# Genericity of homeomorphisms with full mean Hausdorff dimension

Jeovanny M. Acevedo  Jeovanny de Jesus Muentes Acevedo, Facultad de Ciencias
Básicas, Universidad Tecnológica de Bolívar, Cartagena de Indias - Colombia
<EMAIL_ADDRESS>
###### Abstract.

It is well known that the presence of horseshoes leads to positive entropy. If
our goal is to construct a continuous map with infinite entropy, we can
consider an infinite sequence of horseshoes, ensuring an unbounded number of
legs.

Estimating the exact values of both the metric mean dimension and mean
Hausdorff dimension for a homeomorphism is a challenging task. We need to
establish a precise relationship between the sizes of the horseshoes and the
number of appropriated legs to control both quantities.

Let $N$ be an $n$-dimensional compact Riemannian manifold, where $n\geq 2$,
and $\alpha\in[0,n]$. In this paper, we construct a homeomorphism
$\phi:N\rightarrow N$ with mean Hausdorff dimension equal to $\alpha$.
Furthermore, we prove that the set of homeomorphisms on $N$ with both lower
and upper mean Hausdorff dimensions equal to $\alpha$ is dense in
$\text{Hom}(N)$. Additionally, we establish that the set of homeomorphisms
with upper mean Hausdorff dimension equal to $n$ contains a residual subset of
$\text{Hom}(N).$

##  1\. Introduction

Throughout this work, $N$ will be denote a compact topological space with a
fixed metric $d$ and $\phi:N\rightarrow N$ a continuous map. An useful tool to
classify dynamical systems is the topological entropy, introduced by R. L.
Adler, A. G. Konheim and M. H. McAndrew in 1965: if two dynamical systems are
topologically conjugated, then they have the same topological entropy.
However, in (1980), Koichi Yano proved that if $N$ is a compact manifold with
$\text{dim}(N)\geq 2$, then the set $\mathcal{A}$ consisting of homeomorphisms
$\phi:N\rightarrow N$ with infinite topological entropy is residual in
$\text{Hom}(N)$ (see [26]). Hence, topological entropy fails to differentiate
dynamical systems in $\mathcal{A}$.

In 1999, M. Gromov in [11] introduced the notion of mean topological dimension
for continuous maps, which is an invariant under topological conjugacy. If one
dynamical system has positive mean topological dimension, then it has infinite
topological entropy. Therefore, mean topological dimension is an useful tool
to classify dynamical systems with infinite topological entropy. For
instances, the studies presented in [17], [19], [13], [22] and [12] illustrate
that minimal systems with a mean topological dimension less than $\frac{n}{2}$
can be effectively embedded into the left-shift map on
$([0,1]^{n})^{\mathbb{Z}}$. Some applications in information theory can be
found in [16] and [15]. We will denote by $\text{mdim}(N,\phi)$ the mean
topological dimension of $\phi$.

Mean topological dimension is difficult to calculate. Therefore, Lindenstrauss
and Weiss in [17] introduced the notion of metric mean dimension. We will
denote by $\underline{\text{mdim}}_{\text{M}}(N,d,\phi)$ the lower metric mean
dimension of $\phi$ and $\overline{\text{mdim}}_{\text{M}}(N,d,\phi)$ the
upper metric mean dimension of $\phi$. Metric mean dimension depends on the
metric $d$, therefore it is not an invariant under topological conjugacy.
Furthermore, it is zero for any map with finite topological entropy. In [17],
Theorem 4.2, is proved that it is an upper bound for the mean topological
dimension, that is, we have that

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

It is conjectured that, for any dynamical system $\phi:N\rightarrow N$, there
exists a metric $d$ on $N$ such that
$\text{mdim}(N,\phi)=\overline{\text{mdim}}_{\text{M}}(N,d,\phi)$.

The notion of metric mean dimension has been extensively studied, as we can
see in the works [5], [6], [21], [25], among other works. Recently,
Lindenstrauss and Tsukamoto, in [18], defined a new tool which provides a
better upper bound for the mean topological dimension: the mean Hausdorff
dimension. Hence, it is very likely that this will become the subject of
further studies. Mean Hausdorff dimension is not a topological invariant,
because it also depends on the metric $d$. We will denote by
$\underline{\text{mdim}}_{\text{H}}(N,d,\phi)$ and
$\overline{\text{mdim}}_{\text{H}}(N,d,\phi)$ the upper and lower mean
Hausdorff dimension of $\phi$, respectively.

In [18], Proposition 3.2 (see also [23], Proposition 2.1), was proved that

$\text{mdim}(N,\phi)\leq\overline{\text{mdim}}_{\text{H}}(N,d,\phi)\leq\underline{\text{mdim}}_{\text{M}}(N,d,\phi).$
(1.1)

For mean topological dimension and metric mean dimension we have,
respectively, that

${\text{mdim}}(N^{\mathbb{Z}},\sigma)\leq\text{dim}(N)\quad\text{and}\quad\underline{\text{mdim}}_{\text{M}}(N^{\mathbb{Z}},\text{{d}},\sigma)=\text{dim}_{\text{B}}(N,d),$

where $\text{dim}(N)$ is the topological dimension of $N$ and
$\text{dim}_{\text{B}}(N,d)$ is the box dimension of $N$,
$\sigma:N^{\mathbb{Z}}\rightarrow N^{\mathbb{Z}}$ is the left shift map and
${\textbf{d}}(\bar{x},\bar{y})=\underset{j\in\mathbb{Z}}{\sum}\frac{1}{2^{j}}d(x_{k},y_{k})$
for all $\bar{x}=(x_{k}),\bar{y}=(y_{k})\in N^{\mathbb{Z}}$ (see [17], Theorem
3.1, and [24], Theorem 5). In [22], Theorem 1.1 presents the exact value of
${\text{mdim}}(N^{\mathbb{Z}},\sigma)$ depending on the type of space $N$.
Recently, in [3], Theorem 3.5, is proved that

$\text{dim}_{\text{H}}(N,d)\leq\underline{\text{mdim}}_{\text{H}}(N^{\mathbb{Z}},\text{{d}},\sigma),$

where $\text{dim}_{\text{H}}(N,d)$ is the Hausdorff dimension of $N$. Several
interesting results, analog to Furstenberg’s theorem (see [10], Proposition
III.1), which relate the metric mean dimension and mean Hausdorff dimension of
sub-shifts with the topological entropy of $\mathbb{N}^{2}$-actions, are
presented in [23].

In [20], the concepts of mean packing dimension and mean pseudo-packing
dimension for dynamical systems are introduced. The authors provide some basic
properties of these notions and formulas applicable to Cartesian products of
dynamical systems. However, the mean Hausdorff dimension remains a more
accurate approximation of the mean topological dimension (the mean Hausdorff
dimension of a dynamical system is lower than its mean packing dimension and
its mean pseudo-packing dimension).

Obtaining a homeomorphism with mean Hausdorff dimension or metric mean
dimension equal to a fixed $\alpha$ poses a significant challenge and is not a
straightforward task. It requires establishing a precise relationship between
the sizes of horseshoes and the corresponding number of appropriated legs to
effectively control both quantities. Suppose that $N$ is an $n$-dimensional
compact Riemannian manifold and fix $\alpha\in[0,n]$. In [2], we construct
homeomorphisms defined on $N$ with metric mean dimension equal to $\alpha$. In
this paper we construct homeomorphisms defined on $N$ with mean Hausdorff
dimension equal to $\alpha$. Furthermore, we will prove that:

  * •

The set consisting of continuous maps on $[0,1]$ with upper and lower mean
Hausdorff dimension equal to $\alpha\in[0,1]$ is dense in $C^{0}([0,1])$ (see
Theorem 3.1).

  * •

The set consisting of continuous maps on the interval $[0,1]$ with upper mean
Hausdorff dimension equal to $1$ is residual in $C^{0}([0,1])$ (see Theorem
3.4).

  * •

The set consisting of homeomorphisms on $N$, with upper and lower mean
Hausdorff dimension equal to $\alpha\in[0,n]$ is dense in $\text{Hom}(N)$ (see
Theorem 4.4).

  * •

The set consisting of homeomorphisms on $N$ with upper mean Hausdorff
dimension equal to $n$ contains a residual subset of $\text{Hom}(N)$ (see
Theorem 5.4).

