# Unified dynamical systems on coarse spaces

Najmeh Khajoei
Mahani Mathematical Research Center Shahid Bahonar University of Kerman,
Kerman, Iran.
e-mails<EMAIL_ADDRESS>
###### Abstract.

In this paper, we introduce a new class of dynamical systems on a coarse space
with coarse time called, coarse dynamical system. The notion of coarse
conjugacy is studied from coarse geometry point of view. Coarse orbits as
invariant sets under coarse conjugacy are deduced. It is Shown that the
coproduct of two coarse dynamical systems is a coarse dynamical system.
Finally, we define set-value coarse dynamical systems and prove if two coarse
dynamical systems are coarse conjugate, then their corresponding set-value
coarse dynamical systems are coarse conjugate.

Subject Classification: 37B99, 20E99, 54B99

Keywords: coarse space; coarse group; coarse time dynamical system; unified
dynamical system

##  1\. Introduction

The coarse geometry is known as large-scale geometry which studies geometric
objects from a large distance and large-scale features of them. Historically,
coarse geometry goes back from geometric group theory in [10, 18, 19] and
Mostow’s rigidity theorem and its generalization in [12, 13] and [8, 9]
respectively. In [17] Roe considered coarse spaces as a large-scale
counterpart of uniform spaces and in [15] Nowak and Yu studied large-scale
geometry of metric spaces. Large-scale geometry was studied on the group as
well. The first attempt to study coarse groups was pioneered by Protasov in
[16], where this concept was introduced by using balleans. Also, in the same
paper, he expressed coarse groups are uniquely obtained by a specific ideal of
subsets of the group, which is called group ideal. Recently coarse
hyperspaces, i.e., structures induced on power sets, were inspired by metric
and uniform hyperspace are studied in [1, 5].
In coarse geometry, $\mathbb{Z}$ (integer number) and $\mathbb{R}$ (real
number) are coarsely equivalent. This property motivates us to define a coarse
time dynamical system on a coarse space in which the evolution time is a
coarse group and in particular unified dynamical systems with coarse time
$\mathbb{Z}$ or $\mathbb{R}$. Since from dynamical systems viewpoint, we
encounter two classes of dynamical systems i.e., discrete and continuous
dynamical systems our approach changes this aspect by unifying these two
classes due to equivalency in coarse geometry. Therefore from this outlook,
investigating features of dynamical systems does not depend on their time
evolution. In the particular case, hyper semi-dynamical system is studied in
[11].
The paper is organized as follows. In Section $2$, we recall some needed
preliminaries of the coarse geometry which include basic definitions e.g.,
coarse space, the product of coarse structures, subspace coarse structure,
morphisms between coarse spaces, coarse group and asymptotic dimension. In
Section $3$, a new class of dynamical system which is a coarse dynamical
system with coarse time and in particular unified dynamical systems are
presented. By defining conjugate coarse relation on coarse dynamical systems
it is shown that the coarse relation is an equivalence relation and preserves
coarse-orbit. We prove the coproduct of two coarse dynamical systems is a
coarse dynamical system. Also, we define set-value coarse dynamical systems
and prove if two coarse dynamical systems are coarse conjugate, then their
corresponding set-value coarse dynamical systems are coarse conjugate.

##  2\. Preliminaries

In this section, some concepts in coarse geometry e.g., coarse space, coarse
group and asymptotic dimension are presented. In the sequel,
$\mathbb{N},\mathbb{Z}$ and $\mathbb{R}$ denote the sets of natural numbers,
of integers and of real numbers, respectively. The following concepts are
adapted from [2, 3, 17].

###  2.1. Coarse spaces

As mentioned in the previous section coarse geometry studies large-scale
features of spaces, neglecting their local, small-scale ones. Two spaces are
considered equivalent if they are the same as each other for a spectator whose
point of view is getting farther and farther. In this subsection, we briefly
provide some concepts which are needed in the sequel.

######  Definition 2.1.

[17] A coarse space is a pair $(M,\mathcal{E}_{M})$, where $M$ is a set and
$\mathcal{E}_{M}\subseteq P(M\times M)$ a coarse structure on it, which means
that
$(i)$ $\bigtriangleup_{M}:=\\{(m,m)\mid m\in M\\}\in\mathcal{E}_{M}$;
$(ii)$ $\mathcal{E}_{M}$ is closed under taking subsets;
$(iii)$ $\mathcal{E}_{M}$ is closed under finite unions;
$(iv)$ if $E\in\mathcal{E}_{M}$, $E^{-1}=\\{(m_{2},m_{1})\mid(m_{1},m_{2})\in
E\\}\in\mathcal{E}_{M}$;
$(v)$ if $E,F\in\mathcal{E}_{M}$, then $E\circ F=\\{(m_{1},m_{2})\in M\times
M\mid\exists~{}m_{3}\in M:(m_{1},m_{3})\in E,(m_{3},m_{2})\in F\\}$.

