# A note on the differentiability of Palmer’s topological equivalence for
discrete systems

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

A linear system of difference equations and a nonlinear perturbation are
considered. We propose sufficient conditions to ensure that the homeomorphism
of topological equivalence between them is actually a $C^{1}$ diffeomorphism.
These conditions consider that the linear part satisfies a dichotomy on the
positive half-line, while the perturbation satisfies boundness and
Lipschitzness conditions, and anothers that are tailored for our goal. The
family of dichotomies that we study consider (but are not limited to) the
exponential and nonuniform exponential dichotomies. We discuss the possibility
of extending this result to a higher class of differentiability.

###### Key words and phrases:

Nonautonomus hyperbolicity, Nonautonomous difference equation, Smooth
linearization

###### 2010 Mathematics Subject Classification:

34D09, 37C60, 37D25

This research has been partially supported by ANID, Beca Nacional de Magister
22200774.

This research has been partially supported by FONDECYT Regular 1200653

##  1\. Introduction

The concept of topological equivalence between two systems, linear and
nonlinear, has played an important role in the study of differential
equations, mainly due to the difficulty to find analytical solutions for them.
It has been more fructiferous to study a linear, simpler system and from it
obtain information about the nonlinear original system. Some facts in this
analysis are the search for bounded solutions and characterization of their
asymptotic behaviour. The method consists in establishing a continuous
correspondence between solutions of both systems.

The problem of linearization of the flow of solutions of ordinary autonomous
differential equations began in the decade of 1960 with the classical P.
Hartman’s theorem [13], which ensures the existence of a local homeomorphism
between a nonlinear system and and its linearization around a fixed point,
under the assumption that the linear system satisfies an hyperbolicity
condition. During the next decade the result was generalized to the
construction of a global homeomorphism thanks to C. Pugh [26] and A. Reinfeld
[27].

The extension of the previous result is due to result of K. J. Palmer [18],
whom in 1973 studied a couple of continuous systems given by:

(1.1) $\dot{x}=A(t)x$

(1.2) $\dot{y}=A(t)y+f(t,y).$

He used the exponential dichotomy property associated to the nonautonomous
linear system, which mimics the hyperbolocity of the autonomous framework,
namely, establish the existence of stable and unstable manifolds for
solutions. Moreover, Palmer’s result considers a family of nonlinear
perturbations for the system (1.2) that satisfy smallness and Lipschitz
properties. Palmer’s result was the first to establish the topological
equivalence (a more precise definition will be provided later) between the
systems (1.1) and (1.2). However, most often a exponential dichotomy
represents highly restrictive hypothesis, which is why during the next decade
[17], [19] the same author generalized his result to consider a wider family
of dichotomies and perturbations.

Contemporaneously to K. J. Palmer, it was already being developed the
analogous theory in the discrete framework (we refer the reader to C. Coffman
[8]). These contributions were retaken in lately 80’s and early 90’s, when G.
Papaschinopoulos’s work ([20], [21], [22], [23]) was very prominent and helped
to characterize dichotomies and asymptotic behavior of solutions.

In the decade of 1990 many authors came back to study K. J. Palmer’s problem
in its continuous version, admitting wider hypothesis. Among them we recall
the contributions of Z. Lin and Y-X. Lin [15], L. Jiang [14] and J.L. Shi and
K.Q. Xiong [28].

In relation to new dichotomies, M. Pinto and R. Naulin ([24], [16]) stand out
for introducing the concept of $(h,k)-$dichotomies. We also highlight L.
Barreira and C. Valls ([1], [2], [3]), whom in early 2000’s extended the
family of dichotomies, introducing the concept of nonuniform dichotomy. During
the last decade these concepts have been used by many authors in both the
discrete [9], and continuous framework [4], even in integrated contexts as
impulsive equations [12].

Since 2015, authors Á. Castañeda and G. Robledo [7] have studied conclusions
that can be obtained when studying the differentiability of this
homeomorphism, particulary when restricting the problem to the half real line.

Other authors such as D. Dragi$\check{c}$ević et al. [11, 10] have also been
interested in these properties of differentiability in a continuous and
discrete context respectively . Recently, faced with the problem in its
discrete context, the authors on [5] have studied the differentiability of
said homeomorphism under the assumption that the linear system satisfies a
nonuniform contraction.

Novelty of this work. Our work follows the results initiated by Á. Castañeda
and G. Robledo in [7], and is heavily based on their continuation along with
P. González in [5], since we use some of their results and strategies to find
a K.J. Palmer’s homeomorphism between solutions of (2.1) and (2.2), and our
results of topological equivalence are over $J=\mathbb{Z}^{+}$, instead of
$\mathbb{Z}$. In the section of preliminaries we give precise definitions for
these concepts and make explicit which tools and results we are considering
from [5]. We also present the set of conditions under which we will work and
make a brief discussion about their appearances in literature. The main
difference between our work and [5], in terms of differentiability, is that we
allow the existence of nonempty unstable manifolds for the linear system;
furthermore, we do not use spectral theory in order to obtain
$C^{1}-$differentiability such as it is employ in [10].

In the third section, we propose conditions that allows us to prove that K.J.
Palmer’s homeomorphism, which we previously studied, is a
$C^{1}-$diffeomorphism. We give examples in order to achieve these conditions,
including systems that satisfy exponential and nonuniform exponential
dichotomies.

In the fourth section we study a generalization of the previous conditions in
order to obtain higher order derivatives. Finally, in the last section, we
propose an algebraic set of functions and an operator that can be inductively
applied in order to find conditions for arbitrary high order derivatives.

##  2\. Preliminaries

In this work, we study the following nonautonomous discrete systems:

(2.1) $x(k+1)=A(k)x(k)$ (2.2) $y(k+1)=A(k)y(k)+f(k,y(k)),$

where $A\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\to\mathcal{M}_{d\times d}(\mathbb{R})$ and
$f\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$, satisfying
properties that will be given later. Their solutions in each case is a
function $x,y\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\to\mathbb{R}^{d}$, and we denote $n\mapsto
x(n,k,\xi)$ and $n\to y(n,k,\eta)$ for the solutions of (2.1) and (2.2) that
pass through $\xi$ and $\eta$ respectively on $n=k$.