##  2\. Continuous maps on the interval with positive mean Hausdorff dimension

In this section we will construct continuous maps on the interval with mean
Hausdorff dimension equal to a fixed value $\alpha\in[0,1]$. We will recall
the definition of mean Hausdorff dimension. For a continuous map $\phi:N\to N$
and any integer $n\geq 1$ we define the metric

$d_{n}(x,y)=\max\left\\{d(x,y),d(\phi(x),\phi(y)),\dots,d(\phi^{n-1}(x),\phi^{n-1}(y))\right\\},\quad\text{for
}x,y\in N.$ (2.1)

For any $n\in\mathbb{N}$ and $A\subset N$, set $\text{diam}_{d_{n}}(A)$ the
diameter of $A$ with the metric $d_{n}$. For $s\geq 0$ and $\varepsilon>0$,
set

$\text{H}_{\varepsilon}^{s}(N,d_{n},\phi)=\inf\left\\{\sum_{i=1}^{\infty}(\text{diam}_{d_{n}}E_{i})^{s}:N=\bigcup_{i=1}^{\infty}E_{i}\text{
with }\text{diam}_{d_{n}}E_{i}\leq\varepsilon\text{ for all }i\geq
1\right\\}.$

By convention we consider $0^{0}=1$ and
$\text{diam}_{d_{n}}(\emptyset)^{s}=0$. When $N$ is compact, we have

$\text{H}_{\varepsilon}^{s}(N,d_{n},\phi)=\inf\left\\{\sum_{i=1}^{k}(\text{diam}_{d_{n}}E_{i})^{s}:N=\bigcup_{i=1}^{k}E_{i}\text{
with }\text{diam}_{d_{n}}E_{i}\leq\varepsilon\text{ for all }i=1,\dots
k\right\\}$

(see [9], Section 2.4). Take

$\text{dim}_{\text{H}}(N,d_{n},\phi,\varepsilon)=\sup\\{s\geq
0:\text{H}_{\varepsilon}^{s}(N,d_{n},\phi)\geq 1\\}.$

######  Definition 2.1.

The upper mean Hausdorff dimension and lower mean Hausdorff dimension of
$(N,d,\phi)$ are defined respectively as

$\overline{\text{mdim}}_{\text{H}}(N,d,\phi)=\lim_{\varepsilon\rightarrow
0}\left(\limsup_{n\rightarrow\infty}\frac{1}{n}\text{dim}_{\text{H}}(N,d_{n},\phi,\varepsilon)\right),$
$\underline{\text{mdim}}_{\text{H}}(N,d,\phi)=\lim_{\varepsilon\rightarrow
0}\left(\liminf_{n\rightarrow\infty}\frac{1}{n}\text{dim}_{\text{H}}(N,d_{n},\phi,\varepsilon)\right).$

For a fixed $\varepsilon>0$, a subset $A\subset N$ is
$(n,\phi,\varepsilon)$-separated if $d_{n}(x,y)>\varepsilon$, for any two
distinct points $x,y\in A$. Let $\text{sep}(n,\phi,\varepsilon)$ be the
maximal cardinality of any $(n,\phi,\varepsilon)$-separated subset of $N$ and
set

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

The _lower metric mean dimension_ and the _upper metric mean dimension_ of
$(N,d,\phi)$ are defined, respectively, by

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

######  Remark 2.2.

Throughout this work, we will write $\text{mdim}_{\text{M}}(N,d,\phi)$ to
refer to both notions
$\underline{\text{mdim}}_{\text{M}}(N,d,\phi)$ or
$\overline{\text{mdim}}_{\text{M}}(N,d,\phi)$ and, we will write
$\text{mdim}_{\text{H}}(N,d,\phi)$ to refer to both notions
$\underline{\text{mdim}}_{\text{H}}(N,d,\phi)$ or
$\overline{\text{mdim}}_{\text{H}}(N,d,\phi)$.

Next, we present some basic properties of the mean Hausdorff dimension.

  * •

For any $p\in\mathbb{N}$ we have that (see [3], Proposition 3.1)

${\text{mdim}_{\text{H}}}(N,d,\phi^{p})\leq
p\,{\text{mdim}_{\text{H}}}(N,d,\phi).$

  * •

For any two continuous maps $\phi:(N,d)\rightarrow(N,d)$ and
$\psi:(M,d^{\prime})\rightarrow(M,d^{\prime})$ we have that (see [3],
Proposition 3.4)

$\underline{\text{mdim}}_{\text{H}}(N\times M,d\times
d^{\prime},\phi\times\psi)\geq\underline{\text{mdim}}_{\text{H}}(N,d,\phi)+\underline{\text{mdim}}_{\text{H}}(M,d^{\prime},\psi).$

  * •

If $\mathbb{K}$ is $\mathbb{Z}$ or $\mathbb{N}$,
$\sigma:N^{\mathbb{K}}\rightarrow N^{\mathbb{K}}$ is the left shift map and
${\textbf{d}}(\bar{x},\bar{y})=\underset{j\in\mathbb{K}}{\sum}\frac{1}{2^{j}}d(x_{k},y_{k})$
for all $\bar{x}=(x_{k}),\bar{y}=(y_{k})\in N^{\mathbb{K}}$, we have that (see
[3], Theorem 3.5)

$\text{dim}_{\text{H}}(N,d)\leq\underline{\text{mdim}}_{\text{H}}(N^{\mathbb{K}},\text{{d}},\sigma).$

An $s$-horseshoe for a continuous map $\phi:[0,1]\rightarrow[0,1]$ is an
interval $J\subseteq[0,1]$ which has a partition into $s$ subintervals
$J_{1},\dots,J_{s}$ such that $J_{j}^{\circ}\cap J_{i}^{\circ}=\emptyset$ for
$i\neq j$ and $J\subseteq\phi(\overline{J}_{i})$ for each $i=1,\dots,s$. In
[24], Lemma 6, is proved if $I_{k}=[a_{k-1},a_{k}]\subseteq[0,1]$ is an
$s_{k}$-horseshoe for a continuous map $\phi:[0,1]\rightarrow[0,1]$ consisting
of $s_{k}$ subintervals with the same length $I_{k}^{1},\dots,I_{k}^{s_{k}}$,
then

$\text{sep}(\phi,\varepsilon_{k})\geq\log(s_{k}/2)\quad\text{where
}\varepsilon_{k}=\frac{|I_{k}|}{s_{k}}.$

It follows from this fact that

$\overline{\text{mdim}}_{\text{M}}([0,1],|\cdot|,\phi)\geq\underset{k\rightarrow\infty}{\limsup}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|},$ (2.2)

where $|\cdot|$ is the metric induced from the absolute value (see [24],
Proposition 8). Next, we will present some analogous results for mean
Hausdorff dimension.

######  Lemma 2.3.

Suppose for each $k\in\mathbb{N}$ there exists a $s_{k}$-horseshoe for
$\phi\in C^{0}([0,1])$, $I_{k}=[a_{k-1},a_{k}]\subseteq[0,1]$, consisting of
sub-intervals $I_{k}^{1},I_{k}^{2},\dots,I_{k}^{s_{k}}$ with the same length,
where $s_{k}\geq 2$ for all $k\geq 1$. We have

$\overline{\emph{mdim}}_{\emph{H}}([0,1],|\cdot|,\phi)\geq\underset{k\rightarrow\infty}{\limsup}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|}.$ (2.3)

###### Proof.

Note if $\varphi,\psi:[0,1]\rightarrow[0,1]$ are continuous maps such that
$I_{k}\subseteq\varphi(I_{k}^{t})$ and $I_{k}=\psi(I_{k}^{t})$ for all
$k\in\mathbb{N}$ and $t=1,\dots,s_{k}$, then

$\overline{\text{mdim}}_{\text{H}}([0,1],|\cdot|,\varphi)\geq\overline{\text{mdim}}_{\text{H}}([0,1],|\cdot|,\psi).$

Hence, without loss of generality, we can assume that $\phi(I_{k}^{t})=I_{k}$,
for each $k\in\mathbb{N}$ and $i=1,\dots,s_{k}$. We will fix $\varepsilon>0$
and $s\geq 0$. For any $k,n\in\mathbb{N}$, we have

$\text{H}_{\varepsilon}^{s}([0,1],d_{n},\phi)\geq\text{H}_{\varepsilon}^{s}(I_{k},d_{n},\phi|_{I_{k}}).$

Furthermore, if $\delta<\varepsilon,$ we have
$\text{H}_{\varepsilon}^{s}(I_{k},d_{n},\phi|_{I_{k}})\leq\text{H}_{\delta}^{s}(I_{k},d_{n},\phi|_{I_{k}})$.
For any $k\in\mathbb{N}$, set

$\varepsilon_{k}=\frac{|I_{k}|}{s_{k}}=|I_{k}^{t}|\quad\text{for
}t=1,2,\dots,s_{k}.$

Let $k\in\mathbb{N}$ such that
$\varepsilon_{k+1}\leq\varepsilon\leq\varepsilon_{k}$. Hence,

$\text{H}_{\varepsilon_{k+1}}^{s}(I_{k},d_{n},\phi|_{I_{k}})\geq\text{H}_{\varepsilon}^{s}(I_{k},d_{n},\phi|_{I_{k}})\geq\text{H}_{\varepsilon_{k}}^{s}(I_{k},d_{n},\phi|_{I_{k}}).$

We can divide each $I_{k}^{t}$ into $s_{k}^{n}$ subintervals,
$\\{J_{1}^{t},\dots,J_{s_{k}^{n}}^{t}\\}$, such that, for each
$l=1,\dots,s_{k}$, there exist
$m_{1},\dots,m_{s_{k}^{n-1}}\in\\{1,\dots,s_{k}^{n}\\}$ with

