# Groupoid morphisms as an algebraic structure for nonautonomous dynamics

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

We present groupoid morphisms as an algebraic structure for nonautonomous
dynamics, as well as a generalization of group morphisms, which describe
classic dynamical systems. 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. We obtain
results about the differentiability of cotranslations, as well as dimension
invariance and diagonalization (through a generalized notion of kinematic
similarity) for a partial version of them, admitting noninvertible
transformations.

###### Key words and phrases:

Nonautonomus dynamics, Dynamical systems, Groupoids

###### 2020 Mathematics Subject Classification:

18B40, 22A22, 37B55, 37C60.

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

##  1\. Introduction

Consider a category $\mathscr{C}$ (for instance, topological spaces, vector
spaces, Banach spaces, 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, independently of the category $\mathscr{C}$, considers a
group $G$ (with maybe some extra structure) and a group morphism
$\gamma\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{A}_{X}$. Particularly,
in the topological case, where $X$ is a topological space and $G$ a
topological group (for simplicity, let us consider both $X$ and $G$ to be
locally compact and Hausdorff), we set on $\mathbb{A}_{X}$ the compact-open
topology [2, Definition I, p. 301], and 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 (although, if $G$ is Abelian, we can just use $g$ instead of
$g^{-1}$).

Thus, 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.

A very important, on its own right, example of dynamical system is given by
the flow of solutions of an autonomous differential equation (satisfying
standard existence and uniqueness conditions). To rephrase it, an autonomous
differential equation defines a continuous action of $\mathbb{R}$ on (for
instance) $\mathbb{R}^{d}$. We usually call such system an autonomous dynamic.

However, as soon as we consider a nonautonomous differential equations,
dynamical systems (hence group morphisms and actions) no longer describe
properly the dynamics given by the flow of solutions. In other words,
nonautonomous dynamics [5] do not have a widely discussed analogous notion of
action, even less what could be its algebraic counterpart.

The main goal of this paper is to present an algebraic generalization of
dynamical systems, which can be applied to describe the flows given by
nonautonomous equations (both differential and in differences), as well as
generalize the notion of nonautonomous dynamics to general groups, and not
just $\mathbb{R}$ or $\mathbb{Z}$.

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 [1], 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 [7]. 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. The virtue of cotranslations
over skew-products is that more analytic and algebraic properties are easily
deduced from this structure.

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.

##  2\. Skew-product dynamical systems

On 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 [10]
on a symposium on 1976, a (topological) skew-product is constituted from a
locally compact Hausdorff topological space $X$ and a locally compact
Hausdorff topological group $G$, both will be fixed on this section unless
stated otherwise. Thus, in this context, $\mathbb{A}_{X}$ denotes the
topological group of all 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 [4, 9, 10] the groups are
$\mathbb{R}$ or $\mathbb{Z}$, while here we present the construction for any
locally compact Hausdorff topological group.

For $A,B$ locally compact Hausdorff spaces, 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:

  * •

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

  * •

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

  * •

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

which are all continuous [2, Corollary I, p. 303], 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 also continuous [2, Corollary II, p. 303]. For a subspace
$Y\subset\mathcal{C}\big{(}G\\!\times\\!X;X\big{)}$, we define

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

both as topological subspaces. It is easy to see that

$\mathfrak{e}(\psi,g,x)=\mathfrak{e}_{G}\left(\widetilde{\mathfrak{e}}_{X}(\psi,x),g\right)=\mathfrak{e}_{X}\left(\widetilde{\mathfrak{e}}_{G}(\psi,g),x\right),\quad\forall\,\psi\in
Y,g\in G,x\in X.$

######  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 [9], 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 left action $\Theta$ of $G$ on spaces of functions with domain
$G$. In particular, we have $\Theta\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!\mathcal{C}\big{(}G;X\big{)}\to\mathcal{C}\big{(}G;X\big{)}$
given by

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

and an easy topology exercise shows that $\Theta$ is continuous. For an
admissible collection $Y$, the set $Y_{X}$ is contained on
$\mathcal{C}\big{(}G;X\big{)}$, thus we can consider its saturation
$\widetilde{Y}_{X}\mathrel{\mathop{\mathchar
58\relax}}=\Theta\big{(}G\\!\times\\!Y_{X}\big{)}$, which is invariant under
the action $\Theta$, hence $\left(\widetilde{Y}_{X},G,\Theta\right)$ is a
(left) dynamical system.