Furthermore, we will consider the following properties:

  * (d0)

$A\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Z}^{+}\to M_{d\times
d}(\mathbb{R})$ is non singular (i.e. has invertible images) and uniformly
bounded, that is to say, there exists $M>1$ such that:

$\max\left\\{\sup_{k\in\mathbb{Z}^{+}}\mathinner{\\!\left\lVert
A(k)\right\rVert},\sup_{k\in\mathbb{Z}^{+}}\mathinner{\\!\left\lVert
A^{-1}(k)\right\rVert}\right\\}=M.$

  * (d1)

The system (2.1) has a non uniform dichotomy, i.e. there are two invariant
complementary projectors $P(\cdot)$ and $Q(\cdot)$ such that $P(n)+Q(n)=I$ for
every $n\in\mathbb{Z}^{+}$, a non-negative sequence $D$ and a monotone
decreasing sequence convergent to zero $h$ with $h(0)=1$ such that:

$\left\\{\begin{array}[]{lc}\mathinner{\\!\left\lVert\Phi(k,n)P(n)\right\rVert}\leq
D(n)\left({\frac{h(k)}{h(n)}}\right),&\forall k\geq n\geq 0\\\ \\\
\mathinner{\\!\left\lVert\Phi(k,n)Q(n)\right\rVert}\leq
D(n)\left({\frac{h(n)}{h(k)}}\right),&\forall 0\leq k\leq
n,\end{array}\right.$

where $\Phi(k,n)$ is the transition matrix for (2.1).

  * (d2)

There exist $\mu,\gamma\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\to\mathbb{R}^{+}$ sequences such that for every
$k\in\mathbb{Z}^{+}$ and every pair
$(y,\tilde{y})\in\mathbb{R}^{d}\times\mathbb{R}^{d}$ we have:

$|f(k,y)-f(k,\tilde{y})|\leq\gamma(k)|y-\tilde{y}|\text{ ;
}|f(k,y)|\leq\mu(k).$

  * (d3)

The sequence $\mu$ defined above verifies:

$\sum_{j=0}^{\infty}\mathinner{\\!\left\lVert\mathcal{G}(k,j+1)\right\rVert}\mu(j)\leq
p<\infty\text{ for every }k\in\mathbb{Z}^{+},$

where $\mathcal{G}$ is the Green’s operator asociated to the dichotomy
established in (d1).

  * (d4)

The sequence $\gamma$ defined above verifies:

$\sum_{j=0}^{\infty}\mathinner{\\!\left\lVert\mathcal{G}(k,j+1)\right\rVert}\gamma(j)\leq
q<1\text{ for every }k\in\mathbb{Z}^{+}.$

  * (d5)

The sequence $\gamma$ and the matrix function $A$ verify:

$\mathinner{\\!\left\lVert A^{-1}(\ell)\gamma(\ell)\right\rVert}<1\text{ for
every }\ell\in\mathbb{Z}^{+}.$

  * (d6)

The map $u\mapsto f(k,u)$ and its derivates respect to $u$ up to the order $r$
($r\geq 1$) are continuous functions of
$(k,u)\in\mathbb{Z}^{+}\times\mathbb{R}^{d}$ and
$\sup_{u\in\mathbb{R}^{d}}\mathinner{\\!\left\lVert\frac{\partial f}{\partial
u}(k,u)\right\rVert}<+\infty$ is bounded.

  * (d7)

For each fixed $m\in\mathbb{Z}^{+}$, the sequences $D$, $h$ and $\gamma$
satisfy:

${\sum_{j=m}^{\infty}}\left(D(j+1)h(j+1)\gamma(j)\left[{\prod_{p=m}^{j-1}}\mathinner{\\!\left\lVert
A(p)\right\rVert}+\gamma(p)\right]\right)<+\infty.$

######  Remark 2.1.

The transition matrix for (2.1) is the matrix function
$\Phi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\times\mathbb{Z}^{+}\to\mathcal{M}_{d\times
d}(\mathbb{R})$ given by:

$\Phi(k,n)=\left\\{\begin{array}[]{lcc}A(k-1)A(k-2)\cdots A(n)&\text{if
}&k>n\\\ \\\ I&\text{ if }&k=n\\\ \\\ A^{-1}(k)A^{-1}(k+1)\cdots
A^{-1}(n-1)&\text{if }&k<n.\end{array}\right.$

######  Remark 2.2.

Projectors $P(\cdot)$ and $Q(\cdot)$ have been called invariant, which means
that they are under the assumption that for every $k,n\in\mathbb{Z}^{+}$ they
satisfy:

$P(k)\Phi(k,n)=\Phi(k,n)P(n)\text{ and }Q(k)\Phi(k,n)=\Phi(k,n)Q(n).$

######  Remark 2.3.

Green’s operator for the system (2.1) asociated to the dichotomy (d1) is the
matrix function $\mathcal{G}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\times\mathbb{Z}^{+}\to\mathcal{M}_{d\times
d}(\mathbb{R})$ given by:

$\mathcal{G}(k,n)=\left\\{\begin{array}[]{cc}\Phi(k,n)P(n)&\forall k\geq n\geq
0\\\ \\\ -\Phi(k,n)Q(n)&\forall 0\leq k<n.\end{array}\right.$

And it is easily deduced that it satisfies:

$\mathcal{G}(k+1,n)=A(k)\mathcal{G}(k,n)\text{ and
}\mathcal{G}(n,n)=I+A(n-1)\mathcal{G}(n-1,n).$

######  Remark 2.4.

Condition (d5) was used in [5] in order to ensure that solutions for the
system (2.2) can be uniquely backwards continued.

However, we have noted that some authors ([10], for example) assume that this
backwards continuation can be uniquely found even without this condition.

Now, we recall the concept of topological equivalence for discrete systems.

