# Generalized nonautonomous dynamics through groupoid morphisms

Néstor Jara & Emir Molina  Universidad de Chile, Departamento de Matemáticas.
Casilla 653, Santiago, Chile<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We extend the notions of nonautonomous dynamics to arbitrary groups, through
groupoid morphisms. This also presents a generalization of classic dynamical
systems and group actions. We introduce the structure of cotranslations, as a
specific kind of groupoid morphism, and establish a correspondence between
cotranslations and skew-products. We give applications of cotranslations to
nonautonomous equations, both in differences and differential. Our results
delve into the differentiability of cotranslations, along with dimension
invariance and diagonalization, utilizing a generalized notion of kinematic
similarity.

###### Key words and phrases:

Nonautonomous dynamics, Dynamical systems, Groupoids

###### 2020 Mathematics Subject Classification:

37C85, 37B55, 18B40, 37C60.

This research has been partially supported by ANID, Beca de Doctorado Nacional
21220105.

##  1\. Introduction

The theory of dynamical systems is currently one of the most active areas of
research in mathematics, with autonomous dynamics being the primary focus. In
contrast, the field of nonautonomous dynamics has made substantial
advancements in the past two decades, particularly in the study of time-
dependent differential and difference equations (e.g., [1], [4], [6], [11],
[12], [17], [18]). There is a significant disparity between the previously
described concepts, while autonomous dynamics depend only on the time elapsed
from the initial time $t_{0}$, the nonautonomous systems are also dependent on
the initial time $t_{0}$ itself, which has several consequences to
characterize limiting objects. As it was pointed out in [12, Ch.2], the
dynamics arising from nonautonomous differentiable system

(1.1) $\dot{x}=f(t,x)$

can be formally described by two approaches where the above mentioned $t_{0}$
plays a key role: the skew-product semiflows and the process formalism, also
known as the two parameter-$(t,t_{0})$-semigroups.

We emphasize that an autonomous differential equation does indeed define a
continuous group action, i.e. a classic dynamical system, but as soon as we
consider a nonautonomous differential equation as (1.1), these structures no
longer describe properly the dynamics given by the flow of solutions.

###  1.1. Preliminaries and setting

Consider a category $\mathscr{C}$ (topological spaces, Banach spaces, rooted
tree graphs, among others) and an element $X$ on said category. We also
consider

  * •

$\mathbb{B}_{X}$ the collection of all the morphisms of the category
$\mathscr{C}$ of $X$ on itself,

  * •

$\mathbb{A}_{X}$ the collection of all invertible elements of $\mathbb{B}_{X}$
(whose inverse is also in $\mathbb{B}_{X}$). We know that $\mathbb{A}_{X}$ is
a group.

Depending on the category $\mathscr{C}$, both sets $\mathbb{B}_{X}$ and
$\mathbb{A}_{X}$ may have more structure, but for now we keep it general.

A dynamical system, regardless of the category $\mathscr{C}$, involves a group
$G$ (possibly with additional structure) and a group morphism
$\gamma\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{A}_{X}$. In the
topological case, where $X$ is a topological space and $G$ is a topological
group, we equip $\mathbb{A}_{X}$ with the compact-open topology [5, Definition
I, p. 301]. In this case the group morphism $\gamma$ is required to be
continuous.

Equivalently, we may define that a left (topological) dynamical system is a
triple $(X,G,\alpha)$, where $X$ and $G$ satisfy the same conditions as before
and $\alpha$ is a continuous left action of $G$ on $X$, i.e. a map
$\alpha\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!X\to X$ verifying
that each $\alpha(g,\cdot)$ is a homeomorphism of $X$ on itself and

$\alpha(g,\alpha(h,x))=\alpha(gh,x),\qquad\forall\,g,h\in G,\,x\in X.$

It is easy to see that by setting $\hat{\alpha}\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{A}_{X}$ by $\left[\hat{\alpha}(g)\right](x)=\alpha(g,x)$
we obtain a continuous group morphism. On the other hand, we can define right
actions as maps $\beta\mathrel{\mathop{\mathchar 58\relax}}X\times G\to X$
verifying that each $\beta(\cdot,g)$ is a homeomorphism of $X$ on itself and

$\beta(\beta(x,g),h)=\beta(x,gh),\qquad\forall\,g,h\in G,\,x\in X.$

Once more, by setting $\hat{\beta}\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{A}_{X}$ by
$\left[\hat{\beta}(g)\right](x)=\beta(x,g^{-1})$ we obtain a continuous group
morphism. Therefore, both left and right actions describe dynamical systems
and the use of one over the other is just a matter of convenience in notation.
This correspondence between left or right actions and their algebraic
counterpart on group morphisms is a well known fact at the basis of classic
dynamics. Nevertheless, nonautonomous dynamics [12] do not have a widely
discussed analogous notion of action, even less what could be its algebraic
counterpart; therefore, the main objective of this work is to shed light on
how to fill this lack. Through this, we will also generalize nonautonomous
dynamics to arbitrary groups.

###  1.2. Novelty and Structure

The principal contribution of this article is to introduce an algebraic
generalization of dynamical systems. This generalization is applicable to
describe flows arising from nonautonomous equations, encompassing both
differential and difference equations. Notably, it extends the notion of
nonautonomous dynamics beyond the realms of specific number systems such as
$\mathbb{R}$ or $\mathbb{Z}$, bringing it into the realm of general groups.

The paper is organized as follows. In the second section we study the
structure of skew-product, objects which, to the best of our knowledge, emerge
for the first time on 1950 by H. Anzai [2], whom uses them to describe a
certain ergodic dynamic, and then later on 1965 were connected to differential
equations thanks to the work of R. K. Miller [14]. This concept refers to a
generalization of dynamical systems given by left actions, but instead of
considering one action, it uses a family of action-like functions with a
certain compatibility relation (we give more details later). The skew-products
we present here are defined for general groups, not just the usual
$\mathbb{R}$ or $\mathbb{Z}$.

On the third section we present groupoids and groupoid morphisms. We define a
specific type of these groupoid morphisms called cotranslations, which will
give an algebraic structure of the dynamics that can be represented by skew-
products. We also state some results regarding the relation of these groupoid
morphisms to discrete nonautonomous dynamics. We present several examples of
this construction for different groups.

On the fourth section we study differentiable groupoid morphisms, give some
basic properties and give an application to the problem of existence of
solutions to linear nonautonomous differential equations on Banach spaces.

On the final section we study a partial notion of the groupoid morphisms we
presented earlier, mainly on the Euclidean space, and give some of their
algebraic and topological properties, obtaining a generalized notion of
kinematic similarity.

##  2\. Skew-product dynamical systems

In this section we present skew-products dynamical systems. Although we
present them in the topological case, a similar structure can be defined and
studied for objects on different categories. As described by R. J Sacker [20],
a (topological) skew-product is constituted from a topological space $X$ and a
topological group $G$, both will be fixed unless stated otherwise. For
simplicity, we always consider locally compact Hausdorff topologies on $X$ and
$G$. In this context, $\mathbb{A}_{X}$ is the group of all the homeomorphisms
of $X$ on itself, given the compact-open topology. The main characteristic of
this generalization of dynamical systems is that instead of a unique group
action, we consider a family of action-like maps. It is worth noting that in
[8, 19, 20] the groups are $\mathbb{R}$ or $\mathbb{Z}$, while here we present
the construction in general.

For locally compact Hausdorff spaces $A$ and $B$, we set
$\mathcal{C}\big{(}A;B\big{)}$ the space of continuous functions from $A$ to
$B$, given the compact-open topology. Let us consider the evaluation maps:

  * •

$\mathfrak{e}_{G}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{C}\big{(}G;X\big{)}\times G\to X$,
$(\varphi,g)\mapsto\varphi(g)$,

  * •

$\mathfrak{e}_{X}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{C}\big{(}X;X\big{)}\times G\to X$,
$(\varphi,x)\mapsto\varphi(x)$,

as well as the partial evaluation maps:

  * •

$\widetilde{\mathfrak{e}}_{G}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{C}\big{(}G\\!\times\\!X;X\big{)}\times
G\to\mathcal{C}\big{(}X;X\big{)}$, $(\psi,g)\mapsto\psi(g,\cdot)$,

  * •

$\widetilde{\mathfrak{e}}_{X}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{C}\big{(}G\\!\times\\!X;X\big{)}\times
X\to\mathcal{C}\big{(}G;X\big{)}$, $(\psi,x)\mapsto\psi(\cdot,x),$

which are all continuous [5, Corollary I and II, p. 303]. For a subspace
$Y\subset\mathcal{C}\big{(}G\\!\times\\!X;X\big{)}$, define

$Y_{X}\mathrel{\mathop{\mathchar
58\relax}}=\widetilde{e}_{X}(Y\\!\times\\!X)\subset\mathcal{C}\big{(}G;X\big{)}\quad\text{and}\quad
Y_{G}\mathrel{\mathop{\mathchar
58\relax}}=\widetilde{e}_{G}(Y\\!\times\\!G)\subset\mathcal{C}\big{(}X;X\big{)}.$

######  Definition 2.1.

We say that a space of functions
$Y\subset\mathcal{C}\big{(}G\\!\times\\!X;X\big{)}$ is admissible, if:

  * i)

$Y_{G}\subset\mathbb{A}_{X}$

  * ii)

$\psi(e,x)=x$ for every $\psi\in Y$ and $x\in X$, where $e$ is the unit of
$G$.

In [19], such a space $Y$ is called a Hull.

On the other hand, if we consider the right action $\theta$ of $G$ on itself
by translations, i.e.

$\theta\mathrel{\mathop{\mathchar 58\relax}}G\times G\to G,\quad(g,h)\mapsto
gh,$

it lifts to a continuous left action $\Theta\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!\mathcal{C}\big{(}G;X\big{)}\to\mathcal{C}\big{(}G;X\big{)}$
given by

$\left[\Theta(h,\varphi)\right](g)=\varphi\left(\theta(g,h)\right)=\varphi(gh),\quad\forall\,\varphi\in\mathcal{C}\big{(}G;X\big{)},\,g,h\in
G.$

For an admissible $Y$, the set $Y_{X}$ is contained on
$\mathcal{C}\big{(}G;X\big{)}$. If we consider its saturation
$\widetilde{Y}_{X}\mathrel{\mathop{\mathchar
58\relax}}=\Theta\big{(}G\\!\times\\!Y_{X}\big{)}$, it is invariant under the
action $\Theta$, hence $\left(\widetilde{Y}_{X},G,\Theta\right)$ is a
dynamical system.

With this system, we write the map
$\widetilde{\Theta}\mathrel{\mathop{\mathchar
58\relax}}X\\!\times\\!Y\\!\times\\!G\\!\times\\!G\to X$ by

(2.1)
$\widetilde{\Theta}(x,\psi,g,h)=\left[\Theta\left(h,\widetilde{\mathfrak{e}}_{X}(\psi,x)\right)\right](g)=\psi(gh,x).$

Now consider a dynamical system $(Y,G,\sigma)$, given by a continuous left
action $\sigma\mathrel{\mathop{\mathchar 58\relax}}G\times Y\to Y$. The skew-
flow associated to it is the map $\pi\mathrel{\mathop{\mathchar
58\relax}}X\times Y\times G\to X\times Y$ given by

$\pi(x,\psi,h)=\left(\psi(h,x),\sigma(h,\psi)\right),\quad\forall\,\psi\in
Y,\,h\in G,\,x\in X,$

which is how skew-products are depicted on [19]. With this system, we define
the map $\widetilde{\pi}\mathrel{\mathop{\mathchar
58\relax}}X\\!\times\\!Y\\!\times\\!G\\!\times\\!G\to X$ by

(2.2)
$\widetilde{\pi}(x,\psi,g,h)=\left[\sigma\big{(}h,\psi\big{)}\right]\left(g,\psi\big{(}h,x\big{)}\right),\quad\forall\,\psi\in
Y,\,g,h\in G,\,x\in X.$

The key to skew-products is that the systems given by $\Theta$ and $\sigma$
must be compatible. This means that the maps $\widetilde{\Theta}$ and
$\widetilde{\pi}$, on (2.1) and (2.2) respectively, must coincide, i.e.

$\left[\sigma(h,\psi)\right]\left(g,\psi(h,x)\right)=\psi(gh,x),\quad\forall\,\psi\in
Y,\,g,h\in G,\,x\in X.$

In this last statement, it is clear that all the information regarding the
skew-product is contained on the properties of $\sigma$. We formalize this
discussion on the following definition.

######  Definition 2.2.

A skew-product dynamical system is a quadruple $(X,G,Y,\sigma)$, where:

  * i)

$Y\subset\mathcal{C}\big{(}G\\!\times\\!X;X\big{)}$ is admissible,

  * ii)

$(Y,G,\sigma)$ is a dynamical system, where $\sigma\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!Y\to Y$ is a continuous left action,

  * iii)

$\left[\sigma(h,\psi)\right]\left(g,\psi(h,x)\right)=\psi(gh,x)$ for every
$\psi\in Y,\,g,h\in G$ and $\,x\in X$.

######  Remark 2.3.

It is clear that if $(X,G,\alpha)$ is a dynamical system, where $\alpha$ is a
continuous left action, then $(X,G,\\{\alpha\\},\mathfrak{q})$ is a skew-
product, where $\mathfrak{q}\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!\\{\alpha\\}\to\\{\alpha\\}$ is the trivial action. In
other words, a dynamical system is a skew-product where the hull contains only
one function.

Let us illustrate this construction on an example.

######  Example 2.4.

Consider the topological space $\mathbb{R}^{d}$ and its group of
homeomorphisms $\mathbb{A}_{X}$. Set $G=\mathbb{Z}$. Consider the following
nonautonomous difference equation

(2.3) $x(n+1)=F\left(n,x(n)\right),$

where for every $n\in\mathbb{Z}$ we have $F(n,\cdot)\in\mathbb{A}_{X}$. Let
$n\mapsto x(n,m,\xi)$ be the unique solution of (2.3) that verifies
$x(m,m,\xi)=\xi$. Define the collection
$Y=\left\\{\psi_{m}\,\mathrel{\mathop{\mathchar
58\relax}}\,m\in\mathbb{Z}\right\\}$, where
$\psi_{m}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}\\!\times\\!\mathbb{R}^{d}\to\mathbb{R}^{d}$ is given by

$\psi_{m}(n,\xi)=x\big{(}n+m,m,\xi\big{)}.$

Set now the action $\sigma\mathrel{\mathop{\mathchar
58\relax}}Y\\!\times\\!\mathbb{Z}\to Y$ by

$\sigma(n,\psi_{m})=\psi_{n+m},\quad\forall\,n,m\in\mathbb{Z},$

or, evaluating

$\displaystyle\left[\sigma\left(\psi_{m},n\right)\right](p,\xi)=\psi_{n+m}(p,\xi)=x\big{(}p+n+m,n+m,\xi\big{)},\quad\forall\,p,n,m\in\mathbb{Z},\,\xi\in\mathbb{R}^{d},$

and its associated skew-flow $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\\!\times\\!Y\\!\times\\!\mathbb{Z}\to\mathbb{R}^{d}\\!\times\\!Y$
given by
$\pi\left(\xi,\psi_{m},n\right)=\left(\psi_{m}(n,\xi),\psi_{n+m}\right)$.
Evaluating for $p,n,m\in\mathbb{Z}$ and $\xi\in\mathbb{R}^{d}$ we have

$\displaystyle\widetilde{\pi}(\xi,\psi_{m},p,n)$
$\displaystyle=\psi_{n+m}\left(p,\psi_{m}\big{(}n,\xi\big{)}\right)$
$\displaystyle=x\left(p+n+m,n+m,x\big{(}n+m,m,\xi\big{)}\right).$

On the other hand, in this context

$\displaystyle\widetilde{\Theta}(\xi,\psi_{m},p,n)=\psi_{m}(p+n,\xi)=x\big{(}p+n+m,m,\xi\big{)},\quad\forall\,p,n,m\in\mathbb{Z},\,\xi\in\mathbb{R}^{d}.$

Now, by uniqueness of solutions we have

$x\left(p+n+m,n+m,x\big{(}n+m,m,\xi\big{)}\right)=x\big{(}p+n+m,m,\xi\big{)},\quad\forall\,p,n,m\in\mathbb{Z},\,\xi\in\mathbb{R}^{d},$

thus $\widetilde{\Theta}$ and $\widetilde{\pi}$ coincide, which implies that
$\left(\mathbb{R}^{d},\mathbb{Z},\\{\psi_{m}\mathrel{\mathop{\mathchar
58\relax}}m\in\mathbb{Z}\\},\sigma\right)$ is indeed a skew-product.