With this system, we may 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 (topological) 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 $\pi=\mathfrak{e}\times\sigma$, that is

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

which is the way in which skew-products are depicted on [9]. With this system,
we may write the map $\widetilde{\pi}\mathrel{\mathop{\mathchar
58\relax}}X\\!\times\\!Y\\!\times\\!G\\!\times\\!G\to X$ by

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

Now, we have all the components of a skew-product, all that is left is to
state the compatibility of the systems given by $\Theta$ and $\sigma$. 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 topological 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 only possible
action, i.e. 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 $\mathbb{R}^{d}$ as a topological space, hence $\mathbb{A}_{X}$ is
the group of all of its homeomorphisms. 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 its unique solution such that $x(m,m,\xi)=\xi$.
Define $Y=\left\\{\psi_{m}\,\mathrel{\mathop{\mathchar
58\relax}}\,m\in\mathbb{Z}\right\\}$, the collection of all solutions of (2.3)
parameterized by its temporal initial condition, i.e.

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

Set now $\sigma\mathrel{\mathop{\mathchar 58\relax}}Y\\!\times\\!\mathbb{Z}\to
Y$ the action of left translations on the temporal initial condition, that is

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

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, we have

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

On the other hand, note that

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

Now, it is well know that, 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 dynamical
system.

On a presentation given on 2004, S. Elaydi and R. J. Sacker [4] give further
applications of skew-products on the theory of nonautonomous difference
equations, as the search for asymptotically stable solutions for Beverton-Holt
equations [3]. On the other hand, in [9] the authors use this structure to
develop the exponential dichotomy spectrum for nonautonomous linear
differential equations when the admissible space of functions is compact.

##  3\. Groupoid morphisms and cotranslations

In this section we present the structure of cotranslations, which are a
specific kind of groupoid morphisms and give an algebraic counterpart to skew-
products. This gives a generalization of group morphisms, which are known to
describe dynamical systems.

######  Definition 3.1.

[13, 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.

The most trivial example of a groupoid is a group $G$, where $\bullet$ is the
composition law and $\mathfrak{inv}(g)=g^{-1}$. In this case $G^{(2)}=G^{2}$
and $G^{(0)}=\\{e\\}$.

######  Example 3.2.

Consider a topological space $\mathfrak{T}$, the collection
$\widetilde{\Xi}=\left\\{\eta\mathrel{\mathop{\mathchar
58\relax}}[0,1]\to\mathfrak{T}\mathrel{\mathop{\mathchar
58\relax}}\,\eta\text{ is continuous}\right\\}$ and the quotient
$\Xi=\widetilde{\Xi}/\sim$, where $\sim$ is the equivalence relation given by
homotopies that fix start and end. Doting $\Xi$ of

$\Xi^{(2)}\mathrel{\mathop{\mathchar
58\relax}}=\left\\{(\eta,\xi)\in\Xi^{2}\mathrel{\mathop{\mathchar
58\relax}}\eta(1)=\xi(0)\right\\},$

and

$(\eta\bullet\xi)(t)=\left\\{\begin{array}[]{lcc}\eta(2t)&\text{ if }&0\leq
t\leq 1/2\\\ \\\ \xi(2t-1)&\text{ if }&1/2\leq t\leq
1\end{array}\right.,\qquad\mathfrak{inv}(\eta)(t)=\eta(1-t),$

it is easily deduced that $\Xi$ is a groupoid. This is usually called the
paths groupoid with the operation of concatenation. In this case, the units
space corresponds to the collection of the homotopy classes of constant paths.

######  Example 3.3.

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. Analogously we can define a groupoid for right actions.

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 $G$. Note that the groupoid $G\\!\times\\!G$ is never
commutative, even if $G$ is Abelian, since
$\left((g,h),(k,l)\right)\in\left(G\\!\times\\!G\right)^{(2)}$ does not imply
$\left((k,l),(g,h)\right)\in\left(G\\!\times\\!G\right)^{(2)}$.