######  Definition 2.5.

Let $J\subset\mathbb{Z}$ be a interval of integer numbers. The systems (2.1)
and (2.2) are $J$-topologically equivalent if there is a function
$H\mathrel{\mathop{\mathchar 58\relax}}J\times\mathbb{R}^{d}\to\mathbb{R}^{d}$
that satisfies:

  * i)

If $x(k)$ is a solution of (2.1), then $H[k,x(k)]$ is a solution of (2.2).

  * ii)

$H(k,u)-u$ is bounded over $J\times\mathbb{R}^{d}$.

  * iii)

For each fixed $k\in J$, the map $u\mapsto H(k,u)$ is an homeomorphism of
$\mathbb{R}^{d}$.

Moreover, the function $u\mapsto G(k,u)=H^{-1}(k,u)$ satisfies ii) and iii)
and maps solutions of (2.2) into solutions of (2.1).

However, as it was explained before, it turns out to be interesting to study
which properties can be deduced when studying the differentiability of such
homeomorphism. In the discrete case the characterization is even better than
in the continuous framework, because as a difference equation may be
understood as a dynamical system where a discrete group acts on the space
$\mathbb{R}^{d}$, the space can be split on fibers, and so as soon as we find
a diffeomorphism associated to one of such fibers, the other can be replicated
inductively. This idea motivates the following definition:

######  Definition 2.6.

The systems (2.1) and (2.2) are $C^{r}$-topologically equivalent on
$\mathbb{Z}^{+}$ if they are topologically equivalent on $\mathbb{Z}^{+}$ and
$u\mapsto H(k,u)$ is a diffeomorphism of class $C^{r}$, with $r\geq 1$ for
each fixed $k\geq 0$.

Most of the conditions we have stated have been used several times in the
literature. We now highlight some of the results that inspire this work.

######  Theorem 2.7.

[5, Theorem 2.1] If the conditions (d0)-(d5) hold, then the systems (2.1) and
(2.2) are $\mathbb{Z}^{+}-$topologically equivalent.

In the proof of 2.7, some calculation were developed and we highlight some of
them in order to facilitate read of this article. Let us define

$\displaystyle w^{*}(k;(m,\eta))$ $\displaystyle=$
$\displaystyle-\sum_{j=0}^{\infty}\mathcal{G}(k,j+1)f(j,y(j,m,\eta))$
$\displaystyle=$
$\displaystyle-\sum_{j=0}^{k-1}\Phi(k,j+1)P(j+1)f(j,y(j,m,\eta))$
$\displaystyle+\sum_{j=k}^{\infty}\Phi(k,j+1)Q(j+1)f(j,y(j,m,\eta))$

On the other hand, for each $(m,\xi)\in\mathbb{Z}^{+}\times\mathbb{R}^{d}$ we
define the map $\Theta\mathrel{\mathop{\mathchar
58\relax}}\ell^{\infty}(\mathbb{Z}^{+},\mathbb{R}^{d})\to\ell^{\infty}(\mathbb{Z}^{+},\mathbb{R}^{d})$
given by:

$(\Theta\phi)(k;(m,\xi))=\sum_{j=0}^{\infty}\mathcal{G}(k,j+1)f(j,x(j,m,\xi)+\phi(j;(m,\xi)))$

Using Banach’s fixed point Theorem we conclude the existence of a unique fixed
point

(2.4)
$z^{*}(k;(m,\xi)))=\sum_{j=0}^{+\infty}\mathcal{G}(k,j+1)f(j,x(j,m,\xi)+z^{*}(j;(m,\xi))).$

It is easily verified that this map is a solution for the initial values
problem:

(2.5) $\left\\{\begin{array}[]{ccl}z(k+1)&=&A(k)z(k)+f(k,x(k,m,\eta)+z(k))\\\
\\\
z(0)&=&-{\sum}_{j=0}^{\infty}\Phi(0,j+1)Q(j+1)f(j,x(j,m,\eta)+z^{*}(j;(m,\xi))).\end{array}\right.$

By unicity of solutions we have

(2.6) $x(k,m,\xi)=x(k,p,x(p,m,\xi))\text{ , for every
}k,p,m\in\mathbb{Z}^{+}.$

Analogously, we can verify

(2.7) $z^{*}(k;(m,\xi))=z^{*}(k;(p,x(p,m,\xi)))\text{ , for every
}k,p,m\in\mathbb{Z}^{+}.$

For every fixed $k\in\mathbb{Z}^{+}$ we construct the maps
$H(k,\cdot)\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$G(k,\cdot)\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\to\mathbb{R}^{d}$ given by

(2.8)
$\left\\{\begin{array}[]{ccl}H(k,\xi)&=&\xi+{\sum}_{j=0}^{+\infty}\mathcal{G}(k,j+1)f(j,x(j,k,\xi)+z^{*}(j;(k,\xi)))\\\
\\\ &=&\xi+z^{*}(k;(k,\xi))\end{array}\right.$

and

(2.9)
$\left\\{\begin{array}[]{ccl}G(k,\eta)&=&\eta-{\sum}_{j=0}^{+\infty}\mathcal{G}(k,j+1)f(j,y(j,k,\eta))\\\
\\\ &=&\eta+w^{*}(k;(k,\eta)).\end{array}\right.$

It has been proved that $H$ and $G=H^{-1}$ define the homeomorphism that shows
the topological equivalence between (2.1) and (2.2).

