# Mean dimension of Bernstein spaces
and universal real flows

Lei Jin  Lei Jin: Center for Mathematical Modeling, University of Chile and
UMI 2807 - CNRS<EMAIL_ADDRESS>,  Yixiao Qiao  Yixiao Qiao: School of
Mathematical Sciences, South China Normal University, Guangzhou, Guangdong
510631, China<EMAIL_ADDRESS>and  Siming Tu  Siming Tu: School of
Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong 519082, China
<EMAIL_ADDRESS>
###### Abstract.

We study the action of translation on the spaces of uniformly bounded
continuous functions on the real line which are uniformly band-limited in a
compact interval. We prove that two intervals themselves will decide if two
spaces are topologically conjugate, while the length of an interval tells the
mean dimension of a space. We also investigate universal real flows. We
construct a sequence of compact invariant subsets of a space consisting of
uniformly bounded smooth one-Lipschitz functions on the real line, which have
mean dimension equal to one, such that all real flows can be equivariantly
embedded in the translation on their product space. Moreover, we show that the
countable self-product of any among them does not satisfy such a universal
property. This, on the one hand, presents a more reasonable choice of a
universal real flow with a view towards mean dimension, and on the other hand,
clarifies a seemingly plausible impression; meanwhile, it refines the
previously known results in this direction. Our proof goes through an approach
of harmonic analysis. Furthermore, both the universal space that we provide
and an embedding mapping which we build for any real flow are explicit.

###### Key words and phrases:

Hilbert cube; Bernstein space; Mean dimension; Universal real flow;
Equivariant embedding.

###### 2010 Mathematics Subject Classification:

37B05

##  1\. Main results

This paper is devoted to a study of the translation action on Bernstein spaces
and an alternative universal real flow with a view towards mean dimension
theory. By a real flow (or an $\mathbb{R}$-action) we understand a triple
$(X,\mathbb{R},T)$, where $X$ is a compact metric space and

$T:\mathbb{R}\times X\to X,\quad(t,x)\mapsto T_{t}x$

is a continuous mapping satisfying:

$T_{0}x=x,\quad T_{r+s}x=T_{r}T_{s}x,\quad\forall x\in X,\;\forall
r,s\in\mathbb{R}.$

For two real flows $(X,\mathbb{R},T)$ and $(Y,\mathbb{R},S)$ we say that
$(Y,\mathbb{R},S)$ can be embedded in $(X,\mathbb{R},T)$ if there is an
equivariant topological embedding $\phi:Y\to X$, namely a homeomorphism $\phi$
of $Y$ into $X$ satisfying

$\phi(S_{t}y)=T_{t}\phi(y),\quad\forall t\in\mathbb{R},\;\forall y\in Y;$

if in addition, such an equivariant mapping $\phi$ is a homeomorphism of $Y$
onto $X$, then $(X,\mathbb{R},T)$ and $(Y,\mathbb{R},S)$ are said to be
topologically conjugate (to which some researchers may prefer the terminology
“isomorphic”). A real flow $(X,\mathbb{R},T)$ is called universal if all real
flows can be embedded in $(X,\mathbb{R},T)$.

These terminologies for $\mathbb{R}$-actions may be applied similarly to
discrete flows, i.e. $\mathbb{Z}$-actions. A standard universal
$\mathbb{Z}$-action is the Hilbert cube $([-1,1]^{\mathbb{N}})^{\mathbb{Z}}$
under the $\mathbb{Z}$-translation:

$(x_{k})_{k\in\mathbb{Z}}\longmapsto(x_{k+1})_{k\in\mathbb{Z}},\quad\forall
x_{k}\in[-1,1]^{\mathbb{N}}.$

Note that the Hilbert cube $([-1,1]^{\mathbb{N}})^{\mathbb{Z}}$ is a compact
metric space. For $\mathbb{R}$-actions, an analogue of the Hilbert cube
$([-1,1]^{\mathbb{N}})^{\mathbb{Z}}=([-1,1]^{\mathbb{Z}})^{\mathbb{N}}$ is the
function space $C(\mathbb{R})^{\mathbb{N}}$, where $C(\mathbb{R})$ denotes the
space of continuous functions $f:\mathbb{R}\to[-1,1]$ endowed with the
topology of uniform convergence on compact subsets of $\mathbb{R}$, given by
the distance:

(1.1)
$D(f,g)=\sum_{n=1}^{\infty}\frac{|\\!|f-g|\\!|_{L^{\infty}([-n,n])}}{2^{n}}\quad\quad(f,g\in
C(\mathbb{R})).$

Let the group $\mathbb{R}$ act on $C(\mathbb{R})$ continuously by the
translation

(1.2) $\sigma:\mathbb{R}\times C(\mathbb{R})\to
C(\mathbb{R}),\quad\quad(t,f(\cdot))\mapsto f(\cdot+t).$

In the same way as in $\mathbb{Z}$-actions, we can embed all real flows in the
translation on the product space $C(\mathbb{R})^{\mathbb{N}}$ naturally.
Unfortunately, if we try to consider the translation on
$C(\mathbb{R})^{\mathbb{N}}$ as a “universal real flow” then there is a
problem: The space $C(\mathbb{R})^{\mathbb{N}}$ is not compact, nor locally
compact. So it is not a “real flow” in the definition.

We expect to find a universal real flow as simple as possible. Nevertheless,
it would be less interesting if a universal space is “larger” than the
function space $C(\mathbb{R})^{\mathbb{N}}$. This poses the following
question:

  * •

Is there an “explicit” compact invariant subset of
$C(\mathbb{R})^{\mathbb{N}}$ that is universal?

Here “explicitness” means that we may characterize all elements in a chosen
space easily. Answering the above question positively, Gutman and Jin [GJ19]
successfully constructed a countable product of compact invariant subsets of
$C(\mathbb{R})$, which is universal under the translation.

To state this result in a precise way, we briefly recall some necessary
notions and results in Fourier analysis. A rapidly decreasing function is an
infinitely differentiable function $f$ on $\mathbb{R}$ satisfying

$\lim_{|t|\to+\infty}t^{n}f^{(j)}(t)=0,\quad\forall n,j\in\mathbb{N}.$

A tempered distribution on $\mathbb{R}$ is a continuous linear functional on
the space of all rapidly decreasing functions equipped with the topology given
by a family of seminorms as follows:

$|\\!|f|\\!|_{j,n}=\sup_{t\in\mathbb{R}}|t^{n}f^{(j)}(t)|\quad(j,n\in\mathbb{N}).$

The tempered distributions include in particular bounded continuous functions.
For rapidly decreasing functions $f:\mathbb{R}\to\mathbb{C}$ the definition of
the Fourier transforms of $f$ is given by

$\mathcal{F}(f)(\xi)=\int_{-\infty}^{+\infty}e^{-2\pi\sqrt{-1}t\xi}f(t)dt,\quad\overline{\mathcal{F}}(f)(t)=\int_{-\infty}^{+\infty}e^{2\pi\sqrt{-1}t\xi}f(\xi)d\xi.$

The operators $\mathcal{F}$ and $\overline{\mathcal{F}}$ can be extended to
tempered distributions in a standard way (for details we refer to [Sch66,
Chapter 7]).

Let $I$ be a compact subset of $\mathbb{R}$. A bounded continuous function
$f:\mathbb{R}\to\mathbb{C}$ is band-limited in $I$ if
$\operatorname{\mathrm{supp}}(\mathcal{F}(f))\subset I$, meaning that
$\langle\mathcal{F}(f),g\rangle=0$ for all rapidly decreasing functions
$g:\mathbb{R}\to\mathbb{C}$ with $\operatorname{\mathrm{supp}}(g)\cap
I=\emptyset$. We denote by $\mathcal{B}^{\mathbb{C}}(I)$ (resp.
$\mathcal{B}(I)$) the set of continuous functions $f:\mathbb{R}\to\mathbb{C}$
(resp. $f:\mathbb{R}\to\mathbb{R}$) band-limited in $I$ with
$|\\!|f|\\!|_{L^{\infty}(\mathbb{R})}\leq 1$. Clearly, both
$\mathcal{B}^{\mathbb{C}}(I)$ and $\mathcal{B}(I)$ are invariant under the
translation $\sigma$ defined in (1.2). An important and nontrivial fact [GT20,
Lemma 2.3][Sch66, Chapter 7, Section 4] is that if $I\subset\mathbb{R}$ is
compact then $\mathcal{B}^{\mathbb{C}}(I)$ and $\mathcal{B}(I)$ are compact
metric spaces with respect to the distance $D$ given in (1.1) which coincides
with the standard topology of tempered distributions. Thus,
$(\mathcal{B}^{\mathbb{C}}(I),\mathbb{R},\sigma)$ and
$(\mathcal{B}(I),\mathbb{R},\sigma)$ become real flows.

######  Remark 1.1.

The distance $D$ defined in (1.1) and the translation $\sigma$ defined in
(1.2) should be understood a distance and an action of $\mathbb{R}$,
respectively, on the space of continuous functions $f:\mathbb{R}\to\mathbb{C}$
(which is larger than $C(\mathbb{R})$). We do not change the notation here as
it does not cause any confusion.

######  Theorem 1.2 ([GJ19, Theorem 1.2]).

Under the translation $\sigma$ the space
$\prod_{n\in\mathbb{N}}\mathcal{B}([-n,n])$ is universal.

Although Theorem 1.2 provides an affirmative solution to the above question,
it is not so satisfactory because in contrast to the Hilbert cube
$([-1,1]^{\mathbb{Z}})^{\mathbb{N}}$, the universal space
$\prod_{n\in\mathbb{N}}\mathcal{B}([-n,n])$ appearing in Theorem 1.2 is not a
self-product. To proceed, we would like to seek a universal space which is
“closer” to the Hilbert cube $([-1,1]^{\mathbb{Z}})^{\mathbb{N}}$. Hence a
natural question arises as follows:

  * •

Is there an explicit compact invariant subset $F$ of $C(\mathbb{R})$ such that
$F^{\mathbb{N}}$ is universal?

Under this motivation, Jin and Tu [JT19] found that the one-Lipschitz function
space is a solution.

Formally, we let $L(\mathbb{R})$ be the set of all functions
$f:\mathbb{R}\to[-1,1]$ with the following property:

$|f(s)-f(t)|\leq|s-t|,\quad\forall s,t\in\mathbb{R}.$

Obviously, $L(\mathbb{R})$ is an invariant subset of $C(\mathbb{R})$ under the
translation $\sigma$ in (1.2). Moreover, by the Arzela–Ascoli theorem,
$L(\mathbb{R})$ is a compact metric space with respect to the distance $D$ in
(1.1).

######  Theorem 1.3 ([JT19, Theorem 1.2]).

Under the translation $\sigma$ the space $L(\mathbb{R})^{\mathbb{N}}$ is
universal.

However, we are still not satisfied with Theorem 1.3 in spite of the fact that
$L(\mathbb{R})^{\mathbb{N}}$ is a countable self-product of
$L(\mathbb{R})\subset C(\mathbb{R})$, as a deeper observation reveals a more
serious problem: The mean dimension of $L(\mathbb{R})$ (under the translation
$\sigma$) is $+\infty$, while in the Hilbert cube
$([-1,1]^{\mathbb{Z}})^{\mathbb{N}}$ the mean dimension of
$[-1,1]^{\mathbb{Z}}$ (under the $\mathbb{Z}$-translation) is $1$, a finite
number. We shall have a more detailed explanation for mean dimension in
Section 2, Section 3 and Section 4. This is an essential difference between
$L(\mathbb{R})^{\mathbb{N}}$ and the Hilbert cube
$([-1,1]^{\mathbb{Z}})^{\mathbb{N}}$. From this point of view, the “size” of
the space $L(\mathbb{R})$ that we selected in the above solution is indeed too
“large”. Thus, we require a better candidate substantially. More precisely, we
put a further problem:

  * •

Is there an explicit compact invariant subset $F\subset C(\mathbb{R})$ of
finite mean dimension such that $F^{\mathbb{N}}$ is universal?

This problem is temporarily beyond the authors’ reach. The aim of the present
paper is to give a positive answer to a slightly weaker statement which
strengthens Theorem 1.2 with a closer analogue of the Hilbert cube
$([-1,1]^{\mathbb{Z}})^{\mathbb{N}}$ and a more direct construction of an
embedding mapping, where we choose the spaces
$\mathcal{B}([-n/3-1/2,-n/3]\cup[n/3,n/3+1/2])$ (for nonnegative integers $n$)
because their mean dimension are equal to $1$, the same as the mean dimension
of $[-1,1]^{\mathbb{Z}}$. We notice that the mean dimension of those spaces in
Theorem 1.2 are not uniformly bounded by a finite number.