In [8], S. Elaydi and R. J. Sacker explored further applications of skew-
products to the theory of nonautonomous difference equations, as the search of
asymptotically stable solutions for Beverton-Holt equations [7]. Moreover, in
[19], the authors leverage this structure to develop the exponential dichotomy
spectrum for nonautonomous linear differential equations when the hull is
compact.

##  3\. Groupoid morphisms and cotranslations

On this section, we introduce the structure of cotranslations, a specific kind
of groupoid morphisms. Cotranslations serve as an algebraic counterpart to
skew-products, providing a generalization of group morphisms. As group
morphisms are instrumental in describing dynamical systems, this
generalization further extends the applicability of such algebraic concepts.

######  Definition 3.1.

[23, Definition 1.2] We say that a set $\Xi$, doted of a subset
$\Xi^{(2)}\subset\Xi\\!\times\\!\Xi$ (called the collection of composible
pairs) and two maps $\bullet\mathrel{\mathop{\mathchar
58\relax}}\Xi^{(2)}\to\Xi$ given by $(\eta,\xi)\mapsto\eta\bullet\xi$ (called
composition law), and $\mathfrak{inv}\mathrel{\mathop{\mathchar
58\relax}}\Xi\to\Xi$, is a groupoid if the following conditions are verified

  * i)

(associativity) If $(\eta,\xi),\,(\xi,\zeta)\in\Xi^{(2)}$, then
$(\eta\bullet\xi,\zeta),\,(\eta,\xi\bullet\zeta)\in\Xi^{(2)}$ and
$(\eta\bullet\xi)\bullet\zeta=\eta\bullet(\xi\bullet\zeta)$,

  * ii)

(involution) $\mathfrak{inv}(\mathfrak{inv}(\eta))=\eta$ for every
$\eta\in\Xi$,

  * iii)

(identity) for every $\eta\in\Xi$, we have
$\left(\eta,\mathfrak{inv}(\eta)\right)\in\Xi^{(2)}$ and
$(\eta,\xi)\in\Xi^{(2)}$ implies that
$\mathfrak{inv}(\eta)\bullet(\eta\bullet\xi)=\xi$ and
$(\eta\bullet\xi)\bullet\mathfrak{inv}(\xi)=\eta$.

Furthermore, we define $\Xi^{(0)}\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\eta\in\Xi\mathrel{\mathop{\mathchar
58\relax}}\eta=\mathfrak{inv}(\eta)=\eta\bullet\eta\right\\}$ and call it the
units space of the groupoid.

A trivial example of a groupoid is a group $G$. In this case $G^{(2)}=G^{2}$
and $G^{(0)}=\\{e\\}$.

######  Example 3.2.

If a group $G$ acts by the left on a set $M$, the product $M\\!\times\\!G$ has
groupoid structure. Indeed, setting

$\left(M\\!\times\\!G\right)^{(2)}\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\left((x,g),(y,h)\right)\in\left(G\\!\times\\!M\right)^{2}\mathrel{\mathop{\mathchar
58\relax}}x=h\cdot y\right\\},$

and

$\bullet\left((x,g),(y,h)\right)=(y,gh),\quad\mathfrak{inv}(x,g)=(g\cdot
x,g^{-1}),$

the groupoid axioms are easily followed. In this case, the units space
corresponds to the collection of points of the form $(x,e)$, where $e$ is the
group unit.

This example is particularly useful when $G$ acts on itself by left
translations, in which case we call $G\\!\times\\!G$ the left translations
groupoid for $G$.

######  Definition 3.3.

[23, Definition 1.8] If $\Xi$ and $\Upsilon$ are groupoids, a groupoid
morphism is a map $\vartheta\mathrel{\mathop{\mathchar
58\relax}}\Xi\to\Upsilon$ verifying that $(\eta,\xi)\in\Xi^{(2)}$ implies
$\left(\vartheta(\eta),\vartheta(\xi)\right)\in\Upsilon^{(2)}$ and
$\vartheta(\eta\bullet\xi)=\vartheta(\eta)\bullet\vartheta(\xi)$.

Two basic facts about groupoid morphisms are that they preserve involutions
and units spaces. In other words, for a groupoid morphism
$\vartheta\mathrel{\mathop{\mathchar 58\relax}}\Xi\to\Upsilon$ we have
$\mathfrak{inv}(\vartheta(\xi))=\vartheta(\mathfrak{inv}(\xi)$ for all
$\xi\in\Xi$ and $\vartheta(\eta)\in\Upsilon^{(0)}$ for all $\eta\in\Xi^{(0)}$.

######  Example 3.4.

Consider an element $X$ on some category $\mathscr{C}$. Set a group $G$,
$\gamma\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{A}_{X}$ a group
morphism and give $G\\!\times\\!G$ the left translations groupoid structure.
By setting $Z\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$ by $Z(g,h)=\gamma(h)$ we obtain a
groupoid morphism. Indeed:

$Z(g,kh)=\gamma(kh)=\gamma(k)\gamma(h)=Z(hg,k)Z(g,h).$

This example shows that every dynamical system is in particular given by a
groupoid morphism, since a dynamical system is always given by a group
morphism $\gamma\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{A}_{X}$.
Nevertheless, the next example shows that groupoid morphisms describe a more
general kind of dynamics.

######  Example 3.5.

Consider the left translations groupoid structure on
$\mathbb{R}\\!\times\\!\mathbb{R}$. Take $X=\mathbb{R}^{d}$ as a topological
space, thus $\mathbb{A}_{X}$ is the group of all of its homeomorphisms.
Consider a nonautonomous real differential equation $\dot{x}=F(t,x)$ such that
for every $\xi\in\mathbb{R}^{d}$ and every $r\in\mathbb{R}$, there is a unique
and globally defined solution $x_{r,\xi}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\to\mathbb{R}^{d}$ such that $x_{r,\xi}(r)=\xi$. Then the
map $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\\!\times\\!\mathbb{R}\to\mathbb{A}_{X}$ given by
$\left[Z(r,t)\right](\xi)=x_{r,\xi}(t+r)$ is a groupoid morphism. Indeed:

$\left[Z(r,t+s)\right](\xi)=x_{r,\xi}(t+s+r)=x_{s+r,x_{r,\xi}(s+r)}(t+s+r)=\left[Z(s+r,t)\circ
Z(r,s)\right](\xi),$

where the second equality is a well known fact deduced from the uniqueness of
solutions. Moreover, giving $\mathbb{A}_{X}$ the compact-open topology and the
groupoid $\mathbb{R}\\!\times\\!\mathbb{R}$ the product topology, $Z$ turns
out to be continuous.

Given the dynamical relevance shown by the previous examples, we give these
morphisms a distinctive name.

######  Definition 3.6.

Consider a group $G$ and an object $X$ on a category. A cotranslation is a
groupoid morphism $Z\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$, where $G\\!\times\\!G$ has the
left translations groupoid structure.

Now we present the main theorem of this work. Although we state it for the
topological case, it can be generalized for objects and skew-products on
different categories.

######  Theorem 3.7.

Let $X$ be a locally compact Hausdorff topological space and $G$ a locally
compact Hausdorff topological group. Give all function spaces, including
$\mathbb{A}_{X}$, the compact-open topology.

There is a bijective correspondence between skew-product dynamical systems
$(X,G,Y,\sigma)$, where the action $\sigma$ is transitive, and continuous
cotranslations $Z\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$.

###### Proof.

Let $(X,G,Y,\sigma)$ be a skew-product where $\sigma$ is transitive. The
admissibility condition implies $Y_{G}\subset\mathbb{A}_{X}$. On the other
hand, as $\sigma$ is transitive, we can identify $Y$ with the quotient group,
$G/\mathrm{stab}(\sigma)$, where

$\mathrm{stab}(\sigma)=\left\\{g\in G\mathrel{\mathop{\mathchar
58\relax}}\sigma(g,y)=y,\,\forall\,y\in Y\right\\}.$

Hence, we write $Y=\left\\{\psi_{\overline{g}}\mathrel{\mathop{\mathchar
58\relax}}g\in G\right\\}$, where $\overline{g}$ denotes the class of $g$ on
the quotient $G/\mathrm{stab}(\sigma)$. Moreover, the action
$\sigma\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!Y\to Y$ is rewritten
as $\hat{\sigma}\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!Y\to Y$
given by $\hat{\sigma}(\psi_{h,\overline{g}})=\psi_{\overline{hg}}$, that is,
it is identified with the action of left translations of $G$ on
$G/\mathrm{stab}(\sigma)$.

Now, we can define the continuous map $Z\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$ given by

$Z(g,h)=\widetilde{\mathfrak{e}}_{G}(\psi_{\overline{g}},h)=\psi_{\overline{g}}(h,\cdot),$

and we have

$\displaystyle Z(g,kh)=Z(hg,k)\circ Z(g,h)$ $\displaystyle\Leftrightarrow$
$\displaystyle\left[Z(g,kh)\right](x)=\left[Z(hg,k)\circ
Z(g,h)\right](x),\quad\forall\,x\in X$ $\displaystyle\Leftrightarrow$
$\displaystyle\left[\widetilde{\mathfrak{e}}_{G}(\psi_{\overline{g}},kh)\right](x)=\widetilde{\mathfrak{e}}_{G}(\psi_{\overline{hg}},k)\left[\left[\widetilde{\mathfrak{e}}_{G}(\psi_{\overline{g}},h)\right](x)\right],\quad\forall\,x\in
X$ $\displaystyle\Leftrightarrow$
$\displaystyle\psi_{\overline{g}}(kh,x)=\psi_{\overline{hg}}\left(k,\psi_{\overline{g}}(h,x)\right),\quad\forall\,x\in
X$ $\displaystyle\Leftrightarrow$
$\displaystyle\psi_{\overline{g}}(kh,x)=\left[\hat{\sigma}(h,\psi_{\overline{g}})\right]\left(k,\psi_{\overline{g}}(h,x)\right),\quad\forall\,x\in
X,$

and as the last condition is guaranteed by third the axiom of skew-products
(Definition 2.2), then $Z$ is indeed a cotranslation.

Conversely, given a continuous cotranslation $Z\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$, for each $g\in G$ we define
$\psi_{g}\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!X\to X$ given by
$\psi_{g}(h,x)=\left[Z(g,h)\right](x)$. Then, defining
$Y\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\psi_{g}\mathrel{\mathop{\mathchar 58\relax}}g\in
G\right\\}$, it is admissible, since groupoid morphisms preserve units, and
the only unit in $\mathbb{A}_{X}$ is the identity. On the other hand, defining
the action $\sigma\mathrel{\mathop{\mathchar 58\relax}}G\times Y\to Y$ given
by $\sigma(h,\psi_{g})=\psi_{hg}$, or equivalently

$\left[\sigma(h,\psi_{g})\right](k,x)=\left[Z(hg,k)\right](x),$

we obtain, by the same previous argument, that $(X,G,Y,\sigma)$ is a skew-
product dynamical system, where the action $\sigma$ clearly results
transitive. ∎

######  Remark 3.8.

The preceding theorem demonstrates that all generalizations of dynamical
systems derived from skew-products are encompassed by cotranslations. The
distinctive virtue of cotranslations lies in their ability to more clearly
represent the underlying algebraic structure of these dynamics. This parallel
can be drawn to the way group morphisms capture the algebraic structure of
group actions.

Now we give further properties and applications for cotranslations. In the
following we use the notation
$Z^{\mathrm{inv}}(g,h)=\left[Z(g,h)\right]^{-1}$.

######  Lemma 3.9.

Set a cotranslation $Z\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$. If $e\in G$ is the group unit,
then for every $g,h\in G$ we have

$Z(g,e)=\mathrm{Id}\quad\text{and}\quad Z^{\mathrm{inv}}(g,h)=Z(hg,h^{-1}).$

###### Proof.

The first equality follows from the fact that groupoid morphisms preserve
units, and the only unit in $\mathbb{A}_{X}$ is the identity. The second
equality follows from the fact that groupoid morphisms preserve involutions. ∎

######  Proposition 3.10.

Let $Z\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$ be
a cotranslation. Let $\gamma\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{A}_{X}$ be a group morphism such that

$\gamma(k)Z(g,h)=Z(g,h)\gamma(k),\qquad\forall\,g,k,h\in G,$

then $W\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$
given by $W(g,h)=Z(g,h)\gamma(h)$ is a cotranslation.

###### Proof.

It is enough to see that

$\displaystyle W(g,kh)=Z(g,kh)\gamma(kh)$
$\displaystyle=Z(hg,k)Z(g,h)\gamma(k)\gamma(h)$
$\displaystyle=Z(hg,k)\gamma(k)Z(g,h)\gamma(h)$ $\displaystyle=W(hg,k)W(g,h).$

∎

######  Example 3.11.

Consider $X=\mathbb{R}^{d}$ as a Banach space. In this context,
$\mathbb{B}_{X}\cong\mathcal{M}_{d}(\mathbb{R})$ and $\mathbb{A}_{X}\cong
GL_{d}(\mathbb{R})$. Consider a linear nonautonomous differential equation
$\dot{x}=A(t)x(t)$, where $t\mapsto A(t)$ is locally integrable. Denote the
transition matrix for this equation by $\Phi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\\!\times\\!\mathbb{R}\to\mathbb{A}_{X}$. We can see,
analogously to Example 3.5, that $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\\!\times\\!\mathbb{R}\to\mathbb{A}_{X}$, given by
$Z(r,t)=\Phi(r,t+r)$, defines a cotranslation.