######  Definition 3.4.

[13, Definition 1.8] If $\Xi$ and $\Upsilon$ are groupoids with composition
laws $\bullet$ and $\star$ respectively, 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
$\vartheta(\eta\bullet\xi)=\vartheta(\eta)\star\vartheta(\xi)$.

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

######  Example 3.5.

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 $\gamma\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{A}_{X}$. The
next example shows however, that groupoid morphisms describe a more general
kind of dynamics.

######  Example 3.6.

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 3.7.

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 our main theorem. Although we state it for the topological
case, it is easily generalized for objects and skew-products on different
categories.

######  Theorem 3.8.

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 dynamical system with $\sigma$ being
transitive. The admissibility condition implies that
$Y_{G}\subset\mathbb{A}_{X}$. On the other hand, as $\sigma$ is transitive,
basic theory of group actions states that 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 left action
$\sigma\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!Y\to Y$ is rewritten
as the left action $\hat{\sigma}\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!Y\to Y$ 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 the quotient
$G/\mathrm{stab}(\sigma)$.

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 3.9.

The previous theorem illustrates that all the generalizations of dynamical
systems we can obtain from skew-products, are also covered by cotranslations.
However, the virtue of the latter is that they more clearly represent the
algebraic structure behind these dynamics, just as group morphisms represent
the algebraic structure of group actions.

Now we will give further properties and applications for cotranslations. In
the following, we fix a group $G$ and an element $X$ on some category
$\mathscr{C}$.

######  Notation 3.10.

For a cotranslation $Z\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$ we set
$Z^{\mathrm{inv}}\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$, the map given by
$Z^{\mathrm{inv}}(g,h)=\left[Z(g,h)\right]^{-1}$. We do not use $Z^{-1}$ in
order to avoid confusion with a possible inverse function.

######  Lemma 3.11.

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 $g,h\in G$ one has

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

###### Proof.

The first equality follows again 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 and we know the involutions in $G\\!\times\\!G$. ∎

######  Proposition 3.12.

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 3.13.

Consider $X=\mathbb{R}^{d}$ as a Banach space, thus $\mathbb{B}_{X}$ is the
Banach algebra of its linear continuous operators and $\mathbb{A}_{X}$ is the
topological group of all of its homeomorphic isomorphisms, i.e.
$GL_{d}(\mathbb{R})$. Consider a real linear nonautonomous differential
equation $\dot{x}=A(t)x(t)$, where $t\mapsto A(t)$ is locally integrable. Take
$\Phi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\\!\times\\!\mathbb{R}\to\mathbb{A}_{X}$ the transition
matrix associated to this equation. We can see, as in Example 3.6, 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
$\dot{x}=\left[A(t)-\lambda\cdot\mathrm{Id}\right]x(t)$.

Before studying more properties for cotranslations, we will introduce a
specific type of them in order to highlight the difference between this
construction and classic dynamical systems. This will also give emphasis on
the difference between autonomous and nonautonomous equations. This motivates
the following definition:

######  Definition 3.14.

Given a cotranslation $Z\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$, we say it is autonomous if
$Z(g,h)=Z(k,h)$ for every $g,h,k\in G$.

######  Proposition 3.15.

A cotranslation $Z\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$ is autonomous if and only if there
is a group morphism $\gamma\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{A}_{X}$ such that $Z(g,h)=\gamma(h)$.

###### Proof.

We saw on Example 3.5 that if $\gamma\mathrel{\mathop{\mathchar
58\relax}}G\to\mathbb{A}_{X}$ is a group morphism and we define
$Z\mathrel{\mathop{\mathchar 58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$ by
$Z(g,h)=\gamma(h)$ we obtain a cotranslation, which is autonomous by
construction.

Conversely, if $Z$ is a cotranslation, define
$\gamma\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{A}_{X}$ by
$\gamma(g)=Z(e,g)$. It is easy to see that $\gamma$ is a group morphism, since

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

∎

Note once more, as in Remark 3.9, how this proposition illustrates that
cotranslations generalize the notion of dynamical systems on any given
category (as for instance, those given by autonomous differential equations,
hence the name), since those are always given by a group morphism
$\gamma\mathrel{\mathop{\mathchar 58\relax}}G\to\mathbb{A}_{X}$, but clearly
there are much more cotranslations which are not autonomous.