The properties $(ii)$ and $(iii)$ say that $\mathcal{E}_{M}$ is an ideal of
subsets of $M\times M$ and an element $E$ of $\mathcal{E}_{M}$ is called
entourage. The subset $E[m]=\\{m^{\prime}\in M:(m,m^{\prime})\in E\\}$ is
called the ball centred in $m$ with radius $E$. Furthermore, if $N\subseteq
M$, then $E[N]:=\displaystyle\bigcup_{n\in N}E[n]$.
For a set $M$ a base of a coarse structure $\mathcal{B}$ is a family of
entourages such that its completion $\mathcal{CL}(\mathcal{B}):=\\{F\subseteq
B\mid B\in\mathcal{B}\\}$ is a coarse structure. Notice that
$\mathcal{CL}(\mathcal{B})$ is the closure of $\mathcal{B}$ under taking
subsets.
Despite the fact that for every set $M$ we can equip it with two coarse
structure which are the discrete coarse structures
$\mathcal{E}_{M_{dis}}=\mathcal{CL}(\\{\\{\bigtriangleup_{M}\\}\\})$ and the
trivial (or indiscrete) coarse structure $\mathcal{E}_{M_{triv}}=P(M\times
M)$. There is a significant example of coarse structure for metric space which
is called the metric-coarse structure. Consider an extended pseudo-metric $d$
on $M$, i.e., $d:M\times M\rightarrow\mathbb{R}^{+}\cup\\{\infty\\}$ with the
following properties:
$(i)$ $d(m,m)=0$, for every $m\in M$;
$(ii)$ $d(m_{1},m_{2})=d(m_{2},m_{1})$, for every $m_{1},m_{2}\in M$
(symmetry);
$(iii)$ $d(m_{1},m_{2})\leqslant d(m_{1},m_{3})+d(m_{3},m_{2})$; for every
$m_{1},m_{2},m_{3}\in M$ (triangle inequality).
As usual we make the assumption: $a+\infty=\infty+a=\infty$ and
$a\leqslant\infty$ for all $a\in\mathbb{R}^{+}\cup\infty$. For convenience we
indicate $(M,d)$ as a metric space. For every $r>0$, by considering a specific
subsets of the square $M\times M$, which is called the strip of width $r$, as
follows: $S_{r}=\displaystyle\bigcup_{m\in M}\big{(}\\{m\\}\times
B(m,r)\big{)}$, where $B(m,r)$ signifies the open ball centred in $m$ with
radius $r$. The family $\mathcal{B}_{d}=\\{S_{r}\mid r>0\\}$ is a base of the
metric coarse structure $\mathcal{E}_{M_{d}}=\mathcal{CL}(\mathcal{B}_{d})$.
A coarse space $(M,\mathcal{E}_{M})$ is metrisable if there exists an extended
pseudo-metric $d$ on $M$ such that $\mathcal{E}_{M}=\mathcal{E}_{M_{d}}$.

######  Definition 2.2.

[2, 3] Let $\\{(M_{i},\mathcal{E}_{M_{i}})\\}_{i\in I}$ be a family of coarse
spaces and $M=\prod_{i}M_{i}$ and $p_{i}:M\rightarrow M_{i}$, for every $i\in
I$ be the projection maps. Then the product coarse structure
$\mathcal{E}=\prod_{i}\mathcal{E}_{M_{i}}$ is defined by the base
$\mathcal{B}=\big{\\{}\displaystyle\bigcap_{i\in I}(p_{i}\times
p_{i})^{-1}(E_{i})\mid E_{i}\in\mathcal{E}_{M_{i}},\forall i\in I\big{\\}}$

Indeed, products and coproducts are two dual notions in category theory. In
the next definition, we provide coarse structure on coproducts of coarse
spaces.

######  Definition 2.3.

[20, 21] Let $(M,\mathcal{E}_{M})$ and $(\tilde{M},\mathcal{E}_{\tilde{M}})$
be two coarse spaces. The coproduct coarse structure $\mathcal{E}$ on the
disjoint union $M\sqcup\tilde{M}$ is defined by $\mathcal{E}:=\\{(i_{1}\times
i_{1})(X)\cup(i_{2}\times i_{2})(Y)\mid
X\in\mathcal{E}_{M},Y\in\mathcal{E}_{\tilde{Y}}\\}$, where $i_{1}:M\rightarrow
M\sqcup\tilde{M}$ and $i_{2}:\tilde{M}\rightarrow M\sqcup\tilde{M}$ are the
canonical inclusions.

One can easily extend this definition to the coproduct of a finite number of
coarse spaces. As for the infinite case, we refer to [22].

######  Definition 2.4.

[17] If $(M,\mathcal{E}_{M})$ is a coarse space and $N\subset M$, then $N$ can
be equipped with the subspace coarse structure
$\mathcal{E}\mid_{N}=\\{E\cap(N\times N)\mid E\in\mathcal{E}_{M}\\}$. In this
case, the pair $(Y,\mathcal{E}_{M}\mid_{N})$ is called a coarse subspace.

In order to define a map between two coarse spaces, the maps
$f,g:K\rightarrow(M,\mathcal{E}_{M})$ from a set $K$ to a coarse space $M$ are
considered. We say $f$ and $g$ are close, and write $f\sim g$, if
$\\{(f(k),g(k))\mid k\in K\\}\in\mathcal{E}_{M}$. Furthermore, for the map
$f:M\rightarrow N$ between sets, we denote by $f\times f:M\times M\rightarrow
N\times N$ the map defined by $f\times f(m_{1},m_{2})=(f(m_{1}),f(m_{2}))$ for
every $(m_{1},m_{2})\in M\times M$.

######  Definition 2.5.

[2, 3, 21] Let $(M,\mathcal{E}_{M})$ and $(N,\mathcal{E}_{N})$ be two coarse
spaces. A map $f:M\rightarrow N$ is
$(i)$ bornologous if, for every $E\in\mathcal{E}_{M}$, $(f\times
f)(E)\in\mathcal{E}_{N}$;
$(ii)$ effectively proper if, for every $E\in\mathcal{E}_{N}$, $(f\times
f)^{-1}(E)\in\mathcal{E}_{M}$;
$(iii)$ an asymorphism if one of the following equivalent properties is
satisfied:
$\indent(iii_{1})$ $f$ is bijective and both $f$ and $f^{-1}$ are bornologous,
$\indent(iii_{2})$ $f$ is bijective and both bornologous and effectively
proper;
$(iv)$ an asymorphic embedding if one of the following equivalent properties
is satisfied:
$\indent(iv_{1})$ the restriction of $f$ to its image endowed with the
subspace coarse structure is an asymorphism,
$\indent(iv_{2})$ $f$ is injective and both bornologous and effectively
proper;
$(v)$ a coarse equivalence if one of the following equivalent properties is
satisfied:
$\indent(v_{1})$ $f$ is bornologous and there exists another bornologous map
$g:N\rightarrow M$ (called coarse inverse) such that $g\circ f\sim Id_{M}$ and
$f\circ g\sim Id_{N}$,
$\indent(v_{2})$ $f$ is bornologous and effectively proper and $f(M)$ is large
in $N$.

