
inequality $\overline{\text{mdim}}_{\text{M}}(N,d,\phi)\leq n$ always holds
(see [14], Remark 4). ∎

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