Choose $\lambda\in\mathbb{R}$ and define $\gamma\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\to\mathbb{A}_{X}$ by $\gamma(t)=e^{-\lambda
t}\cdot\mathrm{Id}$. It is easily a group morphism which verifies the
conditions of the previous proposition. Hence, $W\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\to\mathbb{A}_{X}$ given by
$W(r,t)=Z(r,t)\gamma(t)$ is a cotranslation. Moreover, in this case $W$ is
once more the cotranslation associated to a linear differential equation,
since it is obtained in the same fashion as $Z$, but regarding the shifted
linear nonautonomous differential equation
$\dot{x}=\left[A(t)-\lambda\cdot\mathrm{Id}\right]x(t)$.

Now we present a result that depicts how a cotranslation on a given space can
be regarded as a group action, i.e. a classic dynamical system, on a different
suitable space. As in Theorem 3.7, we express this result in terms of
topological dynamics.

######  Proposition 3.12.

Let $X$ be a topological space and $G$ a topological group, both endowed with
Hausdorff locally compact topologies. Consider $G\times X$ as topological
space equipped with the product topology, and set on both $\mathbb{A}_{G\times
X}$ and $\mathbb{A}_{X}$ the correspondent compact-open topology. Furthermore,
denote the canonical projections by $\pi_{X}\mathrel{\mathop{\mathchar
58\relax}}G\times X\to X$ and $\pi_{G}\mathrel{\mathop{\mathchar
58\relax}}G\times X\to G$.

There exists a bijective correspondence between continuous cotranslations
$Z\mathrel{\mathop{\mathchar 58\relax}}G\times G\to\mathbb{A}_{X}$ and
continuous group morphisms $W\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{A}_{G\times X}$ such that $(\pi_{G}\circ W(g))(h,x)=gh$
for every $x\in X$. Moreover, the correspondence is given by

$Z(h,g)(x)=\left(\pi_{x}\circ W(g)\right)(h,x),$

and

$W(g)(h,x)=\left(gh,Z(h,g)(x)\right),\qquad\forall\,g,h,\in G,\,x\in X.$

###### Proof.

The continuity of $Z$ follows from $W$ being continuous and vice versa. Take
$g_{1},g_{2},h\in G$ and $x\in X$. If $W$ is a group morphism, then

$\displaystyle\left[Z(g_{1}h,g_{2})\circ Z(h,g_{1})\right](x)$
$\displaystyle=Z(g_{1}h,g_{2})\left(\left[\pi_{X}\circ
W(g_{1})\right](h,x)\right)$ $\displaystyle=\left[\pi_{X}\circ
W(g_{2})\right]\left(g_{1}h,\left[\pi_{X}\circ W(g_{1})\right](h,x)\right)$
$\displaystyle=\left[\pi_{X}\circ W(g_{2})\circ W(g_{1})\right](h,x)$
$\displaystyle=\left[\pi_{X}\circ W(g_{2}g_{1})\right](h,x)$
$\displaystyle=Z(h,g_{2}g_{1})(x),$

thus $Z$ is a cotranslation. On the other hand, if $Z$ is a cotranslation,
then

$\displaystyle\left[W(g_{2})\circ W(g_{1})\right](h,x)$
$\displaystyle=W(g_{2})\left(g_{1}h,Z(h,g_{1})x\right)$
$\displaystyle=\left(g_{2}g_{1}h,\left[Z(g_{1}h,g_{2})\circ
Z(h,g_{1})\right](x)\right)$
$\displaystyle=\left(g_{2}g_{1}h,Z(h,g_{2}g_{1})(x)\right)$
$\displaystyle=W(g_{2}g_{1})(h,x),$

thus $W$ is a group morphism. ∎

It is worth noting that a crucial condition in the previous proposition is
that $G$ is itself a topological space, i.e. an object in the same category as
$X$, so that the object $\mathbb{A}_{G\times X}$ has a clear meaning. Hence,
in order to generalize this conclusion to other categories, we must in general
consider a group object of the desired category (we refer the reader to [3,
Chapter 4]).

Nevertheless, this does not trivialize the construction that cotranslations
present, since by changing the topological space in which the dynamics occurs,
we change qualitative descriptions of said dynamics. For instance, in the case
of a nonautonomous differential equation on $\mathbb{R}^{d}$, say
$\dot{x}=f(t,x)$, the construction on Proposition 3.12 coincides with the
usual definition of an autonomous differential equation on $\mathbb{R}^{d+1}$
by taking $y=(t,x)$ and $\dot{y}=(1,f(y))$. However, it is well known that
qualitative descriptions of the second equation are not well translated to the
first.

Having established the main properties of cotranslations, we now turn our
attention to the specific case when we consider the group $\mathbb{Z}$. This
will shed light on the relevance of cotranslations in the context of
nonautonomous difference equations.

######  Proposition 3.13.

Set $X$ a Banach space, thus $\mathbb{B}_{X}$ is the Banach algebra of linear
continuous operators and $\mathbb{A}_{X}$ is the topological group of its
homeomorphic isomorphisms. There is a bijective correspondence between
cotranslations $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}\times\mathbb{Z}\to\mathbb{A}_{X}$ and nonautonomous
linear difference equations $x(n+1)=A(n)x(n)$, with $\mathbb{Z}\ni n\mapsto
A(n)\in\mathbb{A}_{X}$, where the correspondence is given by

$Z(n,m)=\left\\{\begin{array}[]{lcc}A(n+m-1)A(n+m-2)\cdots A(n)&\text{ if
}&m>0\\\ \\\ \mathrm{Id}&\text{ if }&m=0\\\ \\\ A^{-1}(n+m)A^{-1}(n+m+1)\cdots
A^{-1}(n-1)&\text{ if }&m<0.\end{array}\right.$

###### Proof.

By defining $Z$ as in the statement of the proposition, starting from the
function $n\mapsto A(n)\in\mathbb{A}_{X}$, clearly we obtain a groupoid
morphism. Conversely, if we start with a cotranslation it is enough to define
$A(n)=Z(n,1)$ and we obtain the desired equation. ∎

The next section will present a continuous analogue to the preceding result.
Concluding this section, we provide a generalization of the previous
proposition for finitely generated groups. We omit the proof since it follows
the same steps as before. However, before delving into this generalization, we
introduce an auxiliary definition.

######  Definition 3.14.

Let $G$ be a discrete group with $n$ generators $\\{\xi_{1},\dots,\xi_{n}\\}$
verifying the set of relations $R$. For each word $p\in R$ there is a positive
integer $|p|$ called length of the word and a map
$j_{p}\mathrel{\mathop{\mathchar
58\relax}}\\{1,\dots,|p|\\}\to\\{1,\dots,n\\}$ such that

$p=\prod_{k=1}^{|p|}\xi_{j_{p}(k)}=\xi_{j_{p}(|p|)}\cdot\xi_{j_{p}(|p|-1)}\,\cdots\,\xi_{j_{p}(2)}\cdot\xi_{j_{p}(1)}.$

Let $X$ be a Banach space. We say that a collection of maps
$A_{i}\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{A}_{X}$, $i=1,\dots,n$,
preserves relations if for each $p\in R$ and $\eta\in G$ one has

$\displaystyle\mathrm{Id}_{X}$
$\displaystyle=\prod_{k=1}^{|p|}A_{j_{p}(k)}\left(\left[\prod_{\ell=1}^{k-1}\xi_{j_{p}(\ell)}\right]\eta\right)$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=A_{j_{p}(|p|)}(\xi_{j_{p}(|p|-1)}\cdots\xi_{j_{p}(1)}\eta)\circ\cdots\circ
A_{j_{p}(2)}(\xi_{j_{p}(1)}\eta)\,\circ\,A_{j_{p}(1)}(\eta).$

######  Proposition 3.15.

Let $G$ be a discrete group with $n$ generators $\\{\xi_{1},\dots,\xi_{n}\\}$
and $X$ an object on some category $\mathscr{C}$. A multivariable
nonautonomous difference equation of $G$ on $X$ is an equation of the form

$x(\xi_{i}\eta)=A_{i}(\eta)x(\eta),$

where the collection of maps $A_{i}\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{A}_{X}$, $i=1,\dots,n$ preserves relations. A solution
to this equation is a map $x\mathrel{\mathop{\mathchar 58\relax}}G\to X$.
There is a bijective correspondence between multivariable nonautonomous
difference equations of $G$ on $X$ and cotranslations
$Z\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$, which
is given by

$A_{i}(\eta)=Z\left(\eta,\xi_{i}\right).$

###  3.1. Examples

In the following we present various examples of cotranslations on diverse
groups. These examples possess the characteristic of not corresponding to a
classical dynamical system, yet they still encapsulate dynamics generated from
these groups. Through the examples, the space $\mathbb{R}^{d}$ is provided its
usual topology. On the other hand, for a collection of spaces
$\\{U_{\lambda}\\}_{\lambda\in\Lambda}$, the space
$\amalg_{\lambda\in\Lambda}\,U_{\lambda}$ is provided the disjoint union
topology, while $\Pi_{\lambda\in\Lambda}\,U_{\lambda}$ has the product
topology.

A virtue of the groups we study here is that all of them allow for a canonical
normal form, i.e. a canonical choice of writing of elements corresponding to
their generator set. This normal form on the groups allows for an easier to
verification that the maps we use indeed preserve the group relations.
Finally, the group unit is always denoted by $e$.

We begin by characterizing cotranslations of cyclic groups $C_{n}=\langle
a\,|\,a^{n}=e\rangle$. On any topological space $X$, take
$f_{0},\dots,f_{n-1}$ homeomorphisms of $X$ on itself verifying
$f_{n-1}\circ\cdots\circ f_{0}=\mathrm{Id}_{X}$. Then, defining
$A\mathrel{\mathop{\mathchar 58\relax}}C_{n}\to\mathbb{A}_{X}$ by
$A(a^{k})=f_{k}$, it clearly preserves the group relations, hence by
Proposition 3.15, it defines a cotranslation. Note that the only way that this
cotranslation corresponds to a group morphism is $f_{0}=\cdots=f_{n-1}$, while
the notion of cotranslation allows for a more general choice of
homeomorphisms. In particular, each pair of $f_{k}$, $f_{j}$ does not need to
commute.

######  Example 3.16.

Consider $C_{3}=\langle a\,|\,a^{3}=e\rangle$, the space $X=\mathbb{R}$ and
the functions $f(x)=x+1$ and $g(x)=2x$. Setting $A\mathrel{\mathop{\mathchar
58\relax}}C_{3}\rightarrow\mathbb{A}_{X}$ given by $A(e)(x)=f(x)$,
$A(a)(x)=g(x)$ and $A(a^{2})(x)=(f^{-1}\circ g^{-1})(x)=\frac{x}{2}-1$, we can
define the cotranslation $Z\mathrel{\mathop{\mathchar 58\relax}}C_{3}\times
C_{3}\rightarrow\mathbb{A}_{X}$ given by

$Z(e,e)=\mathrm{Id}_{X}$, | $Z(a,e)=\mathrm{Id}_{X},$ | $Z(a^{2},e)=\mathrm{Id}_{X},$
---|---|---
$Z(e,a)(x)=x+1,$ | $Z(a,a)(x)=2x,$ | $Z(a^{2},a)(x)=\frac{x}{2}-1,$
$Z(e,a^{2})(x)=2x+2,$ | $Z(a,a^{2})(x)=x-1,$ | $Z(a^{2},a^{2})(x)=\frac{x}{2}.$

Remarkably, there are no non-trivial classic group action of $C_{3}$ on
$\mathbb{R}$. Moreover, in this case clearly not all maps commute.

######  Example 3.17.

Consider $C_{n}=\langle\,a\,|\,a^{n}=e\,\rangle$, with $n\geq 2$, and
$X=\mathbb{R}^{d}$. Take $k$ non-empty finite sets of different real numbers
$R_{1}$, $R_{2}$, …$R_{k}$, such that $\Sigma_{r\in R_{i}}\,r\in
2\pi\mathbb{Z}$ for every $i\in\\{1,2,\dots,k\\}$ and $\Sigma_{i=1}^{k}\
m_{i}=n$, where $m_{i}=|R_{i}|$.

Choose $\pi_{1},\dots,\pi_{k}$ planes on $\mathbb{R}^{d}$ containing the
origin and denote by $\theta_{i}(r)$ the counterclockwise rotation of
$\mathbb{R}^{d}$ by the angle $r$ respect to a fixed normal vector to the
plane $\pi_{i}$, for $r$ in $R_{i}$ and $i\in\\{1,2,\dots,k\\}$. Set
$\Theta=\left\\{\theta_{i}(r)\mathrel{\mathop{\mathchar 58\relax}}r\in
R_{i},\,i\in\\{1,\dots,k\\}\right\\}$ and fix a bijection
$A\mathrel{\mathop{\mathchar
58\relax}}C_{n}\rightarrow\Theta\subset\mathbb{A}_{X}$. As every rotation
$\theta_{i}(r)$ commutes with each $\theta_{j}(r^{\prime})$, we have

$A(a^{n-1+t})\circ\dots\circ A(a^{1+t})\circ A(a^{t})=\mathrm{Id}_{X},$

for every $0\leq t\leq n-1$, hence $A$ defines a cotranslation. We name this
dynamics as multi-rotational cotranslations.

######  Example 3.18.

Fix a positive integer $n$ and consider
$C_{3n}=\langle\,a\,|\,a^{3n}=e\,\rangle$ and $X=\amalg_{i=1}^{n}\ S^{1}$. For
each $i\in\\{1,\dots,n\\}$, set a map $A^{i}\mathrel{\mathop{\mathchar
58\relax}}C_{3}=\langle\,r\,|\,r^{3}=e\,\rangle\rightarrow\mathbb{A}_{S_{i}^{1}}$,
where $S_{i}^{1}$ corresponds to the $i$-th copy of $S^{1}$ in $X$, such that
it defines a cotranslation of $C_{3}$ on $S_{i}^{1}$.

Note that each $A^{i}\mathrel{\mathop{\mathchar
58\relax}}C_{3}\rightarrow\mathbb{A}_{S_{i}^{1}}$ defines a map
$\tilde{A}^{i}\mathrel{\mathop{\mathchar
58\relax}}C_{3}\rightarrow\mathbb{A}_{X}$ through
$\left[\tilde{A}^{i}(r^{j})\right](x)=\left[A^{i}(r^{j})\right](x)$ if $x\in
S_{i}^{1}$ and $\left[\tilde{A}^{i}(r^{j})\right](x)=x$ if not. With these
functions we can define $A\mathrel{\mathop{\mathchar
58\relax}}C_{3n}\rightarrow\mathbb{A}_{X}$ given by
$A(a^{3i+j})=\tilde{A}^{i+1}(r^{j})$ for all $i\in\\{0,\dots,n+1\\}$ and
$j\in\\{0,1,2\\}$.

As each $\tilde{A}^{i}(r^{\ell})$ acts non-trivially only on the $i$-th copy
of $S^{1}$, we have
$\tilde{A}^{i}(r^{\ell})\circ\tilde{A}^{j}(r^{k})=\tilde{A}^{j}(r^{k})\circ\tilde{A}^{i}(r^{\ell})$
for all $i,j\in\\{1,\dots,n\\}$ and $\ell,k\in\\{0,1,2\\}$. Therefore, $A$
defines a cotranslation of $C_{3n}$ on $X$. We name this dynamics as disjoint
cotranslations. Note that under a similar construction, we can define a
disjoint cotranslation of $C_{n_{1}+\dots+n_{p}}$ by leveraging cotranslations
of $C_{n_{1}},\dots,C_{n_{p}}$.