It is clear asymorphisms are precisely the bijective coarse equivalences. We
recall that a subset $A$ of coarse space $M$ is called large in $M$ if there
exists $E\in\mathcal{E}_{M}$ such that $E[A]:=\displaystyle\bigcup_{a\in
A}E(a)=M$ (see [4]).

###  2.2. Coarse group

We are interested in using groups with coarse structure as evolution time for
dynamical system on coarse spaces, so some concepts of the coarse group in [2,
3, 21] are presented briefly. In fact, the coarse structures which are agreed
with the algebraic structure of the groups are more remarkable.
Let $G$ is a group and $g\in G$, the left-shift $f_{g}^{l}:G\rightarrow G$ and
the right-shift $f_{g}^{r}:G\rightarrow G$ are defined as follows:
for every $g^{\prime}\in G$ $f_{g}^{l}(g^{\prime})=gg^{\prime}$ and
$f_{g}^{r}(g^{\prime})=g^{\prime}g$.

######  Proposition 2.6.

[2, 3] If $G$ be a group and $\mathcal{E}_{G}$ be a coarse structure on it,
then the following properties are equivalent:
$(i)$ for every $E\in\mathcal{E}$, $GE:=\\{(ge_{1},ge_{2})~{}|~{}g\in
G,(e_{1},e_{2})\in E\\}\in\mathcal{E}_{G}$;
$(ii)$ the family $\mathcal{F}_{g}^{l}=\\{f_{g}^{l}~{}|~{}g\in G\\}$ is
uniformly bornologous, i.e., for every $E\in\mathcal{E}_{G}$ there exsits
$F\in\mathcal{E}_{G}$ such that, for every $g\in G$, $(f_{g}^{l}\times
f_{g}^{l})(E)\subseteq F$.

######  Definition 2.7.

[2, 3] A coarse structure $\mathcal{E}_{G}$ on a group $G$ is said to be a
left group coarse structure if it has the equivalent properties from
Proposition 2.6. A left coarse group is a pair $(G,\mathcal{E}_{G})$ of a
group $G$ and a left group coarse structure $\mathcal{E}_{G}$ on $G$. Right
group coarse structure and right coarse group can be defined in a similar way.

To clarify and make some examples of left(right) group coarse structures and
(left) right coarse groups, the following definitions are needed.

######  Definition 2.8.

[2, 3] Let $G$ be a group. A group ideal $\mathcal{I}$ is a family of subsets
of $G$ containing the singleton $\\{e\\}$, where $e$ is the neutral element of
G, such that:
$(i)$ $\mathcal{I}$ is an ideal;
$(ii)$ $HT:=\\{ht~{}|~{}h\in H,t\in T\\}$, for every $H,T\in\mathcal{I}$;
$(iii)$ $H^{-1}:=\\{h^{-1}~{}|~{}h\in H\\}\in\mathcal{I}$.

$\bigcup\mathcal{I}$ is a subgroup of $G$ if $\mathcal{I}$ is a group ideal on
$G$.

######  Definition 2.9.

[2, 3] Let $G$ be a group and $\mathcal{I}$ be a group ideal. For every
$H\in\mathcal{I}$, we define

$E_{H}^{l}:=G(\\{e\\}\times H):=\displaystyle\bigcup_{g\in G}(\\{g\\}\times
gH).$

The family $\mathcal{E}_{\mathcal{I}}^{l}:=\\{E\subseteq G\times G\mid\exists
H\in\mathcal{I}:E\subseteq E_{H}^{l}\\}$ is a left coarse group structure
which is called left $\mathcal{I}$-group coarse structure, and the pair
$(G,\mathcal{E}_{G}^{l})$ is a left coarse group that is called, left
$\mathcal{I}$-coarse group.
Pay attention the family $\\{E_{H}^{l}\mid H\in\mathcal{I}\\}$ is a base of
the $\mathcal{I}$-group coarse structure. In addition, for every
$H\in\mathcal{I}$ and $g\in G$, $E_{H}^{l}[g]=gH$.
Analogously, the right $\mathcal{I}$-group coarse structure
$\mathcal{E}_{\mathcal{I}}^{r}$ is defined by the base
$\\{E_{H}^{r}|H\in\mathcal{I}\\}$, where

$E_{H}^{l}:=(\\{e\\})\times H)G:=\displaystyle\bigcup_{g\in G}(\\{g\\}\times
Hg).$

The following proposition indicates the left $\mathcal{I}$-group coarse
structure and the right $\mathcal{I}$-group coarse structure are equivalent
for every group $G$ and group ideal $\mathcal{I}$ on it.

######  Proposition 2.10.

[2, 3] If $G$ be a group, $\mathcal{I}$ be a group ideal, and $j:G\rightarrow
G$ defined by $j(g):=g^{-1}$, then
$J:(G,\mathcal{E}_{\mathcal{I}}^{l})\rightarrow(G,\mathcal{E}_{\mathcal{I}}^{r})$
is an asymorphism.