######  Theorem 1.4.

Under the translation $\sigma$ the space

$\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\mathcal{B}([-n/3-1/2,-n/3]\cup[n/3,n/3+1/2])$

is universal.

Furthermore, we have the following refinement which unifies the previously
known results in this direction. Since for any compact $I\subset\mathbb{R}$
both $\mathcal{B}(I)$ and $L(\mathbb{R})$ are compact metric spaces with
respect to the distance $D$, their intersection $\mathcal{B}(I)\cap
L(\mathbb{R})$ is a compact metric space as well. Thus, $\mathcal{B}(I)\cap
L(\mathbb{R})$ becomes also a compact invariant subset of $C(\mathbb{R})$
under the translation $\sigma$.

######  Theorem 1.5 (Main theorem 1).

For any real numbers $0<\alpha<\beta$ the space

$\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\left(\mathcal{B}([-n\alpha-\beta,-n\alpha]\cup[n\alpha,n\alpha+\beta])\cap
L(\mathbb{R})\right)$

is universal under the translation $\sigma$.

######  Remark 1.6.

It is clear that Theorem 1.4 follows directly from Theorem 1.5. As a seemingly
reachable question we may ask if it can be strengthened with a countable self-
product of a member among them. Unfortunately, this is not correct. In fact,
we shall clarify some wrong impression in Section 6:

  * •

Under the translation $\sigma$ the space $\mathcal{B}([-1,1])^{\mathbb{N}}$ is
not universal.

  * •

Under the translation $\sigma$ the space
$\prod_{n=0}^{+\infty}\mathcal{B}([-n/3-1/2,-n/3]\cup[n/3,n/3+1/2])$ is not
universal.

We denote by $C^{\infty}(\mathbb{R})$ the set of smooth (i.e. infinitely
differentiable) functions $f:\mathbb{R}\to[-1,1]$. Note that the space
$C^{\infty}(\mathbb{R})\cap L(\mathbb{R})\subset C(\mathbb{R})$ is compact and
invariant (under the translation $\sigma$).

######  Corollary 1.7.

Under the translation $\sigma$ the space $\left(C^{\infty}(\mathbb{R})\cap
L(\mathbb{R})\right)^{\mathbb{N}}$ is universal.

As a quantitative result complementary to Theorem 1.5, we have the following
classification of the real flows appearing in Theorem 1.4 under topological
conjugacy and mean dimension. Note that for any positive real number $r$ the
mean dimension of the translation on $\mathcal{B}([-r,r])\cap L(\mathbb{R})$
is finite. Therefore Theorem 1.5 provides a more reasonable choice of a
universal real flow.

######  Theorem 1.8 (Main theorem 2).

Let $I,J\subset\mathbb{R}$ be compact intervals. Let $a\leq b$ and $c\geq 0$
be real numbers. The following assertions are true:

  1. (1)

$(\mathcal{B}^{\mathbb{C}}(I),\mathbb{R},\sigma)$ is topologically conjugate
to $(\mathcal{B}^{\mathbb{C}}(J),\mathbb{R},\sigma)$ if and only if $I=J$ or
$I=-J$.

  2. (2)

$(\mathcal{B}(I),\mathbb{R},\sigma)$ is topologically conjugate to
$(\mathcal{B}(J),\mathbb{R},\sigma)$ if and only if $I\cap(-I)=J\cap(-J)$.

  3. (3)

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{R},\sigma)=2(b-a)$.

  4. (4)

$\operatorname{\mathrm{mdim}}(\mathcal{B}([-c,c]),\mathbb{R},\sigma)=2c$.

The definition of mean dimension is located in Section 2. We situate in
Section 3 a short discussion about the universality of $L(\mathbb{R})$ (for
$\mathbb{Z}$-actions), where we will indicate in particular that the mean
dimension of $L(\mathbb{R})$ is $+\infty$. We will prove Theorem 1.8 in
Section 4. In Section 5, we shall give a constructive proof of Theorem 1.5. A
novelty of our method is that it overcomes a shortcoming arising from the
Baire category approach. As presented in our proof, we are able to see an
explicitly constructed embedding mapping of any real flow in the universal
real flow that we suggested (which is also explicit) in the main theorem.
Section 6 contains a collection of explanations in relation to our results
(including the corollary), in which a final remark will end the body of this
paper with the impossibility of improving our main result in a seemingly
achievable direction. An appendix is logically independent of all the
sections.

Acknowledgements. The authors are grateful to the anonymous referee for
his/her insightful comments and helpful suggestions which improve this paper
greatly. L. Jin was supported by Basal Funding AFB 170001 and Fondecyt Grant
No. 3190127, and was partially supported by NNSF of China No. 11971455. Y.
Qiao was supported by NNSF of China No. 11901206. S. Tu was supported by NNSF
of China No. 11801584 and 11871228.

##  2\. Preliminaries for mean dimension

Mean dimension was introduced by Gromov [Gro99] in 1999, and was
systematically studied by Lindenstrauss and Weiss [LW00] in 2000 as a
topological invariant of dynamical systems. We collect in this section
fundamental material (borrowed from [LW00, GJ20]) on mean dimension for
$\mathbb{R}$-actions and $\mathbb{Z}$-actions.

Let $(X,d)$ be a compact metric space and $\epsilon>0$. We denote by
$\operatorname{\mathrm{Widim}}_{\epsilon}(X,d)$ the minimum topological
dimension $\dim(K)$ (i.e. the Lebesgue covering dimension) of a compact
metrizable space $K$ which admits an $\epsilon$-embedding $f:X\to K$ with
respect to the distance $d$, namely, a continuous mapping $f:X\to K$
satisfying that $f(x)=f(x^{\prime})$ implies $d(x,x^{\prime})<\epsilon$ for
all $x,x^{\prime}\in X$. We may easily verify:

$\dim(X)=\lim_{\epsilon\to 0}\operatorname{\mathrm{Widim}}_{\epsilon}(X,d).$

Let $(X,\mathbb{R},T)$ be a real flow and $d$ a compatible metric on $X$. For
each nonnegative real number $r$ we define a compatible metric $d_{r}^{T}$ on
$X$ by

$d_{r}^{T}(x,x^{\prime})=\max_{0\leq t\leq
r}d(T_{t}x,T_{t}x^{\prime}),\quad\forall x,x^{\prime}\in X.$

The mean dimension of $(X,\mathbb{R},T)$ is defined by

$\operatorname{\mathrm{mdim}}(X,\mathbb{R},T)=\lim_{\epsilon\to
0}\lim_{r\to+\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(X,d_{r}^{T})}{r}.$

The limits in the definition exist, and the value
$\operatorname{\mathrm{mdim}}(X,\mathbb{R},T)\in[0,+\infty]$ does not depend
on the choice of a compatible metric $d$ on $X$. Clearly, if
$(Y,\mathbb{R},S)$ can be embedded in $(X,\mathbb{R},T)$ then

$\operatorname{\mathrm{mdim}}(Y,\mathbb{R},S)\leq\operatorname{\mathrm{mdim}}(X,\mathbb{R},T).$

For every $\lambda\in\mathbb{R}$ we denote by $(X,\mathbb{R},T^{\lambda})$ the
real flow with the following action:

$T^{\lambda}:\mathbb{R}\times X\to X,\quad(t,x)\mapsto T_{\lambda t}x.$

######  Lemma 2.1.

For any real flow $(X,\mathbb{R},T)$ and $\lambda\in\mathbb{R}$

$\operatorname{\mathrm{mdim}}(X,\mathbb{R},T^{\lambda})=|\lambda|\cdot\operatorname{\mathrm{mdim}}(X,\mathbb{R},T).$

For a $\mathbb{Z}$-action $(X,\mathbb{Z},\phi)$, i.e. $(X,d)$ is a compact
metric space and $\phi:X\to X$ is a homeomorphism, and a positive integer $n$
we define a compatible metric $d_{n}^{\phi}$ on $X$ by

$d_{n}^{\phi}(x,x^{\prime})=\max_{i\in\mathbb{Z},\,0\leq i\leq
n-1}d(\phi^{i}x,\phi^{i}x^{\prime}),\quad\forall x,x^{\prime}\in X.$

We recall that the mean dimension of $(X,\mathbb{Z},\phi)$ is similarly
defined by

$\operatorname{\mathrm{mdim}}(X,\mathbb{Z},\phi)=\lim_{\epsilon\to
0}\lim_{n\to+\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(X,d_{n}^{\phi})}{n}.$

The $\mathbb{Z}$-translation on $([0,1]^{d})^{\mathbb{Z}}$ (where
$d\in\mathbb{N}\cup\\{+\infty\\}$) is defined by

$\sigma:([0,1]^{d})^{\mathbb{Z}}\to([0,1]^{d})^{\mathbb{Z}},\quad(x_{i})_{i\in\mathbb{Z}}\mapsto(x_{i+1})_{i\in\mathbb{Z}}.$

We do not change the notation $\sigma$ here because there is no ambiguity (as
the acting group is always indicated precisely in the midst of a triple).

######  Lemma 2.2.

For every $d\in\mathbb{N}\cup\\{+\infty\\}$

$\operatorname{\mathrm{mdim}}(([0,1]^{d})^{\mathbb{Z}},\mathbb{Z},\sigma)=d.$

We note that a real flow $(X,\mathbb{R},T)$ naturally induces a
$\mathbb{Z}$-action $(X,\mathbb{Z},T_{1})$.

######  Lemma 2.3.

For any real flow $(X,\mathbb{R},T)$

$\operatorname{\mathrm{mdim}}(X,\mathbb{R},T)=\operatorname{\mathrm{mdim}}(X,\mathbb{Z},T_{1}).$

We would like to remind the reader to keep in mind that all the statements in
this section will be used implicitly in this paper.

##  3\. Universality of $L(\mathbb{R})$ for $\mathbb{Z}$-actions

The purpose of this section is to show that under the translation $\sigma$ the
space $L(\mathbb{R})$ has mean dimension $+\infty$. The main result of this
section is Theorem 3.1.

######  Theorem 3.1.

All $\mathbb{Z}$-actions can be embedded in the $\mathbb{Z}$-translation on
$L(\mathbb{R})$.

In other words, $(L(\mathbb{R}),\mathbb{Z},\sigma_{1})$ is a universal
$\mathbb{Z}$-action. It follows directly from Theorem 3.1 that

$\operatorname{\mathrm{mdim}}(L(\mathbb{R}),\mathbb{R},\sigma)=\operatorname{\mathrm{mdim}}(L(\mathbb{R}),\mathbb{Z},\sigma_{1})=+\infty.$

In order to prove Theorem 3.1, we employ a result for real flows, due to
Gutman, Jin and Tsukamoto [GJT19, Theorem 1.3], which refines the classical
Bebutov–Kakutani dynamical embedding theorem, as follows:

######  Theorem 3.2 ([GJT19]).

A real flow $(X,\mathbb{R},T)$ can be embedded in
$(L(\mathbb{R}),\mathbb{R},\sigma)$ if and only if the set of its fixed points
$\\{x\in X:T_{t}x=x,\,\forall t\in\mathbb{R}\\}$ can be (topologically)
embedded in $[0,1]$.

We remark here that it is also possible to give an elementary proof of Theorem
3.1 (without applying Theorem 3.2). In fact, it suffices to notice that
$[0,1]^{\mathbb{N}}$ is (topologically) embedded in

$L_{0}(\mathbb{R}/\mathbb{Z})=\\{f:\mathbb{R}/\mathbb{Z}\to[0,1]:\,f(0)=0,\,\;|f(x)-f(x^{\prime})|\leq|x-x^{\prime}|,\;\forall
x,x^{\prime}\in\mathbb{R}/\mathbb{Z}\\}.$

However, we go through a more abstract approach as we would like to present a
connection between real flows and $\mathbb{Z}$-actions. Now we prove Theorem
3.1.