Until now, every example corresponds to a 1-generated group, consequently,
according to Proposition 3.15, we only needed to consider a single function
$A\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{A}_{X}$. In the following
we study examples with groups generated by two o more elements.

Consider for instance the dihedral group of $2n$ elements $D_{2n}=\langle
r,s\,|\,r^{n}=s^{2}=(sr)^{2}=e\rangle=\\{e,r,\dots,r^{n-1},s,sr,\dots,sr^{n-1}\\}$.
Let $X$ be some object in a category. In order to define a cotranslation of
$D_{2n}$ on $X$, it is enough to consider two functions
$A_{r},A_{s}\mathrel{\mathop{\mathchar
58\relax}}D_{2n}\rightarrow\mathbb{A}_{X}$ satisfying:

$\displaystyle A_{r}(r^{i+n-1})\circ\dots\circ A_{r}(r^{i+1})\circ
A_{r}(r^{i})$ $\displaystyle=\mathrm{Id}_{X},$ $\displaystyle
A_{r}(sr^{i-n+1})\circ\dots\circ A_{r}(sr^{i-1})\circ A_{r}(sr^{i})$
$\displaystyle=\mathrm{Id}_{X},$ $\displaystyle A_{s}(sr^{i})\circ
A_{s}(r^{i})=A_{s}(r^{i})\circ A_{s}(sr^{i})$ $\displaystyle=\mathrm{Id}_{X},$
$\displaystyle A_{s}(sr^{i})\circ A_{r}(sr^{i+1})\circ A_{s}(r^{i+1})\circ
A_{r}(r^{i})$ $\displaystyle=\mathrm{Id}_{X},$ $\displaystyle
A_{s}(r^{i})\circ A_{r}(r^{i-1})\circ A_{s}(sr^{i-1})\circ A_{r}(sr^{i})$
$\displaystyle=\mathrm{Id}_{X},$

for all $i\in\\{0,1,\dots,n-1\\}$.

######  Example 3.19.

Take the group $D_{6}=\langle r,s\,|\,r^{3}=s^{2}=(sr)^{2}=e\rangle$ and three
real numbers $\alpha,$ $\beta$ and $\gamma$ such that $\alpha+\beta+\gamma\in
2\pi\mathbb{Z}$. For a real number $\zeta$, the counterclockwise rotation of
$\mathbb{R}^{2}$ by the angle $\zeta$ is denoted by $\theta_{\zeta}$. Let us
consider $X=\amalg_{k\in\mathbb{Z}}\ \mathbb{R}^{2}\times\\{k\\}$. Given a
real number $\zeta$, set $R_{e,\zeta}$ and $R_{o,\zeta}$ the homeomorphisms of
$X$ given by

$R_{e,\zeta}(v,k)=\left\\{\begin{array}[]{ccc}(\theta_{\zeta}(v),k)&\text{if}&k\text{
is even,}\\\ (v,k)&\text{if}&k\text{ is odd,}\end{array}\right.$

and

$R_{o,\zeta}(v,k)=\left\\{\begin{array}[]{ccc}(v,k)&\text{if}&k\text{ is
even,}\\\ (\theta_{\zeta}(v),k)&\text{if}&k\text{ is odd.}\end{array}\right.$

Furthermore, let us denote by $\sigma$ the homeomorphism of $X$ given by
$\sigma(v,k)=(v,k-1)$ for every $v\in\mathbb{R}^{2}$ and $k\in\mathbb{Z}$.
Now, we define the functions $A_{r},A_{s}\mathrel{\mathop{\mathchar
58\relax}}D_{6}\rightarrow\mathbb{A}_{X}$ given by

$A_{r}(e)=R_{e,\alpha}$ | $A_{r}(r)=R_{e,\beta}$ |  $A_{r}(r^{2})=R_{e,\gamma}$,
---|---|---
$A_{r}(s)=R_{o,-\gamma}$ | $A_{r}(sr)=R_{o,-\beta}$ | $A_{r}(sr^{2})=R_{o,-\alpha},$

as well as $A_{s}(e)=A_{s}(r)=A_{s}(r^{2})=\sigma$ and
$A_{s}(s)=A_{s}(sr)=A_{s}(sr^{2})=\sigma^{-1}$. It is easy to check that
$A_{r}$ and $A_{s}$ satisfy the conditions above, therefore define a
cotranslation of $D_{6}$ on $X$. Note that this cotranslation corresponds to a
group action only if $\alpha=\beta=\gamma$.

######  Example 3.20.

More generally, suppose that a map $A\mathrel{\mathop{\mathchar
58\relax}}C_{n}\to\mathbb{A}_{X}$ defines a cotranslation of the cyclic group
$C_{n}=\langle\,a\,|\,a^{n}=e\,\rangle$ on a topological space $X$. Then, take
the space $Y=\amalg_{k\in\mathbb{Z}}\,X$, with elements denoted by
$y=(y_{k})_{k\in\mathbb{Z}}$ and once again consider $\sigma$ the shift on $Y$
given by $\sigma(y)_{k}=y_{k-1}$ for $k\in\mathbb{Z}$ and $y\in Y$. Define the
functions $A_{e},A_{o}\mathrel{\mathop{\mathchar
58\relax}}C_{n}\rightarrow\mathbb{A}_{Y}$ given, for $g\in C_{n}$, by

$(A_{e}(g)(y))_{(k)}=\left\\{\begin{array}[]{ccc}A(g)(y_{k})&\text{if}&k\text{
is even},\\\ y_{k}&\text{if}&k\text{ is odd},\end{array}\right.$

and

$(A_{o}(g)(y))_{(k)}=\left\\{\begin{array}[]{ccc}y_{k}&\text{if}&k\text{ is
even},\\\ A(g)(y_{k})&\text{if}&k\text{ is odd}.\end{array}\right.$

Finally, considering $D_{2n}=\langle r,s\,|\,r^{n}=s^{2}=(sr)^{2}=e\rangle$,
define the maps $B_{r},B_{s}\mathrel{\mathop{\mathchar
58\relax}}D_{2n}\rightarrow\mathbb{A}_{Y}$ by

$B_{r}(r^{i})=A_{e}(a^{i}),\quad B_{r}(sr^{i})=(A_{o}(a^{n-1-i}))^{-1},\quad
B_{s}(r^{i})=\sigma\quad\text{and}\quad B_{s}(sr^{i})=\sigma^{-1},$

for $i\in\\{0,1,\dots,n-1\\}$. It can be verified that these maps preserve the
relations of $D_{2n}$, therefore generating a cotranslation of $D_{2n}$ on
$Y$.

Let us now study the infinite dihedral group
$D_{\infty}=\langle\,a,b\,|\,a^{2}=b^{2}=e\,\rangle$. According to Proposition
3.15, to define a cotranslation on a space $X$ we need two functions
$A_{a},A_{b}\mathrel{\mathop{\mathchar
58\relax}}D_{\infty}\rightarrow\mathbb{A}_{X}$ verifying $A_{a}(aw)\circ
A_{a}(w)=\mathrm{Id}_{X}$ and $A_{b}(bw)\circ A_{b}(w)=\mathrm{Id}_{X}$ for
every $w\in D_{\infty}$.

For this, choose $\\{g_{n}\\}_{n\geq 0}$ and $\\{h_{n}\\}_{n\geq 0}$ two
arbitrary sequences of homeomorphisms in $\mathbb{A}_{X}$. Then, we can define
the functions $A_{a}$ and $A_{b}$ by $A_{a}(au)=g_{|u|}$ and
$A_{a}(u)=g_{|u|}^{-1}$, where $u$ is a word not beginning with $a$, and
$A_{b}(bv)=h_{|v|}$ and $A_{b}(v)=h_{|v|}^{-1}$, where $v$ is a word not
beginning with $b$. The dynamics emerging from this construction describe a
classic dynamical system only if the sequences $\\{g_{n}\\}_{n\geq 0}$ and
$\\{h_{n}\\}_{n\geq 0}$ are constant and each one is a fixed homeomorphism
which is its own inverse.

######  Example 3.21.

Consider the infinite dihedral group $D_{\infty}$ and $X=\mathbb{R}^{2}$.
Consider the homeomorphism $f(x,y)=(x+1,y)$ and the real numbers sequence
$\\{\zeta_{n}=\frac{\pi}{2^{n+1}}\\}_{n\geq 0}$. Again, let us denote by
$\theta_{\zeta}$ the counterclockwise rotation of $\mathbb{R}^{2}$ by the
angle $\zeta$.

We define the sequences $\\{g_{n}\\}_{n\geq 0}$, $\\{h_{n}\\}_{n\geq 0}$ by
$g_{n}=f^{n}$ and $h_{n}=\theta_{\zeta_{n}}$ for all $n\geq 0$. If we define
the maps $A_{a}$ and $A_{b}$ as above, we generate a cotranslation
$Z\mathrel{\mathop{\mathchar 58\relax}}D_{\infty}\times
D_{\infty}\rightarrow\mathbb{A}_{X}$ as in Proposition 3.15. Explicitly, and
among others, we have

$Z(bab,ab)=g_{|b|}\circ h_{|ab|}=f\circ\theta_{\zeta_{2}},$
$Z(abab,ab)=g_{|babab|}^{-1}\circ
h_{|abab|}^{-1}=f^{-5}\circ\theta_{-\zeta_{4}},$
$Z(ab,ababa)=g_{|ba|}^{-1}\circ h_{|a|}^{-1}\circ g_{|1|}^{-1}\circ
h_{|1|}\circ
g_{|b|}=f^{-2}\circ\theta_{-\zeta_{1}}\circ\mathrm{Id}_{X}\circ\theta_{\zeta_{0}}\circ
f=f^{-2}\circ\theta_{\zeta_{0}-\zeta_{1}}\circ f.$

######  Example 3.22.

Let $X=\mathbb{R}^{2}$, consider a sequence $\\{L_{n}\\}_{n\geq 0}$ of
straight lines going through the origin and a sequence $\\{\zeta_{n}\\}_{n\geq
0}$ of real numbers such that $0<\zeta_{n}\leq\pi$ for all $n\geq 0$. In order
to construct a cotranslation, as previously, we define the sequences
$\\{g_{n}\\}_{n\geq 0}$ and $\\{h_{n}\\}_{n\geq 0}$ of homeomorphisms, where
$g_{n}$ denotes the symmetry of $\mathbb{R}^{2}$ respect to $L_{n}$, while
$h_{n}$ denotes the rotation $\theta_{\zeta_{n}}$, with the same notation as
before, for all $n\geq 0$. Then, the functions
$A_{a},A_{b}\mathrel{\mathop{\mathchar
58\relax}}D_{\infty}\rightarrow\mathbb{A}_{X}$ generate a cotranslation of
$D_{\infty}$ on $X=\mathbb{R}^{2}$ as above. Note that the only case in which
this dynamics are given by a classic action is if all the lines are the same
and $\zeta_{n}=\pi$ for every $n\in\mathbb{N}$.

A group with a similar structure to the infinite dihedral is $C_{2}\ast
C_{3}=\langle\,a,b\,|\,a^{2}=b^{3}=e\,\rangle$, the free product of $C_{2}$
and $C_{3}$. On this group every non-trivial element $w$ can be written
uniquely in one of the forms below:

$ab^{j}\cdots ab^{k}a,\quad ab^{j}\cdots ab^{k},\quad b^{j}\cdots
ab^{k}\quad\text{or}\quad b^{j}\cdots ab^{k}a,$

where $j,k\in\\{0,1,2\\}$. Therefore, every non-trivial element $w$ satisfies
either $w=au$ for a unique $u$ not beginning with $a$, or $w=b^{j}v$ for
$j\in\\{1,2\\}$ and a unique element $v$ not beginning with a power of $b$.

According to Proposition 3.15, a cotranslation of $C_{2}\ast C_{3}$ on $X$ is
defined by two functions $A_{a},A_{b}\mathrel{\mathop{\mathchar
58\relax}}C_{2}\ast C_{3}\rightarrow\mathbb{A}_{X}$ such that $A_{a}(aw)\circ
A_{a}(w)=\mathrm{Id}_{X}$ and $A_{b}(b^{2}w)\circ A_{b}(bw)\circ
A_{b}(w)=\mathrm{Id}_{X}$ for all $w$ in $C_{2}\ast C_{3}$. Thus, in order to
define a cotranslation, we only need a sequence $\\{g_{n}\\}_{n\geq 0}$ in
$\mathbb{A}_{X}$ and a sequence $\\{h_{0,n},\ h_{1,n},\ h_{2,n}\\}_{n\geq 0}$
of automorphisms set such that $h_{2,n}\circ h_{1,n}\circ
h_{0,n}=\mathrm{Id}_{X}$ for every $n\geq 0$.

Hence, we can define functions $A_{a}$ and $A_{b}$ given by
$A_{a}(au)=g_{|u|}$, $A_{a}(u)=g_{|u|}^{-1}$ for every $u$ not beginning with
$a$ and $A_{b}(b^{2}v)=h_{2,|v|}$, $A_{b}(bv)=h_{1,|v|}$, $A_{b}(v)=h_{0,|v|}$
for all elements $v$ which do not begin with a power of $b$.

######  Example 3.23.

Take the set 3$\mathrel{\mathop{\mathchar 58\relax}}=\\{0,1,2\\}$ and consider
the space $X=\textbf{3}^{\mathbb{N}_{0}}=\Pi_{n\in\mathbb{N}_{0}}\textbf{3}$
provided the prodiscrete topology. We denote the elements of $X$ by
$x=(x_{k})_{k}$. On 3, we denote the permutations $(0\ 1)$, $(1\ 2)$ and $(0\
1\ 2)$ by $\sigma_{0}$, $\sigma_{1}$ and $\sigma_{2}$, respectively. Let us
define homeomorphisms
$\Sigma_{0},\Sigma_{1},\Sigma_{2}\mathrel{\mathop{\mathchar
58\relax}}X\rightarrow X$ given by $\Sigma_{j}(x)_{0}=\sigma_{j}(x_{0})$ and
$\Sigma_{j}(x)_{k}=x_{k}$ for every $k\geq 1$ and $j\in\\{0,1,2\\}$.

In addition, let us denote the permutation $(0\ n)$ in $\mathbb{N}_{0}$ by
$\tau_{n}$, for all $n\geq 1$. Furthermore, we define, for all $n\geq 1$, the
functions $\rho_{n}\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow X$ given
by $\rho_{n}(x)_{k}=x_{\tau_{n}(k)}$ for all $k\geq 0$. Each $\rho_{n}$ is
also a homeomorphism of $X$.