$\phi^{n}(J_{m_{h}}^{t})=I_{k}^{l},\quad\text{for each
}t=1,\dots,s_{k},h=1,\dots,s_{k}^{n-1}.$

Hence, for each $m\in\\{1,\dots,s_{k}^{n}\\}$, we have

$\text{diam}_{d_{n}}(J_{m}^{t})=\varepsilon_{k}\quad\text{ and therefore
}\quad\text{H}_{\varepsilon_{k}}^{s}(I_{k},d_{n},\phi|_{I_{k}})\leq\sum_{t=1}^{s_{k}}\sum_{i=1}^{s_{k}^{n}}(\text{diam}_{d_{n}}(J_{i}^{t}))^{s}=s_{k}^{n+1}(\varepsilon_{k})^{s}.$

Now, give that $\phi(I_{k}^{t})=I_{k}$, we have
$|\phi(I_{k}^{t})|=s_{k}|I_{k}^{t}|=s_{k}\varepsilon_{k}.$ Hence, for each
$n\in\mathbb{N}$, we need at least $s_{k}^{n}$ subintervals with diameter less
that $\varepsilon_{k}$ to cover each $I_{k}^{t}$ and, therefore, $s_{k}^{n+1}$
subintervals with diameter less that $\varepsilon_{k}$ to cover each $I_{k}$.
Since every $J_{m}^{t}$ has the same $d_{n}$-diameter, we have

$\text{H}_{\varepsilon_{k}}^{s}(I_{k},d_{n},\phi|_{I_{k}})=s_{k}^{n+1}(\varepsilon_{k})^{s}.$
(2.4)

Next, if $s\leq-\frac{(n+1)\log(s_{k})}{\log(\varepsilon_{k})}$, then
$s_{k}^{n+1}(\varepsilon_{k})^{s}\geq 1$. Thus,

$\text{dim}_{\text{H}}(I_{k},d_{n},\phi,\varepsilon_{k})=\frac{(n+1)\log(s_{k})}{\log(\frac{1}{\varepsilon_{k}})}.$

Consequently,

$\displaystyle\overline{\text{mdim}}_{\text{H}}([0,1],|\cdot|,\phi)$
$\displaystyle=\lim_{\varepsilon\rightarrow
0}\left(\limsup_{n\rightarrow\infty}\frac{1}{n}\text{dim}_{\text{H}}([0,1],d_{n},\phi,\varepsilon)\right)$
$\displaystyle\geq\limsup_{k\rightarrow\infty}\left(\limsup_{n\rightarrow\infty}\frac{1}{n}\text{dim}_{\text{H}}(I_{k},d_{n},\phi,\varepsilon_{k})\right)$
$\displaystyle\geq\underset{k\rightarrow\infty}{\limsup}\left(\limsup_{n\rightarrow\infty}\frac{1}{n}\frac{(n+1)\log(s_{k})}{\log(\frac{1}{\varepsilon_{k}})}\right)=\underset{k\rightarrow\infty}{\limsup}\left(\frac{\log(s_{k})}{\log(\frac{s_{k}}{|I_{k}|})}\right)$
$\displaystyle=\underset{k\rightarrow\infty}{\limsup}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|},$

which proves the lemma. ∎

Next lemma provides examples of continuous maps on the interval with mean
Hausdorff dimension equal to fixed value in $[0,1]$.

######  Lemma 2.4.

Suppose for each $k\in\mathbb{N}$ there exists a $s_{k}$-horseshoe for
$\phi\in C^{0}([0,1])$, $I_{k}=[a_{k-1},a_{k}]\subseteq[0,1]$, consisting of
sub-intervals with the same length $I_{k}^{1},I_{k}^{2},\dots,I_{k}^{s_{k}}$
and $[0,1]=\cup_{k=1}^{\infty}I_{k}$. We can rearrange the intervals and
suppose that $2\leq s_{k}\leq s_{k+1}$ for each $k$. If each
$\phi|_{I_{k}^{i}}:I_{k}^{i}\rightarrow I_{k}$ is a bijective affine map for
all $k$ and $i=1,\dots,s_{k}$, we have

  1. i.

${\underline{\emph{mdim}}_{\emph{H}}}([0,1],|\cdot|,\phi)=\underset{k\rightarrow\infty}{\liminf}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|}.$

  2. ii.

If the limit
$\underset{k\rightarrow\infty}{\lim}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|}$ exists, then
${\emph{mdim}}_{\emph{H}}([0,1],|\cdot|,\phi)=\underset{k\rightarrow\infty}{\lim}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|}.$

###### Proof.

In [1], Theorem 3.3, is proved that
${\underline{\text{mdim}}_{\text{M}}}([0,1],|\cdot|,\phi)\leq\underset{k\rightarrow\infty}{\liminf}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|}.$ Therefore,

${\underline{\text{mdim}}_{\text{H}}}([0,1],|\cdot|,\phi)\leq{\underline{\text{mdim}}_{\text{M}}}([0,1],|\cdot|,\phi)\leq\underset{k\rightarrow\infty}{\liminf}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|}.$

Next, we will prove
${\underline{\text{mdim}}_{\text{H}}}([0,1],|\cdot|,\phi)\geq\underset{k\rightarrow\infty}{\liminf}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|}$. We will fix $\varepsilon>0$ and $s\geq 0$. For any
$k\in\mathbb{N}$, set

$\varepsilon_{k}=\frac{|I_{k}|}{s_{k}}=|I_{k}^{t}|\quad\text{for
}t=1,2,\dots,s_{k}.$

Let $k\in\mathbb{N}$ such that
$\varepsilon_{k+1}\leq\varepsilon\leq\varepsilon_{k}$. We have (see (2.4))

$\text{H}_{\varepsilon}^{s}([0,1],d_{n},\phi)\geq\text{H}_{\varepsilon_{k}}^{s}(I_{k},d_{n},\phi|_{I_{k}})=s_{k}^{n+1}(\varepsilon_{k})^{s}.$

Next, if $s\leq-\frac{(n+1)\log(s_{k})}{\log(\varepsilon_{k})}$, then
$s_{k}^{n+1}(\varepsilon_{k})^{s}\geq 1$. Thus,

${\text{dim}_{\text{H}}([0,1],d_{n},\phi,\varepsilon)}\geq\text{dim}_{\text{H}}(I_{k},d_{n},\phi,\varepsilon_{k})=\frac{(n+1)\log(s_{k})}{\log(\frac{1}{\varepsilon_{k}})}.$

Consequently,

$\displaystyle\underline{\text{mdim}}_{\text{H}}([0,1],|\cdot|,\phi)$
$\displaystyle=\lim_{\varepsilon\rightarrow
0}\left(\liminf_{n\rightarrow\infty}\frac{1}{n}\text{dim}_{\text{H}}([0,1],d_{n},\phi,\varepsilon)\right)$
$\displaystyle\geq\liminf_{k\rightarrow\infty}\left(\liminf_{n\rightarrow\infty}\frac{1}{n}\text{dim}_{\text{H}}(I_{k},d_{n},\phi,\varepsilon_{k})\right)$
$\displaystyle=\underset{k\rightarrow\infty}{\liminf}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|},$

which proves i.

ii. In [1], Theorem 3.3, is proved if the limit
$\underset{k\rightarrow\infty}{\lim}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|}$ exists, then

$\overline{\text{mdim}}_{\text{M}}([0,1],|\cdot|,\phi)=\underset{k\rightarrow\infty}{\lim}\frac{1}{\left|1-\frac{\log|I_{k}|}{\log
s_{k}}\right|}.$

Therefore, ii. follows from this fact and i. ∎

Next examples follows from the above facts (see [1], Examples 3.1, 3.5 and
3.6).