The fundamental result in this subsection is the following proposition which
indicates every left coarse group can be achieved as Definition 2.9

######  Proposition 2.11.

[14] Let $G$ be a group and $\mathcal{E}_{G}$ be a coarse structure on it.
Then the following properties are equivalent:
$(i)$ $(G,\mathcal{E}_{G})$ is a left coarse group;
$(ii)$ $\mathcal{E}_{G}=\mathcal{E}_{\mathcal{I}}^{l}$ where
$\mathcal{I}:=\\{E[e]:E\in\mathcal{E}_{G}\\}$.

In view of Proposition 2.11 coarse groups are equivalently determined by group
ideals and this proposition satisfies for right coarse structures. In the
sequel, left group coarse structures (and left coarse groups), are referred to
as group coarse structures (and coarse groups). In the next ramark some
examples of coarse groups are provided.

######  Remark 2.12.

[2, 3] Let $G$ be a group.
$(i)$ There is two trivial groups ideal $\\{\\{e\\}\\}$ and $P(G)$.
$\\{\\{e\\}\\}$-group coarse structure is called the discrete coarse
structure, which is containing only the subsets of the diagonal and group
ideal $P(G)$, creates the bounded coarse structure that means every subset of
$G\times G$ is an entourage. We recall that, for a coarse space
$(M,\mathcal{E}_{M})$ and $m\in M$, a subset $N$ of $M$ is bounded from $m$ if
there exists an entourage $E\in\mathcal{E}_{M}$ such that $N\subseteq E[m]$. A
subset is called bounded if it is bounded from a point.
$(ii)$ The family $[G]^{<\omega}$ contain all finite subsets of $G$ is a group
ideal. $[G]^{<\omega}$-coarse structure is called finitary-group coarse
structure.
$(iii)$ If $d$ be a left-invariant pseudo-metric on $G$, the family
$\mathcal{B}_{d}:=\\{H\subseteq G\mid\exists R\geqslant 0:H\subseteq
B_{d}(e,R)\\}$ is a group ideal and $\mathcal{B}_{d}$-group coarse structure
is called metric-group coarse structure. Note that
$\mathcal{E}_{\mathcal{B}_{d}}=\mathcal{E}_{G{{}_{d}}}$.

###  2.3. Asymptotic dimension

One of the extremely significant coarse invariant is the asymptotic dimension
that associates a value to coarse spaces which is invariant under coarse
equivalences. It is the large scale counterpart of the classical
$\breve{C}$ech-Lebesgue covering dimension which was introduced by Gromov [7].
We provide the necessary outline about the notion of asymptotic dimension of
coarse spaces.

######  Definition 2.13.

[2] A family $\mathcal{U}$ of subsets of a coarse space $(M,\mathcal{E}_{M})$
is uniformly bounded if there exists $E\in\mathcal{E}_{M}$ such that, for
every $U\in\mathcal{U}$ and $u\in U,U\subseteq E[u]$.

######  Definition 2.14.

[17] A coarse space $(M,\mathcal{E}_{M})$ has asymptotic dimension at most
$n\in\mathbb{N}$, where $n\in\mathbb{N}$ and we write
$asdim(M,\mathcal{E}_{M})\leqslant n$, if, for every $E\in\mathcal{E}_{M}$
there exists a uniformly bounded cover
$\mathcal{U}=\mathcal{U}_{0}\cup...\cup\mathcal{U}_{n}$ such that, for every
$i=0,...,n$ and every $U,V\in\mathcal{U}_{i}$, $E[U]\cap V=\varnothing$ (i.e.,
$\mathcal{U}_{i}$ is $E$-separated, for every $i=0,...,n)$. If
$asdim(M,\mathcal{E}_{M})\leqslant n$ and $asdim(M,\mathcal{E}_{M})>n-1$, we
write $asdim(M,\mathcal{E}_{M})=n$, where $n\in\mathbb{N}$. Finally, if, for
every $n\in\mathbb{N}$, $asdim(M,\mathcal{E}_{M})>n$, then
$asdim(M,\mathcal{E}_{M})=\infty$.

There are the similar definitions for uniformly bounded and asymptotic
dimension of coarse groups which are presented in the two following
definitions.

######  Definition 2.15.

[2] A family $\mathcal{U}$ of subsets of a coarse group
$(G,\mathcal{E}_{\mathcal{I}})$ is uniformly bounded if and only if there
exists $H\in\mathcal{I}$ such that $U\subseteq E_{H}[u]=uH$, for every
$U\in\mathcal{U}$ and $u\in U$.

It is said that $\mathcal{U}$ is $S$-disjoint if, for every pair of distinct
elements $U,V\in\mathcal{U}$, $U\cap E_{S}[V]=U\cap VS=\varnothing$ where
$S\in\mathcal{I}$ (see [2]).

######  Definition 2.16.

[2] A coarse group $(G,\mathcal{E}_{I})$ has asymptotic dimension at most $n$
(asdim$(G,\mathcal{E}_{\mathcal{I}})\leqslant n$), where $n\in\mathbb{N}$, if,
for every $S\in\mathcal{I}$, there exists a uniformly bounded cover
$\mathcal{U}=\mathcal{U}_{0}\cup...\cup\mathcal{U}_{n}$ such that, for every
$i=0,...,n$, $\mathcal{U}_{i}$ is $S$-disjoint. The asymptotic dimension of
$(G,\mathcal{E}_{\mathcal{I}})$ is $n$ if
asdim$(G,\mathcal{E}_{\mathcal{I}})\leqslant n$ and
asdim$(G,\mathcal{E}_{\mathcal{I}})>n-1$. Finally,
asdim$(G,\mathcal{E}_{\mathcal{I}})=\infty$ if, for every $n\in\mathbb{N}$,
asdim$(G,\mathcal{E}_{\mathcal{I}})>n$.