Then, we can consider functions $A_{a},A_{b}\mathrel{\mathop{\mathchar
58\relax}}C_{2}\ast C_{3}\rightarrow\mathbb{A}_{X}$ given by
$A_{a}(au)=\tau_{|u|}$, $A_{a}(u)=\tau_{|u|}^{-1}$ for every $u$ not beginning
with $a$ and $A_{b}(b^{j}v)=\Sigma_{j}$ for $v$ not beginning with a power of
$b$, for $j\in\\{0,1,2\\}$. Then, the maps $A_{a}$ and $A_{b}$ define a
cotranslation of $C_{2}\ast C_{3}$ on $X$.

Note that we can analogously define a cotranslation of the group $C_{2}\ast
C_{n}$ on the space $\textbf{n}^{\mathbb{N}_{0}}\mathrel{\mathop{\mathchar
58\relax}}=\\{0,\dots,n-1\\}^{\mathbb{N}_{0}}$.

Other family of finitely generated groups with great relevance on dynamics are
free groups. Consider for instance the free group of rank 2, denoted by
$F_{2}$ with generators $a$ and $b$. As this group admits a presentation
without non-trivial relations, then any couple of functions
$A_{a},A_{b}\mathrel{\mathop{\mathchar 58\relax}}F_{2}\to\mathbb{A}_{X}$ can
define a cotranslation, according to Proposition 3.15.

In the following examples we explore dynamics of $F_{2}$ on the category of
rooted tree graphs.

Let $L$ be a set of $d$ elements. We denote by $L^{*}$ the set of all words of
finite length on alphabet $L$. We regard $L^{*}$ as a rooted tree graph, as in
[15, Chapter 1]. A function $f\mathrel{\mathop{\mathchar
58\relax}}L^{*}\rightarrow L^{*}$ is an endomorphism if it preserves adjacency
of the vertices. If $f$ is also a bijective function, it is called
automorphism.

Given a word $w$ in $L^{*}$, we define the subtree
$wL^{*}\mathrel{\mathop{\mathchar 58\relax}}=\\{wv\ |\ v\in L^{*}\\}$. It
admits a canonical injective morphism $\phi_{w}\mathrel{\mathop{\mathchar
58\relax}}L^{*}\rightarrow wL^{*}$, given by $\phi_{w}(v)=wv$ for every $v\in
L^{*}$. Having said that and considering an automorphism $g$ of $L^{*}$, we
define the automorphism $\tilde{g}_{(w)}$ such that

$\tilde{g}_{(w)}(v)=\left\\{\begin{array}[]{ccc}v&\text{if}&v\not\in
wL^{*},\\\ wg(v^{\prime})&\text{if}&v=wv^{\prime}\in
wL^{*}.\end{array}\right.$

######  Example 3.24.

Let us consider $\textbf{2}\mathrel{\mathop{\mathchar 58\relax}}=\\{0,1\\}$,
the binary tree $\textbf{2}^{*}$ and the graph morphism
$\phi\mathrel{\mathop{\mathchar
58\relax}}\\{a^{\pm},b^{\pm}\\}^{*}\rightarrow\textbf{2}^{*}$ which is induced
by $\phi(a)=\phi(a^{-1})=1$ and $\phi(b)=\phi(b^{-1})=0$. Let us also consider
$\sigma$ in $\mathbb{A}_{\textbf{2}^{*}}$ defined by $\sigma(0v)=1v$ and
$\sigma(1v)=0v$ for all $v$ in $\textbf{2}^{*}$.

Consider the free group $F_{2}=\langle\,a,b\,|\ \rangle$. Then, we define maps
$A_{a},A_{b}\mathrel{\mathop{\mathchar
58\relax}}F_{2}\rightarrow\mathbb{A}_{\textbf{2}^{*}}$ by
$A_{a}(w)=A_{b}(w)=\tilde{\sigma}_{(\phi(w))}$ for every $w\in F_{2}$. As
$F_{2}$ does not admit any non-trivial relation, these functions define a
cotranslation of $F_{2}$ on the binary tree, as desired.

The previous example has an clear disadvantage in comparison to others: the
cotranslation defined is not injective. In particular, we can generate the
same tree automorphism from different elements in the groupoid.

######  Example 3.25.

Consider again the group $F_{2}$ with the previous presentation and the tree
$L^{*}$, where $L=\\{a^{\pm},b^{\pm}\\}$. Let $\sigma_{a}$ and $\sigma_{b}$ be
a couple of permutations of $L$. Given these permutations, we can define the
maps $A_{a},A_{b}\mathrel{\mathop{\mathchar
58\relax}}F_{2}\rightarrow\mathbb{A}_{L^{*}}$ given by
$A_{a}(w)=\tilde{\sigma_{a}}_{(w)}$ and $A_{b}(w)=\tilde{\sigma_{b}}_{(w)}$
for all $w\in F_{2}$. Once again, these functions allows us define a
cotranslation of $F_{2}$ on the tree $L^{*}$. Besides, as every element in
$F_{2}$ is in correspondence with a unique vertex in $L^{*}$, the functions
$A_{a}$ and $A_{b}$ are injective.

######  Example 3.26.

More generally, let us consider the free group of rank $n$, which we denote by
$F_{n}$ with free generators $a_{1}$, $a_{2}$, …$a_{n}$, and the $2n$-ary tree
$L^{*}$, where $L=\\{a_{1}^{\pm},a_{2}^{\pm},\dots,a_{n}^{\pm}\\}$. Given a
finite sequence $\sigma_{1}$, $\sigma_{2}$, …, $\sigma_{n}$ of permutations of
$L$, we can define the functions $A_{i}\mathrel{\mathop{\mathchar
58\relax}}F_{n}\rightarrow\mathbb{A}_{L^{*}}$ given by
$A_{i}(w)=\tilde{\sigma_{i}}_{(w)}$ for all $w\in F_{n}$ and
$i\in\\{1,2,\dots,n\\}$.

Finally, the functions $\\{A_{i}\\}_{i=1}^{n}$ define a cotranslation of
$F_{n}$ on the $2n$-ary tree $L^{*}$.

It should be emphasized that every time that a group $G$ acts on the binary
tree through automorphisms, it induces a unique continuous action of $G$ on
the topological Cantor space $\textbf{2}^{\mathbb{N}}$ [15, p. 2]. In the same
fashion, all the examples provided here for cotranslations of free groups can
be translated from the category of rooted trees to the category of topological
spaces.

In the reminder of this section we delve into the problem of defining new
cotranslations from others previously known. For this purpose, we leverage the
concept of free product of groups [10, Chapter 2]. Particularly, we highlight
that given two groups, with presentation $G=\langle\,S\,|\,R\,\rangle$ and
$H=\langle\,T\,|\,Q\,\rangle$, their free product admits the presentation
$G\ast H=\langle\,S\cup T\,|\,R\cup Q\,\rangle$. [13, Chapter 4]. For
instance, we have $D_{\infty}=C_{2}\ast C_{2}$ and
$F_{2}=\mathbb{Z}\ast\mathbb{Z}$.

Furthermore, the free product $G\ast H$ supports two canonical injective maps
$\iota_{G}\mathrel{\mathop{\mathchar 58\relax}}G\rightarrow G\ast H$ and
$\iota_{H}\mathrel{\mathop{\mathchar 58\relax}}H\rightarrow G\ast H$, and from
its universal property [10], we have that given a group $K$ and two morphisms
$\phi_{G}\mathrel{\mathop{\mathchar 58\relax}}G\rightarrow K$,
$\phi_{H}\mathrel{\mathop{\mathchar 58\relax}}H\rightarrow K$, there is a
unique morphism $\phi\mathrel{\mathop{\mathchar 58\relax}}G\ast H\rightarrow
K$ such that $\phi_{G}=\phi\circ\iota_{G}$ and $\phi_{H}=\phi\circ\iota_{H}$.

In particular, if we consider $K=G$, we denote by
$\pi_{G}\mathrel{\mathop{\mathchar 58\relax}}G\ast H\rightarrow G$ the unique
morphism such that $\mathrm{Id}_{G}=\pi_{G}\circ\iota_{G}$ and
$\rho_{G,H}=\pi_{G}\circ\iota_{H}$, where $\rho_{G,H}$ denotes the trivial
morphism from $G$ to $H$. The map $\pi_{H}$ is defined analogously when $K=H$.

Let us now establish a result regarding cotranslations of free product groups.

######  Proposition 3.27.

Let $G=\langle\,S\,|\,R\,\rangle$, $H=\langle\,T\,|\,Q\,\rangle$ be finitely
presented groups and $X$ a topological space. According to Proposition 3.15,
let us suppose the functions $\\{A_{s}\mathrel{\mathop{\mathchar
58\relax}}G\rightarrow\mathbb{A}_{X}\\}_{s\in S}$ and
$\\{A_{t}\mathrel{\mathop{\mathchar
58\relax}}H\rightarrow\mathbb{A}_{X}\\}_{t\in T}$ define cotranslations of $G$
and $H$ respectively on $X$. Now, for every $s\in S$ and $t\in T$, let us
define $\tilde{A}_{s}\mathrel{\mathop{\mathchar 58\relax}}G\ast
H\rightarrow\mathbb{A}_{X}$, $\tilde{A}_{t}\mathrel{\mathop{\mathchar
58\relax}}G\ast H\rightarrow\mathbb{A}_{X}$ by
$\tilde{A}_{s}(w)=A_{s}(\pi_{G}(w))$ and $\tilde{A}_{t}(w)=A_{t}(\pi_{H}(w))$
for every element $w$.

Then, the functions $\\{\tilde{A}_{s}\\}_{s\in S}$ and
$\\{\tilde{A}_{t}\\}_{t\in T}$ define a cotranslation of $G\ast H$ on $X$.

###### Proof.

It is enough to verify that functions $\\{\tilde{A}_{s}\\}_{s\in S}$ and
$\\{\tilde{A}_{t}\\}_{t\in T}$ preserve the relations of $G\ast H$.

Let $r=s_{n}\dots s_{2}s_{1}$ be a relation in $R$. By hypothesis,

$A_{s_{n}}(s_{n-1}\dots s_{1}g)\circ\dots\circ A_{s_{2}}(s_{1}g)\circ
A_{s_{1}}(g)=\mathrm{Id}_{X},$

for every $g\in G$. Now, consider an arbitrary $w\in G\ast H$. Since
$\pi_{G}(s)=s$ for every $s\in S$, we have

$\displaystyle\tilde{A}_{s_{n}}(s_{n-1}\dots
s_{1}w)\circ\dots\circ\tilde{A}_{s_{1}}(w)$
$\displaystyle=A_{s_{n}}(\pi_{G}(s_{n-1}\dots s_{1}w))\circ\dots\circ
A_{s_{1}}(\pi_{G}(w))$ $\displaystyle=A_{s_{n}}(s_{n-1}\dots
s_{1}\pi_{G}(w))\circ\dots\circ A_{s_{1}}(\pi_{G}(w))$
$\displaystyle=\mathrm{Id}_{X},$

because $\pi_{G}(w)$ is an element of $G$. Since $r$ is an arbitrary relation
in $R$, we conclude that functions $\\{\tilde{A}_{s}\\}_{s\in S}$ and
$\\{\tilde{A}_{t}\\}_{t\in T}$ preserve every relation in $R$. Likewise, we
can conclude that $\\{\tilde{A}_{s}\\}_{s\in S}$ and
$\\{\tilde{A}_{t}\\}_{t\in T}$ respect every relation in $Q$. ∎

To conclude this section, let us develop one last observation.

Let $G$ be a group provided by the presentation $\langle\,S\,|\,R\,\rangle$.
Suppose that there is a cotranslation $Z$ of $G$ on some space $X$, defined by
functions $\\{A_{s}\mathrel{\mathop{\mathchar
58\relax}}G\rightarrow\mathbb{A}_{X}\\}_{s\in S}$. If we consider a non-void
subset $R_{0}$ of $R$, it is possible to define the group
$H=\langle\,S\,|\,R_{0}\,\rangle$. There exist a canonical morphism
$\pi\mathrel{\mathop{\mathchar 58\relax}}H\rightarrow G$ such that $\pi(s)=s$
for each generator $s$. It is clear that $\pi$ is a surjective map, therefore
$G$ is isomorphic to $H/\ker(\pi)$. Moreover, we can define the map
$\tilde{A}_{s}\mathrel{\mathop{\mathchar 58\relax}}=A_{s}\circ\pi$ for all $s$
in $S$.

If we consider a relation $r=s_{n}\dots s_{2}s_{1}$ in $R_{0}$, then we have

$A_{s_{n}}(s_{n-1}\dots s_{1}g)\circ\dots\circ A_{s_{2}}(s_{1}g)\circ
A_{s_{1}}(g)=\mathrm{Id}_{X},$

for all $g$ in $G$, because the family $\\{A_{s}\\}_{s\in S}$ preserves the
relations $R$. Thus, for an arbitrary element $h$ in $H$, we have

$\displaystyle\tilde{A}_{s_{n}}(s_{n-1}\dots
s_{1}h)\circ\dots\circ\tilde{A}_{s_{2}}(s_{1}h)\circ\tilde{A}_{s_{1}}(h)$
$\displaystyle=A_{s_{n}}(\pi(s_{n-1}\dots s_{1}h))\circ\dots\circ
A_{s_{2}}(\pi(s_{1}h))\circ A_{s_{1}}(\pi(h))$
$\displaystyle=A_{s_{n}}(s_{n-1}\dots s_{1}\pi(h))\circ\dots\circ
A_{s_{2}}(s_{1}\pi(h))\circ A_{s_{1}}(\pi(h))$
$\displaystyle=\mathrm{Id}_{X},$

due to $\pi(h)$ being in $G$. As $r$ is an arbitrary relation in $R_{0}$, we
conclude that the functions $\\{\tilde{A}_{s}\mathrel{\mathop{\mathchar
58\relax}}H\rightarrow\mathbb{A}_{X}\\}_{s\in S}$ preserve the relations of
$H$. In conclusion, this defines a cotranslation of $H$ on the space $X$.

From this observation, we conclude the following particular claim: If we
consider the groups $D_{6}=\langle\,r,s\,|\,r^{3}=s^{2}=(sr)^{2}=e\,\rangle$,
$C_{3}\ast C_{2}=\langle\,r,s\,|\,r^{3}=s^{2}=e\,\rangle$ and
$F_{2}=\langle\,r,s\,|\,\rangle$, we clearly have
$\emptyset\subset\\{r^{3},s^{2}\\}\subset\\{r^{3},s^{2},(sr)^{2}\\}$.
Therefore, all cotranslations of $D_{6}$ on some space $X$ are inherited in
this manner to $C_{3}\ast C_{2}$, in the same way that all cotranslation of
$C_{3}\ast C_{2}$ on $X$ are inherited to $F_{2}$.

##  4\. Differentiable cotranslations on Banach spaces

In this section we study the differentiability of cotranslations and analyze
their relation to differential equations. During this section we fix a Banach
space $X$ over the field $\mathbb{K}$, which can be $\mathbb{R}$ or
$\mathbb{C}$. We set

  * •

$\mathbb{L}_{X}$ the collection of all densely defined linear operators on
$X$,

  * •

$\mathbb{B}_{X}$ the algebra of all continuous elements of $\mathbb{L}_{X}$,
given the strong topology,

  * •

$\mathbb{A}_{X}$ the group of all invertible elements of $\mathbb{B}_{X}$
whose inverse is also in $\mathbb{B}_{X}$.