The following proposition shows the direct relation of cotranslations with
group $\mathbb{Z}$ and nonautonomous difference equations.

######  Proposition 3.16.

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. ∎

In the next section we will show a continuous analogous to the previous
proposition. We finish this section by giving a generalization of the
preceding proposition for finitely generated groups (for which we do not give
a proof, since it follows the same steps as before), but first we need an
auxiliary definition.

######  Definition 3.17.

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 3.18.

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

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

##  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 might be $\mathbb{R}$ or
$\mathbb{C}$. We set

  * •

$\mathbb{L}_{X}\mathrel{\mathop{\mathchar 58\relax}}=\mathbb{L}(X)$ the
collection of all linear operators $T\mathrel{\mathop{\mathchar 58\relax}}X\to
X$,

  * •

$\mathbb{B}_{X}\mathrel{\mathop{\mathchar 58\relax}}=\mathbb{B}(X)$ the Banach
algebra of all continuous elements of $\mathbb{L}_{X}$, given the operator
norm,

  * •

$\mathbb{A}_{X}\mathrel{\mathop{\mathchar 58\relax}}=\mathbb{A}(X)$ the group
of all invertible elements of $\mathbb{B}_{X}$ whose inverse is also in
$\mathbb{B}_{X}$, as a topological subspace of $\mathbb{B}_{X}$.

If $X$ has finite dimension $d$, then $\mathbb{L}_{X}=\mathbb{B}_{X}$ and
$\mathbb{A}_{X}\cong GL_{d}(\mathbb{K})$.

The group for the cotranslations we study in this section is the field
$\mathbb{K}$, but the formalism presented here allows generalizations of the
results to other non commutative by Lie groups.

We use Banach spaces in this section because we want $\mathbb{B}_{X}$ to have
the structure of a Banach algebra, thus we will be able to study derivatives
for groupoid morphisms in the sense of Gâteaux. However, it is easy to see
that these constructions apply just as well in other categories that admit
such sense of derivatives. For instance, an alternative is to consider the
strong topology in $\mathbb{B}_{X}$ (as usually studied on [8]) instead of the
operator norm.

It is worth noting that for groups it is well known that continuous morphisms
between Lie groups are immediately smooth [6, Problem 20-11], however, for
cotranslations (i.e. groupoid morphisms) there are differences to be
considered.

######  Notation 4.1.

Consider $\varphi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{B}_{X}$ and
$\psi\mathrel{\mathop{\mathchar 58\relax}}\mathbb{K}\to\mathbb{B}_{X}$. We set
the following notation for these limits (independently if they 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) will be useful when we want to apply derivatives to functions
obtained as composition of other functions or when we want to make an emphasis
on the variable respect to which we want to derivate, while notations i) y ii)
are meant to make emphasis on the position of the variable respect to which we
want to derivate.

######  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 $t\in\mathbb{K}$ 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 easily obtained with an analogous proof.

######  Lemma 4.3.

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 4.4.

Let $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$ be a continuous
cotranslation. Suppose that for every $t\in\mathbb{K}$ 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 4.2), 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 4.5.

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 show results which show how to deduce derivability respect to one
coordinate when we have information about the other.

######  Lemma 4.6.

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

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

∎

We would like to give a reciprocal result to Lemma 4.6, but we have not yet
been able to prove or find a counterexample for this. As a summary of the
previous lemmas we state the following:

######  Corollary 4.7.

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 $t\in\mathbb{K}$.

Ideally we would show that a cotranslation which is (separately) derivable
respect to both variables is differentiable as a two-variable function.
However, as the following lemma shows, an extra condition is needed for this.

######  Lemma 4.8.

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

$(r,t)\mapsto\partial_{2}Z(r,t)\,,$

is continuous, then $Z$ is differentiable as a two-variable function.