###### Proof.

We take a $\mathbb{Z}$-action $(X,\mathbb{Z},\phi)$. Let $(S(X),\mathbb{R},T)$
be the suspension over $(X,\mathbb{Z},\phi)$ generated by the constant
function $1$, namely,

$S(X)=(X\times[0,1])/\sim$

where $\sim$ is the equivalence relation given by $(x,1)\sim(\phi(x),0)$,

$T:\mathbb{R}\times S(X)\to
S(X),\quad(t,(x,s))\mapsto(\phi^{n}(x),s^{\prime})$

where $n\in\mathbb{N}$ and $s^{\prime}\in[0,1]$ satisfy $n+s^{\prime}=t+s$.

It is clear that the real flow $(S(X),\mathbb{R},T)$ has no fixed points. By
Theorem 3.2 we know that $(S(X),\mathbb{R},T)$ can be embedded in
$(L(\mathbb{R}),\mathbb{R},\sigma)$. Thus, $(X,\mathbb{Z},\phi)$ can be
embedded in $(L(\mathbb{R}),\mathbb{Z},\sigma_{1})$. ∎

######  Remark 3.3.

Theorems 1.3, 3.1 and 3.2 reveal a difference between $\mathbb{R}$-actions and
$\mathbb{Z}$-actions. The space $L(\mathbb{R})$ (under the
$\mathbb{Z}$-translation) encompasses all $\mathbb{Z}$-actions, while (under
the $\mathbb{R}$-translation) it is far from universal for
$\mathbb{R}$-actions. As we see in Theorem 3.2, the fixed-point set of an
$\mathbb{R}$-action is the only obstacle and decides its embeddability in
$L(\mathbb{R})$. In particular, any $\mathbb{R}$-action containing no fixed
points can be embedded in $L(\mathbb{R})$. Unfortunately, this could never
happen between $\mathbb{Z}$-actions and the Hilbert cube $[0,1]^{\mathbb{Z}}$
(under the $\mathbb{Z}$-translation). Moreover it turns out that mean
dimension [Gro99, LW00] becomes crucial for the embedding problem of
$\mathbb{Z}$-actions. As we mentioned previously, a key difference is that
$L(\mathbb{R})$ has infinite mean dimension while the Hilbert cube
$[0,1]^{\mathbb{Z}}$ has finite mean dimension. We do not plan to involve more
detailed explanation in this paper because it is not the main topic in
relation to our purpose here. For the latest progress in this direction we
refer to [GT20, GQT19]. For $\mathbb{R}$-actions we can prove (with an
argument essentially the same as in [GJT19, JT19]) the following theorem:

  * •

A real flow $(X,\mathbb{R},T)$ can be embedded in
$(L(\mathbb{R})^{N},\mathbb{R},\sigma)$, where
$N\in\mathbb{N}\cup\\{+\infty\\}$, if and only if the set of its fixed points
$\\{x\in X:T_{t}x=x,\,\forall t\in\mathbb{R}\\}$ can be (topologically)
embedded in $[0,1]^{N}$.

##  4\. Mean dimension of Bernstein spaces

The goal of this section is to show that under the translation $\sigma$ the
space $\mathcal{B}([-r,r])$, and hence the space $\mathcal{B}([-r,r])\cap
L(\mathbb{R})$, has finite mean dimension (where $r$ is a nonnegative real
number). The main theorem of this section is Theorem 4.4.

Let us start with necessary notions as a continuation of Section 1. For a
tempered distribution $f$ its Fourier transforms $\mathcal{F}(f)$ and
$\overline{\mathcal{F}}(f)$ are defined by

$\langle\mathcal{F}(f),g\rangle=\langle
f,\overline{\mathcal{F}}(g)\rangle,\quad\langle\overline{\mathcal{F}}(f),g\rangle=\langle
f,\mathcal{F}(g)\rangle,$

where $g$ ranges over all rapidly decreasing functions. For example, we have
$\mathcal{F}(0)=0$, $\mathcal{F}(e^{2\pi\sqrt{-1}\tau\cdot})=\delta_{\tau}$
(i.e. the delta probability measure at the point $\tau\in\mathbb{R}$), and
$\overline{\mathcal{F}}(\mathcal{F}(f))=\mathcal{F}(\overline{\mathcal{F}}(f))=f$.

We would like to remark here that for any real-valued bounded continuous
function $f$ on $\mathbb{R}$ we may verify that
$\operatorname{\mathrm{supp}}(\mathcal{F}(f))$ must be a symmetric subset of
$\mathbb{R}$. In fact, for a compact interval $I\subset\mathbb{R}$ and any
$f\in\mathcal{B}(I)$ we have

$\operatorname{\mathrm{supp}}(\mathcal{F}(f))=\operatorname{\mathrm{supp}}(\mathcal{F}(\overline{f}))=-\operatorname{\mathrm{supp}}(\mathcal{F}(f)).$

Namely $\mathcal{B}(I)$ is equal to $\mathcal{B}(I\cap(-I))$. So it is better
to fix the notation $\mathcal{B}([-c,c])$ (where $c\geq 0$) for simplicity,
and apparently, Theorem 4.4 and Theorem 1.8 are equivalent.

######  Remark 4.1.

For a compact interval $[a,b]\subset\mathbb{R}$ we usually call the space of
(complex-valued) bounded continuous functions on $\mathbb{R}$ band-limited in
$[a,b]$ a Bernstein space. This is a Banach space (with respect to the
$L^{\infty}$-norm over $\mathbb{R}$). Strictly speaking, the title of this
section is somewhat misleading, as $\mathcal{B}^{\mathbb{C}}([a,b])$ and
$\mathcal{B}([-c,c])$ ($c\geq 0$) themselves are not Bernstein spaces. But
they are compact subsets of a Bernstein space. It is worth mentioning that we
are interested in the compact metric spaces $\mathcal{B}^{\mathbb{C}}([a,b])$
and $\mathcal{B}([-c,c])$ because they have a background of deep applications
to dynamical systems, and were heavily used in the embedding problem of
$\mathbb{Z}^{k}$-actions (for details see [GT20, GQT19]). Giving a dynamical
classification is valuable. The value of their mean dimension was first
announced in [GT20] (without a proof).

We shall need brief preparations in front of the main result of this section.
Our tools (borrowed from [GT20] with a slight modification in the statement)
are sampling (Lemma 4.2) and interpolation (Lemma 4.3).

######  Lemma 4.2.

Suppose that two positive real numbers $a$ and $d$ satisfy $2ad<1$ and
$f\in\mathcal{B}([-a,a])$. If $f(dn)=0$ for all $n\in\mathbb{Z}$, then
$f\equiv 0$.

######  Lemma 4.3.

For every $\epsilon>0$ there exists a rapidly decreasing function
$f:\mathbb{R}\to\mathbb{R}$ band-limited in $[-(1+\epsilon)/2,(1+\epsilon)/2]$
such that $f(0)=1$ and $f(n)=0$ for all nonzero $n\in\mathbb{Z}$.

######  Theorem 4.4.

Let $I,J,K,H\subset\mathbb{R}$ be compact intervals, where $K$ and $H$ are
symmetric (i.e. $K=-K$ and $H=-H$). Let $a\leq b$ and $c\geq 0$ be real
numbers. The following assertions are true:

  1. (1)

$(\mathcal{B}^{\mathbb{C}}(I),\mathbb{R},\sigma)$ is topologically conjugate
to $(\mathcal{B}^{\mathbb{C}}(J),\mathbb{R},\sigma)$ if and only if $I=J$ or
$I=-J$.

  2. (2)

$(\mathcal{B}(K),\mathbb{R},\sigma)$ is topologically conjugate to
$(\mathcal{B}(H),\mathbb{R},\sigma)$ if and only if $K=H$.

  3. (3)

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{R},\sigma)=\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{Z},\sigma_{1})=2(b-a)$.

  4. (4)

$\operatorname{\mathrm{mdim}}(\mathcal{B}([-c,c]),\mathbb{R},\sigma)=\operatorname{\mathrm{mdim}}(\mathcal{B}([-c,c]),\mathbb{Z},\sigma_{1})=2c$.

###### Proof.

In the beginning of the proof we would like to remind the reader that the
proofs of (4) and (3) are independent of (2) and (1), and we shall use (4) and
(3) when proving (2) and (1).

(2): The “if” part “$\Longleftarrow$” of the statement is obvious. To see the
“only if” part “$\Longrightarrow$”, we note that the topological conjugacy
implies that the two real flows have the same mean dimension. By (4) and the
fact that both $K$ and $H$ are symmetric compact intervals, we deduce $K=H$.
The proof for (2) is simple (and moreover, it holds if and only if $|K|=|H|$
if and only if
$\operatorname{\mathrm{mdim}}(\mathcal{B}(K),\mathbb{R},\sigma)=\operatorname{\mathrm{mdim}}(\mathcal{B}(H),\mathbb{R},\sigma)$),
because a symmetric compact interval is fully decided by its length. However,
this will be a problem for an arbitrary compact interval in the proof of (1).

(1): We first note that the “if” part “$\Longleftarrow$” is easy, because if
$I=J$ or $I=-J$ then $(\mathcal{B}^{\mathbb{C}}(I),\mathbb{R},\sigma)$ is
topologically conjugate to $(\mathcal{B}^{\mathbb{C}}(J),\mathbb{R},\sigma)$
with the mapping: $f\mapsto f$ or $f\mapsto\overline{f}$. So we now prove the
“only if” part “$\Longrightarrow$”.

We suppose that $(\mathcal{B}^{\mathbb{C}}(I),\mathbb{R},\sigma)$ is
topologically conjugate to $(\mathcal{B}^{\mathbb{C}}(J),\mathbb{R},\sigma)$.
It follows that the two real flows must have the same value of mean dimension.
According to (3) the compact intervals $I$ and $J$ must have the same length
$|I|=|J|$. Let us assume $I\neq J$ and $I\neq-J$. This implies that there
exist, without loss of generality, a positive real number $\tau\in I$ and a
nonnegative real number $\gamma\in J$ satisfying that $\tau>\gamma$ and
$J\subset[-\gamma,\gamma]$.

We consider the function $f(\cdot)=e^{2\pi\sqrt{-1}\tau\cdot}$ defined on
$\mathbb{R}$. Note that $\mathcal{F}(f)=\delta_{\tau}$ which implies
$f\in\mathcal{B}^{\mathbb{C}}(I)$. Clearly, $f$ is a periodic function, and
its (fundamental) period is $T=1/\tau>0$. Thus, by equivariance, its image
(under an embedding mapping) $g\in\mathcal{B}^{\mathbb{C}}(J)$ is also a
periodic function such that $g(t+T)=g(t)$ for all $t\in\mathbb{R}$. We write
the complex-valued function $g$ on $\mathbb{R}$ as $g=h+k\cdot\sqrt{-1}$,
where $h$ and $k$ are real-valued functions on $\mathbb{R}$. Since
$h=(g+\overline{g})/2$, we have
$h\in\mathcal{B}(J\cup(-J))\subset\mathcal{B}([-\gamma,\gamma])$. The function
$k=(g-\overline{g})/2\sqrt{-1}$ also belongs to
$\mathcal{B}([-\gamma,\gamma])$. So the following argument, which deals with
$h$, applies to $k$ as well.

Since $h\in\mathcal{B}([-\gamma,\gamma])$ and since $h(t+T)=h(t)$ for all
$t\in\mathbb{R}$, the Fourier series representation of the periodic function
$h$ on $\mathbb{R}$ (being in particular a restriction of a holomorphic
function on $\mathbb{C}$)

$h(t)=\sum_{n=-\infty}^{+\infty}c_{n}\cdot e^{2\pi\sqrt{-1}nt/T},\quad\forall
t\in\mathbb{R},$

with $c_{-n}=\overline{c_{n}}$ for any $n\in\mathbb{Z}$, converges uniformly
to $h$. In fact, we use here a generalization (for bounded continuous
functions on the real line) of the classical Paley–Wiener theorem (we refer to
Section 6 for its precise statement). It follows that

$\mathcal{F}(h)=c_{0}\mathcal{F}(1)+\sum_{n=1}^{+\infty}c_{n}\mathcal{F}(e^{2\pi\sqrt{-1}n\cdot/T})+\overline{c_{n}}\mathcal{F}(e^{-2\pi\sqrt{-1}n\cdot/T}).$

Since $h$ is band-limited in $[-\gamma,\gamma]$, we have $c_{n}=0$ for any
$n\in\mathbb{Z}$ with $|n|>\gamma T$. Since $0\leq\gamma T=\gamma/\tau<1$, we
finally deduce $c_{n}=0$ for all nonzero $n\in\mathbb{Z}$. This implies that
$h$ is a constant function.