The group associated with the cotranslations studied in this section is the
field $\mathbb{K}$. Nevertheless, the formalism presented here is versatile,
enabling generalizations of the results to other non-commutative Lie groups.
We use Banach spaces so that $\mathbb{B}_{X}$ has the algebra structure, thus
we are able to study derivatives for groupoid morphisms through limits.

Consider $\varphi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{B}_{X}$ and
$\psi\mathrel{\mathop{\mathchar 58\relax}}\mathbb{K}\to\mathbb{B}_{X}$. In
this section we use the following notation (independently if these limits
exist or not):

  * i)

$\partial_{1}\varphi(r,t)=\lim_{h\to 0}\frac{\varphi(r+h,t)-\varphi(r,t)}{h}$,

  * ii)

$\partial_{2}\varphi(r,t)=\lim_{h\to 0}\frac{\varphi(r,t+h)-\varphi(r,t)}{h}$,

  * iii)

$\frac{d}{du}\left[\psi(u)\right]=\lim_{h\to 0}\frac{\psi(u+h)-\psi(u)}{h}$.

Notation iii) proves useful when applying derivatives to functions obtained as
compositions of other functions or when emphasizing the variable with respect
to which we intend to differentiate. On the other hand, notations i) and ii)
are designed to underscore the position of the variable with respect to which
we seek differentiation.We begin by stating basic results .

######  Lemma 4.1.

Let $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$ be a continuous
cotranslation. If for every $t\in\mathbb{K}$ the map $r\mapsto Z(r,t)$ is
derivable at $r=r_{0}$, for some $r_{0}\in\mathbb{K}$, then the map $r\mapsto
Z^{\mathrm{inv}}(r,t)$ is derivable at $r=r_{0}$ and

$\partial_{1}Z^{\mathrm{inv}}(r_{0},t)=-Z^{\mathrm{inv}}(r_{0},t)\Big{[}\partial_{1}Z(r_{0},t)\Big{]}Z^{\mathrm{inv}}(r_{0},t).$

###### Proof.

By the following

$\displaystyle Z(r,t)Z^{\mathrm{inv}}(r,t)=\mathrm{Id}$
$\displaystyle\Rightarrow$
$\displaystyle\frac{d}{dr}\left[Z(r,t)Z^{\mathrm{inv}}(r,t)\right]=0$
$\displaystyle\Leftrightarrow$ $\displaystyle\Big{[}\partial_{1}Z(r,t)\Big{]}\
Z^{\mathrm{inv}}(r,t)+Z(r,t)\left[\partial_{1}Z^{\mathrm{inv}}(r,t)\right]=0$
$\displaystyle\Leftrightarrow$
$\displaystyle\partial_{1}Z^{\mathrm{inv}}(r,t)=-Z^{\mathrm{inv}}(r,t)\Big{[}\partial_{1}Z(r,t)\Big{]}Z^{\mathrm{inv}}(r,t),$

it is easy to see that the derivatives $\partial_{1}$ at $r=r_{0}$ exist
simultaneously for both $r\mapsto Z(r,t)$ and $r\mapsto
Z^{\mathrm{inv}}(r,t)$. ∎

The following result is obtained with an analogous proof.

######  Lemma 4.2.

Let $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$ be a continuous
cotranslation. If for every $r\in\mathbb{K}$ the map $t\mapsto Z(r,t)$ is
derivable at $t_{0}$, for some $t_{0}\in\mathbb{K}$, then the map $t\mapsto
Z^{\mathrm{inv}}(r,t)$ is derivable at $t=t_{0}$ and

$\partial_{2}Z^{\mathrm{inv}}(r,t_{0})=-Z^{\mathrm{inv}}(r,t_{0})\Big{[}\partial_{2}Z(r,t_{0})\Big{]}Z^{\mathrm{inv}}(r,t_{0}).$

######  Lemma 4.3.

Let $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$ be a continuous
cotranslation. Suppose that for every $t\in\mathbb{K}$ the map $r\mapsto
Z(r,t)$ is derivable at $r=0$. Then, for every $t\in\mathbb{K}$ the function
$r\mapsto Z(r,t)$ is derivable and

$\partial_{1}Z(r,t)=\Big{[}\partial_{1}Z(0,t+r)\Big{]}Z(r,-r)-Z(r,t)\Big{[}\partial_{1}Z(0,r)\Big{]}Z(r,-r).$

###### Proof.

We suppose the following limit exists

$\lim_{h\to 0}\frac{Z(0+h,t)-Z(0,t)}{h}.$

We have

$\displaystyle\frac{Z(r+h,t)-Z(r,t)}{h}$ $\displaystyle=$
$\displaystyle\frac{Z(h,t+r)Z^{\mathrm{inv}}(h,r)-Z(0,r+t)Z^{\mathrm{inv}}(0,r)}{h}$
$\displaystyle=$
$\displaystyle\frac{Z(h,t+r)-Z(0,r+t)}{h}Z^{\mathrm{inv}}(h,r)$
$\displaystyle+Z(0,r+t)\frac{Z^{\mathrm{inv}}(h,r)-Z^{\mathrm{inv}}(0,r)}{h}.$

Both addends at the right hand side have limit when $h\to 0$ by hypothesis
(and Lemma 4.1), thus the left hand side as limit as well, hence $r\mapsto
Z(r,t)$ is derivable at every point for every fixed $t\in\mathbb{K}$ and

$\displaystyle\partial_{1}Z(r,t)$ $\displaystyle=$
$\displaystyle\Big{[}\partial_{1}Z(0,t+r)\Big{]}Z^{\mathrm{inv}}(0,r)+Z(0,r+t)\Big{[}\partial_{1}Z^{\mathrm{inv}}(0,r)\Big{]}$
$\displaystyle=$
$\displaystyle\Big{[}\partial_{1}Z(0,t+r)\Big{]}Z(r,-r)-Z(0,r+t)Z(r,-r)\Big{[}\partial_{1}Z(0,r)\Big{]}Z(r,-r)$
$\displaystyle=$
$\displaystyle\Big{[}\partial_{1}Z(0,t+r)\Big{]}Z(r,-r)-Z(r,t)\Big{[}\partial_{1}Z(0,r)\Big{]}Z(r,-r).$

∎

Analogously we have the following result.

######  Lemma 4.4.

Let $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$ be a continuous
cotranslation. Suppose that for every $r\in\mathbb{K}$ the function $t\mapsto
Z(r,t)$ is derivable at $t=0$. Then, for every $r\in\mathbb{K}$ the function
$t\mapsto Z(r,t)$ is derivable and

$\partial_{2}Z(r,t)=\Big{[}\partial_{2}Z(r+t,0)\Big{]}Z(r,t).$

###### Proof.

Suppose that for every $r\in\mathbb{K}$ the following limit exists

$\lim_{h\to 0}\frac{Z(r,h)-Z(r,0)}{h}.$

We have

$\displaystyle\frac{Z(r,t+h)-Z(r,t)}{h}$ $\displaystyle=$
$\displaystyle\frac{Z(r+t,h)Z(r,t)-Z(r,t)}{h}$ $\displaystyle=$
$\displaystyle\frac{Z(r+t,h)-\mathrm{Id}}{h}Z(r,t)$ $\displaystyle=$
$\displaystyle\frac{Z(r+t,h)-Z(r+t,0)}{h}Z(r,t).$

By taking limits $h\to 0$ we obtain the desired identity. ∎

Now, we present a result demonstrating how to deduce derivability with respect
to one coordinate when we have information about the other.

######  Lemma 4.5.

Let $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$ be a continuous
cotranslation such that for every $r\in\mathbb{K}$ the map $t\mapsto Z(r,t)$
is derivable. Then, the map $r\mapsto Z(r,t)$ is derivable for every
$t\in\mathbb{K}$ and

$\partial_{1}Z(r,t)=\partial_{2}Z(r,t)-Z(r,t)\Big{[}\partial_{2}Z(r,0)\Big{]}.$

###### Proof.

Suppose that for every $r,t\in\mathbb{K}$ the following limit exists

$\lim_{h\to 0}\frac{Z(r,t+h)-Z(r,t)}{h}.$

Note that

$\displaystyle\frac{Z(r,t+h)-Z(r,t)}{h}$ $\displaystyle=$
$\displaystyle\frac{Z(r+h,t)Z(r,h)-Z(r,t)}{h}$ $\displaystyle=$
$\displaystyle\frac{Z(r+h,t)Z(r,h)-Z(r,t)Z(r,h)}{h}+\frac{Z(r,t)Z(r,h)-Z(r,t)}{h}$
$\displaystyle=$
$\displaystyle\frac{Z(r+h,t)-Z(r,t)}{h}Z(r,h)+Z(r,t)\frac{Z(r,h)-Z(r,0)}{h}.$

from where, reorganizing terms and taking limits we obtain

$\displaystyle\lim_{h\to 0}\frac{Z(r+h,t)-Z(r,t)}{h}$ $\displaystyle=$
$\displaystyle\Big{[}\partial_{2}Z(r,t)\Big{]}Z(r,0)-Z(r,t)\Big{[}\partial_{2}Z(r,0)\Big{]}$
$\displaystyle=$
$\displaystyle\partial_{2}Z(r,t)-Z(r,t)\Big{[}\partial_{2}Z(r,0)\Big{]}.$

∎

As a summary of the previous lemmas we state the following:

######  Corollary 4.6.

For a continuous cotranslation $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$, the following
statements are equivalent:

  * i)

$t\mapsto Z(r,t)$ is derivable at $t=0$, for every $r\in\mathbb{K}$.

  * ii)

$t\mapsto Z(r,t)$ is derivable for every $r\in\mathbb{K}$.

Moreover, each one of them implies the following statements, which are
equivalent:

  * iii)

$r\mapsto Z(r,t)$ is derivable at $r=0$, for every $t\in\mathbb{K}$.

  * iv)

$r\mapsto Z(r,t)$ is derivable for every $t\in\mathbb{K}$.

From now on, we say that a cotranslation is differentiable if it verifies i)
or ii) on the previous corollary. In the following we study the relation of
differentiable cotranslations and linear nonautonomous differential equations.
Even in the autonomous case, the existence of solutions is a non trivial
problem when $X$ is infinite dimensional. An alternative is to look for
solutions with a restricted domain as $[0,\infty)$ or $[0,t]$ (we refer the
reader to [16, Chapter 4]).

On the other hand, the nonautonomous case presents even greater difficulties.
On finite dimension, it is known that it is enough to ask $t\mapsto A(t)$ to
be locally integrable in order to guarantee the existence and uniqueness of
solutions. On infinite dimensional spaces, the problem is much harder. A
partial result stays that if $t\mapsto A(t)\in\mathbb{B}_{X}$ is continuous
under the uniform operator norm, then we have the existence and uniqueness of
solutions defined on a bounded interval [16, Theorem 5.1.1].

In general, the problem of the existence and uniqueness of globally defined
solutions is not fully resolved. In the concluding part of this section we
address this issue by leveraging the structure of cotranslations.

######  Proposition 4.7.

Set $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$ a differentiable
cotranslation. Define $A\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\to\mathbb{L}_{X}$ and the operator
$\Psi\mathrel{\mathop{\mathchar 58\relax}}\mathbb{K}^{2}\to\mathbb{A}_{X}$ by

$A(u)\mathrel{\mathop{\mathchar
58\relax}}=\partial_{2}Z(u,0),\qquad\Psi(u,v)\mathrel{\mathop{\mathchar
58\relax}}=Z(v,u-v),$

then, the following are verified:

  * i)

$\Psi(u,v)\Psi(v,w)=\Psi(u,w)$ for every $u,v,w\in\mathbb{K}$,

  * ii)

$\frac{d\Psi}{du}=A(u)\Psi(u,v)$,

  * iii)

$\frac{d\Psi}{dv}=-\Psi(u,v)A(v)$,

  * iv)

for every $\xi\in X$, the map $\psi_{v,\xi}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\to X$, given by
$\psi_{v,\xi}(u)=\left[\Psi(u,v)\right](\xi)$ verifies $\psi_{v,\xi}(v)=\xi$
and is a solution to the equation

(4.1) $\frac{dx}{du}=A(u)x(u).$

###### Proof.

It is easy to see that each $A(u)$ is a linear transformation of $X$ (although
we cannot ensure in general that it is continuous). Similarly, we cannot in
general state that $u\mapsto A(u)$ is continuous. To verify i) it is enough to
see:

$\Psi(u,v)\Psi(v,w)=Z(v,u-v)Z(w,v-w)=Z\left(w,(v-w)+(u-v)\right)=Z(w,u-w)=\Psi(u,w).$

On the other hand, on Lemma 4.4 we proved the identity

$\partial_{2}Z(r,t)=\Big{[}\partial_{2}Z(r+t,0)\Big{]}Z(r,t),$

hence

$\frac{d\Psi}{du}(u,v)=\frac{d}{du}\left[Z(v,u-v)\right]=\partial_{2}Z(v,u-v)=\Big{[}\partial_{2}Z(v+u-v,0)\Big{]}Z(v,u-v)=A(u)\Psi(u,v),$

from where ii) follows. Then, trivially $\psi_{v,\xi}$ is a solution to (4.1)
and

$\psi_{v,\xi}(v)=\left[\Psi(v,v)\right](\xi)=\left[Z(v,v-v)\right](\xi)=\mathrm{Id}(\xi)=\xi,$

thus verifying iv). Finally, note that

$\Psi(u,v)=Z(v,u-v)=Z(0,u)Z(v,-v)=Z(0,u)Z^{\mathrm{inv}}(0,v),$

hence, using the identity from Lemma 4.2 we obtain

$\displaystyle\frac{d\Psi}{dv}(u,v)=Z(0,u)\frac{d}{dv}\left[Z^{\mathrm{inv}}(0,v)\right]$
$\displaystyle=Z(0,u)\Big{[}\partial_{2}Z^{\mathrm{inv}}(0,v)\Big{]}$
$\displaystyle=-Z(0,u)Z^{\mathrm{inv}}(0,v)\Big{[}\partial_{2}Z(0,v)\Big{]}Z^{\mathrm{inv}}(0,v)$
$\displaystyle=-\Psi(u,v)\Big{[}\partial_{2}Z(v,0)\Big{]}$
$\displaystyle=-\Psi(u,v)A(v),$

where the second to last equality follows from the identify of Lemma 4.4, thus
proving iii). ∎

The existence of a function with the properties of $\Psi$ as described in the
preceding is precisely what is required to articulate globally defined
solutions for every initial condition. We define this concept as follows:

######  Definition 4.8.

[16, Definition 5.1.3] A map $\Psi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{B}_{X}$ is an evolution
operator (or evolution system) if the following conditions are verified:

  * i)

$\Psi(r,r)=\mathrm{Id}$, $\Psi(r,t)\Psi(t,s)=\Psi(r,s)$ for every
$r,s,t\in\mathbb{K}$,

  * ii)

$(r,t)\to\Psi(r,t)\in\mathbb{B}_{X}$ is strongly continuous.