###### Proof.

From Lemma 4.6 we know that as for every $r\in\mathbb{K}$ the map $t\mapsto
Z(r,t)$ is derivable, then $r\mapsto Z(r,t)$ is derivable for every
$t\in\mathbb{R}$. Hence, for $h_{1},h_{2}\in\mathbb{K}$, with $h_{2}\neq 0$
and $h_{1}+h_{2}\neq 0$, we have

$\displaystyle\frac{Z(r+h_{1},t+h_{2})-Z(r,t)}{|h_{1}|+|h_{2}|}=$
$\displaystyle\frac{Z(r+h_{1}+h_{2},t)-Z(r,t)}{|h_{1}|+|h_{2}|}Z(r+h_{1},h_{2})$
$\displaystyle+Z(r,t)\frac{Z(r+h_{1},h_{2})-\mathrm{Id}}{|h_{1}|+|h_{2}|}$
$\displaystyle=$
$\displaystyle\frac{h_{1}+h_{2}}{|h_{1}|+|h_{2}|}\frac{Z(r+h_{1}+h_{2},t)-Z(r,t)}{h_{1}+h_{2}}Z(r+h_{1},h_{2})$
$\displaystyle+Z(r,t)\frac{Z(r+h_{1},h_{2})-Z(r+h_{1},0)}{h_{2}}\frac{h_{2}}{|h_{1}|+|h_{2}|}.$

All elements at the right hand side on the last equality have limit when
$(h_{1},h_{2})\to 0$, since $Z$ is derivable respect to each of its variables
and $(r,t)\mapsto\partial_{2}Z(r,t)$ is continuous. If either $h_{2}=0$ or
$h_{1}+h_{2}=0$, some terms nullify at the second equality. ∎

######  Remark 4.9.

Given the identity found on Lemma 4.5, in order to satisfy the continuity of
$(r,t)\mapsto\partial_{2}Z(r,t)$ required on the previous lemma it is enough
to verify that $r\mapsto\partial_{2}Z(r,0)$ is continuous.

In the following we study the relation of cotranslations and nonautonomous
differential equations. As we want to consider more general spaces than
$\mathbb{R}^{d}$, let us consider the following definition:

######  Definition 4.10.

A linear nonautonomous differential equation on $X$ is an equation of the form

$\frac{dx}{dt}=A(t)x(t),$

where $t\mapsto A(t)\in\mathbb{L}_{X}$, and its solutions are differentiable
maps $x\mathrel{\mathop{\mathchar 58\relax}}\mathbb{K}\to X$.

Even in the autonomous case (i.e., $t\mapsto A(t)$ is constant) the existence
of solutions of these kind of equations can be a hard problem when $X$ is an
infinite dimensional space. On the other hand, an usual generalization is to
look for solutions only partially defined, for instance with domain
$[0,\infty)\subset\mathbb{R}$. Moreover, on infinite dimensional spaces it may
be interesting to study such equations where $A$ is not globally defined, but
only on a dense subspace, in which case it can be difficult enough to find
solutions defined on a compact interval like $[0,t]$, in contrast to the
solutions defined on the whole group $\mathbb{K}$ as we propose. For more
details we refer the reader to [8, Chapter 4]).

On the other hand, the nonautonomous case presents even greater difficulties.
On finite dimension, it is well known that it is enough to ask the function
$t\mapsto A(t)$ to be locally integrable in order to guarantee the existence
and uniqueness of solutions, i.e., the existence of an evolution matrix. On
infinite dimensional spaces, the problem for the existence of globally defined
solutions 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
[8, Theorem 5.1.1].

In general, the problem of existence and uniqueness of globally defined
solutions has not been solved, but we only know specific conditions under
which we have this, as hyperbolicity or parabolicity (for more details we
refer the reader to [8, Chapter 5]). We dedicate the end of this section to
give an answer to this problem using the structure of cotranslations.