In order to study additional properties of $G$, note that for $n<k$ we have:

(2.10)
$y(n,k,\eta)=\Phi(n,k)\eta-\sum_{j=n}^{k-1}\Phi(n,j+1)f(j,y(j,k,\eta)).$

Which implies that

$\displaystyle\Phi(k,n)y(n,k,\eta)$ $\displaystyle=$
$\displaystyle\eta-\sum_{j=n}^{k-1}\Phi(k,j+1)f(j,y(j,k,\eta))$
$\displaystyle=$
$\displaystyle\eta-\sum_{j=n}^{k-1}\Phi(k,j+1)P(j+1)f(j,y(j,k,\eta))$
$\displaystyle-\sum_{j=n}^{k-1}\Phi(k,j+1)Q(j+1)f(j,y(j,k,\eta)).$

In particular, for $n=0$ we have

$\displaystyle\Phi(k,0)y(0,k,\eta)$ $\displaystyle=$
$\displaystyle\eta-\sum_{j=0}^{k-1}\Phi(k,j+1)P(j+1)f(j,y(j,k,\eta))$
$\displaystyle-\sum_{j=0}^{k-1}\Phi(k,j+1)Q(j+1)f(j,y(j,k,\eta))$
$\displaystyle=$
$\displaystyle\eta-\sum_{j=0}^{k-1}\Phi(k,j+1)P(j+1)f(j,y(j,k,\eta))$
$\displaystyle+\sum_{j=k}^{\infty}\Phi(k,j+1)Q(j+1)f(j,y(j,k,\eta))$
$\displaystyle-\sum_{j=0}^{\infty}\Phi(k,j+1)Q(j+1)f(j,y(j,k,\eta))$
$\displaystyle=$
$\displaystyle\eta-\sum_{j=0}^{\infty}\mathcal{G}(k,j+1)f(j,y(j,k,\eta))-\sum_{j=0}^{\infty}\Phi(k,j+1)Q(j+1)f(j,y(j,k,\eta))$
$\displaystyle=$ $\displaystyle
G(k,\eta)-\Phi(k,0)\sum_{j=0}^{\infty}\Phi(0,j+1)Q(j+1)f(j,y(j,k,\eta)).$

Hence, using the definition of $k\mapsto w^{*}(0;(k,\eta))$ we conclude

$\displaystyle G(k,\eta)$ $\displaystyle=$
$\displaystyle\Phi(k,0)\left\\{y(0,k,\eta)+\sum_{j=0}^{\infty}\Phi(0,j+1)Q(j+1)f(j,y(j,k,\eta))\right\\}$
$\displaystyle=$
$\displaystyle\Phi(k,0)\left\\{y(0,k,\eta)+w^{*}(0;(k,\eta))\right\\}.$

Nevertheless, in order to prove the $C^{r}-$topologically equivalence, those
authors had to impose stronger conditions than before, having to remain to the
stable manifold, i.e., supposing the system (2.1) admits a nonuniform
contraction. Thus, condition (d1) is called as:

  * (D1)

The system (2.1) has a nonuniform dichotomy with trivial projector, i.e. there
exist a nonnegative sequence $D$ and a monotone decreasing sequence convergent
to zero $h$, with $h(0)=1$, such that

$\mathinner{\\!\left\lVert\Phi(k,n)\right\rVert}\leq
D(n)\frac{h(k)}{h(n)}\text{ , }\forall k\geq n\geq 0.$

Furthermore, (d6) is also considered, hence obtaining the next result:

######  Theorem 2.8.

[5, Lemma 2.3] If the conditions (d0), (D1) and (d2)-(d6) hold, then the
systems (2.1) and (2.2) are $C^{r}-$topologically equivalent on
$\mathbb{Z}^{+}$.

######  Remark 2.9.

Note that when replacing (d1) with (D1), the form of Green’s operator is
simpler,and hence conditions (d3) and (d4) are also considerably easy to
handle. Also, in this case, the map $k\mapsto w^{*}(0;(k,\eta))$ nullifies,
which allowed the authors to prove more easily the differentiability of
$u\mapsto G(k,u)$.

######  Remark 2.10.

Note that Theorem 2.8 and the expression (2), implies that under conditions
(d0), (d1)-(d6), the map $\eta\mapsto y(0,k,\eta)$ is $C^{r}$ for every
$k\in\mathbb{Z}^{+}$.

##  3\. Diffeomorphism of discrete topological equivalence under a dichotomy

In this section, we study the differentiability properties of the function
$\eta\mapsto w^{*}(0;(m,\eta)),$ in order to obtain that the topological
equivalence is of class $C^{1}.$

######  Lemma 3.1.

If conditions (d0)-(d7) hold, then $\eta\mapsto w^{*}(0;(m,\eta))$ is a
$C^{1}$ map.

###### Proof.

Fix $\eta\in\mathbb{R}^{d}$ and let
$(\delta_{n})_{n\in\mathbb{N}}\subset\mathbb{R}^{d}$ be a properly convergent
to zero sequence. Choose and fix $m\in\mathbb{Z}^{+}$, we define

$\psi_{n}(j)=\mathcal{G}(0,j+1)\frac{f(j,y(j,m,\eta+\delta_{n}))-f(j,y(j,m,\eta))-\frac{\partial
f}{\partial u}(j,y(j,m,\eta))\frac{\partial
y}{\partial\eta}(j,m,\eta)\delta_{n}}{|\delta_{n}|},$

note that as $\eta\mapsto y(j,m,\eta)$ is continuous, then
$\lim_{n\to\infty}y(j,m,\eta+\delta_{n})=y(j,m,\eta)$. Applying (d6), we have:

$\lim_{n\to\infty}\psi_{n}(j)=0.$

In the proof of Theorem 2.7, in [7], authors define for $j<m$:

$\mathcal{C}_{m}(j)={\prod_{i=j}^{m-1}}\frac{\mathinner{\\!\left\lVert
A^{-1}(i)\right\rVert}}{1-\mathinner{\\!\left\lVert
A^{-1}(i)\gamma(i)\right\rVert}}.$

Thus, for $j<m,$ it is proved that

$|y(j,m,\eta)-y(j,m,\tilde{\eta})|\leq\mathcal{C}_{m}(j)|\eta-\tilde{\eta}|.$

On the other hand, we know

$\displaystyle y(m+1,m,\eta)=A(m)\eta+f(m,\eta),$

hence

$\displaystyle
y(m+1,m,\eta)-y(m+1,m,\tilde{\eta})=A(m)(\eta-\tilde{\eta})+f(m,\eta)-f(m,\tilde{\eta}).$