Similarly, so is $k$. Thus, we conclude that $g=h+k\cdot\sqrt{-1}$ (which is
the image of $f$) must be a constant function. This, however, contradicts the
injectivity of an embedding mapping.

Here we would like to remark shortly that by the sampling lemma we may show
that the space $\mathcal{B}([-\gamma,\gamma])$ does not contain a function
whose fundamental period is strictly less than $T/2$. But this is far from
adequate for our argument (as the coefficient $1/2$ with $T$ is unpleasant).
So we have to go through Fourier series representations of those functions in
$\mathcal{B}([-\gamma,\gamma])$ in the proof.

(4) & (3): First of all, we should note that both
$\mathcal{B}^{\mathbb{C}}(\\{0\\})$ and $\mathcal{B}(\\{0\\})$ contain only
constant functions, and thus, their mean dimension (under the translation
$\sigma$) are equal to zero. As follows we build three lemmas which will
reduce the statement of (4) and (3) to a standard case.

######  Lemma 4.5.

For any $c>0$ and any $\lambda>0$

$\operatorname{\mathrm{mdim}}(\mathcal{B}([-\lambda c,\lambda
c]),\mathbb{R},\sigma)=\lambda\cdot\operatorname{\mathrm{mdim}}(\mathcal{B}([-c,c]),\mathbb{R},\sigma).$

###### Proof.

We omit the proof of this lemma because it is absolutely the same as the lemma
below. A bridge is
$\operatorname{\mathrm{mdim}}(\mathcal{B}([-c,c]),\mathbb{R},\sigma^{\lambda})$.
∎

######  Lemma 4.6.

For any $a<b$ and any $\lambda>0$

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([\lambda a,\lambda
b]),\mathbb{R},\sigma)=\lambda\cdot\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{R},\sigma).$

###### Proof.

For each $a<b$ and each $\lambda>0$ we consider the mapping:

$\rho:\mathcal{B}^{\mathbb{C}}([a,b])\to\mathcal{B}^{\mathbb{C}}([\lambda
a,\lambda b]),\quad f(t)\mapsto f(\lambda t).$

Note here that for any bounded continuous function $f:\mathbb{R}\to\mathbb{C}$
band-limited in $[a,b]$ we have

$\operatorname{\mathrm{supp}}(\mathcal{F}(f(\lambda\cdot)))\subset\lambda\operatorname{\mathrm{supp}}(\mathcal{F}(f))\subset[\lambda
a,\lambda b].$

Clearly, the mapping $\rho$ is continuous and injective, and hence (by
compactness) is a topological embedding mapping. Moreover, $\rho$ is
equivariant because we have for any $f\in\mathcal{B}^{\mathbb{C}}([a,b])$ and
any $r\in\mathbb{R}$

$\rho\circ(\sigma^{\lambda})_{r}f(t)=\rho\circ\sigma_{\lambda r}f(t)=f(\lambda
t+\lambda r)=f(\lambda(t+r))=\sigma_{r}\circ\rho f(t),\quad\forall
t\in\mathbb{R}.$

It follows that
$(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{R},\sigma^{\lambda})$ can be
embedded in $(\mathcal{B}^{\mathbb{C}}([\lambda a,\lambda
b]),\mathbb{R},\sigma)$. This implies that

$\lambda\cdot\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{R},\sigma)=\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{R},\sigma^{\lambda})\leq\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([\lambda
a,\lambda b]),\mathbb{R},\sigma).$

This inequality also applies to $\lambda a<\lambda b$ and $1/\lambda>0$
(instead of $a<b$ and $\lambda>0$, respectively). Thus we conclude

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([\lambda a,\lambda
b]),\mathbb{R},\sigma)=\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{R},\sigma^{\lambda})=\lambda\cdot\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{R},\sigma).$

∎

######  Lemma 4.7.

For every $a<b$ and every $\tau\in\mathbb{R}$

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a+\tau,b+\tau]),\mathbb{R},\sigma)=\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{R},\sigma).$

###### Proof.

If $\tau=0$ then the statement is trivial. So we fix $a<b$ and a nonzero real
number $\tau$ arbitrarily. We first notice that for any
$f\in\mathcal{B}^{\mathbb{C}}([a,b])$ the function $e^{2\pi\sqrt{-1}\cdot}f$
is band-limited in

$\operatorname{\mathrm{supp}}(\mathcal{F}(e^{2\pi\sqrt{-1}\cdot})\ast\mathcal{F}(f))\subset\operatorname{\mathrm{supp}}(\delta_{1})+\operatorname{\mathrm{supp}}(\mathcal{F}(f))\subset[1+a,1+b].$

Since $(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{Z},\sigma_{1})$ can be
embedded in $(\mathcal{B}^{\mathbb{C}}([a+1,b+1]),\mathbb{Z},\sigma_{1})$
(which are $\mathbb{Z}$-actions) with the mapping

$\mathcal{B}^{\mathbb{C}}([a,b])\to\mathcal{B}^{\mathbb{C}}([a+1,b+1]),\quad
f(t)\mapsto e^{2\pi\sqrt{-1}t}f(t),$

which also applies to $a+1<b+1$ and $-1$ (instead of $a<b$ and $1$,
respectively), we deduce

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{Z},\sigma_{1})=\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a+1,b+1]),\mathbb{Z},\sigma_{1})$

which implies

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{R},\sigma)=\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a+1,b+1]),\mathbb{R},\sigma).$

In contrast to the above two lemmas (where the proof showed an
$\mathbb{R}$-equivariant embedding mapping), here the point is that
$e^{2\pi\sqrt{-1}\cdot}$ is a periodic function and its fundamental period is
$1$, which allows us to obtain an embedding mapping which is
$\mathbb{Z}$-equivariant rather than $\mathbb{R}$-equivariant. Without loss of
generality we now assume $\tau>0$. Applying the above equality together with
the previous lemma we conclude

$\displaystyle\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a+\tau,b+\tau]),\mathbb{R},\sigma)$
$\displaystyle=$
$\displaystyle\tau\cdot\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([1+a/\tau,1+b/\tau]),\mathbb{R},\sigma)$
$\displaystyle=$
$\displaystyle\tau\cdot\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a/\tau,b/\tau]),\mathbb{R},\sigma)$
$\displaystyle=$
$\displaystyle\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([a,b]),\mathbb{R},\sigma).$

∎

Thus, it suffices to prove the following assertion:

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([0,1/2]),\mathbb{Z},\sigma_{1})=\operatorname{\mathrm{mdim}}(\mathcal{B}([-1/2,1/2]),\mathbb{Z},\sigma_{1})=1.$

So in the remaining part of the proof we are going to show the following three
inequalities, respectively, which will end our argument:

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([0,1/2]),\mathbb{Z},\sigma_{1})\leq\operatorname{\mathrm{mdim}}(\mathcal{B}([-1/2,1/2]),\mathbb{Z},\sigma_{1}),$
$\operatorname{\mathrm{mdim}}(\mathcal{B}([-1/2,1/2]),\mathbb{Z},\sigma_{1})\leq
1,$
$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([0,1/2]),\mathbb{Z},\sigma_{1})\geq
1.$

To show the first inequality, we take $0<\epsilon<1/2$ arbitrarily. We
consider a continuous mapping:

$H_{\epsilon}:\mathcal{B}^{\mathbb{C}}([\epsilon,1/2])\to\mathcal{B}([-1/2,1/2]),\quad
f\mapsto\frac{1}{2}(f+\overline{f}).$

Note that for any $f\in\mathcal{B}^{\mathbb{C}}([\epsilon,1/2])$ the function
$(f+\overline{f})/2$ is real-valued and is band-limited in

$\operatorname{\mathrm{supp}}(\mathcal{F}(f)+\mathcal{F}(\overline{f}))\subset\operatorname{\mathrm{supp}}(\mathcal{F}(f))\cup(-\operatorname{\mathrm{supp}}(\mathcal{F}(f)))\subset[-1/2,-\epsilon]\cup[\epsilon,1/2].$

Clearly, $H_{\epsilon}$ is equivariant, i.e. it satisfies
$H_{\epsilon}\circ\sigma_{1}=\sigma_{1}\circ H_{\epsilon}$. For every
$f,g\in\mathcal{B}^{\mathbb{C}}([\epsilon,1/2])$ with
$H_{\epsilon}(f)=H_{\epsilon}(g)$ we have $f-g=\overline{g-f}$. Since
$\operatorname{\mathrm{supp}}(\mathcal{F}(f-g))\subset[\epsilon,1/2]$ and
$\operatorname{\mathrm{supp}}(\mathcal{F}(\overline{g-f}))\subset[-1/2,-\epsilon]$,
we get $\mathcal{F}(f-g)=0$ and hence $f=g$. Therefore $H_{\epsilon}$ is
injective. This shows that
$(\mathcal{B}^{\mathbb{C}}([\epsilon,1/2]),\mathbb{Z},\sigma_{1})$ can be
embedded in $(\mathcal{B}([-1/2,1/2]),\mathbb{Z},\sigma_{1})$ (by the mapping
$H_{\epsilon}$). It follows that

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([\epsilon,1/2]),\mathbb{Z},\sigma_{1})\leq\operatorname{\mathrm{mdim}}(\mathcal{B}([-1/2,1/2]),\mathbb{Z},\sigma_{1})$

which implies

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([0,(1-2\epsilon)/2]),\mathbb{Z},\sigma_{1})\leq\operatorname{\mathrm{mdim}}(\mathcal{B}([-1/2,1/2]),\mathbb{Z},\sigma_{1}).$

Thus,

$(1-2\epsilon)\cdot\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([0,1/2]),\mathbb{Z},\sigma_{1})\leq\operatorname{\mathrm{mdim}}(\mathcal{B}([-1/2,1/2]),\mathbb{Z},\sigma_{1}).$

Since $0<\epsilon<1/2$ is arbitrary, we get the first inequality.

In order to show the second inequality, we need employ the sampling lemma. We
fix an arbitrary $0<b<1/2$ and consider a continuous mapping:

$\Lambda_{b}:\mathcal{B}([-b,b])\to[-1,1]^{\mathbb{Z}},\quad f\mapsto
f|_{\mathbb{Z}}.$

Obviously, $\Lambda_{b}$ is $\mathbb{Z}$-equivariant, i.e. satisfying
$\Lambda_{b}\circ\sigma_{1}=\sigma\circ\Lambda_{b}$. By Lemma 4.2, the mapping
$\Lambda_{b}$ is injective. Thus,
$(\mathcal{B}([-b,b]),\mathbb{Z},\sigma_{1})$ can be embedded in
$([-1,1]^{\mathbb{Z}},\mathbb{Z},\sigma)$ (with the mapping $\Lambda_{b}$). It
follows that

$\operatorname{\mathrm{mdim}}(\mathcal{B}([-b,b]),\mathbb{Z},\sigma_{1})\leq\operatorname{\mathrm{mdim}}([-1,1]^{\mathbb{Z}},\mathbb{Z},\sigma)$

which implies

$2b\cdot\operatorname{\mathrm{mdim}}(\mathcal{B}([-1/2,1/2]),\mathbb{Z},\sigma_{1})\leq\operatorname{\mathrm{mdim}}([-1,1]^{\mathbb{Z}},\mathbb{Z},\sigma)=1.$

Since $0<b<1/2$ is arbitrary, we obtain the second inequality.

Finally, to prove the third inequality we shall need the interpolation lemma.
We fix an arbitrary $\epsilon>0$. By Lemma 4.3, there is a rapidly decreasing
function $f:\mathbb{R}\to\mathbb{R}$ band-limited in
$[-(1+\epsilon)/2,(1+\epsilon)/2]$ satisfying that $f(0)=1$ and $f(n)=0$ for
all $n\in\mathbb{Z}\setminus\\{0\\}$. In particular,

$|f(t)|\leq\frac{C}{1+t^{2}},\quad\forall t\in\mathbb{R},$

for some constant $C>0$. Set

$K=\max_{t\in\mathbb{R}}\sum_{n\in\mathbb{Z}}\frac{C}{1+(t-n)^{2}}<+\infty.$

We define a mapping as follows:

$G_{\epsilon}:([0,1]^{2})^{\mathbb{Z}}\to\mathcal{B}^{\mathbb{C}}([-(1+\epsilon)/2,(1+\epsilon)/2]),$
$a=(a_{1,n},a_{2,n})_{n\in\mathbb{Z}}\mapsto G_{\epsilon}(a),$
$G_{\epsilon}(a)(t)=\frac{1}{2K}\sum_{n\in\mathbb{Z}}(a_{1,n}+a_{2,n}\sqrt{-1})f(t-n),\quad\forall
t\in\mathbb{R}.$

It is clear that $G_{\epsilon}$ is $\mathbb{Z}$-equivariant, namely
$G_{\epsilon}\circ\sigma=\sigma_{1}\circ G_{\epsilon}$.