######  Theorem 2.17.

[6] Let $G$ be an abelian group. Then $asdimG=r_{0}(G)$, where $G$ is endowed
with the finitary-group coarse structure .

Let us recall that for an abelian group $G$, $r_{0}(G)$, is its free-rank
which is the cardinality of the maximal independent subset of $G$.

##  3\. Coarse time dynamical systems on coarse space

In this section, we introduce dynamical systems on a coarse space with coarse
time evolution. In coarse geometry, $\mathbb{Z}$ and $\mathbb{R}$ are coarsely
equivalent groups. Because of this property we call the dynamical system on a
coarse space, with the time evolution of $\mathbb{Z}$ or $\mathbb{R}$, a
unified dynamical system on coarse apace. We define coarse conjugacy and
coarse orbits for this new class of dynamical system. Also, it is shown that
the coproduct of two coarse dynamical systems is a coarse dynamical system.

######  Definition 3.1.

A coarse dynamical system with coarse time is a triple
$[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$, where
$(M,\mathcal{E}_{M})$ is a non-empty coarse space, $(G,\mathcal{E}_{G},\ast)$
is a coarse group and $\phi$ is a set of asymorphism mappings
$\phi^{g}:M\rightarrow M$ where $g\in G$ with the following properties:
$(i)$ If $g_{1},g_{2}\in G$, and $m\in M$, then
$\phi^{g_{1}}\circ\phi^{g_{2}}\in\phi^{g_{1}\ast g_{2}}(m)$, where
$\phi^{g_{1}\ast g_{2}}(m)=\\{\phi^{g}(m):g\in g_{1}\ast g_{2}\\}$;
$(ii)$ $\phi^{e}=Id_{M}$ where $e$ is the identity element of $G$.

We call the mapping $\phi^{g}:M\rightarrow M$ a coarse evolution operator.

######  Remark 3.2.

$(i)$ If $G=\mathbb{Z}$ or $G=\mathbb{R}$ we call a triple
$[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$, unified dynamical
system.
Let us clarify this denomination from coarse geometry point of view. If we
consider $\mathbb{Z}$ and $\mathbb{R}$ as a metric space with usual metric
$d(a,b)=|a-b\mid$ is left-invariant, then $\mathbb{Z}$ and $\mathbb{R}$ are
coarsely equivalent. It is enough to take the inclusion
$i:\mathbb{Z}\hookrightarrow\mathbb{R}$ and the floor map
$\lfloor.\rfloor:\mathbb{R}\rightarrow\mathbb{Z}$ such that, for every
$r\in\mathbb{R}$, $\lfloor r\rfloor=max\\{z\in\mathbb{Z}:z\leqslant r\\}$. If
we look at $\mathbb{Z}$ and $\mathbb{R}$ as a group with the binary operation
$``+"$ by the statement $(iii)$ of Remark 2.12 they are coarsely equivalent
with their group-coarse structure. In fact, $\mathbb{Z}$ and $\mathbb{R}$ are
similar to each other from the coarse geometry point of view. One can imagine
$\mathbb{Z}$ whose points get closer and closer until the whole space will
simulate a line which is $\mathbb{R}$.
We ask the reader to pay attention to this point if we consider the finitary-
group coarse structure on $\mathbb{Z}$ and $\mathbb{R}$ the situation is very
different, as the two coarse groups are not coarsely equivalent. Since by
Theorem 2.17 $r_{0}(\mathbb{Z})=1$, while $r_{0}(\mathbb{R})=\infty$, and
therefore they can not be coarsely equivalent.
$(ii)$ Consider a coarse dynamical system
$[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$. Let $N\subset M$ and
$H\leqslant G$ i.e. $H$ is a subgroup of $G$. Then we can make a sub-coarse
dynamical system $[(N,\mathcal{E}_{N}),\phi\mid_{N},(H,\mathcal{E}_{H},\ast)]$
by taking subspace coarse structure on $N$ and $H$.

######  Definition 3.3.

Two coarse dynamical systems
$[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$ and
$[(\tilde{M},\mathcal{E}_{\tilde{M}}),\tilde{\phi},(\tilde{G},\mathcal{E}_{\tilde{G}},\tilde{\ast})]$
are called $(f,h)$-coarse conjugate if there exist an asymorphism
$f:(M,\mathcal{E}_{M})\rightarrow(\tilde{M},\mathcal{E}_{\tilde{M}})$ between
coarse spaces and an asymorphism
$h:(G,\mathcal{E}_{G})\rightarrow(\tilde{G},\mathcal{E}_{\tilde{G}})$ such
that the following two axioms hold.
$(i)$ $h(g_{1}\ast g_{2})=h(g_{1})\tilde{\ast}h(g_{2})$ for all
$g_{1},g_{2}\in G$;
$(ii)$ $f\circ\phi^{g}=\tilde{\phi}^{h(g)}\circ f$ for all $g\in G$ (see the
following diagram).

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{g}}$$\scriptstyle{f}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\tilde{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\phi}^{h(g)}}$$\textstyle{\tilde{M}}$

The next theorem implies that conjugate coarse relation is an equivalence
relation on coarse dynamical systems.