If furthermore there is a map $t\mapsto A(t)\in\mathbb{L}_{X}$ such that

  * iii)

$\partial_{1}\Psi(r,t)=A(r)\Psi(r,t)$ and
$\partial_{2}\Psi(r,t)=-\Psi(r,t)A(t)$, we say that $\Psi$ is the evolution
operator associated to the differential equation $\dot{x}=A(t)x(t)$.

We know [16, Theorem 4.1.3] that an autonomous linear differential equation
$\dot{x}=Ax$ on a Banach space has uniquely defined solutions on $[0,\infty)$
for every initial condition if and only if $A$ is the infinitesimal generator
[9, Definition 1.2, p.49] of a $C_{0}$-semigroup. Inspired by this, we present
the following definition and theorem.

######  Definition 4.9.

For a differentiable cotranslation $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$, its
infinitesimal generator is the function $A\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\to\mathbb{L}_{X}$ given by $A(t)=\partial_{2}Z(t,0)$.

Note that the infinitesimal generator of a cotranslation corresponds to the
derivative respect to the second coordinate evaluated on the unit space
$\left(\mathbb{K}\\!\times\\!\mathbb{K}\right)^{(0)}$ of the groupoid.

######  Theorem 4.10.

A linear nonautonomous differential equation on a Banach space $X$

(4.2) $\dot{x}(t)=A(t)x(t),$

has an evolution operator associated to it (i.e. unique and globally defined
solution for every initial condition on $X$) if and only if $A$ is the
infinitesimal generator of a differentiable cotranslation
$Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$.

###### Proof.

Proposition 4.7 states that if $A(t)=\partial_{2}Z(t,0)$ for some
cotranslation, then there is an evolution operator associated to (4.2).

On the other hand, if (4.2) has an evolution operator associated, say $\Psi$,
then set $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$ by
$Z(r,t)=\Psi(t+r,r)$. It is easy to see that such $Z$ is a differentiable
groupoid morphism and furthermore, by Lemma 4.4, we conclude
$A(t)=\partial_{2}Z(t,0)$. ∎

##  5\. Partial cotranslations

In this section, we focus on a partial version of cotranslations. We fix a
topological group $G$ and consider the left translations groupoid structure in
$G\times G$. Additionally, we work with a Banach space $X$ over the field
$\mathbb{K}$. While some results presented in this section apply to general
Banach spaces, the key conclusions require $X$ to be finite-dimensional.
Towards the end of the section, the notion of an orthonormal basis is
entailed, so in essence, our main results are established when considering the
space $X=\mathbb{K}^{d}$ with its Euclidean norm. In this case,
$\mathbb{B}_{X}\cong\mathcal{M}_{d}(\mathbb{K})$ and $\mathbb{A}_{X}\cong
GL_{d}(\mathbb{K})$.

######  Definition 5.1.

  * a)

We say a function $W\mathrel{\mathop{\mathchar 58\relax}}G\times
G\to\mathbb{B}_{X}$ is a partial cotranslation if

$W(g,kh)=W(hg,k)\circ W(g,h),\quad\text{ for all }g,h,k\in G.$

  * b)

For two partial cotranslations $W,V\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{B}_{x}$, we say they are mutually
orthogonal if $W(hg,k)V(g,h)=V(hg,k)W(g,h)=0$ for all $g,h,k\in G$.

Unlike a cotranslation, a partial cotranslation is not necessarily a groupoid
morphism, since its codomain is not a groupoid (at least not if we define
$\mathbb{B}_{X}^{(2)}=\mathbb{B}_{X}^{2}$). Now we study some basic properties
for cotranslations.

######  Lemma 5.2.

If $W\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$ is
a partial cotranslation, then $W(g,e)$ is idempotent for all $g\in G$.

###### Proof.

By definition, we have $W(g,e)W(g,e)=W(g,e\cdot e)=W(g,e)$ for all $g\in G$. ∎

######  Lemma 5.3.

Let $W\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$ be
a partial cotranslation. For all $g,h,k\in G$ one gets $\ker W(h,g)=\ker
W(h,k)$. In particular, if $X$ is finite dimensional, then
${\mathrm{rank}}W(h,g)={\mathrm{rank}}W(h,k)$.

###### Proof.

From the identity

$W(h,k)=W(gh,kg^{-1})W(h,g)$

one gets $\xi\in\ker W(h,g)\Rightarrow\xi\in\ker W(h,k)$, thus $\ker
W(h,g)\leq\ker W(h,k)$ and by symmetry of the argument we obtain the equality.
The second conclusion is trivial if $X$ is finite dimensional. ∎

A fundamental result in linear algebra that we require is that for two
transformations $A$ and $B$ in $\mathbb{B}_{X}$, if $AB$ has the same kernel
as $B$, then $\dim\ker A\leq\dim\ker B$.

######  Proposition 5.4.

Let $W\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$ be
a partial cotranslation, where $X$ is finite dimensional. Then
${\mathrm{rank}}W(g,h)={\mathrm{rank}}W(k,\ell)$ for every $g,h,k,\ell\in G$.

###### Proof.

Starting from the identity

$W(hg,h^{-1})W(g,h)=W(g,e),$

as Lemma 5.3 indicates $W(g,h)$ and $W(g,e)$ have the same kernel, then the
previous observation implies $\dim\ker W(hg,h^{-1})\leq\dim\ker W(g,h)$.

Now, from the identity

$W(g,h)W(hg,h^{-1})=W(hg,e),$

as Lemma 5.3 showed that $W(hg,h^{-1})$ and $W(hg,e)$ have the same kernel,
then $\dim\ker W(g,h)\leq\dim\ker W(hg,h^{-1})$. In conclusion, $\dim\ker
W(hg,h^{-1})=\dim\ker W(g,h)$, thus
${\mathrm{rank}}W(hg,h^{-1})={\mathrm{rank}}W(g,h)$.

Taking $h=kg^{-1}$ in the last equality we obtain
${\mathrm{rank}}W(k,gk^{-1})={\mathrm{rank}}W(g,kg^{-1})$, but for arbitrary
$h,\ell\in G$ we know, from Lemma 5.3 that
${\mathrm{rank}}W(g,kg^{-1})={\mathrm{rank}}W(g,h)$ and
${\mathrm{rank}}W(k,gk^{-1})={\mathrm{rank}}W(k,\ell)$, thus
${\mathrm{rank}}W(g,h)={\mathrm{rank}}W(k,\ell)$. ∎

In view of the above, we can define without ambiguity:

######  Definition 5.5.

Let $X$ be finite dimensional. For a partial cotranslation
$W\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$, the
rank of the partial cotranslation is
${\mathrm{rank}}\,W\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{rank}}\,W(g,h)$, for arbitrary $g,h\in G$.

As every cotranslation is in particular a partial cotranslation, the following
definition applies as well for cotranslations.

######  Definition 5.6.

Given a partial cotranslation $V\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$, we say a function
$\mathrm{P}\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{B}_{X}$ is:

  * a)

projector if $\mathrm{P}(g)$ is idempotent on $\mathbb{B}_{X}$ for every $g\in
G$.

  * b)

invariant projector associated to $V$ if it is a projector and

$\mathrm{P}(hg)V(g,h)=V(g,h)\mathrm{P}(g),\qquad\forall\,g,h\in G.$

  * c)

For other invariant projectors $\mathrm{Q}\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{B}_{X}$, we say $\mathrm{P}$ and $\mathrm{Q}$ are
mutually orthogonal if
$\mathrm{P}(g)\mathrm{Q}(g)=\mathrm{Q}(g)\mathrm{P}(g)=0$ for every $g\in G$.

It is easy to see that if $\mathrm{P}$ is an invariant projector for a partial
cotranslation, then $\mathrm{Id}-\mathrm{P}$ is invariant as well, and they
are mutually orthogonal. In the following we examine properties related to
invariant projectors.

######  Lemma 5.7.

If $W\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$ is
a partial cotranslation and we define $\mathrm{P}\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{B}_{X}$ by $\mathrm{P}(g)=W(g,e)$, called its projector
of the units space, we obtain that $\mathrm{P}$ is an invariant projector
associated to $W$. If $W$ is continuous, then its projector of the units space
is continuous as well.

###### Proof.

We know from Lemma 5.2 that it is indeed a projector. To obtain its invariance
it is enough to see

$\displaystyle
W(g,h)\mathrm{P}(g)=W(g,h)W(g,e)=W(g,h)=W(hg,e)W(g,h)=\mathrm{P}(hg)W(g,h).$

The last statement is trivial. ∎

######  Proposition 5.8.

If $V\mathrel{\mathop{\mathchar 58\relax}}G\times G\to\mathbb{B}_{X}$ is a
partial cotranslation and $\mathrm{P}$ is an associated invariant projector,
then $W\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$
given by $W(g,h)=V(g,h)\mathrm{P}(g)$ is a partial cotranslation.

If moreover both $\mathrm{P}$ and $V$ are continuous, then $W$ is continuous
as well.

###### Proof.

It is enough to see that

$\displaystyle W(g,kh)=V(g,kh)\mathrm{P}(g)$ $\displaystyle=$ $\displaystyle
V(g,kh)\mathrm{P}(g)^{2}$ $\displaystyle=$ $\displaystyle
V(hg,k)V(g,h)\mathrm{P}(g)^{2}$ $\displaystyle=$ $\displaystyle
V(hg,k)\mathrm{P}(hg)V(g,h)\mathrm{P}(g)$ $\displaystyle=$ $\displaystyle
W(hg,k)W(g,h).$

Finally, continuity follows trivially. ∎

######  Example 5.9.

Consider the linear nonautonomous differential equation $\dot{x}=A(t)x$ with
transition matrix $\Phi$. We know from Example 3.5 that $Z(s,t)=\Phi(t+s,s)$
defines a continuous cotranslation with $X=\mathbb{R}^{d}$ and $G=\mathbb{R}$.
Consider now a continuous invariant projector $\mathrm{P}$ for this
cotranslation (in the sense of Definition 5.6). Proposition 5.8 shows that
$W(s,t)=Z(s,t)\mathrm{P}(s)$ defines a continuous partial cotranslation.

Moreover, in this case $\mathrm{P}$ is also an invariant projector for this
differential equation, in the usual sense for nonautonomous dynamics [21, p.
246]. Indeed:

$\mathrm{P}(t)\Phi(t,s)=\mathrm{P}(t)Z(s,t-s)=\mathrm{P}\left((t-s)+s\right)Z(s,t-s)=Z(s,t-s)\mathrm{P}(s)=\Phi(t,s)\mathrm{P}(s).$

Note that $W$ can also be written $W(s,t)=\Phi(t+s,s)\mathrm{P}(s)$, which is
a matrix function that describes the solutions to the equation $\dot{x}=A(t)x$
whose graph lies in the linear integral manifold [21, Definition 2.2]
associated to the image of this projector [21, Lemma 2.1].

######  Lemma 5.10.

If $W,V\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$
are mutually orthogonal partial cotranslations, then
$W+V\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$ is a
partial cotranslation. If both $W$ and $V$ are continuous, then $W+V$ is as
well.

###### Proof.

Note that

$\displaystyle W(g,kh)+V(g,kh)$ $\displaystyle=$ $\displaystyle
W(hg,k)W(g,h)+V(hg,k)V(g,h)$ $\displaystyle=$ $\displaystyle
W(hg,k)W(g,h)+W(hg,k)V(g,h)$ $\displaystyle+V(hg,k)W(g,h)+V(hg,k)V(g,h)$
$\displaystyle=$
$\displaystyle\left(W(hg,k)+V(hg,k)\right)\left(W(g,h)+V(g,h)\right).$

Once again, continuity follows trivially. ∎

######  Lemma 5.11.

If $W,V\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$
are mutually orthogonal partial cotranslations, then the unit space projector
of $W$ is invariant for $W+V$. Moreover, the partial cotranslation obtained
through $W+V$ and the projector of the units space of $W$ (following the
construction on Proposition 5.8) is $W$.

###### Proof.

It is enough to note that

$\displaystyle W(hg,e)\left(W(g,h)+V(g,h)\right)$
$\displaystyle=W(hg,e)W(g,h)+W(hg,e)V(g,h)$ $\displaystyle=W(g,h)W(g,e)$
$\displaystyle=W(g,h)W(g,e)+V(g,h)W(g,e)$
$\displaystyle=\left(W(g,h)+V(g,h)\right)W(g,e).$

For the second statement, note that

$W(g,h)=W(g,h)W(g,e)=\left(W(g,h)+V(g,h)\right)W(g,e).$

∎

######  Lemma 5.12.

If $\mathrm{P}$ and $\mathrm{Q}$ are mutually orthogonal invariant projectors
for a partial cotranslation $V\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$, then the partial cotranslations
$V_{\mathrm{P}},V_{\mathrm{Q}}\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$, given by
$V_{\mathrm{P}}(g,h)=V(g,h)\mathrm{P}(g)$ and
$V_{\mathrm{Q}}(g,h)=V(g,h)\mathrm{Q}(g)$ are mutually orthogonal.

###### Proof.

We know both $V_{\mathrm{P}}$ and $V_{\mathrm{Q}}$ are partial cotranslations
from Proposition 5.8. Moreover

$V_{P}(hg,k)V_{\mathrm{Q}}(g,h)=V(hg,k)\mathrm{P}(hg)V(g,h)\mathrm{Q}(g)=V(hg,k)V(g,h)\mathrm{P}(g)\mathrm{Q}(g)=0,$

and the other composition follows similarly. ∎

Now we present a mechanism to relate two different partial cotranslations.

######  Definition 5.13.

Given two partial cotranslations $W,V\mathrel{\mathop{\mathchar
58\relax}}G\times G\to\mathbb{B}_{X}$, we say they are conjugated if there is
a map $T\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{A}_{X}$ such that

$T(hg)V(g,h)=W(g,h)T(g),\qquad\forall\,g,h\in G.$

In that case we call $T$ a conjugation between $W$ and $V$. If $T$, $W$ and
$V$ are continuous, we say they are continuously conjugated. On the other
hand, if

$\sup_{g\in G}\left\\{\mathinner{\\!\left\lVert
T(g)\right\rVert},\mathinner{\\!\left\lVert{T(g)^{-1}}\right\rVert}\right\\}<\infty,$

then we say $W$ and $V$ are boudedly conjugated

The significance of conjugating two partial cotranslations lies in extracting
information from one based on the other. Furthermore, as we progress through
this section, we leverage this concept to demonstrate that every partial
cotranslation can be represented as a cotranslation multiplied by an invariant
projector.

In addition, the pursuit of continuous conjugations is motivated by the
preservation of topological properties, while bounded conjugations are sought
for their ability to preserve asymptotic behavior.

######  Lemma 5.14.

Let $W\mathrel{\mathop{\mathchar 58\relax}}G\times G\to\mathbb{B}_{X}$ be a
partial cotranslation and let $T\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{A}_{X}$ be an un arbitrary map. Defining
$W_{T}\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{B}_{X}$ by

$W_{T}(g,h)=T(hg)^{-1}W(g,h)T(g),$

we obtain that $W_{T}$ is a partial cotranslation. Moreover, if $X$ is finite
dimensional, then ${\mathrm{rank}}W_{T}={\mathrm{rank}}W$. Finally, if $T$ and
$W$ are continuous, then $W_{T}$ is continuous as well.