So, we conclude

$\displaystyle|y(m+1,m,\eta)-y(m+1,m,\tilde{\eta})|$ $\displaystyle\leq$
$\displaystyle\mathinner{\\!\left\lVert
A(m)\right\rVert}|\eta-\tilde{\eta}|+|f(m,\eta)-f(m,\tilde{\eta})|$
$\displaystyle\leq$ $\displaystyle\left(\mathinner{\\!\left\lVert
A(m)\right\rVert}+\gamma(m)\right)|\eta-\tilde{\eta}|.$

For $j>m$, we define

(3.2) $\mathcal{B}_{m}(j)\mathrel{\mathop{\mathchar
58\relax}}={\prod_{i=m}^{j-1}}\mathinner{\\!\left\lVert
A(i)\right\rVert}+\gamma(i),$

which, by (3), allows us to write for $j\geq m$

$|y(j,m,\eta)-y(j,m,\tilde{\eta})|\leq\mathcal{B}_{m}(j)|\eta-\tilde{\eta}|.$

Now, we define

$\mathcal{A}_{m}(j)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\begin{array}[]{lc}\mathcal{C}_{m}(j)&j<m\\\ \\\ 1&j=m\\\
\\\ \mathcal{B}_{m}(j)&j>m,\end{array}\right.$

hence, for every $j\in\mathbb{Z}^{+}$

$|y(j,m,\eta)-y(j,m,\tilde{\eta})|\leq\mathcal{A}_{m}(j)|\eta-\tilde{\eta}|.$

So, by continuity of the norm

$\displaystyle\mathinner{\\!\left\lVert\frac{\partial
y}{\partial\eta}(j,m,\eta)\right\rVert}$ $\displaystyle=$
$\displaystyle\lim_{\delta\to
0}\frac{|y(j,m,\eta+\delta)-y(j,m,\eta)|}{|\delta|}$ $\displaystyle\leq$
$\displaystyle\lim_{\delta\to 0}\mathcal{A}_{m}(j)=\mathcal{A}_{m}(j).$

Similarly

$\displaystyle\mathinner{\\!\left\lVert\frac{\partial f}{\partial
u}(j,u)\right\rVert}$ $\displaystyle=$ $\displaystyle\lim_{\delta\to
0}\frac{|f(j,u+\delta)-f(j,u)|}{|\delta|}$ $\displaystyle\leq$
$\displaystyle\lim_{\delta\to 0}\gamma(j)=\gamma(j).$

Hence

$\displaystyle|\psi_{n}(j)|$ $\displaystyle\leq$
$\displaystyle\mathinner{\\!\left\lVert\mathcal{G}(0,j+1)\right\rVert}\frac{|f(j,y(j,m,\eta+\delta_{n}))-f(j,y(j,m,\eta))|+\left|\frac{\partial
f}{\partial u}(j,y(j,m,\eta))\frac{\partial
y}{\partial\eta}(j,m,\eta)\delta_{n}\right|}{|\delta_{n}|}$
$\displaystyle\leq$
$\displaystyle\mathinner{\\!\left\lVert\mathcal{G}(0,j+1)\right\rVert}\frac{\gamma(j)|y(j,m,\eta+\delta_{n})-y(j,m,\eta)|+\gamma(j)\mathinner{\\!\left\lVert\frac{\partial
y}{\partial\eta}(j,m,\eta)\right\rVert}|\delta_{n}|}{|\delta_{n}|}$
$\displaystyle\leq$
$\displaystyle\mathinner{\\!\left\lVert\mathcal{G}(0,j+1)\right\rVert}\gamma(j)\left(\frac{|y(j,m,\eta+\delta_{n})-y(j,m,\eta)|}{|\delta_{n}|}+\mathinner{\\!\left\lVert\frac{\partial
y}{\partial\eta}(j,m,\eta)\right\rVert}\right)$ $\displaystyle\leq$
$\displaystyle
2\mathinner{\\!\left\lVert\mathcal{G}(0,j+1)\right\rVert}\gamma(j)\mathcal{A}_{m}(j).$

On the other hand, by using (d4) and (d7) we have that

$\displaystyle\sum_{j=0}^{\infty}\mathinner{\\!\left\lVert\mathcal{G}(0,j+1)\right\rVert}\gamma(j)\mathcal{A}_{m}(j)$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{m-1}\mathinner{\\!\left\lVert\mathcal{G}(0,j+1)\right\rVert}\gamma(j)\mathcal{C}_{m}(j)+\sum_{j=m}^{\infty}\mathinner{\\!\left\lVert\mathcal{G}(0,j+1)\right\rVert}\gamma(j)\mathcal{B}_{m}(j)$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{m-1}\mathinner{\\!\left\lVert\mathcal{G}(0,j+1)\right\rVert}\gamma(j)\max_{i<m}\mathcal{C}_{m}(i)+\sum_{j=m}^{\infty}D(j+1)h(j+1)\gamma(j)\mathcal{B}_{m}(j)$
$\displaystyle\leq$ $\displaystyle
q\max_{i<m}\mathcal{C}_{m}(i)+\sum_{j=m}^{\infty}D(j+1)h(j+1)\gamma(j)\mathcal{B}_{m}(j)<+\infty.$

So finally, using Lebesgue’s dominated convergence theorem we obtain

$\begin{array}[]{ccc}&\displaystyle\lim_{n\to\infty}\frac{w^{*}(0;(m,\eta+\delta_{n}))-w^{*}(0;(m,\eta))+\left[\sum_{j=0}^{\infty}\mathcal{G}(0,j+1)\frac{\partial
f}{\partial u}(j,y(j,m,\eta))\frac{\partial
y}{\partial\eta}(j,m,\eta)\right]\delta_{n}}{|\delta_{n}|}&\\\ \\\
&=-\lim_{n\to\infty}\sum_{j=0}^{\infty}\psi_{n}(j)&\\\ \\\
&=-\sum_{j=0}^{\infty}\left(\lim_{n\to\infty}\psi_{n}(j)\right)=0,&\end{array}$

which implies $\eta\mapsto w^{*}(0;(m,\eta))$ is differentiable. ∎

######  Theorem 3.2.

If conditions (d0)-(d7) hold, and (d6) is satisfied with $r=1$, then (2.1) and
(2.2) are $C^{1}$-topologically equivalent on $\mathbb{Z}^{+}$.