######  Theorem 3.4.

Let $(f,h)$ be a coarse conjugate relation between
$[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$ and
$[(\tilde{M},\mathcal{E}_{\tilde{M}}),\tilde{\phi},(\tilde{G},\mathcal{E}_{\tilde{G}},\tilde{\ast})]$,
and let $(k,l)$ be a coarse conjugate relation between
$[(\tilde{M},\mathcal{E}_{\tilde{M}}),\tilde{\phi},(\tilde{G},\mathcal{E}_{\tilde{G}},\tilde{\ast})]$
and
$[(\bar{M},\mathcal{E}_{\bar{M}}),\bar{\phi},(\bar{G},\mathcal{E}_{\bar{G}},\bar{\ast})]$.
Then
$(i)$ $(f^{-1},h^{-1})$ is a coarse conjugate relation between
$[(\tilde{M},\mathcal{E}_{\tilde{M}}),\tilde{\phi},(\tilde{G},\mathcal{E}_{\tilde{G}},\tilde{\ast})]$
and $[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$.
$(ii)$ $(k\circ f,l\circ h)$ is a coarse conjugate relation between
$[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$ and
$[(\tilde{M},\mathcal{E}_{\tilde{M}}),\tilde{\phi},(\tilde{G},\mathcal{E}_{\tilde{G}},\tilde{\ast})]$.

###### Proof.

Proof (i). Since $f$ and $h$ are asymorphisms. It is enough to prove two
properties in Definition 3.3 are satisfied.
If $\tilde{g},\tilde{s}\in\tilde{G}$, then
$h^{-1}(\tilde{g}{\ast}\tilde{s})=h^{-1}(h(g\ast
s))=h^{-1}(h(g)\tilde{\ast}h(s))=h^{-1}(h(g))\ast
h^{-1}(h(s))=h^{-1}(\tilde{g})\ast h^{-1}(\tilde{s})$.
For $\tilde{g}\in\tilde{G}$ we have $f\circ\phi^{h^{-1}(\tilde{g})}\circ
f^{-1}=\tilde{\phi}^{h\circ h^{-1}(\tilde{g})}\circ f\circ
f^{-1}=\tilde{\phi}^{\tilde{g}}$. Therefore $\phi^{h^{-1}(\tilde{g})}\circ
f^{-1}=f^{-1}\circ\tilde{\phi}^{\tilde{g}}$.
Proof (ii). $l\circ h(g_{1}\ast g_{2})=l(h(g_{1}\ast
g_{2}))=l(h(g_{1})\tilde{\ast}h(g_{2}))=l\circ h(g_{1})\bar{\ast}l\circ
h(g_{2})$ for $g_{1},g_{2}\in G$ and
$\bar{\phi}^{l\circ h(g)}\circ(k\circ f)=k\circ\tilde{\phi}^{h(g)}\circ
f=k\circ(f\circ\phi^{g})=(k\circ f)\phi^{g}$. ∎

######  Definition 3.5.

The set $O^{G}(m)=\\{\phi^{g}(m):g\in G\\}$ is called coarse-orbit for $m\in
M$.

It is clear $O^{G(m)}$ is a coarse group. Because for any
$\phi^{g}(m),\phi^{\tilde{g}}(m)\in O^{G}(m)$ we have
$\phi^{g}(m)\ast\phi^{{\tilde{g}}^{-1}}(m)=\phi^{g\ast\tilde{g}^{-1}}(m)\in
O^{G}(m)$ where $g,\tilde{g}\in G$. So as we mentioned in Section $(2)$ it can
be equipped to a coarse structure.
In the next theorem we show the coarse conjugate relation preserves coarse-
orbit.

######  Theorem 3.6.

If $[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$ and
$[(\tilde{M},\mathcal{E}_{\tilde{M}}),\tilde{\phi},(\tilde{G},\mathcal{E}_{\tilde{G}},\tilde{\ast})]$
are coarse conjugate under a coarse conjugate $(f,h)$, then
$f(O^{G}(m))=O^{\tilde{G}}(f(m))$.

###### Proof.

Let $s\in f(O^{G}(m))$. Then $s=f(\phi^{g}(m))=(\tilde{\phi}^{h(g)}\circ
f)(m)=\tilde{\phi}^{h(g)}(f(m))\in O^{\tilde{G}}(f(m))$. So
$f(O^{G}(m))\subseteq O^{\tilde{G}}(f(m))$. Because coarse conjugate relation
is an equivalence relation. One can easily check
$f^{-1}(O^{\tilde{G}}(f(m))\subseteq O^{G}(m)$. Therefore
$O^{\tilde{G}}(f(m)\subseteq f(O^{G}(m))$. ∎

######  Definition 3.7.

If $\phi_{1}:M_{1}\rightarrow\tilde{M}_{1}$ and
$\phi_{2}:M_{2}\rightarrow\tilde{M}_{2}$ are two maps, a map between disjoint
unions can be defined as follows: $\phi_{1}\sqcup\phi_{2}:M_{1}\sqcup
M_{2}\rightarrow\tilde{M}_{1}\sqcup\tilde{M}_{2}$ for every $i_{j}(m)\in
M_{1}\sqcup M_{2}$, $\phi_{1}\sqcup\phi_{2}(i_{j}(m)):=i_{j}(\phi_{j}(m))$ for
$j=1,2$.

In the sequel, the members of the disjoint union of $M$ and $\tilde{M}$ are
considered as follows:
$M\sqcup\tilde{M}=\\{\big{(}(1,m)(2,\tilde{m})\big{)}:m\in
M,\tilde{m}\in\tilde{M}\\}$.

######  Remark 3.8.

If $(G,\ast)$ and $(\tilde{G},\tilde{\ast})$ be two groups. Then
$(G\sqcup\tilde{G},\ast\sqcup\tilde{\ast})$ is group where
$\big{(}(1,g_{1})(2,\tilde{g}_{1})\big{)}\ast\sqcup\tilde{\ast}\big{(}(1,g_{2})(2,\tilde{g}_{2})\big{)}:=\big{(}(1,g_{1}\ast
g_{2})(2,\tilde{g}_{1}\tilde{\ast}\tilde{g}_{2})\big{)}$ for all
$\big{(}(1,g_{1})(2,\tilde{g}_{1})\big{)},\big{(}(1,g_{2})(2,\tilde{g}_{2})\big{)}\in
G\sqcup\tilde{G}$. Because
$\big{(}(1,g_{1})(2,\tilde{g}_{1})\big{)}\ast\sqcup\tilde{\ast}\big{(}(1,g_{2}^{-1})(2,\tilde{g}_{2}^{-1})\big{)}=\big{(}(1,g_{1}\ast
g_{2}^{-1})(2,\tilde{g}_{1}\tilde{\ast}\tilde{g}_{2}^{-1})\big{)}\in
G\sqcup\tilde{G}$.