######  Example 2.5.

Fix $r\in(0,\infty)$. Set $a_{0}=0$ and
$a_{n}=\sum_{i=0}^{n-1}\frac{C}{3^{ir}}$ for $n\geq 1$, where
$C=\frac{1}{\sum_{i=0}^{\infty}\frac{1}{3^{ir}}}=\frac{3^{r}-1}{3^{r}}$. For
each $n\geq 0$, let $T_{n}:I_{n}:=[a_{n},a_{n+1}]\rightarrow[0,1]$ be the
unique increasing affine map from $I_{n}$ onto $[0,1]$. For $s\in\mathbb{N}$,
set $\phi_{s,r}:[0,1]\rightarrow[0,1]$, given by
$\phi_{s,r}|_{I_{n}}=T_{n}^{-1}\circ g^{s(n+1)}\circ T_{n}$ for any $n\geq 0$,
where $g:[0,1]\rightarrow[0,1]$, is defined by $x\mapsto|1-|3x-1||$. It
follows from Lemmas 2.3 and 2.4 that

${{\text{mdim}}_{\text{H}}}([0,1],|\cdot|,\phi_{s,r})=\frac{s}{r+s}\quad\text{for
any }s\in\mathbb{N}.$

######  Example 2.6.

Set $a_{0}=0$ and $a_{n}=\sum_{i=1}^{n}\frac{6}{\pi^{2}i^{2}}$ for $n\geq 1$.
Set $I_{n}:=[a_{n-1},a_{n}]$ for any $n\geq 1$. Let $\varphi\in C^{0}([0,1])$
be defined by $\varphi|_{I_{n}}=T_{n}^{-1}\circ g^{n}\circ T_{n}$ for any
$n\geq 1$, where $T_{n}$ and $g$ are as in Example 2.5. For $\varphi^{s}$,
with $s\in\mathbb{N},$ we have $s_{k}=3^{sk}$ for each $k\in\mathbb{N}$. We
have

${{\text{mdim}}_{\text{H}}}([0,1],|\cdot|,\varphi^{s})=1\quad\text{for any
}s\in\mathbb{N}.$

######  Example 2.7.

Take $I_{n}=[a_{n-1},a_{n}]$ as in the above example. Divide each interval
$I_{n}$ into $2n+1$ sub-intervals with the same lenght, $I_{n}^{1}$, $\dots$,
$I_{n}^{2n+1}$. For $k=1,3,\dots,2n+1$, let
$\psi|_{I_{n}^{k}}:I_{n}^{k}\rightarrow I_{n}$ be the unique increasing affine
map from $I_{n}^{k}$ onto $I_{n}$ and for $k=2,4,\dots,2{n}$, let
$\psi|_{I_{n}^{k}}:I_{n}^{k}\rightarrow I_{n}$ be the unique decreasing affine
map from $I_{n}^{k}$ onto $I_{n}$. We have

${\text{mdim}}_{\text{H}}([0,1],|\cdot|,\psi^{s})=\frac{s}{s+2}.$

##  3\. Density of continuous maps on the interval with positive mean
Hausdorff dimension

On $C^{0}([0,1])$ we will consider the metric

$\hat{d}(\phi,\varphi)=\max_{x\in[0,1]}|\phi(x)-\varphi(x)|\quad\quad\text{
for any }\phi,\varphi\in C^{0}([0,1]).$

In [1], Theorem 4.1 (see also [8], Theorem C), it is proved that

$M_{a}([0,1])=\\{\phi\in
C^{0}([0,1]):\overline{\text{mdim}}_{\text{M}}([0,1],|\cdot|,\phi)=\underline{\text{mdim}}_{\text{M}}([0,1],|\cdot|,\phi)=a\\}$

is dense in $C^{0}([0,1])$. Following the same steps that in [1], Theorem 4.1,
we prove the next result.

######  Theorem 3.1.

For any $a\in[0,1]$, the set

$H_{a}([0,1])=\\{\phi\in
C^{0}([0,1]):\overline{\emph{mdim}}_{\emph{H}}([0,1],|\cdot|,\phi)=\underline{\emph{mdim}}_{\emph{H}}([0,1],|\cdot|,\phi)=a\\}$

is dense in $C^{0}([0,1])$.

###### Proof.

The set $C^{1}([0,1])$ consisting of all $C^{1}$-map defined on $[0,1]$ is
dense in $C^{0}([0,1])$ and any $\varphi\in C^{1}([0,1])$ has mean Hausdorff
dimension equal to 0, because it has finite topological entropy. Therefore,
$H_{0}([0,1])$ is dense in $C^{0}([0,1]).$

Next, fix $\phi_{0}\in H_{0}([0,1])$ and let $p$ be a fixed point of
$\phi_{0}$. Take $\varepsilon>0$. Choose $\delta>0$ such that
$|\phi_{0}(x)-\phi_{0}(p)|<\varepsilon/2$ for any $x$ with $|x-p|<\delta$.
Take $\phi_{a}\in H_{a}([0,1])$ for some $a\in(0,1]$. Set $J_{1}=[0,p]$,
$J_{2}=[p,p+\delta/2]$, $J_{3}=[p+\delta/2,p+\delta]$ and
$J_{4}=[p+\delta,1]$. Set $\psi_{a}\in C^{0}([0,1])$ defined as

$\psi_{a}(x)=\begin{cases}\phi_{0}(x),&\text{ if }x\in J_{1}\cup J_{4},\\\
T_{2}^{-1}\phi_{a}T_{2}(x),&\text{ if }x\in J_{2},\\\ T_{3}(x),&\text{ if
}x\in J_{3},\end{cases}$

where $T_{2}:J_{2}\rightarrow I$ is the affine map such that $T_{2}(p)=0$ and
$T_{2}(p+\delta/2)=1$, and
$T_{3}:J_{3}\rightarrow[p+\delta/2,\phi_{0}(p+\delta)]$ is the affine map such
that $T_{3}(p+\delta/2)=p+\delta/2$ and $T_{3}(p+\delta)=\phi_{0}(p+\delta)$.
We have that $\hat{d}(\psi_{a},\phi_{0})<\varepsilon.$ Set
$A=\cup_{i=0}^{\infty}\psi_{a}^{-i}(J_{2})$ and $B=A^{c}$. We have that

$\text{mdim}_{\text{H}}(A,|\cdot|,\psi_{a}|_{A})=\text{mdim}_{\text{H}}(J_{2},|\cdot|,\psi_{a}|_{J_{2}}),\quad\text{mdim}_{\text{H}}(B,|\cdot|,\psi_{a}|_{B})=0$

and hence

$\displaystyle\text{mdim}_{\text{H}}([0,1],|\cdot|,\psi_{a})$
$\displaystyle=\max\\{\text{mdim}_{\text{H}}(A,|\cdot|,\psi_{a}|_{A}),\text{mdim}_{\text{H}}(B,|\cdot|,\psi_{a}|_{B})\\}$
$\displaystyle=\text{mdim}_{\text{H}}(J_{2},|\cdot|,\psi_{a})=\text{mdim}_{\text{H}}(J_{2},|\cdot|,\phi_{a})=a.$

This fact proves the theorem. ∎

Next, in [8], Theorem C (see also [1], Theorem 4.6), it is proved the set
consisting of continuous maps $\phi:[0,1]\rightarrow[0,1]$ with upper metric
mean dimension equal to 1 is dense in $C^{0}([0,1])$. We will prove the set
$\mathcal{H}_{1}=\\{\phi\in
C^{0}([0,1]):\overline{\text{mdim}}_{\text{H}}([0,1],|\cdot|,\phi)=1\\}$ is
residual in $C^{0}([0,1])$. In order to prove this fact we need to present the
next definition.

######  Definition 3.2.

For any closed interval $J=[a,b]$, let
$\hat{J}=[\frac{2a+b}{3},\frac{a+2b}{3}]$, that is, the second third of $J$.
For $\epsilon\in(0,1)$ and $k\in\mathbb{N}$, we say that $J=[a,b]$ is a strong
$(\epsilon,k)$-horseshoe of a continuous map $\phi:[0,1]\rightarrow[0,1]$ if
$|J|>\epsilon$ and it contains $k$ closed intervals
$J_{1},\dots,J_{k}\subseteq J$, with
$(J_{s})^{\circ}\cap(J_{r})^{\circ}=\emptyset$ for $s\neq r$, such that
$|J_{i}|>\frac{|J|}{2{k}}$ and $J\subset(\phi(\hat{J}_{i}))^{\circ}$ for any
$i=1,\dots,k$.

For $\epsilon>0$ and $k\in\mathbb{N}$, set

  * •

$H(\epsilon,k)=\\{\phi\in C^{0}([0,1]):\phi\text{ has a strong
}(\epsilon,k)\text{-horseshoe}\\};$

  * •

$H(k)=\bigcup_{i\in\mathbb{N}}H\left(\frac{1}{i^{2}},3^{k\,i}\right);$

  * •

$\mathcal{H}=\overset{\infty}{\underset{k=1}{\bigcap}}H(k).$

######  Lemma 3.3.

$\mathcal{H}$ is residual in $C^{0}([0,1])$.

###### Proof.

It is clear that, for any $\epsilon\in(0,1)$ and $k\in\mathbb{N}$, the set
$H(\epsilon,k)$ is not empty (we can construct a continuous function on
$H(\epsilon,k)$ similarly to what was done in Theorem 3.1).

$H(\epsilon,k)$ is open in $C^{0}([0,1])$: if $\phi\in H(\epsilon,k)$ and $J$
is a strong $(\epsilon,k)$-horseshoe of $\phi$ we can take a small enough open
neighborhood $U$ of $\phi$ such that for any $\psi\in U$ we have the same $J$
is a strong $(\epsilon,k)$-horseshoe of $\psi$.

$H(k)$ is dense in $C^{0}(N)$: fix $\psi\in C^{0}(N)$ with a $s$-periodic
point. Every small neighborhood of the orbit of this point can be perturbed in
order to obtain a strong $\left(\frac{1}{i^{2}},3^{k\,i}\right)$ horseshoe for
a $\phi$ close to $\psi$ for a large enough $i$.