###### Proof.

By Theorem 2.7 we know they are topologically equivalent. By Lemma 3.1
$\eta\mapsto w^{*}(0;(k,\eta))$ is $C^{1}$ differentiable for every
$k\in\mathbb{Z}^{+}$, and, as stated in Remark 2.10, $\eta\mapsto y(0,k,\eta)$
is as well. By using (2) allows us to conclude $\eta\mapsto G(k,\eta)$ is
$C^{1}$ differentiable.

Furthermore, as $G$ is a topological equivalence, we know $\xi\mapsto
G(k,\xi)-\xi$ is bounded and so $G(k,\xi)\to\infty$ when $|\xi|\to\infty$.
This, combined with the previous expression, implies by Corollary 2.1 of [25],
that $\xi\mapsto G(k,\xi)$ is a diffeomorphism. Moreover, as
$G(k,H(k,\xi))=\xi$, we have

$\frac{\partial G}{\partial\xi}(k,H(k,\xi))\frac{\partial
H}{\partial\xi}(k,\xi)=I,$

and so $\frac{\partial H}{\partial\xi}(k,\xi)=\left[\frac{\partial
G}{\partial\xi}(k,H(k,\xi))\right]^{-1}$, which completes the proof. ∎

######  Corollary 3.3.

Suppose systems (2.1) and (2.2) satisfy (d0) and (d6) with $r=1$. Moreover,
suppose (d1) is fulfilled with a classical exponential dichotomy, which means,
$P$ and $Q$ are complementary constant projectors, $D(n)=D>0$ for every
$n\in\mathbb{Z}^{+}$ and $h(n)=\theta^{n}$, with $\theta\in(0,1)$, and (d2) is
fulfilled with $\gamma(n)=\gamma>0$ and $\mu(n)=\mu>0$ for every
$n\in\mathbb{Z}^{+}$. Then, if $M\gamma<1$, $\frac{D\gamma}{1-\theta}<1$ and
$\theta(M+\gamma)<1$, the systems (2.1) and (2.2) are $C^{1}$-topologically
equivalent on $\mathbb{Z}^{+}$.

###### Proof.

Note that for an arbitrary $n\in\mathbb{Z}^{+}$ we have:

$\mathinner{\\!\left\lVert A^{-1}(n)\gamma\right\rVert}\leq M\gamma<1,$

hence (d5) is satisfied. On the other hand, for an arbitrary
$k\in\mathbb{Z}^{+}$ we have

$\displaystyle\sum_{j=0}^{\infty}\mathinner{\\!\left\lVert\mathcal{G}(k,j+1)\right\rVert}\mu$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{k-1}\mathinner{\\!\left\lVert\mathcal{G}(k,j+1)\right\rVert}\mu+\sum_{j=k}^{\infty}\mathinner{\\!\left\lVert\mathcal{G}(k,j+1)\right\rVert}\mu$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{k-1}\mathinner{\\!\left\lVert\Phi(k,j+1)P\right\rVert}\mu+\sum_{j=k}^{\infty}\mathinner{\\!\left\lVert\Phi(k,j+1)Q\right\rVert}\mu$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{k-1}\frac{\theta^{k}}{\theta^{j+1}}D\mu+\sum_{j=k}^{\infty}\frac{\theta^{j+1}}{\theta^{k}}D\mu$
$\displaystyle=$
$\displaystyle\theta^{k-1}D\mu\frac{1-\left(\frac{1}{\theta}\right)^{k}}{1-\frac{1}{\theta}}+\theta
D\mu\frac{1}{1-\theta}$ $\displaystyle=$ $\displaystyle
D\mu\frac{1+\theta-\theta^{k}}{1-\theta}\leq\frac{D\mu}{1-\theta}.$

Thus (d3) is satisfied. Analogously

$\sum_{j=0}^{\infty}\mathinner{\\!\left\lVert\mathcal{G}(k,j+1)\right\rVert}\gamma\leq\frac{D\gamma}{1-\theta}<1.$

Hence (d4) is satisfied. Finally, for an arbitrary $m\in\mathbb{Z}^{+}$ we
have

$\displaystyle{\sum_{j=m}^{\infty}}D\gamma\left(h(j+1)\left[{\prod_{p=m}^{j-1}}\mathinner{\\!\left\lVert
A(p)\right\rVert}+\gamma\right]\right)$ $\displaystyle\leq$
$\displaystyle{\sum_{j=m}^{\infty}}D\gamma\left(\theta^{j+1}\left[{\prod_{p=m}^{j-1}}M+\gamma\right]\right)$
$\displaystyle\leq$ $\displaystyle
D\gamma{\sum_{j=m}^{\infty}}\theta^{j+1}(M+\gamma)^{j-m}$ $\displaystyle\leq$
$\displaystyle\frac{D\gamma\theta}{(M+\gamma)^{m}}{\sum_{j=m}^{\infty}}\left(\theta(M+\gamma)\right)^{j}<\infty,$

hence (d7) is satisfied. So, by applying Theorem 3.2, the systems are
$C^{1}$-topologically equivalent on $\mathbb{Z}^{+}$. ∎

######  Remark 3.4.

The conditions imposed in corollary 3.3 are easily reachable with $\theta$
being small enough. For example, set $\theta=0.1$, $M=2$ and $\gamma=0.4$.