###### Proof.

Note that

$\displaystyle W_{T}(g,kh)$ $\displaystyle=T(khg)^{-1}W(g,kh)T(g)$
$\displaystyle=T(khg)^{-1}W(hg,k)W(g,h)T(g)$
$\displaystyle=T(khg)^{-1}W(hg,k)T(hg)T(hg)^{-1}W(g,h)T(g)$
$\displaystyle=W_{T}(hg,k)W_{T}(g,h).$

The second statement follows since $T(g)$ is invertible. ∎

######  Lemma 5.15.

If $W,V\mathrel{\mathop{\mathchar 58\relax}}G\times G\to\mathbb{B}_{X}$ are
two mutually orthogonal partial cotranslations and
$T\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{A}_{X}$ is an arbitrary
map, then $W_{T}$ and $V_{T}$ (as defined on Lemma 5.14) are mutually
orthogonal.

###### Proof.

It is enough to see that

$\displaystyle W_{T}(hg,k)V_{T}(g,h)$
$\displaystyle=T(khg)^{-1}W(hg,k)T(hg)T(hg)^{-1}V(g,h)T(g)^{-1}$
$\displaystyle=T(khg)^{-1}W(hg,k)V(g,h)T(g)^{-1}$ $\displaystyle=0,$

and the other composition follows similarly. ∎

From now on we fix $X=\mathbb{K}^{d}$ with the Euclidean norm. Thus, we
replace $\mathbb{B}_{X}$ by $\mathcal{M}_{d}(\mathbb{K})$ and $\mathbb{A}_{X}$
by $GL_{d}(\mathbb{K})$.

######  Proposition 5.16.

Every partial cotranslation $W\mathrel{\mathop{\mathchar 58\relax}}G\times
G\to\mathcal{M}_{d}(\mathbb{K})$ is conjugated to a partial cotranslation
$\widehat{W}\mathrel{\mathop{\mathchar 58\relax}}G\times
G\to\mathcal{M}_{d}(\mathbb{K})$ whose projector of the units space is
constant and orthogonal, i.e.

$\widehat{W}(g,e)=\begin{pmatrix}\mathrm{Id}_{{\mathrm{rank}}W}&0\\\
0&0\end{pmatrix},\qquad\forall\,g\in G.$

where $\mathrm{Id}_{{\mathrm{rank}}W}$ is the identity on
$\mathbb{K}^{{\mathrm{rank}}W}$.

###### Proof.

For every $g\in G$ we know $W(g,e)$ is an idempotent (Lemma 5.2) of rank
${\mathrm{rank}}W$ (Proposition 5.4). Then, it follows that there exists some
$T(g)\in GL_{d}(\mathbb{K})$ such that

$T(g)^{-1}W(g,e)T(g)=\begin{pmatrix}\mathrm{Id}_{{\mathrm{rank}}W}&0\\\
0&0\end{pmatrix},$

which defines a map $T\mathrel{\mathop{\mathchar 58\relax}}G\to
GL_{d}(\mathbb{K})$. Defining $\widehat{W}\mathrel{\mathop{\mathchar
58\relax}}G\times G\to\mathbb{B}_{X}$ by $\widehat{W}=W_{T}$ as in Lemma 5.14,
we obtain a partial cotranslation which is by construction conjugated to $W$
and verifies

$\widehat{W}(g,e)=T(g)^{-1}W(g,e)T(g)=\begin{pmatrix}\mathrm{Id}_{{\mathrm{rank}}W}&0\\\
0&0\end{pmatrix}.$

∎

Note that the map $T$ we defined on the preceding proposition is not unique.
We know eigenvectors (thus diagonalizations) of a continuous matrix function
are not in general continuous or bounded. We dedicate the end of this section
to deal with this fact.

######  Proposition 5.17.

Fix $\mathrm{P}\mathrel{\mathop{\mathchar
58\relax}}G\to\mathcal{M}_{d}(\mathbb{K})$ a projector of constant rank $n$
for which exists $M\geq 1$ such that

$\sup_{g\in
G}\left\\{\mathinner{\\!\left\lVert\mathrm{P}(g)\right\rVert},\mathinner{\\!\left\lVert\mathrm{Id}-\mathrm{P}(g)\right\rVert}\right\\}<M,$

then, there exists a map $T\mathrel{\mathop{\mathchar 58\relax}}G\to
GL_{d}(\mathbb{K})$ such that

(5.1) $T(g)^{-1}\mathrm{P}(g)T(g)=\begin{pmatrix}\mathrm{Id}_{n}&0\\\
0&0\end{pmatrix},\qquad\forall\,g\in G,$

and

$\sup_{g\in G}\left\\{\mathinner{\\!\left\lVert
T(g)\right\rVert},\mathinner{\\!\left\lVert
T(g)^{-1}\right\rVert}\right\\}<\infty.$

###### Proof.

Set $\xi_{1},\dots,\xi_{d}$ the canonical basis of $\mathbb{K}^{d}$. For each
$g\in G$, there is an orthonormal basis of $\mathrm{im}\,\mathrm{P}(g)$ given
by $\\{\eta_{1}^{g},\dots,\eta_{n}^{g}\\}$ and an orthonormal basis of
$\ker\mathrm{P}(g)$ given by $\\{\eta_{n+1}^{g},\dots,\eta_{d}^{g}\\}$. If we
define a linear transformation $T(g)\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}^{d}\to\mathbb{K}^{d}$ by

$T(g)\xi_{i}=\eta_{i}^{g},$

evidently it verifies (5.1). Let $\zeta\in\mathbb{K}^{d}$ with
$\mathinner{\\!\left\lVert\zeta\right\rVert}=1$. There are unique
$\alpha_{1}^{\zeta},\dots,\alpha_{d}^{\zeta}\in\mathbb{K}$ with
$|\alpha_{i}^{\zeta}|\leq 1$ such that

$\zeta=\sum_{i=1}^{d}\alpha_{i}^{\zeta}\xi_{i},$

then

$\displaystyle\mathinner{\\!\left\lVert
T(g)\zeta\right\rVert}=\mathinner{\\!\left\lVert
T(g)\left(\sum_{i=1}^{d}\alpha_{i}^{\zeta}\xi_{i}\right)\right\rVert}\leq\sum_{i=1}^{d}|\alpha_{i}^{\zeta}|\mathinner{\\!\left\lVert
T(g)\xi_{i}\right\rVert}\leq\sum_{i=1}^{d}\mathinner{\\!\left\lVert\eta_{i}^{g}\right\rVert}=d,$

hence $\mathinner{\\!\left\lVert T(g)\right\rVert}\leq d$. On the other hand,
there exists unique
$\beta_{1}^{\zeta,g},\dots,\beta_{d}^{\zeta,g}\in\mathbb{K}$ such that

$\zeta=\sum_{i=1}^{d}\beta_{i}^{\zeta,g}\eta_{i}^{g},$

hence

$\mathrm{P}(g)\zeta=\sum_{i=1}^{n}\beta_{i}^{\zeta,g}\eta_{i}^{g},\qquad[\mathrm{Id}-\mathrm{P}(g)]\zeta=\sum_{i=n+1}^{d}\beta_{i}^{\zeta,g}\eta_{i}^{g}.$

As $\mathinner{\\!\left\lVert\mathrm{P}(g)\right\rVert}\leq M$, then
$\mathinner{\\!\left\lVert\mathrm{P}(g)\zeta\right\rVert}\leq M$, and as
$\\{\eta_{1}^{g},\dots,\eta_{n}^{g}\\}$ is an orthonormal basis, then
$|\beta_{i}^{\zeta,g}|\leq M$ for every $i=1,\dots,n$. Analogously, as
$\mathinner{\\!\left\lVert\mathrm{Id}-\mathrm{P}(g)\right\rVert}\leq M$ and
$\\{\eta_{n+1}^{g},\dots,\eta_{d}^{g}\\}$ is an orthonormal basis, we obtain
$|\beta_{i}^{\zeta,g}|\leq M$ for every $i=n+1,\dots,d$. Thus

$\displaystyle\mathinner{\\!\left\lVert
T(g)^{-1}\zeta\right\rVert}=\mathinner{\\!\left\lVert
T(g)^{-1}\left(\sum_{i=1}^{d}\beta_{i}^{\zeta,g}\eta_{i}^{g}\right)\right\rVert}$
$\displaystyle\leq\sum_{i=1}^{d}|\beta_{i}^{\zeta,g}|\mathinner{\\!\left\lVert
T(g)^{-1}\eta_{i}^{g}\right\rVert}$
$\displaystyle\leq\sum_{i=1}^{d}M\cdot\mathinner{\\!\left\lVert\xi_{i}\right\rVert}=dM,$

hence $\mathinner{\\!\left\lVert T(g)^{-1}\right\rVert}\leq dM$. ∎

The next corollary follows trivially from Propositions 5.16 and 5.17.

######  Corollary 5.18.

If $W\mathrel{\mathop{\mathchar 58\relax}}G\times
G\to\mathcal{M}_{d}(\mathbb{K})$ is a partial cotranslation whose projector of
the units space and its compliment are uniformly bounded, i.e. there exist
$M\geq 1$ such that

(5.2) $\sup_{g\in G}\left\\{\mathinner{\\!\left\lVert
W(g,e)\right\rVert},\mathinner{\\!\left\lVert\mathrm{Id}-W(g,e)\right\rVert}\right\\}<M,$

then $W$ is boundedly conjugated to a partial cotranslation whose units space
projector is constant and orthogonal.

To continue, we present a conjecture:

######  Conjecture 5.19.

Denote the canonical basis of $\mathbb{K}^{d}$ by
$\\{\xi_{1},\dots,\xi_{d}\\}$. If $\mathrm{P}\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{B}_{X}$ is a continuous projector, then for every $g\in
G$ there exists an orthonormal basis of $\mathrm{im}\,\mathrm{P}(g)$ given by
$\\{\eta_{1}^{g},\dots,\eta_{n}^{g}\\}$ and an orthonormal basis of
$\ker\mathrm{P}(g)$ given by $\\{\eta_{n+1}^{g},\dots,\eta_{d}^{g}\\}$ such
that defining $T\mathrel{\mathop{\mathchar 58\relax}}G\to GL_{d}(\mathbb{K})$
by

$T(g)\xi_{i}=\eta_{i}^{g},$

we obtain that $T$ is continuous.

Once again a corollary follows trivially from Proposition 5.16 and the known
fact that inversion of linear transformations is continuous. The second
statement in the following is deduced trivially from Corollary 5.18.

######  Corollary 5.20.

Suppose Conjecture 5.19 is true. Then, every continuous partial cotranslation
$W\mathrel{\mathop{\mathchar 58\relax}}G\times G\to\mathbb{B}_{X}$ is
continuously conjugated to a partial cotranslation whose projector of the
units space is constant and orthogonal.

Moreover, if the projector of the units space of $W$ and its compliment is
uniformly bounded, i.e. it verifies (5.2), then $W$ is continuously and
boundedly conjugated to a partial cotranslation whose projector of the units
space is constant and orthogonal.

######  Remark 5.21.

The preceding corollary, if Conjecture 5.19 is true, presents a non-
commutative and non-differentiable extension of S. Siegund’s reducibility
result for nonautonomous differential equations [22, Theorem 3.2]. The key
distinction lies in the fact that, in [22], all invariant projectors are
derived from a dichotomy.

Furthermore, it’s noteworthy that when two partial cotranslations are
boundedly and continuously conjugated, we are essentially describing a
generalized notion of ’kinematic similarity’ (see, for instance, [22,
Definition 2.1]), irrespective of whether the involved group is commutative or
possesses a differential structure.

Remarkably, for discrete groups, the generalization holds trivially, as
continuity is not a concern.

To conclude, we state a theorem and consequent corollary to summarize the
results of this section.

######  Theorem 5.22.

Every partial cotranslation is completable to a cotranslation, i.e. for every
partial cotranslation $W\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathcal{M}_{d}(\mathbb{K})$ there exists a partial
cotranslation $V\mathrel{\mathop{\mathchar 58\relax}}G\times
G\to\mathcal{M}_{d}(\mathbb{K})$ such that $W$ and $V$ are mutually orthogonal
and $W+V$ has rank $d$.

Moreover, if Conjecture 5.19 is true and $W$ is continuous, then $V$ can also
be chosen continuous.

###### Proof.

Define the constant partial cotranslation
$\widehat{V}\mathrel{\mathop{\mathchar 58\relax}}G\times
G\to\mathcal{M}_{d}(\mathbb{K})$ by

$\widehat{V}(g,h)=\begin{pmatrix}0&0\\\
0&\mathrm{Id}_{d-{\mathrm{rank}}W}\end{pmatrix}.$

We affirm $\widehat{V}$ is mutually orthogonal to $\widehat{W}$ (defined as in
Proposition 5.16). Indeed:

$\displaystyle\widehat{W}(hg,k)\widehat{V}(g,h)$
$\displaystyle=\widehat{W}(hg,k)\widehat{W}(hg,e)\widehat{V}(g,h)=\widehat{W}(hg,k)\begin{pmatrix}\mathrm{Id}_{{\mathrm{rank}}W}&0\\\
0&0\end{pmatrix}\begin{pmatrix}0&0\\\
0&\mathrm{Id}_{d-{\mathrm{rank}}W}\end{pmatrix}=0,$

and the other composition follows analogously. Choose $T$ as in Proposition
5.16. Denoting $T^{\mathrm{inv}}\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{A}_{X}$ by $T^{\mathrm{inv}}(g)=T(g)^{-1}$, it follows
by Lemma 5.15 that $W=\widehat{W}_{T^{\mathrm{inv}}}$ and
$V\mathrel{\mathop{\mathchar 58\relax}}=\widehat{V}_{T^{\mathrm{inv}}}$ are
mutually orthogonal. Hence, by Lemma 5.10, $W+V$ is also a partial
cotranslation. Moreover, for an arbitrary $g\in G$:

${\mathrm{rank}}\left(\widehat{W}+\widehat{V}\right)={\mathrm{rank}}\left(\widehat{W}(g,e)+\widehat{V}(g,e)\right)={\mathrm{rank}}\left(\begin{pmatrix}\mathrm{Id}_{{\mathrm{rank}}W}&0\\\
0&0\end{pmatrix}+\begin{pmatrix}0&0\\\
0&\mathrm{Id}_{d-{\mathrm{rank}}W}\end{pmatrix}\right)=d,$

thus, by Lemma 5.14 we have

${\mathrm{rank}}(W+V)={\mathrm{rank}}\left(\widehat{W}+\widehat{V}\right)_{T^{\mathrm{inv}}}={\mathrm{rank}}\left(\widehat{W}+\widehat{V}\right)=d.$

The second statement follows trivially from Corollary 5.20. ∎

######  Corollary 5.23.

Every partial cotranslation is a cotranslation multiplied with an invariant
projector.

###### Proof.

The statement follows trivially from Theorem 5.22 and Lemma 5.11. ∎

##  6\. Statements and Declarations

There is no competing interest related to this project. This research has been
partially supported by ANID, Beca de Doctorado Nacional 21220105

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