The above facts prove that $\mathcal{H}$ is residual.∎

######  Theorem 3.4.

For any $\phi\in\mathcal{H}$ we have
$\overline{\emph{mdim}}_{\emph{H}}([0,1],|\cdot|,\phi)=1$. Therefore,

$\mathcal{H}_{1}=\\{\phi\in
C^{0}([0,1]):\overline{\emph{mdim}}_{\emph{H}}([0,1],|\cdot|,\phi)=1\\}$

contains a residual subset of $C^{0}([0,1])$.

###### Proof.

Take $\phi\in\mathcal{H}$. We have $\phi\in H(k)$ 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
$\left(\frac{1}{i_{k}^{2}},3^{k\,i_{k}}\right)$-horseshoe $J_{i_{k}}$,
consisting of $3^{ki_{k}}$ intervals
$J_{i_{k}}^{1},\dots,J_{i_{k}}^{3^{ki_{k}}}$, such that
$J_{i_{k}}\subset(\phi(\hat{J}_{i_{k}}^{t}))^{\circ}$ for each
$t=1,\dots,3^{ki_{k}}$. Without loss of generality, we can assume that
$J_{i_{k}}=\phi({J}_{i_{k}}^{t})$ for each $t=1,\dots,3^{ki_{k}}$. For any
$k\in\mathbb{N}$, set

$\varepsilon_{k}=\max\\{|{J}_{i_{k}}^{t}|:t=1,2,\dots,3^{ki_{k}}\\}.$

We can divide each $J_{i_{k}}^{t}$ into $3^{nki_{k}}$ subintervals,
$\\{T_{1}^{t},\dots,T_{3^{nki_{k}}}^{t}\\}$, such that, for each
$l=1,\dots,3^{ki_{k}}$, there exist
$m_{1},\dots,m_{3^{(n-1)ki_{k}}}\in\\{1,\dots,3^{nki_{k}}\\}$ with

$\phi^{n}(T_{m_{h}}^{t})=J_{i_{k}}^{l},\quad\text{for each
}t=1,\dots,3^{ki_{k}},h=1,\dots,3^{(n-1)ki_{k}}.$

Hence, for each $m=1,\dots,3^{nki_{k}}$, we have

$\text{diam}_{d_{n}}(T_{m}^{t})=\varepsilon_{k}\quad\text{ and therefore
}\quad\text{H}_{\varepsilon_{k}}^{s}(I_{k},d_{n},\phi|_{I_{k}})\leq\sum_{t=1}^{3^{ki_{k}}}\sum_{i=1}^{3^{nki_{k}}}(\text{diam}_{d_{n}}(T_{i}^{t}))^{s}=3^{(n+1)ki_{k}}(\varepsilon_{k})^{s}.$

Now, give that $\phi(J_{i_{k}}^{t})=J_{i_{k}}$, we have
$|\phi(J_{i_{k}}^{t})|=3^{ki_{k}}|J_{i_{k}}^{t}|=3^{ki_{k}}\varepsilon_{k}.$
Hence, for each $n\in\mathbb{N}$, we need at least $3^{nki_{k}}$ subintervals
with diameter less that $\varepsilon_{k}$ to cover each $J_{i_{k}}^{t}$ and,
therefore, $3^{(n+1)ki_{k}}$ subintervals with diameter less that
$\varepsilon_{k}$ to cover each $J_{i_{k}}$. Since every $J_{m}^{t}$ has the
same $d_{n}$-diameter, we have

$\text{H}_{\varepsilon_{k}}^{s}(J_{i_{k}},d_{n},\phi|_{J_{i_{k}}})=3^{(n+1)ki_{k}}(\varepsilon_{k})^{s}.$

Next, if $s\leq-\frac{(n+1)\log 3^{ki_{k}}}{\log(\varepsilon_{k})}$, then
$3^{(n+1)ki_{k}}(\varepsilon_{k})^{s}\geq 1$. Thus,

$\text{dim}_{\text{H}}(J_{i_{k}},d_{n},\phi,\varepsilon_{k})=\frac{(n+1)\log(3^{ki_{k}})}{\log(\frac{1}{\varepsilon_{k}})}.$

Consequently,

$\displaystyle\overline{\text{mdim}}_{\text{H}}([0,1],|\cdot|,\phi)$
$\displaystyle\geq\underset{k\rightarrow\infty}{\limsup}\left(\liminf_{n\rightarrow\infty}\frac{1}{n}\frac{(n+1)\log(3^{ki_{k}})}{\log(\frac{1}{\varepsilon_{k}})}\right)=\underset{k\rightarrow\infty}{\limsup}\left(\frac{\log(3^{ki_{k}})}{\log(\frac{3^{ki_{k}}}{|I_{k}|})}\right)$
$\displaystyle=\underset{k\rightarrow\infty}{\limsup}\frac{1}{\left|1-\frac{\log|J_{i_{k}}|}{\log
3^{ki_{k}}}\right|},$

which proves the lemma. ∎

##  4\. Homeomorphisms on manifold with positive mean Hausdorff dimension

Suppose that $\text{dim}(N)=m\geq 2$ and fix $\alpha\in[0,m]$. In [8] and [24]
are presented techniques to obtain homeomorphisms $\phi:N\rightarrow N$ with
upper metric mean dimension equal to $\text{dim}(N)$. In [2], Definition 2.1,
was constructed an $m$-dimensional $(2k+1)^{m-1}$-horseshoe in order to obtain
a homeomorphism on the cube $[a,b]^{m}$ with upper and lower metric mean
dimension equal to $\alpha$. In the next definition, we will make an
appropriate modification to that definition in order to obtain a homeomorphism
on the cube $[a,b]^{m}$ with lower and upper mean Hausdorff dimension equal to
$\alpha$.

Figure 4.1. 3-dimensional 9-horseshoe

######  Definition 4.1 ($m$-dimensional $(2k+1)^{m-1}$-horseshoe).

Fix $m\geq 2$. Take $E=[a,b]^{m}$ 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

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

Take $a=s_{0}<s_{1}<\cdots<s_{2(2k+1)^{m-1}-2}<s_{2(2k+1)^{m-1}-1}=b$ and
consider

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

We will require that:

$|s_{l}-s_{l-1}|=\frac{b-a}{4k+1}\quad\text{for
}l=1,\dots,4k\quad\text{and}\quad|s_{a}-s_{a-1}|=|s_{b}-s_{b-1}|\quad\text{for
}a,b\geq 4k+1.$

See Figure 4.1. We say that $E\subseteq A\subseteq\mathbb{R}^{m}$ is an
$m$-dimensional $(2k+1)^{m-1}$-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 $H_{i_{1},i_{2},\dots,i_{m-1}}$, with $i_{j}\in\\{1,3,\dots,4k+1\\}$,
there exists some $l\in\\{1,3,\dots,2(2k+1)^{m-1}-1\\}$ with

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

  * •

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

Note that

$\tilde{V}_{4k+1}:=\bigcup_{l=4k+1}^{2(2k+1)^{m-1}-1}V_{l}=\left[b-\frac{b-a}{4k+1},b\right]\times[a,b]^{m-1}.$

######  Lemma 4.2.

Let $E^{\prime}:=[a^{\prime},b^{\prime}]^{m}$ and
$E:=[a,b]^{m}\subseteq\mathbb{R}^{m}$ be closed $m$-cubes with
$E^{\prime}\subsetneqq(E)^{\circ}$ and fix any $k\in\mathbb{N}$. There exists
a homeomorphism $\psi:E\longrightarrow E$ such that
$\psi|_{E^{\prime}}:E^{\prime}\longrightarrow E^{\prime}$ is an
$m$-dimensional $(2k+1)$-horseshoe, and $\phi:=\psi^{2}$ satisfies
$\phi|_{\partial E}\equiv Id$ and
$h_{\emph{top}}(\phi)=h_{\emph{top}}(\phi|_{E^{\prime}\cap\phi(E^{\prime})})$,
where $\partial A$ is the boundary of a subset $A$ and $Id$ is the identity
map.

###### Proof.

For the sake of clarity in both presentation and illustration, we will
demonstrate the lemma specifically for the case where $m=2$, although it is
important to note that the same principle applies for any $m\geq 2$.

Suppose that $\psi:E\rightarrow E$ has an $m$-dimensional $(2k+1)$-horseshoe
$E^{\prime}$. In Figure 2(a) we show the the image
$\phi(E^{\prime})=\psi^{2}(E^{\prime})$ of an 2-dimensional 3-horseshoe for
$\psi.$ We will extend $\psi$ with the conditions in the lemma.