######  Proposition 4.11.

Let $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{K}\\!\times\\!\mathbb{K}\to\mathbb{A}_{X}$ a continuous
cotranslation. Suppose the map $t\mapsto Z(r,t)$ is derivable for every
$r\in\mathbb{R}$. 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 $X$ (however,
we cannot ensure in general that it is continuous). Similarly, we cannot in
general state that $u\mapsto A(u)$ is continuous (even when $\mathbb{L}_{X}$
has a topology that extends that of $\mathbb{B}_{X}$). 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 4.5 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

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

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 4.3 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 4.5, thus
proving iii). ∎

The existence of a function with the properties of $\Psi$ on the previous
proposition is exactly what is needed to describe globally defined solutions,
for every initial condition, to a linear differential equation. It is easy to
see that they are generalization of the well known transition matrices on
finite dimension. To formalize this notion we state the following definition,
adapted from [8, Definition 5.1.3].

######  Definition 4.12.

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, i.e.,
$(r,t)\to\left[\Psi(r,t)\right](x)\in X$ is continuous for every $x\in X$.

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

  * iii)

$\partial_{1}\Psi(r,t)=A(r)\Psi(r,t)$ y
$\partial_{2}\Psi(r,t)=-\Psi(r,t)A(t)$,

we say that $\Psi$ is the evolution operator associated to the differential
equation $\dot{x}=A(t)x(t)$.

On [8, Theorem 4.1.3] it is proved that a real autonomous linear differential
equation $\dot{x}=Ax$ on a Banach space has a solution defined on $[0,\infty)$
and uniquely defined for every initial condition on $X$ if and only if $A$ is
a linear operator, defined on a dense subspace of $X$ which is obtained as the
strong derivative at the origin of a semigroup morphism
$T\mathrel{\mathop{\mathchar 58\relax}}[0,\infty)\to\mathbb{B}_{X}$. Inspired
by this, we present the following definition and theorem.

######  Definition 4.13.

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 4.14.

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 4.11 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 4.5, we conclude
$A(t)=\partial_{2}Z(t,0)$. ∎

##  5\. Partial cotranslations

In this section we consider a partial version of cotranslations. We fix a
group $G$ and consider in $G\\!\times\\!G$ the left translations groupoid
structure. When we consider a topology in $G$, we assume it to be Hausdorff
and locally compact and in $G\\!\times\\!G$ we consider the product topology.
We also consider a Banach space $X$ over the field $\mathbb{K}$. Some of the
results we present in this section apply for general Banach spaces, but the
main conclusions need $X$ to be finite dimensional. Moreover, at the end of
the section we need the notion of orthonormal basis, thus $X$ will need to be
also a Hilbert space. In other words, our main results hold when we consider
the space $X=\mathbb{K}^{d}$, with its Euclidean norm. In this case we obtain
$\mathbb{B}_{X}=\mathcal{M}_{d}(\mathbb{K})$, the space of square
$d\\!\times\\!d$ matrices, and $\mathbb{A}_{X}=GL_{d}(\mathbb{K})$ the general
linear group of dimension $d$.

######  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
$\mathbb{B}_{X}^{(2)}=\mathbb{B}_{X}^{2}$).

######  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 basic fact of linear algebra we need is that for two transformations
$A,B\in\mathbb{B}_{X}$, 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.

######  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. ∎

######  Lemma 5.8.

Let $X$ be finite dimensional. If $\mathrm{P}$ is an invariant projector
associated to a (non-partial) cotranslation $Z\mathrel{\mathop{\mathchar
58\relax}}G\\!\times\\!G\to\mathbb{A}_{X}$, then
${\mathrm{rank}}\,\mathrm{P}(g)$ is a constant (for all $g\in G$). Moreover,
if $Z$ is continuous, then $g\mapsto\mathrm{P}(g)$ is continuous as well.

###### Proof.

By definition we have

$\mathrm{P}(e)Z(g,g^{-1})=Z(g,g^{-1})\mathrm{P}(g),$

thus
$\mathrm{P}(g)=Z^{\mathrm{inv}}(g,g^{-1})\mathrm{P}(e)Z(g,g^{-1})=Z(e,g)\mathrm{P}(e)Z(g,g^{-1})$,
which shows that ${\mathrm{rank}}\mathrm{P}(g)={\mathrm{rank}}\mathrm{P}(e)$
for all $g\in G$. If $Z$ is continuous, this identity also shows $\mathrm{P}$
is continuous. ∎

The previous result is not true for partial cotranslations. Consider for
instance $X=\mathbb{R}^{2}$, $G=\mathbb{Z}$, and the partial cotranslation
given by $W(m,n)=\begin{pmatrix}0&0\\\ 0&2^{n}\end{pmatrix}$. The projector
given by $\mathrm{P}(n)=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$ for even $n$
and $\mathrm{P}(n)=\mathrm{Id}$ for odd $n$, is clearly invariant, but its
rank is not constant.