To prove the injectivity of $G_{\epsilon}$, we take
$a=(a_{1,n},a_{2,n})_{n\in\mathbb{Z}},b=(b_{1,n},b_{2,n})_{n\in\mathbb{Z}}\in([0,1]^{2})^{\mathbb{Z}}$
and assume $G_{\epsilon}(a)=G_{\epsilon}(b)$ which means

$\sum_{n\in\mathbb{Z}}\left((a_{1,n}-b_{1,n})+(a_{2,n}-b_{2,n})\sqrt{-1}\right)\cdot
f(t-n)=0,\quad\forall t\in\mathbb{R}.$

For every $m\in\mathbb{Z}$ by letting $t=m$ in the above equality we have
$a_{1,m}=b_{1,m}$ and $a_{2,m}=b_{2,m}$. Thus we conclude $a=b$, which shows
that $G_{\epsilon}$ is injective.

To see the continuity of $G_{\epsilon}$, we fix
$a=(a_{1,n},a_{2,n})_{n\in\mathbb{Z}}\in([0,1]^{2})^{\mathbb{Z}}$. We take a
sequence
$\\{a^{(k)}=(a^{(k)}_{1,n},a^{(k)}_{2,n})_{n\in\mathbb{Z}}\\}_{k\in\mathbb{N}}$
in $([0,1]^{2})^{\mathbb{Z}}$ and suppose $a^{(k)}\to a$ as $k\to\infty$. We
fix $N\in\mathbb{N}$ arbitrarily. For any $\delta>0$ there exists
$m\in\mathbb{N}$ sufficiently large such that

$\frac{C}{K}\cdot\sum_{|n|>m}\frac{1}{1+(t-n)^{2}}<\frac{\delta}{2},\quad\forall
t\in[-N,N].$

For such an $m\in\mathbb{N}$ there is $l\in\mathbb{N}$ sufficiently large
satisfying

$|\\!|a^{(k)}-a|\\!|_{l^{\infty}([-m,m])}<\delta$

for all $k\geq l$. Therefore we have for any $k\geq l$ and any $t\in[-N,N]$

$\displaystyle\left|G_{\epsilon}(a^{(k)})(t)-G_{\epsilon}(a)(t)\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{2K}\sum_{n\in\mathbb{Z}}|(a^{(k)}_{1,n}-a_{1,n})+(a^{(k)}_{2,n}-a_{2,n})\sqrt{-1}|\cdot|f(t-n)|$
$\displaystyle\leq$
$\displaystyle\frac{1}{2K}\cdot|\\!|a^{(k)}-a|\\!|_{l^{\infty}([-m,m])}\cdot\sum_{|n|\leq
m}|f(t-n)|+\frac{1}{K}\cdot\sum_{|n|>m}|f(t-n)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2K}\cdot|\\!|a^{(k)}-a|\\!|_{l^{\infty}([-m,m])}\cdot\sum_{n\in\mathbb{Z}}\frac{C}{1+(t-n)^{2}}+\frac{C}{K}\cdot\sum_{|n|>m}\frac{1}{1+(t-n)^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\cdot|\\!|a^{(k)}-a|\\!|_{l^{\infty}([-m,m])}+\frac{\delta}{2}<\delta.$

This implies

$\lim_{k\to\infty}|\\!|G_{\epsilon}(a^{(k)})-G_{\epsilon}(a)|\\!|_{L^{\infty}([-N,N])}=0.$

Since $N\in\mathbb{N}$ is arbitrary, we deduce that $G_{\epsilon}$ is
continuous.

Thus, $(([0,1]^{2})^{\mathbb{Z}},\mathbb{Z},\sigma)$ can be embedded in
$(\mathcal{B}^{\mathbb{C}}([-(1+\epsilon)/2,(1+\epsilon)/2]),\mathbb{Z},\sigma_{1})$
(with the mapping $G_{\epsilon}$). It follows that

$\operatorname{\mathrm{mdim}}(([0,1]^{2})^{\mathbb{Z}},\mathbb{Z},\sigma)\leq\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([-(1+\epsilon)/2,(1+\epsilon)/2]),\mathbb{Z},\sigma_{1}).$

As an immediate consequence

$\displaystyle
2(1+\epsilon)\cdot\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([0,1/2]),\mathbb{Z},\sigma_{1})$
$\displaystyle=$ $\displaystyle
2(1+\epsilon)\cdot\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([-1/4,1/4]),\mathbb{Z},\sigma_{1})$
$\displaystyle=$
$\displaystyle\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([-(1+\epsilon)/2,(1+\epsilon)/2]),\mathbb{Z},\sigma_{1})$
$\displaystyle\geq$
$\displaystyle\operatorname{\mathrm{mdim}}(([0,1]^{2})^{\mathbb{Z}},\mathbb{Z},\sigma)=2.$

Since $\epsilon>0$ is arbitrary, we obtain

$\operatorname{\mathrm{mdim}}(\mathcal{B}^{\mathbb{C}}([0,1/2]),\mathbb{Z},\sigma_{1})\geq
1.$

This completes the proof. ∎

##  5\. Construction of an explicit embedding mapping

###  5.1. Overview

The aim of this section is to prove Theorem 1.5. The proof goes through an
approach of harmonic analysis. Our strategy demonstrates an application of
classical analysis to topological dynamical systems. It would be worth
mentioning that convolution will be used frequently in our method, which
brings about a dynamical interaction, with visible embedding mappings to the
reader, between abstract topological objects and concrete function spaces
possessed of some good nature (i.e. expressing all the abstract flows as a
family of uniformly bounded analytic functions having a uniformly band-limited
property and sharing a uniform Lipschitz constant). Our technique has some
novelty substantially different from the Baire category framework (which, in
particular, was extensively applied when embedding a class of actions in a
finite mean dimensional space), and applies to an example of a countable
(infinite) product of finite mean dimensional spaces (which, however, seems to
be hopeless at dealing with a finite product of finite mean dimensional
spaces).

To begin with, let us fix a real flow $(X,\mathbb{R},T)$. We shall embed
$(X,\mathbb{R},T)$ in the universal real flow
$\left(\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\left(\mathcal{B}([-n/2-1,-n/2]\cup[n/2,n/2+1])\cap
L(\mathbb{R})\right),\mathbb{R},\sigma\right)$ with an explicitly built
embedding mapping. We assume here $\alpha=1/2$ and $\beta=1$ only for the sake
of convenience, as to which the proof for any positive real numbers
$\alpha<\beta$ is being similar.

The procedure is going to be fulfilled within three steps. We shrink in each
step the universal space, while preserving all the required properties,
established in the previous step. The task of each step is indicated precisely
in its title. The following diagram is a sketch of the route:

$X\longrightarrow\textrm{ (a compact invariant subset of)
}C(\mathbb{R})^{\mathbb{N}}$
$\longrightarrow\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\mathcal{B}([-n/2-1,-n/2]\cup[n/2,n/2+1])$
$\longrightarrow\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\left(\mathcal{B}([-n/2-1,-n/2]\cup[n/2,n/2+1])\cap
L(\mathbb{R})\right)$ Figure 1: Outline.

Setting. We denote by D a compatible metric on the product space
$C(\mathbb{R})^{\mathbb{N}}$:

$\textbf{D}(f,g)=\sum_{n\in\mathbb{N}}\sum_{N\in\mathbb{N}}\frac{|\\!|f_{n}-g_{n}|\\!|_{L^{\infty}([-N,N])}}{2^{n+N}},$

where $f=(f_{n})_{n\in\mathbb{N}},g=(g_{n})_{n\in\mathbb{N}}\in
C(\mathbb{R})^{\mathbb{N}}$. We note that each element in
$C(\mathbb{R})^{\mathbb{N}}$ is identified with a continuous function
$f:\mathbb{R}\to[-1,1]^{\mathbb{N}}$. Moreover, when dealing with complex-
valued continuous functions on $\mathbb{R}$ we shall automatically adapt the
distance D to the complex context.

###  5.2. Embedding $(X,\mathbb{R},T)$ in
$(C(\mathbb{R})^{\mathbb{N}},\mathbb{R},\sigma)$

The way to this target is standard. We reproduce it in this step for
completeness.

Let $\phi:X\to\phi(X)\subset[0,1]^{\mathbb{N}}$ be a homeomorphism (i.e. a
topological embedding) of the compact metric space $X$ into
$[0,1]^{\mathbb{N}}$. We define a mapping

$\Phi_{1}:X\to C(\mathbb{R})^{\mathbb{N}},\quad
x\mapsto\Phi_{1}(x);\quad\quad\Phi_{1}(x)(t)=\phi(T_{t}x),\quad\forall
t\in\mathbb{R}.$

Note here that for every $x\in X$ its image
$\Phi_{1}(x):\mathbb{R}\to[0,1]^{\mathbb{N}}$ is indeed a continuous function.

Obviously, the mapping $\Phi_{1}:X\to C(\mathbb{R})^{\mathbb{N}}$ is
injective. To see that $\Phi_{1}$ is continuous, we take a sequence of points
$x_{n}$ ($n\in\mathbb{N}$) in $X$, tending to some $x\in X$ as $n\to+\infty$,
and a compact subset $A$ of $\mathbb{R}$. When $n\in\mathbb{N}$ is large
enough, the distance between any two points $(t,x_{n})$ and $(t,x)$ in
$A\times X$ is sufficiently close to zero. Since $A\times X$ and $X$ are
compact, $T$ and $\phi$ are uniformly continuous on $A\times X$ and $X$,
respectively. It follows that the distance between $T_{t}x_{n}$ and $T_{t}x$
in $X$ is sufficiently close to zero as well, for all $t\in A$. This implies
that $\Phi_{1}(x_{n})(t)=\phi(T_{t}x_{n})$ is sufficiently close to
$\Phi_{1}(x)(t)=\phi(T_{t}x)$ for all $t\in A$. Thus, the sequence of
continuous functions $\Phi_{1}(x_{n})$ converges uniformly to $\Phi_{1}(x)$ on
$A\subset\mathbb{R}$ as $n\to+\infty$, which shows that the mapping $\Phi_{1}$
is continuous. Since $X$ is compact and since $\Phi_{1}$ is continuous and
one-to-one, $\Phi_{1}:X\to\Phi_{1}(X)$ is a homeomorphism.

For every $r\in\mathbb{R}$ and every $x\in X$ we have

$\Phi_{1}(T_{r}x)(t)=\phi(T_{t}(T_{r}x))=\phi(T_{t+r}x)=\Phi_{1}(x)(t+r)=\sigma_{r}(\Phi_{1}(x))(t)$

for all $t\in\mathbb{R}$, which means $\Phi_{1}\circ
T_{r}=\sigma_{r}\circ\Phi_{1}$ for any $r\in\mathbb{R}$, i.e. $\Phi_{1}$ is
equivariant. Therefore $(X,\mathbb{R},T)$ can be embedded in
$(C(\mathbb{R})^{\mathbb{N}},\mathbb{R},\sigma)$ with the mapping $\Phi_{1}$.

###  5.3. Embedding the translation on any compact invariant subset of
$C(\mathbb{R})^{\mathbb{N}}$ in
$\left(\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\mathcal{B}([-n/2-1,-n/2]\cup[n/2,n/2+1]),\mathbb{R},\sigma\right)$

This step will be accomplished with the following construction and lemmas. As
the space $C(\mathbb{R})^{\mathbb{N}}$ is not compact, we shall embed any of
its compact invariant subsets (namely $\Phi_{1}(X)$) rather than itself. In
fact, embedding $C(\mathbb{R})^{\mathbb{N}}$ itself is also achievable with a
little bit more effort. Nevertheless, we need not deal with it. Now let us fix
a compact invariant subset of $C(\mathbb{R})^{\mathbb{N}}$, which we denote
still by $X$ (instead of $\Phi_{1}(X)$).