(a) $\phi(E)=\psi^{2}(E)$

(b) $K_{1},\dots,K_{4}$

(c) $Q_{1},\dots,Q_{4}$

Figure 4.2. 3-dimensional 9-horseshoe

Next,

  * •

let $K_{1}$ be the polygonal region with vertices
$(a,a),(a^{\prime},a^{\prime}),(a^{\prime},b^{\prime}),(a,b);$

  * •

let $K_{2}$ be the polygonal region with vertices
$(a,b),(a^{\prime},b^{\prime}),(b^{\prime},b^{\prime}),(b,b);$

  * •

let $K_{3}$ be the polygonal region with vertices
$(b,b),(b^{\prime},b^{\prime}),(b^{\prime},a^{\prime}),(b,a);$

  * •

let $K_{4}$ be the polygonal region with vertices
$(a,a),(a^{\prime},b^{\prime}),(b^{\prime},a^{\prime}),(b,a).$

These regions are shown in Figure 2(b). Now,

  * •

let $Q_{1}$ be region bounded by the next curves: a curve $L_{1}$ from $(a,a)$
to $\phi(a^{\prime},a^{\prime})$ contained in $K_{1}$ and such that does not
intersect $\phi(E)$; the segment from $\phi(a^{\prime},a^{\prime})$ to
$(a^{\prime},b^{\prime});$ the segment from $(a^{\prime},b^{\prime})$ to
$(a,b);$ the segment from $(a,b)$ to $(a,a).$

  * •

let $Q_{2}$ be the region bounded by the next curves: the segment from $(a,b)$
to $(a^{\prime},b^{\prime})$;
$\phi(\\{b^{\prime}\\}\times[a^{\prime},b^{\prime}])$; a curve $L_{2}$ from
$\phi(b^{\prime},b^{\prime})$ to $(b,b)$ contained in $K_{3}$ and such that
does not intersect $\phi(E)$; the segment from $(b,b)$ to $(a,b).$

  * •

let $Q_{3}$ be the region bounded by the next curves: $L_{2}$; the segment
from $(b,b)$ to $(b,a)$; the segment from $(b,a)$ to
$(b^{\prime},a^{\prime})$; the segment from $(b^{\prime},a^{\prime})$ to
$\phi(b^{\prime},b^{\prime})$.

  * •

let $Q_{4}$ be the region bounded by the next curves: the segment from $(b,a)$
to $(b^{\prime},a^{\prime})$;
$\phi(\\{a^{\prime}\\}\times[a^{\prime},b^{\prime}])$; $L_{1}$; the segment
from $(a,a)$ to $(b,a)$.

These regions are shown in Figure 2(c). The annular regions $K=K_{1}\cup
K_{2}\cup K_{3}\cup K_{4}$ and $Q=Q_{1}\cup Q_{2}\cup Q_{3}\cup Q_{4}$ are
homeomorphic. We can consider a homeomorphism $\varphi:K\to Q$ such that
$\varphi(K_{i})=Q_{i}$, $\varphi|_{\partial E}=\phi|_{\partial E}$, and
$\varphi|_{\partial E^{\prime}}=Id|_{\partial E^{\prime}}$. Therefore, we can
to extend $\phi$ to $E$, setting $\phi|_{K}=\varphi$.

Next, notice that
$h_{\text{top}}(\phi)=h_{\text{top}}(\phi|_{E^{\prime}\cap\phi(E^{\prime})})$,
because $\Omega(\phi)\subseteq(E^{\prime}\cap\phi(E^{\prime}))\cup\partial E$
and $\phi|_{\partial E}$ is the identity. The above facts prove the lemma. ∎

On any $m$-cube $E\subseteq\mathbb{R}^{m}$, we will consider the metric
$\rho(\cdot,\cdot)$, given by

$\rho((x_{1},\dots,x_{m}),(y_{1},\dots,y_{m}))=\max\\{|x_{1}-y_{1}|,\dots,|x_{m}-y_{m}|\\}.$

######  Lemma 4.3.

Let $\phi:{\mathcal{C}}:=[0,1]^{m}\rightarrow{\mathcal{C}}$ be a
homeomorphism, $E_{k}=[a_{k},b_{k}]^{m}$ 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 $m$-dimensional $3^{k(m-1)}$-horseshoe for $\phi$;

  * C4.

For each $k$, $\phi|_{E_{k}^{\prime}}:E_{k}^{\prime}\rightarrow
E_{k}^{\prime}$ satisfies the properties in Lemma 4.2;

  * 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

$\emph{mdim}_{\emph{H}}({\mathcal{C}},\rho,\phi^{2})=\frac{m}{r+1}.$

  * (ii)

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

$\emph{mdim}_{\emph{H}}({\mathcal{C}},\rho,\phi^{2})=m.$

###### Proof.

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

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

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

$\text{H}_{\varepsilon_{k}}^{s}(N,\rho_{m},\varphi)\geq\text{H}_{\varepsilon_{k}}^{s}(E_{k}\cap\Omega(\varphi),\rho_{m},\varphi|_{E_{k}\cap\Omega(\varphi)})\quad\text{for
any }m\geq 1.$

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

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

and

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

as in Definition 4.1. Set

$U_{l}^{k}=V_{l}^{k}\quad\text{for }l=1,\dots,2(3^{k})-2\quad\text{and}\quad
U_{2(3^{k})-1}^{k}=\bigcup_{l=2(3^{k})-1}^{2(3^{k})^{m-1}-1}V_{l}^{k}.$

For each $j=1,\dots,m$, let $i_{j}\in\\{1,\dots,2(3^{k})-1\\}$ and take

$C^{k}_{i_{1},\dots,i_{m}}=H^{k}_{i_{1},i_{2},\dots,i_{m-1}}\cap
U^{k}_{i_{m}}.$

For $t=1,\dots,n$, let $i_{1}^{(t)},$
$\dots,i_{m}^{(t)}\in\\{1,3,5,\dots,2(3^{k})-1\\}$ and set

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

From the definition of $\varphi$, these sets are non-empty. Furthermore,

$\text{diam}_{n}\left(C^{k}_{i_{1}^{(n)},\dots,i_{m}^{(n)},\dots,i_{1}^{(1)},\dots,i_{m}^{(1)}}\right)=\varepsilon_{k},\quad\text{
if }{i_{t}^{(j)}\in\\{{1},3,5,\dots,2({3^{k}})-1\\}},$

We have $3^{knm}$ sets of this form. Hence,

$\text{H}_{\varepsilon_{k}}^{s}(E_{k}\cap\Omega(\varphi),\rho_{n},\varphi|_{E_{k}\cap\Omega(\varphi)})\leq\sum_{t=1}^{3^{knm}}\left(\frac{B}{(2(3^{k})-1)3^{kr}}\right)^{s}=3^{knm}\left(\frac{B}{(2(3^{k})-1)3^{kr}}\right)^{s}.$

We have

$\text{H}_{\varepsilon_{k}}^{s}(E_{k}\cap\Omega(\varphi),\rho_{n},\varphi|_{E_{k}\cap\Omega(\varphi)})=3^{knm}\left(\frac{B}{(2(3^{k})-1)3^{kr}}\right)^{s},$

because
$\text{diam}_{n}\left(C_{i_{1}^{(n)},\dots,i_{m}^{(n)},\dots,i_{1}^{(1)},\dots,i_{m}^{(1)}}\right)=\varepsilon_{k}$.
Next,

$3^{knm}\left(\frac{B}{(2(3^{k})-1)3^{kr}}\right)^{s}\geq 1\Longleftrightarrow
3^{knm}\geq\left(\frac{(2(3^{k})-1)3^{kr}}{B}\right)^{s}\Longleftrightarrow\frac{\log
3^{knm}}{\log\left(\frac{(2(3^{k})-1)3^{kr}}{B}\right)}\geq s.$

Therefore,

$\displaystyle\text{dim}_{\text{H}}(E_{k}\cap\Omega(\varphi),\rho_{n},\varepsilon_{k})=\frac{\log
3^{knm}}{\log\left(\frac{(2(3^{k})-1)3^{kr}}{B}\right)}$

and hence

$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\text{dim}_{\text{H}}(N,\rho_{n},\varepsilon)\geq\lim_{n\rightarrow\infty}\frac{1}{n}\text{dim}_{\text{H}}(E_{k}\cap\Omega(\varphi),\rho_{n},\varepsilon_{k})=\frac{\log
3^{km}}{\log\left(\frac{(2(3^{k})-1)3^{kr}}{B}\right)}.$

Thus,

$\text{mdim}_{\text{H}}(N,\rho,\varphi)\geq\lim_{k\rightarrow\infty}\frac{\log
3^{km}}{\log\left(\frac{(2(3^{k})-1)3^{kr}}{B}\right)}=\frac{m}{1+r}.$ (4.1)

Next, we will prove that
${\overline{\text{mdim}}_{\text{M}}}(\mathcal{C},\rho,\varphi)\leq\frac{m}{r+1}$.
Note that $\frac{\log
3^{km}}{\log[4(2(3^{k})-1)3^{kr}B^{-1}]}\rightarrow\frac{m}{1+r}$ as
$k\rightarrow\infty$. Hence, for any $\delta>0$, there exists $k_{0}\geq 1$,
such that, for any $k>k_{0}$, we have $\frac{\log
3^{km}}{\log[4(2(3^{k})-1)3^{kr}B^{-1}]}<\frac{m}{1+r}+\delta$. 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