######  Theorem 3.9.

Let $[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$ and
$[(\tilde{M},\mathcal{E}_{\tilde{M}}),\tilde{\phi},(\tilde{G},\mathcal{E}_{\tilde{G}},\tilde{\ast})]$
be two coarse dynamical systems. Then
$[(M\sqcup\tilde{M},\mathcal{E}),\phi\sqcup\tilde{\phi},(G\sqcup\tilde{G},\ast\sqcup\tilde{\ast})]$
is a coarse dynamical system.

###### Proof.

In view of Definition 2.3 both $M\sqcup\tilde{M}$ and $G\sqcup\tilde{G}$ are
equipped by coarse structure. So we show two properties in Definition 3.1 are
satisfied.
$(i)$
$(\phi^{g_{1}}\sqcup\tilde{\phi}^{\tilde{g}_{1}})\circ(\phi^{g_{2}}\sqcup\tilde{\phi}^{\tilde{g}_{2}})\big{(}(1,m)(2,\tilde{m})\big{)}=(\phi^{g_{1}}\circ\phi^{g_{2}})(1,m)\sqcup(\tilde{\phi}^{\tilde{g}_{1}}\circ\tilde{\phi}^{\tilde{g}_{2}})(2,\tilde{m})\in\phi^{g_{1}\ast
g_{2}}(1,m)\sqcup\tilde{\phi}^{\tilde{g}_{1}\tilde{\ast}\tilde{g}_{2}}(2,\tilde{m})$
, where $\phi^{g_{1}\ast
g_{2}}(1,m)\sqcup\tilde{\phi}^{\tilde{g}_{1}\tilde{\ast}\tilde{g}_{2}}(2,\tilde{m})=\\{\big{(}(1,\phi^{s}(m))(2,\tilde{\phi}^{t}(\tilde{m}))\big{)}\mid
s\in g_{1}\ast g_{2},t\in\tilde{g}_{1}\tilde{\ast}\tilde{g}_{2}\\}$ if
$g_{1},g_{2}\in G$, $\tilde{g}_{1}$, $\tilde{g}_{2}\in\tilde{G}$, $m\in M$ and
$\tilde{m}\in\tilde{M}$.
$(ii)$
$\phi^{e}\sqcup\tilde{\phi}^{\tilde{e}}\big{(}(1,m)(2,\tilde{m})\big{)}=\big{(}(1,\phi^{e}(m))(2,\tilde{\phi}^{\tilde{e}}(\tilde{m})\big{)}=\big{(}(1,Id_{M})(2,Id_{\tilde{M}})\big{)}=Id_{M}\sqcup
Id_{\tilde{M}}$. ∎

######  Theorem 3.10.

If $(f,h)$ be a coarse conjugate relation between
$[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$ and
$[(\tilde{M},\mathcal{E}_{\tilde{M}}),\tilde{\phi},(\tilde{G},\mathcal{E}_{\tilde{G}},\tilde{\ast})]$,
and let $(k,l)$ be a coarse conjugate relation between
$[(\hat{M},\mathcal{E}_{\hat{M}}),\hat{\phi},(\hat{G},\mathcal{E}_{\hat{G}},\hat{\ast})]$
and
$[(\bar{M},\mathcal{E}_{\bar{M}}),\bar{\phi},(\bar{G},\mathcal{E}_{\bar{G}},\bar{\ast})]$,
then two coarse dynamical systems
$[M\sqcup\hat{M},\phi\sqcup\hat{\phi},(G\sqcup\hat{G},\ast\sqcup\hat{\ast})]$
and
$[\tilde{M}\sqcup\bar{M},\tilde{\phi}\sqcup\bar{\phi},(\tilde{G}\sqcup\bar{G},\tilde{\ast}\sqcup\bar{\ast})]$
are $(f\sqcup k,h\sqcup l)$ coarse conjugate dynamical systems.

###### Proof.

Firstly, since $f$ and $k$ are asymorphisms. Then $f^{-1}\sqcup k^{-1}$ is
bornologous map and $(f\sqcup k)\circ(f^{-1}\sqcup k^{-1})=(Id_{M}\sqcup
Id_{\tilde{M}})$, $(f^{-1}\sqcup k^{-1})\circ(f\sqcup k)=(Id_{\tilde{M}}\sqcup
Id_{M})$.
Secondly, it is clear $h\sqcup l$ is an asymorphism from $G\sqcup\hat{G}$ to
$\tilde{G}\sqcup\bar{G}$.
Thirdly, $h\sqcup
l[\big{(}(1,g_{1})(2,\hat{g}_{1})\big{)}\ast\sqcup\hat{\ast}\big{(}(1,g_{2})(2,\hat{g})\big{)}]=h(g_{1})\tilde{\ast}h(g_{2})\sqcup
l(\hat{g}_{2})\bar{\ast}l(\hat{g}_{2})=h(g_{1})\sqcup
l(\hat{g}_{1})\tilde{\ast}\sqcup\bar{\ast}h(g_{2})\sqcup
l(\hat{g}_{2})=h\sqcup
l[\big{(}(1,g_{1})(2,\hat{g}_{1})\big{)}]\tilde{\ast}\sqcup\bar{\ast}h\sqcup
l[\big{(}(1,g_{2}),(2,\hat{g}_{2})\big{)}]$.
Finally, to complete the proof we show the following diagram is commutative.