Nevertheless, the projector of the units space of a partial cotranslation does
indeed have constant rank.

######  Proposition 5.9.

If $V\mathrel{\mathop{\mathchar 58\relax}}G\times G\to\mathbb{B}_{X}$ is a
partial cotranslation and $\mathrm{P}$ is an invariant associated 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 5.10.

Consider the linear nonautonomous differential equation $\dot{x}=A(t)x$ with
transition matrix $\Phi$. We know from Example 3.6 that $Z(s,t)=\Phi(t+s,s)$
defines a cotranslation with $X=\mathbb{R}^{d}$ and $G=\mathbb{R}$. Consider
now an invariant projector $\mathrm{P}$ for this cotranslation (in the sense
of Definition 5.6). By Lemma 5.8, $\mathrm{P}$ is continuous. Proposition 5.9
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 equations [11, 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 [11, Definition 2.2]
associated to the image of this projector [11, Lemma 2.1].

######  Lemma 5.11.

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 5.12.

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 5.9) 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 5.13.

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 5.9. 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. ∎

######  Definition 5.14.

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 interest for conjugate two partial cotranslations is to obtain information
of one of them from the other. Moreover, through the end of this section we
use this to show that every partial cotranslation is indeed a cotranslation
multiplied by some invariant projector.

The interest of finding continuous conjugations is that they would preserve
topological properties, while bounded conjugations would preserve asymptotic
behavior.

######  Lemma 5.15.

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 is trivial, since every $T(g)$ is invertible. The third
statement is trivial. ∎

######  Lemma 5.16.

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 5.15) are mutually
orthogonal.

###### Proof.

It is enough to see that

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

and the other composition follows similarly. ∎

From now on we fix $X=\mathbb{K}^{d}$ with the Euclidean norm. Thus, we
replace $\mathbb{B}_{X}$ with $\mathcal{M}_{d}(\mathbb{K})$, the space of
square $d\times d$ matrices with entries on the field $\mathbb{K}$, and
$\mathbb{A}_{X}$ with $GL_{d}(\mathbb{K})$ the general linear group.

######  Proposition 5.17.

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 5.15,
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 5.18.

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

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 exists 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 5.17 and 5.18.

######  Corollary 5.19.

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.

In this point we present a conjecture that so far we have not been able to
proof nor to contradict.

######  Conjecture 5.20.

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 5.17 and the known
fact that inversion of linear transformations is a continuous map. The second
statement in the following corollary follows trivially from Corollary 5.19.

######  Corollary 5.21.

Suppose Conjecture 5.20 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 5.22.

Note that the previous corollary provides, if Conjecture 5.20 is true, a non
commutative, non invertible and non differentiable generalization of S.
Siegund’s result of reducibility for nonautonomous differential equations [12,
Theorem 3.2], since we already showed (Example 5.10) that an evolution
operator multiplied by an invariant proyector defines a partial cotranslation.
The only difference is that in [12], all invariant projectors come from a
dichotomy.

Note that if two partial cotranslations are boundedly and continuously
conjugated, we are describing a generalized notion of kinematic similarity
(see for instance [12, Definition 2.1]), independently if the group involved
is commutative or has a differential structure. Remarkably, for discrete
groups, as continuity is trivial, the generalization holds.

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

######  Theorem 5.23.

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 5.20 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 5.17). Indeed:

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

and the other composition follows analogously. Choose $T$ as in Proposition
5.17. 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 5.16 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 5.11, $W+V$ 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 5.15 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 5.21. ∎

######  Corollary 5.24.

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

###### Proof.

The statement follows trivially from Theorem 5.23 and Lemma 5.12. ∎

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