######  Corollary 3.5.

Suppose conditions (d0) and (d5) are satisfied. Also, consider system (2.1)
admits a nonuniform exponential dichotomy, which means there are two
complementary invariant projectors $P(\cdot)$ and $Q(\cdot)$ and constants
$C,\lambda,\varepsilon>0$ such that:

$\left\\{\begin{array}[]{lc}\mathinner{\\!\left\lVert\Phi(k,n)P(n)\right\rVert}\leq
Ce^{-\lambda(k-n)+\varepsilon n},&\forall k\geq n\geq 0\\\ \\\
\mathinner{\\!\left\lVert\Phi(k,n)Q(n)\right\rVert}\leq
Ce^{\lambda(k-n)+\varepsilon n},&\forall 0\leq k\leq n.\end{array}\right.$

Furthermore, suppose that for every $k\in\mathbb{Z}^{+}$ $u\mapsto f(k,u)$ is
a $C^{1}$ function such that $u\mapsto\frac{\partial f}{\partial u}(k,u)$ is a
bounded map that satisfies:

(3.3) $\mathinner{\\!\left\lVert\frac{\partial f}{\partial
u}(k,u)\right\rVert}\leq\nu e^{-\varepsilon(k+1)}$

and

(3.4) $|f(k,u)|\leq\kappa e^{-\tau(k+1)}$

for given $\nu,\kappa>0$ and $\tau>\varepsilon-\lambda$. Then, if
$Me^{-\lambda}<1$, for sufficiently small $\nu>0$, systems (2.1) and (2.2) are
$C^{1}$-topologically equivalent on $\mathbb{Z}^{+}.$

###### Proof.

Condition (d1) is easily verified with $D(n)=Ce^{\varepsilon n}$ and
$h(n)=e^{-\lambda n}$. Note that using generalized intermediate point Theorem,
condition (3.3) implies:

$|f(k,y)-f(k,\tilde{y})|\leq\nu e^{-\varepsilon(k+1)}|y-\tilde{y}|,$

so condition (d2) is verified with $\gamma(k)=\nu e^{-\varepsilon(k+1)}$ and
$\mu(k)=\kappa e^{-\tau(k+1)}$. Hence condition (d3) is immediately satisfied
and, provided $\nu$ is sufficiently small, so is (d4) in a similar fashion as
in the previous Corollary. Condition (d6) is immediately satisfied by
hypothesis, with $r=1$. Now, denote

$\Psi_{m}(j)={\prod_{p=m}^{j-1}}\mathinner{\\!\left\lVert
A(p)\right\rVert}+\gamma(p).$

It is easy to see $m\geq n$ implies $\Psi_{m}(j)\leq\Psi_{n}(j)$, hence for a
fixed $m\in\mathbb{Z}^{+}$:

$\Psi_{m}(j)\leq\Psi_{0}(j)\leq\left(M+\nu e^{-\varepsilon}\right)^{j}$

So we have

$\displaystyle\sum_{j=m}^{\infty}D(j+1)h(j+1)\gamma(j)\Psi_{m}(j)$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{\infty}D(j+1)h(j+1)\gamma(j)\Psi_{0}(j)$
$\displaystyle\leq$ $\displaystyle C\nu
e^{-\lambda}\sum_{j=0}^{\infty}\left[e^{-\lambda}\left(M+\nu
e^{-\varepsilon}\right)\right]^{j},$

which, as $Me^{-\lambda}<1$, is finite provided $\nu$ is sufficiently small,
so condition (d7) is satisfied. Applying Theorem 3.2, the corollary is proved.
∎

######  Remark 3.6.

Note that for condition (3.3) to be satisfied, we only have $\tau$ to be
bigger than $\varepsilon-\lambda$, regardless of the sign of such $\tau$. This
could mean that, in the case $\lambda>\varepsilon$, this dichotomy even admits
a non bounded perturbation with exponential growth, it just has to have a
increasing ratio smaller than the decay ratio of the dichotomy.

######  Remark 3.7.

Corollary 3.5 has been inspired by [10, Theorem 2]. Our approach and that
result consider conditions (d0) and (3.3), the same dichotomy and both results
work for $\nu$ sufficiently small. Furthermore, we imposed the extra
hypothesis $Me^{-\lambda}<1$ and conditions (3.4) and (d5). However, as stated
on Remark 2.4, we think this last hypothesis may be dropped.

Moreover, our result gives $C^{1}$-topologically equivalence on
$\mathbb{Z}^{+}$, in contrast to the mentioned theorem, which gives it on the
whole $\mathbb{Z}$. Nevertheless, our approach does not rely on spectral
properties of the dichotomy, which is the core of that result.

######  Example 3.8.

Let
$(a_{n})_{n\in\mathbb{N}},(b_{n})_{n\in\mathbb{N}},(c_{n})_{n\in\mathbb{N}}$
three sequences such that

(3.5) $0<\alpha\leq a_{n},b_{n}\leq c_{n}^{-1}\leq 1\leq c_{n}\leq M,$

for given $M,\alpha>0$, and let $(c_{n})_{n\in\mathbb{N}}$ be monotone
increasing. Consider $A(n)\in\mathcal{M}_{3\times 3}(\mathbb{R})$ the diagonal
matrix given by

(3.6) $A(n)=\begin{pmatrix}a_{n}&0&0\\\ 0&b_{n}&0\\\ 0&0&c_{n}\end{pmatrix},$

and consider the system (2.1) associated to this sequence of matrices. Let
$(r_{n})_{n\in\mathbb{N}}$ be a sumable non negative sequence and define
$(\gamma_{n})_{n\in\mathbb{N}}$ the sequence given by

$\gamma_{n}=\frac{r_{n}c_{n}}{r_{n-1}+\dots+r_{1}+1}.$

Let $g\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{3}\to\mathbb{R}^{3}$ be
a bounded, differentiable Lipschitz (with constant $\leq 1$) map, define
$f\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\times\mathbb{R}^{3}\to\mathbb{R}^{3}$ given by
$f(k,x)=\gamma_{k}g(x)$ and consider the the system (2.2) associated to this
perturbation and the previous linear system. Then, if
$\sup_{k\in\mathbb{Z}^{+}}\frac{c_{k-1}r_{k-1}}{r_{k-2}+r_{k-3}+\cdots+1}+\sum_{j=1,j\neq
k-1}^{\infty}r_{j}<1$, systems (2.1) and (2.2) are $C^{1}$-topologically
equivalent on $\mathbb{Z}^{+}$.