For every $n\in\mathbb{Z}$ we take a continuous function
$\xi_{n}:\mathbb{R}\to[0,1]$ defined by

$\xi_{n}(t+n/2)=\begin{cases}0,\quad&t\in(-\infty,-1/2)\cup(1/2,+\infty)\\\
1+2t,\quad&t\in[-1/2,0]\\\ 1-2t,\quad&t\in[0,1/2]\end{cases}$

and let $\varphi_{n}=\overline{\mathcal{F}}(\xi_{n})$. For any integer $n$ it
is clear that $\varphi_{n}:\mathbb{R}\to\mathbb{C}$ is a bounded continuous
function and satisfies

$\operatorname{\mathrm{supp}}(\mathcal{F}(\varphi_{n}))=\operatorname{\mathrm{supp}}(\mathcal{F}(\overline{\mathcal{F}}(\xi_{n})))=\operatorname{\mathrm{supp}}(\xi_{n})=[(n-1)/2,(n+1)/2].$

Moreover, we may verify

$\sum_{n=-\infty}^{+\infty}\xi_{n}(t)=1,\quad\forall t\in\mathbb{R}.$

We set for each $n\in\mathbb{Z}$

$k_{n}=\int_{-\infty}^{+\infty}\left|\varphi_{n}(s)\right|ds=\int_{-\infty}^{+\infty}\left|\overline{\mathcal{F}}(\xi_{n})(s)\right|ds.$

Notice that

$0<k_{n}<+\infty,\quad\forall n\in\mathbb{Z}.$

We guarantee that each $k_{n}$ is positive because $\xi_{n}\in C(\mathbb{R})$
is not the constant function zero, while each $k_{n}$ is a finite number
because the continuous function $\xi_{n}:\mathbb{R}\to[0,1]$ is supported in a
compact interval.

Note that here we stick to the case $\beta=2\alpha>0$. In general, for
$0<\alpha<\beta$ the intervals $(n\alpha,n\alpha+\beta)$, where $n$ ranges
over $\mathbb{Z}$, still form a countable open cover of the real line
$\mathbb{R}$. If $\beta<2\alpha$ then the roof of any tent function $\xi_{n}$
that we chose should be replaced by a segment (instead of a point) at the
level $1$. If $\beta>2\alpha$, then there are removable elements in the
countable open cover of $\mathbb{R}$, and we thus select a subcover of
$\mathbb{R}$, from which, we need ensure that no members can be removed.

As we mentioned previously, convolution will be used frequently in our method.
Such an operation is (quite) mild and is (in some sense) easy to control. In
particular, it has some “low-pass filter” nature. A simple but important fact
is that it is able to dominate the support of the Fourier transform (under
this operation) with that of the participants. Besides, we recall that for any
bounded continuous function $f:\mathbb{R}\to\mathbb{R}$ the set
$\operatorname{\mathrm{supp}}(\mathcal{F}(f))$ must be symmetric in
$\mathbb{R}$. So we adopt a process with the help of
$\mathcal{B}^{\mathbb{C}}(\cdot)$ which is more flexible than
$\mathcal{B}(\cdot)$.

######  Lemma 5.1.

For any $n\in\mathbb{Z}$ and any $h\in C(\mathbb{R})$ we have

$\frac{h\ast\varphi_{n}}{k_{n}}\in\mathcal{B}^{\mathbb{C}}([(n-1)/2,(n+1)/2]),$

where $h\ast\varphi_{n}$ denotes the convolution:

$h\ast\varphi_{n}(t)=\int_{-\infty}^{+\infty}h(t-s)\varphi_{n}(s)ds,\quad\forall
t\in\mathbb{R}.$

###### Proof.

We fix an integer $n$ and a continuous function $h:\mathbb{R}\to[-1,1]$.
Clearly, the function $h\ast\varphi_{n}:\mathbb{R}\to\mathbb{C}$ is
continuous. Since $|\\!|h|\\!|_{L^{\infty}(\mathbb{R})}\leq 1$, we have for
all $t\in\mathbb{R}$

$\left|h\ast\varphi_{n}(t)\right|=\left|\int_{-\infty}^{+\infty}h(t-s)\varphi_{n}(s)ds\right|\leq\int_{-\infty}^{+\infty}|\varphi_{n}(s)|ds=k_{n}$

which implies

$\left|\\!\left|\frac{h\ast\varphi_{n}}{k_{n}}\right|\\!\right|_{L^{\infty}(\mathbb{R})}\leq
1.$

Since
$\operatorname{\mathrm{supp}}(\mathcal{F}(\varphi_{n}))=[(n-1)/2,(n+1)/2]$, we
deduce

$\operatorname{\mathrm{supp}}(\mathcal{F}(h\ast\varphi_{n}))\subset\operatorname{\mathrm{supp}}(\mathcal{F}(h))\cap\operatorname{\mathrm{supp}}(\mathcal{F}(\varphi_{n}))\subset[(n-1)/2,(n+1)/2],$

as required. ∎

We now define a mapping

$\Phi_{2}:C(\mathbb{R})^{\mathbb{N}}\to\prod_{(l,n)\in\mathbb{N}\times\mathbb{Z}}\mathcal{B}^{\mathbb{C}}([(n-1)/2,(n+1)/2]),$
$(f_{l})_{l\in\mathbb{N}}\mapsto\left(\frac{f_{l}\ast\varphi_{n}}{k_{n}}\right)_{(l,n)\in\mathbb{N}\times\mathbb{Z}}.$

We shall prove that $\Phi_{2}:X\to\Phi_{2}(X)$ is an equivariant
homeomorphism, which will imply that $(X,\mathbb{R},\sigma)$ can be embedded
in
$\left(\prod_{(l,n)\in\mathbb{N}\times\mathbb{Z}}\mathcal{B}^{\mathbb{C}}([(n-1)/2,(n+1)/2]),\mathbb{R},\sigma\right)$
with the mapping $\Phi_{2}$.

######  Lemma 5.2.

The mapping $\Phi_{2}:X\to\Phi_{2}(X)$ is an equivariant homeomorphism.

###### Proof.

For any $n\in\mathbb{Z}$, any $h\in C(\mathbb{R})$ and any $r\in\mathbb{R}$

$\displaystyle\sigma_{r}(h\ast\varphi_{n})(t)$
$\displaystyle=\int_{-\infty}^{+\infty}h(t+r-s)\varphi_{n}(s)ds$
$\displaystyle=\int_{-\infty}^{+\infty}\sigma_{r}h(t-s)\varphi_{n}(s)ds$
$\displaystyle=(\sigma_{r}h)\ast\varphi_{n}(t)$

for all $t\in\mathbb{R}$. This shows that
$\sigma_{r}\circ\Phi_{2}=\Phi_{2}\circ\sigma_{r}$ for every $r\in\mathbb{R}$.

We note that to show that $\Phi_{2}:X\to\Phi_{2}(X)$ is a homeomorphism, it
suffices to prove that $\Phi_{2}$ is continuous and injective (because $X$ is
a compact metric space).

To see that $\Phi_{2}$ is continuous, we take $f=(f_{j})_{j\in\mathbb{N}}\in
C(\mathbb{R})^{\mathbb{N}}$. We fix $(i,n)\in\mathbb{N}\times\mathbb{Z}$
arbitrarily. For any $\epsilon>0$ we choose:

  * •

$A>0$ sufficiently large such that

$\int_{\mathbb{R}\setminus[-A,A]}\left|\varphi_{n}(t)\right|dt<\epsilon/8;$

  * •

$N\in\mathbb{N}$ sufficiently large such that

$|\\!|u-v|\\!|_{L^{\infty}([-N,N])}<\epsilon/2\Longrightarrow
D(u,v)<\epsilon,\quad\forall u,v\in C(\mathbb{R});$

  * •

$\delta>0$ sufficiently small such that

$\textbf{D}(f,g)<\delta\Longrightarrow|\\!|f_{i}-g_{i}|\\!|_{L^{\infty}([-A-N,A+N])}<\epsilon/4k_{n},\quad\forall
g=(g_{j})_{j\in\mathbb{N}}\in C(\mathbb{R})^{\mathbb{N}}.$

It follows that if $g=(g_{j})_{j\in\mathbb{N}}\in C(\mathbb{R})^{\mathbb{N}}$
satisfies $\textbf{D}(f,g)<\delta$ then for any $t\in[-N,N]$ we have

$\displaystyle\left|f_{i}\ast\varphi_{n}(t)-g_{i}\ast\varphi_{n}(t)\right|$
$\displaystyle\leq\int_{-\infty}^{+\infty}\left|f_{i}(t-s)-g_{i}(t-s)\right|\cdot\left|\varphi_{n}(s)\right|ds$
$\displaystyle\leq
2\int_{\mathbb{R}\setminus[-A,A]}\left|\varphi_{n}(s)\right|ds+\int_{-A}^{A}\left|f_{i}(t-s)-g_{i}(t-s)\right|\cdot\left|\varphi_{n}(s)\right|ds$
$\displaystyle\leq
2\cdot(\epsilon/8)+(\epsilon/4)\cdot\frac{1}{k_{n}}\int_{-A}^{A}\left|\varphi_{n}(s)\right|ds<\epsilon/2$

which implies

$|\\!|f_{i}\ast\varphi_{n}-g_{i}\ast\varphi_{n}|\\!|_{L^{\infty}([-N,N])}<\epsilon/2.$

Thus,

$D(f_{i}\ast\varphi_{n},g_{i}\ast\varphi_{n})<\epsilon.$

Since $(i,n)\in\mathbb{N}\times\mathbb{Z}$ and the function $f\in
C(\mathbb{R})^{\mathbb{N}}$ are arbitrary, we deduce that $\Phi_{2}$ is
continuous.

To verify that $\Phi_{2}$ is injective, we take
$f=(f_{j})_{j\in\mathbb{N}},g=(g_{j})_{j\in\mathbb{N}}\in
C(\mathbb{R})^{\mathbb{N}}$ and assume $\Phi_{2}(f)=\Phi_{2}(g)$. It follows
that $f_{l}\ast\varphi_{n}=g_{l}\ast\varphi_{n}$ for all
$(l,n)\in\mathbb{N}\times\mathbb{Z}$. This implies

$(f_{l}-g_{l})\ast\sum_{n=-\infty}^{+\infty}\varphi_{n}=0,\quad\forall
l\in\mathbb{N}.$

Since $\sum_{n=-\infty}^{+\infty}\xi_{n}=1$ and
$\overline{\mathcal{F}}(1)=\delta_{0}$ (i.e. the delta probability measure at
the origin), we conclude $f_{l}-g_{l}=0$ for any $l\in\mathbb{N}$. Thus,
$f=g$. ∎

Next we indicate that
$\left(\prod_{(l,n)\in\mathbb{N}\times\mathbb{Z}}\mathcal{B}^{\mathbb{C}}([(n-1)/2,(n+1)/2]),\mathbb{R},\sigma\right)$
can be embedded in
$\left(\prod_{(l,n)\in\mathbb{N}\times\mathbb{Z}}\mathcal{B}([-(n+1)/2,-(n-1)/2]\cup[(n-1)/2,(n+1)/2])^{2},\mathbb{R},\sigma\right)$
with the mapping $\Phi_{3}$ defined as follows:

$\prod_{(l,n)\in\mathbb{N}\times\mathbb{Z}}\mathcal{B}^{\mathbb{C}}\left(\left[\frac{n-1}{2},\frac{n+1}{2}\right]\right)\to\prod_{(l,n)\in\mathbb{N}\times\mathbb{Z}}\mathcal{B}\left(\left[-\frac{n+1}{2},-\frac{n-1}{2}\right]\bigcup\left[\frac{n-1}{2},\frac{n+1}{2}\right]\right)^{2},$
$\left(g_{(l,n)}\right)_{(l,n)\in\mathbb{N}\times\mathbb{Z}}\mapsto\left(\frac{g_{(l,n)}+\overline{g_{(l,n)}}}{2},\frac{g_{(l,n)}-\overline{g_{(l,n)}}}{2}\right)_{(l,n)\in\mathbb{N}\times\mathbb{Z}}.$

Thus, this step (subsection) concludes by reenumerating the indices of the
(latter) product space as well as the embedding mapping $\Phi_{3}$ affiliated.

###  5.4. Embedding
$\left(\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\mathcal{B}([-n/2-1,-n/2]\cup[n/2,n/2+1]),\mathbb{R},\sigma\right)$
in the universal real flow
$\left(\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\left(\mathcal{B}([-n/2-1,-n/2]\cup[n/2,n/2+1])\cap
L(\mathbb{R})\right),\mathbb{R},\sigma\right)$

This is the final step.