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

If $x\in\tilde{\Omega}$, then $x$ belongs to some
$C^{k}_{i_{1}^{(n)},\dots,i_{m}^{(n)},\dots,i_{1}^{(1)},\dots,i_{m}^{(1)}}$,
where ${i_{t}^{(j)}\in\\{{1},3,5,\dots,2({3^{k}})-1\\}}$. Furthermore,
$\rho_{n}(x,y)\leq 4\varepsilon_{k}$ for any $y\in
C^{k}_{i_{1}^{(n)},\dots,i_{m}^{(n)},\dots,i_{1}^{(1)},\dots,i_{m}^{(1)}}$.
Hence, if $Y_{k}=\cup_{j=1}^{k}E_{j}$, for every $n\geq 1$, we have

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

Therefore,

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

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

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

Hence,
${\overline{\text{mdim}}_{\text{M}}}(\mathcal{C},\rho,\varphi)\leq\frac{m}{r+1}$.
From (1.1) we have

${\text{mdim}}_{\text{H}}(\mathcal{C},\rho,\varphi)\leq\overline{\text{mdim}}_{\text{M}}(\mathcal{C},\rho,\varphi),$
(4.2)

therefore, from (4.1) and (4.2) it follows that
${\text{mdim}}_{\text{M}}(\mathcal{C},\rho,\varphi)={\text{mdim}}_{\text{H}}(\mathcal{C},\rho,\varphi)=\frac{m}{r+1}.$
∎

On $\text{Hom}(N)$ we will consider the metric

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

Take $\alpha\in[0,m]$ and set

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

In [2], Theorem 3.1, is proved the set consisting of homeomorphisms
$\phi:N\rightarrow N$ with upper and lower metric mean dimension equal to
$\alpha$ is dense in $\text{Hom}(N)$. We adapt that proof in order to show the
analog result for mean Hausdorff dimension.

######  Theorem 4.4.

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

###### Proof.

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 set $P^{r}(N)$ consisting of all $C^{r}$-diffeomorphisms on $N$ with a
periodic point is $C^{0}$-dense in $\text{Hom}(N)$ (see [4], [14]). 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 $k$. Take $\beta>0$, small enough, such that

$[-\beta,\beta]^{m}\subset B_{\frac{\varepsilon}{2}}(0)\,\text{ and
}\,C_{i+1}:=\psi(\exp_{\psi^{i}(p)}((-\beta,\beta)^{m})))\subset
D_{i+1}:=\exp_{\psi^{i+1}(p)}(B_{\frac{\varepsilon}{2}}(0_{\psi^{i+1}(p)})),$

for each $i=0,\dots,k-1.$ Notice that

$p=\psi^{k}(p)=\text{exp}_{p}(0_{p})\in C_{k}.$

Now, let $\lambda\in(0,\frac{\beta}{4})$ be such that

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

Figure 4.3. $E_{i}\subset C_{i}\subset D_{i}$.

(see Figure 4.3). The annular regions

$[-\beta,\beta]^{m}\setminus(-\lambda,\lambda)^{m}\quad\text{ and
}\quad\exp^{-1}_{\psi^{i+1}(p)}(C_{i+1}\setminus E_{i+1}^{\circ})$

are homeomorphic. Consider a homeomorphism

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

such that

$H_{i}|_{\partial[-\lambda,\lambda]^{m}}=Id\ \ \text{and}\ \
H_{i}|_{\partial[-\beta,\beta]^{m}}=\exp^{-1}_{\psi^{i+1}(p)}\circ\psi\circ\exp_{\psi^{i}(p)}.$

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

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

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

${{\text{mdim}}_{\text{H}}}([-\lambda,\lambda]^{m},\rho,\phi_{\alpha})=\alpha,$

(see Lemma 4.3). Set $K:=\bigcup_{i=0}^{k-1}Q_{i}$. Note that $N\setminus K$
is $\varphi_{\alpha}$ invariant and

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

Hence,

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

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

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

Hence, $D\subseteq[-\lambda,\lambda]^{m}$ has $\rho_{n}$-diameter less than
$\epsilon\in(0,\delta_{N})$ if and only if
$\text{exp}_{\psi^{i}(p)}(D)\subseteq Q_{i}$ has $d_{n}$ diameter less than
$\epsilon\in(0,\delta_{N})$ for all $i=0,\dots,k-1$. Since
$K=\bigcup_{i=0}^{k-1}Q_{i}$, we have for any $j\in\\{0,\dots,k-1\\}$ that

$\displaystyle\text{mdim}_{\text{H}}(N,d,\varphi_{\alpha})$
$\displaystyle=\text{mdim}_{\text{H}}(K,d,\varphi_{\alpha}|_{K})=\underset{i=0,\dots,k-1}{\max}\\{\text{mdim}_{\text{H}}(Q_{i},d,\varphi_{\alpha})\\}$
$\displaystyle=\text{mdim}_{\text{H}}(Q_{j},d,\varphi_{\alpha})=\text{mdim}_{\text{H}}([-\lambda,\lambda]^{n},\rho,\phi_{\alpha})=\alpha.$

It is clear that $\hat{d}(\varphi_{\alpha},\psi)<\varepsilon$, which proves
the theorem. ∎

##  5\. Tipical homeomorphism on a manifold has maximal mean Hausdorff
dimension

Suppose $\text{dim}(N)=m\geq 2$, and let $\alpha\in[0,m]$ be fixed. In [8], it
was demonstrated that the collection of all homeomorphisms on $N$ with an
upper metric mean dimension equal to $m$ forms a residual set in
$\text{Hom}(N)$ (also see [2] and [24]). The case for continuous maps (which
need not be homeomorphisms) was established in [1]. In this section, we will
establish that the set of all homeomorphisms on $N$ with an upper mean
Hausdorff dimension equal to $m$ constitutes a residual set in
$\text{Hom}(N)$.

######  Definition 5.1 ($m$-dimensional strong horseshoe).

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

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

Take $a=s_{0}<s_{1}<\cdots<s_{2(2k+1)^{m-1}-2}<s_{2(2k+1)^{m-1}-1}=b$ and
consider

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

We will require that:

$|s_{l}-s_{l-1}|=\frac{b-a}{4k+1}\quad\text{for
}l=1,\dots,4k\quad\text{and}\quad|s_{a}-s_{a-1}|=|s_{b}-s_{b-1}|\quad\text{for
}a,b\geq 4k+1.$

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

  * •

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

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

  * •

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

######  Definition 5.2.

Let $N$ be an $m$-dimensional Riemannian manifold and fix $k\geq 1$. We say
that $\phi\in\text{Hom}(N)$ has an $m$-dimensional strong
$(\epsilon,(2k+1)^{m-1})$-horseshoe $E=[a,b]^{m}$, 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 $m$-dimensional strong $(\epsilon,(2k+1)^{m-1})$-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

  * •

$G(\epsilon,k)=\\{\phi^{2}\in\text{Hom}(N):\phi\text{ has a strong
}(\epsilon,k)\text{-horseshoe}\\};$

  * •

$G(k)=\bigcup_{i\in\mathbb{N}}H\left(\frac{1}{i^{2}},3^{ki(m-1)}\right);$

  * •

$\mathcal{G}=\overset{\infty}{\underset{k=1}{\bigcap}}G(k).$

A similar subset is defined in [1], Section 4, and it is proved that is
residual and all homeomorphism in that set has metric mean dimension equal to
$m$ (see Lemma 4.4 in [1]). We can to adapt that proof in order to show that
$\mathcal{G}$ is residual and furthermore we prove that all homeomorphism in
$\mathcal{G}$ has mean Hausdorff dimension equal to $m$. First, we can show
the following lemma analogously to Lemma 3.3, and therefore we omit the proof.

######  Lemma 5.3.

$\mathcal{G}$ is residual in $\emph{Hom}(N)$.

######  Theorem 5.4.

For any $\phi\in\mathcal{G}$ we have
$\overline{\emph{mdim}}_{\emph{H}}(N,d,\phi)=m$. Therefore,

$\mathcal{G}_{m}=\\{\phi\in\emph{Hom}(N):\overline{\emph{mdim}}_{\emph{H}}(N,d,\phi)=m\\}$

contains a residual subset of $\emph{Hom}(N)$.