$\textstyle{M\sqcup\hat{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{g}\sqcup\hat{\phi}^{\hat{g}}}$$\scriptstyle{f\sqcup
k}$$\textstyle{M\sqcup\hat{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\sqcup
k}$$\textstyle{\tilde{M}\sqcup\bar{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\phi}^{\tilde{g}}\sqcup\bar{\phi}^{\bar{g}}}$$\textstyle{\tilde{M}\sqcup\bar{M}}$

So we have $(f\sqcup
k)\circ(\phi^{g}\sqcup\hat{\phi}^{\hat{g}})=(f\circ\phi^{g})\sqcup(k\circ\hat{\phi}^{\hat{g}})=(\tilde{\phi}^{\tilde{g}}\circ
f)\sqcup(\bar{\phi}^{\bar{g}}\circ
k)=(\tilde{\phi}^{\tilde{g}}\sqcup\bar{\phi}^{\bar{g}})\circ(f\sqcup k)$. ∎

######  Remark 3.11.

The Theorems 3.9 and 3.10 are satisfied in the case of considering product
instead of coproduct of coarse dynamical systems. We leave it to the reader to
prove similar results.

We are going to define a set-value coarse dynamical system. So some concepts
of coarse structure on the power set of the coarse space $(M,\mathcal{E}_{M})$
are provided. The following definition tells us how to make a coarse structure
on $P(M)$ i.e., the power set of the coarse space $(M,\mathcal{E}_{M})$.

######  Definition 3.12.

[21] Let $M$ be a set and $E$ be an entourge of $M$. Consider
$E^{*}:=\\{(K,L)\in P(M)\times P(M)\mid K\subseteq E[L],L\subseteq E[K]\\}.$
For coarse space $(M,\mathcal{E}_{M})$, the family
$B_{\mathcal{E}_{M}}^{*}=\\{X^{*}:X\in\mathcal{E}_{M}\\}$ creates a base of
coarse structure $\exp\mathcal{E}_{M}$ on $P(M)$ and
$(P(M),\exp\mathcal{E}_{M})$ is called coarse hyperspace.

######  Proposition 3.13.

[21] If $(M,\mathcal{E}_{M})$ is a coarse space, then $\exp\mathcal{E}_{M}$ is
a coarse structure.

######  Proposition 3.14.

[21] Let $f:(M,\mathcal{E}_{M})\rightarrow(\tilde{M},\mathcal{E}_{\tilde{M}})$
be a map between coarse spaces. Then
$(i)$ $f$ is bornologous if and only if $\exp f$ bornologous;
$(ii)$ $f$ effectively proper if and only if $\exp f$ effectively proper;
$(iii)$ $f$ an asymorphism if and only if $\exp f$ is an asymorphism;
$(iv)$ $f$ a coarse equivalence if and only if $\exp f$ is a coarse
equivalence.

######  Remark 3.15.

Due to Proposition 3.14. In Definition 3.1 the asymorphism mapping
$\phi^{g}:(M,\mathcal{E}_{M})\rightarrow(M,\mathcal{E}_{M})$ between coarse
spaces induces the asymorphism mapping
$\exp\phi^{g}:(P(M),\exp\mathcal{E}_{M})\rightarrow(P(M),\exp\mathcal{E}_{M})$.
So we can define the set-value coarse dynamical system.

######  Definition 3.16.

Consider a coarse dynamical system
$[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$.
A set-value coarse dynamical system with caorse time which is induced by
$[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$ is a triple
$[(P(M),\exp\mathcal{E}_{M}),\exp\phi,(G,\mathcal{E}_{G},\ast)]$ with the
following properties:
$(i)$ If $g_{1},g_{2}\in G$, and $N\in P(M)$, then
$\exp\phi^{g_{1}}\circ\exp\phi^{g_{2}}\in\exp\phi^{g_{1}\ast g_{2}}(N)$, where
$\exp\phi^{g_{1}\ast g_{2}}(N)=\\{\exp\phi^{g}(N):g\in g_{1}\ast g_{2}\\}$;
$(ii)$ $\exp\phi^{e}=Id_{P(M)}$ where $e$ is the identity element of $G$.

The mapping $\exp\phi^{g}:P(M)\rightarrow P(M)$ is called a set-value coarse
evolution operator.

######  Theorem 3.17.

Let two coarse dynamical systems
$[(M,\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$ and
$[(\tilde{M},\mathcal{E}_{\tilde{M}}),\tilde{\phi},(\tilde{G},\mathcal{E}_{\tilde{G}},\tilde{\ast})]$
be $(f,h)$-coarse conjugate. Then
$[(P(M),\exp\mathcal{E}_{M}),\phi,(G,\mathcal{E}_{G},\ast)]$ and
$[(P(\tilde{M}),\exp\mathcal{E}_{\tilde{M}}),\tilde{\phi},(\tilde{G},\mathcal{E}_{\tilde{G}},\tilde{\ast})]$
are $(\exp f,h)$ coarse conjugate.

###### Proof.

In view of Proposition 3.14 the map $f$ is an asymorphism if and only if $\exp
f$ is an asymorphism and $\phi^{g},\tilde{\phi}^{h(g)}$ are asymorphism if and
only if $\exp\phi^{g},\exp\tilde{\phi}^{h(g)}$ are asymorphism. It is clear
that $f\circ\phi^{g}=\tilde{\phi}^{h(g)}\circ f$ if and only if $\exp
f\circ\exp\phi^{g}=\exp\tilde{\phi}^{h(g)}\circ\exp f$ for all $g\in G$. ∎

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