Throughout this subsection we put $q_{j}=1/(j+1)$ for each $j\in\mathbb{N}$.
In order to make all those functions attain to the Lipschitz constant $1$, we
define for every $(l,j)\in\mathbb{N}\times\mathbb{N}$ a mapping
$H_{l}^{j}:C(\mathbb{R})^{\mathbb{N}}\to L(\mathbb{R})$ by

$H_{l}^{j}(f)(t)=\frac{1}{2}\int_{t}^{t+q_{j}}f_{l}(s)ds,\quad\forall
f=(f_{l})_{l\in\mathbb{N}}\in C(\mathbb{R})^{\mathbb{N}},\quad\forall
t\in\mathbb{R}.$

Note here that for any $(l,j)\in\mathbb{N}\times\mathbb{N}$ and any
$f=(f_{l})_{l\in\mathbb{N}}\in C(\mathbb{R})^{\mathbb{N}}$ we have indeed
$|\\!|H_{l}^{j}(f)|\\!|_{L^{\infty}(\mathbb{R})}<1$ and

$\left|H_{l}^{j}(f)(s)-H_{l}^{j}(f)(t)\right|=\frac{1}{2}\left|\int_{s}^{t}f_{l}(u)du-\int_{s+q_{j}}^{t+q_{j}}f_{l}(u)du\right|\leq\left|s-t\right|,\quad\forall
s,t\in\mathbb{R},$

which implies $H_{l}^{j}(f)\in L(\mathbb{R})$.

We list three properties of the mapping $H_{l}^{j}$ (for an arbitrary
$(l,j)\in\mathbb{N}\times\mathbb{N}$) as follows:

  * •

$H_{l}^{j}:C(\mathbb{R})^{\mathbb{N}}\to L(\mathbb{R})$ is equivariant;

  * •

$H_{l}^{j}:C(\mathbb{R})^{\mathbb{N}}\to L(\mathbb{R})$ is continuous;

  * •

$\operatorname{\mathrm{supp}}(\mathcal{F}(H_{l}^{j}(f)))\subset\operatorname{\mathrm{supp}}(\mathcal{F}(f_{l}))$
for any $f=(f_{l})_{l\in\mathbb{N}}\in C(\mathbb{R})^{\mathbb{N}}$.

In fact, the first assertion follows from the equality

$H_{l}^{j}(\sigma_{r}f)(t)=\frac{1}{2}\int_{t}^{t+q_{j}}f_{l}(s+r)ds=\frac{1}{2}\int_{t+r}^{t+r+q_{j}}f_{l}(s)ds=\sigma_{r}H_{l}^{j}(f)(t)$

for all $f=(f_{l})_{l\in\mathbb{N}}\in C(\mathbb{R})^{\mathbb{N}}$ and all
$t,r\in\mathbb{R}$.

To verify the second assertion, we fix a compact interval
$[a,b]\subset\mathbb{R}$ arbitrarily. We take a sequence of functions
$g^{(i)}=(g^{(i)}_{l})_{l\in\mathbb{N}}\in C(\mathbb{R})^{\mathbb{N}}$
converging to $h=(h_{l})_{l\in\mathbb{N}}\in C(\mathbb{R}^{\mathbb{N}})$
uniformly on the interval $[a,b+1]$ as $i\to+\infty$. This implies that
$g^{(i)}_{l}\in C(\mathbb{R})$ converges to $h_{l}\in C(\mathbb{R})$ uniformly
on $[a,b+1]$ as $i\to+\infty$. It follows that the sequence of functions
$H_{l}^{j}(g^{(i)})\in L(\mathbb{R})$ converges to $H_{l}^{j}(h)\in
L(\mathbb{R})$ uniformly on the interval $[a,b]$ as $i\to+\infty$. This shows
the continuity of $H_{l}^{j}$.

To see the third property, it suffices to observe that

$2H_{l}^{j}(f)=\chi_{[-q_{j},0]}\ast f_{l},$

where $\chi_{[-q_{j},0]}:\mathbb{R}\to\\{0,1\\}$ is defined by

$\chi_{[-q_{j},0]}(t)=\begin{cases}1,\quad&t\in[-q_{j},0]\\\
0,\quad&t\notin[-q_{j},0]\end{cases}.$

The above statements allow us to define a continuous and equivariant mapping

$\Phi_{4}:\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\mathcal{B}\left(\left[-\frac{n}{2}-1,-\frac{n}{2}\right]\cup\left[\frac{n}{2},\frac{n}{2}+1\right]\right)$
$\to\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\left(\mathcal{B}\left(\left[-\frac{n}{2}-1,-\frac{n}{2}\right]\cup\left[\frac{n}{2},\frac{n}{2}+1\right]\right)\cap
L\left(\mathbb{R}\right)\right),$
$f=\left(f_{l}\right)_{l\in\mathbb{N}}\mapsto\left(H_{l}^{j}(f)\right)_{(l,j)\in\mathbb{N}\times\mathbb{N}}.$

Note here that we have reenumerated the indices of the product spaces (both
the former and the latter, respectively) in the definition of the mapping
$\Phi_{4}$. More precisely, we need regard the former product space as a
compact invariant subset of $C(\mathbb{R})^{\mathbb{N}}$, with a single index
$l$ ranging over $\mathbb{N}$ instead, to which we apply the mapping
$\Phi_{4}$ (as we wrote in the “$\mapsto$” line), and reenumerate the indices
$(l,j)\in\mathbb{N}\times\mathbb{N}$ of the resulting product space, which is
considered finally as the latter.

To show that
$\left(\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\mathcal{B}([-n/2-1,-n/2]\cup[n/2,n/2+1]),\mathbb{R},\sigma\right)$
can be embedded in
$\left(\prod_{m=0}^{+\infty}\prod_{n=0}^{m}\left(\mathcal{B}([-n/2-1,-n/2]\cup[n/2,n/2+1])\cap
L(\mathbb{R})\right),\mathbb{R},\sigma\right)$ with the mapping $\Phi_{4}$, we
have (by compactness) just its injectivity left over.

In fact, we take $g=(g_{l})_{l\in\mathbb{N}},h=(h_{l})_{l\in\mathbb{N}}\in
C(\mathbb{R})^{\mathbb{N}}$ and assume $g\neq h$. Without loss of generality
there exist $p\in\mathbb{N}$ and two real numbers $a<b$ such that
$g_{p}(s)>h_{p}(s)$ for all $s\in[a,b]$. We choose $t\in\mathbb{R}$ and
$j\in\mathbb{N}$ such that $a<t<t+q_{j}<b$. This results in
$H_{p}^{j}(g)(t)>H_{p}^{j}(h)(t)$ which implies $\Phi_{4}(g)\neq\Phi_{4}(h)$.
Therefore $\Phi_{4}$ is injective.

So we have shown what we stated in the title of this subsection. Thus, the
whole proof is eventually completed.

##  6\. Further remarks

This section aims to make a clearer picture of universal properties (for those
function spaces mentioned in the previous sections) with a finite sequence of
short remarks clarifying some potential deliberation.

######  Remark 6.1.

We clarify that under the translation $\sigma$ the space

$\prod_{n=0}^{+\infty}\mathcal{B}([-n/3-1/2,-n/3]\cup[n/3,n/3+1/2])$

is not universal. Furthermore, we may prove a variant of the Bebutov–Kakutani
dynamical embedding theorem similar to Theorem 3.2 as follows: Let
$0<\alpha<\beta$ be two real numbers. A real flow $(X,\mathbb{R},T)$ can be
embedded in

$\left(\prod_{n=0}^{+\infty}\mathcal{B}([-n\alpha-\beta,-n\alpha]\cup[n\alpha,n\alpha+\beta]),\mathbb{R},\sigma\right)$

if and only if the set of its fixed points $\\{x\in X:T_{t}x=x,\,\forall
t\in\mathbb{R}\\}$ can be (topologically) embedded in $[0,1]$. We give a
sketch of a proof: For the “if” part “$\Longleftarrow$” we employ Theorem 3.2
and follow the argument in the proof of the first main theorem, which allows
us to embed $\left(L(\mathbb{R}),\mathbb{R},\sigma\right)$ in
$\left(\prod_{n=0}^{+\infty}\mathcal{B}([-n\alpha-\beta,-n\alpha]\cup[n\alpha,n\alpha+\beta]),\mathbb{R},\sigma\right)$.
For the “only if” part “$\Longrightarrow$” we note that for every positive
integer $n$ the fixed-point set of the real flow
$\left(\mathcal{B}([-n\alpha-\beta,-n\alpha]\cup[n\alpha,n\alpha+\beta]),\mathbb{R},\sigma\right)$
contains only one element, i.e. the constant function $0$, while for $n=0$ its
fixed-point set consists of all constant functions $\mathbb{R}\to[-1,1]$,
which is homeomorphic to $[0,1]$.

######  Remark 6.2.

We indicate that under the translation $\sigma$ the space
$\mathcal{B}([-1,1])^{\mathbb{N}}$ is not universal. The outline of a proof is
similar to our second main theorem. We take a (non-constant) bounded
continuous function on $\mathbb{R}$ having a sufficiently small (positive)
fundamental period, and consider its image in the space
$\mathcal{B}([-1,1])^{\mathbb{N}}$, under an embedding mapping (if we assume
the embeddability). Applying the sampling lemma (Lemma 4.2) we will deduce
that only constant functions in $\mathcal{B}([-1,1])$ may attain such a small
period (and thus, by equivariance, are the only possible candidates for the
embedding image), which contradicts the injectivity of an embedding mapping.
However, we notice that
$\left(\mathcal{B}([-1,1])^{\mathbb{N}},\mathbb{Z},\sigma_{1}\right)$ is
universal for $\mathbb{Z}$-actions. This fact follows from the interpolation
lemma (Lemma 4.3).

######  Remark 6.3.

In this remark we prove the corollary: Under the translation $\sigma$ the
space $\left(C^{\infty}(\mathbb{R})\cap L(\mathbb{R})\right)^{\mathbb{N}}$ is
universal. This corollary follows from Theorem 1.5. In fact, a generalized
Paley–Wiener theorem [GT20, Lemma 2.2][Sch66, Chapter 7, Section 8] asserts
that a bounded continuous function $f:\mathbb{R}\to\mathbb{C}$ satisfies
$\operatorname{\mathrm{supp}}(\mathcal{F}(f))\subset[-r,r]$ for some $r>0$ if
and only if $f$ can be extended to a holomorphic function on $\mathbb{C}$ such
that $|f(x+y\sqrt{-1})|\leq e^{2\pi
r|y|}\cdot|\\!|f|\\!|_{L^{\infty}(\mathbb{R})}$. Thus, for any positive real
number $r$ we have in particular that all the functions in
$\mathcal{B}([-r,r])$ must be analytic, and hence the space
$\mathcal{B}([-r,r])\cap L(\mathbb{R})$ is a compact invariant subset of
$C^{\infty}(\mathbb{R})\cap L(\mathbb{R})$.

######  Remark 6.4.

It is also possible to give a direct proof of the corollary. We describe a
sketch of the proof as follows. As we mentioned in Section 5, it suffices to
build a mapping
$\Theta:C(\mathbb{R})^{\mathbb{N}}\to\left(C^{\infty}(\mathbb{R})\cap
L(\mathbb{R})\right)^{\mathbb{N}}$ which is equivariant, continuous and
injective. In order to make all the functions in $C(\mathbb{R})$ infinitely
differentiable, we take a function $\theta$ on $\mathbb{R}$ as follows:

$\theta(t)=\begin{cases}c\cdot e^{1-t^{2}},&|t|<1\\\ 0,&|t|\geq 1\end{cases}$

where the constant $c$ should be chosen to ensure

$\int_{-\infty}^{+\infty}\theta(t)\mathrm{d}t=1.$

It is clear that $\theta$ is a nonnegative and smooth function on
$\mathbb{R}$, supported in the compact interval $[-1,1]$, and it satisfies
$\theta(t)=\theta(-t)$ for all $t\in\mathbb{R}$. For each $n\in\mathbb{N}$ we
set $\theta_{n}(t)=n\theta(nt)$ for any $t\in\mathbb{R}$. Note that every
$\theta_{n}$ has almost the same properties as $\theta$’s, but is supported in
$[-n,n]$. Moreover, we can show that for any $f\in C(\mathbb{R})$ and
$n\in\mathbb{N}$ the function $f\ast\theta_{n}$ is smooth, and for any
$N\in\mathbb{N}$ we have $|\\!|f\ast\theta_{n}-f|\\!|_{L^{\infty}([-N,N])}\to
0$ as $n\to\infty$. To conclude we apply the construction with the argument in
Section 5 (twice) to the following diagram:

$C(\mathbb{R})^{\mathbb{N}}\longrightarrow
C^{\infty}(\mathbb{R})^{\mathbb{N}}\longrightarrow\left(C^{\infty}(\mathbb{R})\cap
L(\mathbb{R})\right)^{\mathbb{N}}.$

More precisely, the former arrow corresponds to the embedding mapping

$(f_{i})_{i\in\mathbb{N}}\mapsto\left(\left(f_{i}\ast\theta_{j}\right)_{i=1}^{j}\right)_{j=1}^{+\infty}$

while the latter arrow corresponds to the embedding mapping which we
constructed in (the final step of) Section 5.