###### Proof.

We prove that $\overline{\text{mdim}}_{\text{H}}(N,d,\phi)=m$ for any
$\phi\in\mathcal{G}$. For this sake, take $\varphi=\phi^{2}\in\mathcal{G}$. We
have $\varphi\in G(k)$ 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
$\left(\frac{1}{i_{k}^{2}},3^{m\,k\,i_{k}}\right)$-horseshoe
$E_{{k}}=[a_{k},b_{k}]^{m}$, such that $|E_{{k}}|>\frac{1}{i_{k}^{2}}$,
consisting of rectangles

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

and

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

with

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

We can assume that $E_{k}=\frac{1}{i_{k}^{2}}$ and
$H^{{k}}_{j_{1},j_{2},\dots,j_{m-1}}=\phi(V^{{k}}_{l}).$ Set

$U_{l}^{k}=V_{l}^{k}\quad\text{for
}l=1,\dots,2(3^{ki_{k}})-2\quad\text{and}\quad
U_{2(3^{k})-1}^{k}=\bigcup_{l=2(3^{k})-1}^{2(3^{ki_{k}({m-1})})-1}V_{l}^{k}.$

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

$C^{k}_{i_{1},\dots,i_{m}}=H^{k}_{i_{1},i_{2},\dots,i_{m-1}}\cap
U^{k}_{i_{m}}.$

For $t=1,\dots,n$, let $i_{1}^{(t)},$
$\dots,i_{m}^{(t)}\in\\{1,3,5,\dots,2(3^{ki_{k}})-1\\}$ and set

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

Furthermore, set

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

For any $k\geq 1$, taking
$\varepsilon_{k}=\frac{1}{i_{k}^{2}(2(3^{k\,i_{k}})-1)}$, we have

$\text{diam}_{n}\left(\tilde{C}^{k}_{i_{1}^{(n)},\dots,i_{m}^{(n)},\dots,i_{1}^{(1)},\dots,i_{m}^{(1)}}\right)=\varepsilon_{k},\quad\text{
if }{i_{t}^{(j)}\in\\{{1},3,5,\dots,2({3^{ki_{k}}})-1\\}},$

We have $3^{ki_{k}nm}$ sets of this form. Hence,

$\text{H}_{\varepsilon_{k}}^{s}(E_{k}\cap\Omega(\varphi),d_{n},\varphi|_{E_{k}\cap\Omega(\varphi)})\leq\sum_{t=1}^{3^{ki_{k}nm}}\left(\frac{1}{i_{k}^{2}(2(3^{k\,i_{k}})-1)}\right)^{s}=3^{ki_{k}nm}\left(\frac{1}{i_{k}^{2}(2(3^{k\,i_{k}})-1)}\right)^{s}.$

We have

$\text{H}_{\varepsilon_{k}}^{s}(E_{k}\cap\Omega(\varphi),d_{n},\varphi|_{E_{k}\cap\Omega(\varphi)})=3^{ki_{k}nm}\left(\frac{1}{i_{k}^{2}(2(3^{k\,i_{k}})-1)}\right)^{s},$

because
$\text{diam}_{n}\left(\tilde{C}_{i_{1}^{(n)},\dots,i_{m}^{(n)},\dots,i_{1}^{(1)},\dots,i_{m}^{(1)}}\right)=\varepsilon_{k}$.
Next,

$3^{ki_{k}nm}\left(\frac{1}{i_{k}^{2}(2(3^{k\,i_{k}})-1)}\right)^{s}\geq
1\Longleftrightarrow
3^{ki_{k}nm}\geq\left({i_{k}^{2}(2(3^{k\,i_{k}})-1)}\right)^{s}\Longleftrightarrow\frac{\log
3^{ki_{k}nm}}{\log\left({i_{k}^{2}(2(3^{k\,i_{k}})-1)}\right)}\geq s.$

Therefore,

$\displaystyle\text{dim}_{\text{H}}(E_{k}\cap\Omega(\varphi),d_{n},\varepsilon_{k})=\frac{\log
3^{ki_{k}nm}}{\log\left({i_{k}^{2}(2(3^{ki_{k}\,i_{k}})-1)}\right)}$

and hence

$\displaystyle\limsup_{n\rightarrow\infty}\frac{1}{n}\text{dim}_{\text{H}}(N,d_{n},\varepsilon)\geq\lim_{n\rightarrow\infty}\frac{1}{n}\text{dim}_{\text{H}}(E_{k}\cap\Omega(\varphi),d_{n},\varepsilon_{k})=\frac{\log
3^{ki_{k}n}}{\log\left(i_{k}^{2}(2(3^{k\,i_{k}})-1)\right)}.$

Thus,

$\overline{\text{mdim}}_{\text{H}}(N,d,\varphi)\geq\lim_{k\rightarrow\infty}\frac{\log
3^{ki_{k}m}}{\log\left(i_{k}^{2}(2(3^{k\,i_{k}})-1)\right)}=\lim_{k\rightarrow\infty}\frac{\log
3^{ki_{k}m}}{\log 3^{k\,i_{k}}}=m,$

therefore $\overline{\text{mdim}}_{\text{H}}(N,d,\varphi)=m$, which proves the
first part of the theorem. It follows from Lemma 5.3 that $\mathcal{G}_{m}$
contains a residual subset. ∎

## References

  * [1] Acevedo, J.M. “Genericity of continuous maps with positive metric mean dimension”. Results Math 77, 2 (2022).
  * [2] Acevedo, J. D. J. M., Ibarra, S. R., & Cantillo, R. A. “Density of the level sets of the metric mean dimension for homeomorphisms”. Journal of Dynamics and Differential Equations (2024).
  * [3] Acevedo, Jeovanny M., Baraviera, A., Becker, A. J., & Scopel, É. “Metric mean dimension and mean Hausdorff dimension varying the metric.” (2022).
  * [4] Artin, Michael, and Barry Mazur. “On periodic points.” Annals of Mathematics (1965): 82-99.
  * [5] Backes, Lucas, and Fagner B. Rodrigues. “A Variational Principle for the Metric Mean Dimension of Level Sets.” IEEE Transactions on Information Theory, vol. 69, no. 9, pp. 5485-5496.
  * [6] Cheng, Dandan, Zhiming Li, and Bilel Selmi. “Upper metric mean dimensions with potential on subsets.” Nonlinearity 34.2 (2021): 852.
  * [7] Dou, Dou. “Minimal subshifts of arbitrary mean topological dimension.” Discrete and Continuous Dynamical Systems 37.3 (2016): 1411-1424.
  * [8] Carvalho, Maria, Fagner B. Rodrigues, and Paulo Varandas. “Generic homeomorphisms have full metric mean dimension.” Ergodic Theory and Dynamical Systems: 1-25.
  * [9] Falconer, Kenneth. Fractal geometry: mathematical foundations and applications. John Wiley & Sons, 2004.
  * [10] Furstenberg, Harry. “Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation.” Mathematical systems theory 1.1 (1967): 1-49.
  * [11] Gromov, Misha. “Topological invariants of dynamical systems and spaces of holomorphic maps: I.” Mathematical Physics, Analysis and Geometry 2.4 (1999): 323-415
  * [12] Gutman, Yonatan. “Embedding topological dynamical systems with periodic points in cubical shifts.” Ergodic Theory and Dynamical Systems 37.2 (2017): 512-538.
  * [13] Gutman, Yonatan, Yixiao Qiao, and Masaki Tsukamoto. “Application of signal analysis to the embedding problem of ${\mathbb{Z}}^{k}$-actions.” Geometric and Functional Analysis 29.5 (2019): 1440-1502.
  * [14] Hurley, Mike. “On proofs of the $C^{0}$ general density theorem.” Proceedings of the American Mathematical Society 124.4 (1996): 1305-1309.
  * [15] Lindenstrauss, Elon, and Masaki Tsukamoto. “Double variational principle for mean dimension.” Geometric and Functional Analysis 29.4 (2019): 1048-1109.
  * [16] Lindenstrauss, Elon, and Masaki Tsukamoto. “From rate distortion theory to metric mean dimension: variational principle.” IEEE Transactions on Information Theory 64.5 (2018): 3590-3609.
  * [17] Lindenstrauss, Elon, and Benjamin Weiss. “Mean topological dimension.” Israel Journal of Mathematics 115.1 (2000): 1-24.
  * [18] Lindenstrauss, Elon, and Masaki Tsukamoto. “Double variational principle for mean dimension.” Geometric and Functional Analysis 29.4 (2019): 1048-1109.
  * [19] Lindenstrauss, Elon. “Mean dimension, small entropy factors and an embedding theorem.” Inst. Hautes Etudes Sci. Publ. Math. 89 (1999), 227–262.
  * [20] Liu, Yu, Bilel Selmi, and Zhiming Li. “On the mean fractal dimensions of the Cartesian product sets.” Chaos, Solitons & Fractals 180 (2024): 114503.
  * [21] Ma, Xianfeng, Junqi Yang, and Ercai Chen. “Mean topological dimension for random bundle transformations”. Ergodic Theory and Dynamical Systems. 2019;39(4):1020-1041.
  * [22] Tsukamoto, Masaki. “Mean dimension of full shifts.” Israel Journal of Mathematics 230.1 (2019): 183-193.
  * [23] Tsukamoto, Masaki. “Mean Hausdorff dimension of some infinite dimensional fractals.” arXiv preprint arXiv:2209.00512 (2022).
  * [24] Velozo, Anibal, and Renato Velozo. “Rate distortion theory, metric mean dimension and measure theoretic entropy.” arXiv preprint arXiv:1707.05762 (2017).
  * [25] Yang, Rui, Ercai Chen, and Xiaoyao Zhou. “Bowen’s equations for upper metric mean dimension with potential.” Nonlinearity 35.9 (2022): 4905.
  * [26] Yano, Koichi. “A remark on the topological entropy of homeomorphisms.” Inventiones mathematicae 59.3 (1980): 215-220.