######  Remark 6.5.

We would like to remark here that the following refinement of our main theorem
is impossible. Let $C^{\infty}_{c}(\mathbb{R})$ be the set of all functions
$f\in C^{\infty}(\mathbb{R})$ supported in a compact subset of $\mathbb{R}$.
It is classically known that a continuous function can be written as a limit
(in some sense) of a sequence of functions chosen in $C^{\infty}(\mathbb{R})$;
and further, it is also feasible to require an approximation sequence of
functions coming from $C^{\infty}_{c}(\mathbb{R})$. This fact leads naturally
to a seemingly plausible question as follows:

  * •

Is the space $\left(L(\mathbb{R})\cap
C^{\infty}_{c}(\mathbb{R})\right)^{\mathbb{N}}$ universal under the
translation $\sigma$?

However, it turns out that we cannot expect this space to be universal under
the translation. In fact, such a space is (very) non-interesting for
embedding. For example, if we choose a real flow possessing at least two
distinct fixed points, then its (embedding) image must contain a nonzero
constant function which thus does not have a compact support.

## Appendix

The appendix is logically independent of the body of this paper. The only
ingredient is to point out that the injectivity of an embedding mapping
appearing in the Bebutov–Kakutani dynamical embedding theorem (in relation to
Theorem 3.2 affiliated to Section 3) can be observed from finitely many points
in the real line provided the phase space is finite dimensional and possesses
no periodic points.

In 1981 Takens established a well-known theorem in differential dynamical
systems, a variant of which we may obtain within a Baire category framework
essentially the same as the classical method in this direction.

  * •

Suppose that $(X,\mathbb{R},T)$ is a real flow containing no periodic points.
If $\dim(X)=d<+\infty$ then for every $(2d+1)$ distinct real numbers
$r_{0},r_{1},\dots,r_{2d}$ there exists an embedding mapping $F:X\to
C(\mathbb{R})$ which embeds $(X,\mathbb{R},T)$ in the translation on
$C(\mathbb{R})$ and satisfies that for any two distinct points
$x,x^{\prime}\in X$ there is some integer $0\leq n\leq 2d$ such that
$F(x)(r_{n})\neq F(x^{\prime})(r_{n})$.

This is an immediate consequence of the following statement:

  * •

Suppose that $(X,\mathbb{R},T)$ is a real flow containing no periodic points.
If $\dim(X)=d<+\infty$ then for every $(2d+1)$ distinct real numbers
$r_{0},r_{1},\dots,r_{2d}$ there exists a continuous mapping $f:X\to[0,1]$
such that

$X\to[0,1]^{2d+1},\quad
x\mapsto(f(T_{r_{0}}x),f(T_{r_{1}}x),\dots,f(T_{r_{2d}}x))$

is a topological embedding mapping.

We sketch the outline of its proof. For a more detailed and technical
treatment we refer to [Gut15, GQS18]. We denote $\Delta_{X}=\\{(x,x):x\in
X\\}$. The strategy is to find for every pair $(x,x^{\prime})\in X\times
X\setminus\Delta_{X}$ an open neighbourhood $U_{(x,x^{\prime})}\subset X\times
X\setminus\Delta_{X}$ of $(x,x^{\prime})$ satisfying that the set

$D_{U_{(x,x^{\prime})}}^{r_{0},\dots,r_{2d}}=\\{f\in
C(X,[0,1]):f_{r_{0},\dots,r_{2d}}(y)\neq
f_{r_{0},\dots,r_{2d}}(y^{\prime}),\,\forall(y,y^{\prime})\in\overline{U_{(x,x^{\prime})}}\\}$

is open (which is easier and which we do not plan to explain here) and dense
(which is harder and for which we will put an explanation in a moment) in the
space $C(X,[0,1])$, where the continuous mapping $f_{r_{0},\dots,r_{2d}}$ is
defined by

$f_{r_{0},\dots,r_{2d}}:X\to[0,1]^{2d+1},\quad
x\mapsto(f(T_{r_{0}}x),f(T_{r_{1}}x),\dots,f(T_{r_{2d}}x)).$

Since $\\{U_{(x,x^{\prime})}:(x,x^{\prime})\in X\times
X\setminus\Delta_{X}\\}$ is an open cover of $X\times X\setminus\Delta_{X}$
which is a Lindelöf space (namely, any of its open covers admits a countable
subcover), there is a countable open cover
$\\{U_{(x_{m},x^{\prime}_{m})}:m\in\mathbb{N}\\}$ of $X\times
X\setminus\Delta_{X}$. By the Baire category theorem there exists a continuous
mapping
$f\in\bigcap_{m\in\mathbb{N}}D_{U_{(x_{m},x^{\prime}_{m})}}^{r_{0},\dots,r_{2d}}$.
Thus, the mapping $f_{r_{0},\dots,r_{2d}}:X\to[0,1]^{2d+1}$ is injective.

To make $D_{U_{(x,x^{\prime})}}^{r_{0},\dots,r_{2d}}$ dense in $C(X,[0,1])$,
let us suppose $x^{\prime}\neq T_{t}x$ for all $t\in\mathbb{R}$. We notice
that the case $x^{\prime}=T_{t}x$ for some $t\in\mathbb{R}\setminus\\{0\\}$ is
highly similar to what we assumed here, but should be with a more careful
construction of perturbations. We take open neighbourhoods $U_{x}$ and
$U_{x^{\prime}}$ of $x$ and $x^{\prime}$, respectively, such that the sets
$T_{r_{0}}(\overline{U_{x}}),\dots,T_{r_{2d}}(\overline{U_{x}})$,
$T_{r_{0}}(\overline{U_{x^{\prime}}}),\dots,T_{r_{2d}}(\overline{U_{x^{\prime}}})$
are pairwise disjoint, and set $U_{(x,x^{\prime})}=U_{x}\times
U_{x^{\prime}}$. So we have already defined
$D_{U_{(x,x^{\prime})}}^{r_{0},\dots,r_{2d}}$. Now we fix $f\in C(X,[0,1])$
and $\epsilon>0$ arbitrarily. Noting that $\dim(X)=d<+\infty$ we need choose
finite open covers $\alpha_{x}$ and $\alpha_{x^{\prime}}$ of
$\overline{U_{x}}$ and $\overline{U_{x^{\prime}}}$, respectively, which are
sufficiently fine, such that $\mathrm{diam}(f(T_{r_{n}}(V)))<\epsilon/2$ for
each $V\in\alpha_{x}\cup\alpha_{x^{\prime}}$ and each integer $0\leq n\leq
2d$. Let $w\in\\{x,x^{\prime}\\}$. We take a partition of unity
$\\{\psi_{V}^{w}\\}_{V\in\alpha_{w}}$ of $\overline{U_{w}}$ subordinate to
$\alpha_{w}$, namely, a family of continuous functions
$\psi_{V}^{w}:\overline{U_{w}}\to[0,1]$ satisfying:

$\sum_{V\in\alpha_{w}}\psi_{V}^{w}(z)=1,\quad\forall\,z\in\overline{U_{w}};\quad\quad\mathrm{supp}(\psi_{V}^{w})\subset
V,\quad\forall\,V\in\alpha_{w}.$

Without loss of generality we can choose pairwise distinct points
$p_{V}^{w}\in V$, for every $V\in\alpha_{w}$ and for $w\in\\{x,x^{\prime}\\}$,
with $\psi_{V}^{w}(p_{V}^{w})=1$. We set for any $V\in\alpha_{w}$ a vector

$u_{V}^{w}=\left(f(T_{r_{0}}p_{V}^{w}),\dots,f(T_{r_{2d}}p_{V}^{w})\right)\;\in[0,1]^{2d+1}.$

We need find $q_{V}^{w}\in[0,1]^{2d+1}$, for each $V\in\alpha_{w}$, with
$|\\!|q_{V}^{w}-u_{V}^{w}|\\!|_{\infty}<\epsilon/2$, such that any $(2d+2)$
pairwise distinct vectors in the family
$\\{q_{V}^{w}:V\in\alpha_{w},w\in\\{x,x^{\prime}\\}\\}$ are affinely
independent in $\mathbb{R}^{2d+1}$. Next we define a continuous mapping

$k_{w}:\overline{U_{w}}\to[0,1]^{2d+1},\quad
z\mapsto\sum_{V\in\alpha_{w}}\psi_{V}^{w}(z)q_{V}^{w}.$

We put
$W=\bigcup_{n=0}^{2d}\bigcup_{w\in\\{x,x^{\prime}\\}}T_{r_{n}}(\overline{U_{w}})$
and let $g:W\to[0,1]$ be a continuous function defined by
$g(T_{r_{n}}z)=\mathsf{proj}_{n}(k_{w})(z)$ for any integer $0\leq n\leq 2d$
and any $z\in\overline{U_{w}}$, where $w$ ranges over $\\{x,x^{\prime}\\}$. We
may verify $|\\!|g-f|_{W}|\\!|_{\infty}<\epsilon$. Finally it suffices to
extend $g:W\to[0,1]$ to a continuous function $h:X\to[0,1]$ (i.e. $h|_{W}=g$)
with $|\\!|f-h|\\!|_{\infty}<\epsilon$, and to show $h\in
D_{U_{(x,x^{\prime})}}^{r_{0},\dots,r_{2d}}$, as required.

## References

  * [Beu89] A. Beurling. Collected works of Arne Beurling, Vol. 2, Harmonic analysis, edited by L. Carleson, P. Malliavan, J. Neuberger, J. Wermer. Birkhäuser, Boston–Basel–Berlin, 1989.
  * [Gro99] M. Gromov. Topological invariants of dynamical systems and spaces of holomorphic maps: I. Math. Phys. Anal. Geom. 2 (1999), 323–415.
  * [GJ19] Y. Gutman, L. Jin. An explicit compact universal space for real flows. Journal of Differential Equations 266(6) (2019), 3438–3449.
  * [GJ20] Y. Gutman, L. Jin. Mean dimension and an embedding theorem for real flows. Fundamenta Mathematicae 251 (2020), 161–181.
  * [GJT19] Y. Gutman, L. Jin, M. Tsukamoto. A Lipschitz refinement of the Bebutov–Kakutani dynamical embedding theorem. Journal of Dynamics and Differential Equations 31 (2019), 81–91.
  * [GQS18] Y. Gutman, Y. Qiao, G. Szabó. The embedding problem in topological dynamics and Takens’ theorem. Nonlinearity 31 (2018), 597–620.
  * [GQT19] Y. Gutman, Y. Qiao, M. Tsukamoto. Application of signal analysis to the embedding problem of $\mathbb{Z}^{k}$-actions. Geometric and Functional Analysis 29 (2019), 1440–1502.
  * [GT20] Y. Gutman, M. Tsukamoto. Embedding minimal dynamical systems into Hilbert cubes. Inventiones Mathematicae 221 (2020), 113–166.
  * [Gut15] Y. Gutman. Mean dimension and Jaworski-type theorems. Proc. Lond. Math. Soc. 111 (2015), 831–850.
  * [JT19] L. Jin, S. Tu. A new universal real flow of the Hilbert-cubical type. Dynamical Systems 34 (2019), 234–238.
  * [LW00] E. Lindenstrauss, B. Weiss. Mean topological dimension. Israel Journal of Mathematics 115 (2000), 1–24.
  * [Sch66] L. Schwartz. Théorie des distributions. Hermann, Paris, 1966.