In fact, since $0\leq a_{n},b_{n}\leq c_{n}$, then $\mathinner{\\!\left\lVert
A(n)\right\rVert}=c_{n}\leq M$ and $\mathinner{\\!\left\lVert
A^{-1}(n)\right\rVert}\leq\alpha^{-1}$, hence (d0) is fulfilled. Furthermore,
let $P(\cdot)$ and $Q(\cdot)$ be

(3.7) $P(n)=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&0\end{pmatrix}\text{ and
}Q(n)=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&0&1\end{pmatrix}.$

Then, if we consider $\Phi(m,n)$ be the transition matrix for (2.1), it is
clear that

(3.8)
$\left\\{\begin{array}[]{lc}\mathinner{\\!\left\lVert\Phi(k,n)P(n)\right\rVert}\leq\prod_{j=n}^{k-1}\max\\{a_{j},b_{j}\\},&\forall
k\geq n\geq 0\\\ \\\
\mathinner{\\!\left\lVert\Phi(k,n)Q(n)\right\rVert}\leq\prod_{j=k}^{n-1}c_{j}^{-1},&\forall
0\leq k\leq n.\end{array}\right.$

Hence, setting $h(n)=\prod_{p=1}^{n-1}c_{p}^{-1}$ and $D(n)=1$, (d1) follows.
Note that (d2) and (d6) follow immediately by hypothesis. Moreover

$\mathinner{\\!\left\lVert
A(n)^{-1}\gamma_{n}\right\rVert}\leq\alpha^{-1}\frac{r_{n}c_{n}}{r_{n-1}+\dots+r_{1}+1}\leq
M\alpha^{-1}r_{n}.$

As $r_{n}\to 0$, we obtain (d5). It is easy to inductively prove that

$\gamma_{j}=r_{j}c_{j}\left(\prod_{p=1}^{j-1}1+\frac{\gamma_{p}}{c_{p}}\right)^{-1}=r_{j}c_{j}\prod_{p=1}^{j-1}\frac{c_{p}}{\gamma_{p}+c_{p}}.$

Now, denote

$\Psi_{m}(j)={\prod_{p=m}^{j-1}}\mathinner{\\!\left\lVert
A(p)\right\rVert}+\gamma_{p}.$

It is easy to see $m\geq n$ implies $\Psi_{m}(j)\leq\Psi_{n}(j)$, hence for a
fixed $m\in\mathbb{Z}^{+}$

$\Psi_{m}(j)\leq\Psi_{1}(j)\leq\prod_{p=1}^{j-1}c_{p}+\gamma_{p}.$

So we have

$\displaystyle\sum_{j=m}^{\infty}D(j+1)h(j+1)\gamma(j)\Psi_{m}(j)$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{\infty}h(j+1)\gamma(j)\Psi_{1}(j)$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{\infty}\gamma(j)\prod_{p=1}^{j}c_{p}^{-1}\prod_{p=1}^{j-1}(c_{p}+\gamma_{p})$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{\infty}\gamma_{j}c_{j}^{-1}\left(\prod_{p=1}^{j-1}1+\frac{\gamma_{p}}{c_{p}}\right)$
$\displaystyle=$ $\displaystyle\mathinner{\\!\left\lVert
r\right\rVert}_{1}<\infty.$

Hence (d7) is fulfilled. Finally note that for a fixed $k\in\mathbb{Z}^{+}$

$\displaystyle\sum_{j=0}^{\infty}\mathinner{\\!\left\lVert\mathcal{G}(k,j+1)\right\rVert}\gamma_{j}$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{k-1}\frac{h(k)}{h(j+1)}\gamma_{j}+\sum_{j=k}^{\infty}\frac{h(j+1)}{h(k)}\gamma_{j}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{k-1}\left(\prod_{p=j+1}^{k-1}c_{p}^{-1}\right)\frac{c_{j}r_{j}}{r_{j-1}+\cdots
r_{1}+1}$
$\displaystyle+\sum_{j=k}^{\infty}r_{j}c_{j}\left(\prod_{p=k}^{j}c_{p}^{-1}\right)\left(\prod_{p=1}^{j-1}1+\frac{\gamma_{p}}{c_{p}}\right)^{-1}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{k-1}\left(\prod_{p=j+1}^{k-1}c_{p}^{-1}\right)\frac{c_{k-1}r_{j}}{r_{j-1}+\cdots
r_{1}+1}$
$\displaystyle+\sum_{j=k}^{\infty}r_{j}\left(\prod_{p=0}^{k-1}1+\frac{\gamma_{p}}{c_{p}}\right)^{-1}\left(\prod_{p=k}^{j-1}\frac{1}{c_{p}+\gamma_{p}}\right)$
$\displaystyle<$
$\displaystyle\frac{c_{k-1}r_{k-1}}{r_{k-2}+r_{k-3}+\cdots+1}+\sum_{j=1,j\neq
k-1}^{\infty}r_{j}.$

Hence it satisfies (d4) and (d3) with
$\mu_{j}=\gamma_{j}\mathinner{\\!\left\lVert g\right\rVert}_{\infty}$.
Applying Theorem 3.2, we show our claim.

######  Remark 3.9.

Condition
$\sup_{k\in\mathbb{Z}^{+}}\frac{c_{k-1}r_{k-1}}{r_{k-2}+r_{k-3}+\cdots+1}+\sum_{j=1,j\neq
k-1}^{\infty}r_{j}<1$ is obtained, for example, if the increasing rate of
$(c_{n})$ is smaller than the growing rate of the partial sums of $(r_{n})$
and $\mathinner{\\!\left\lVert r\right\rVert}_{1}<1$. Another simpler case is
if $M\mathinner{\\!\left\lVert r\right\rVert}_{1}<1$.

######  Example 3.10.

Let
$(a_{n})_{n\in\mathbb{N}},(b_{n})_{n\in\mathbb{N}},(c_{n})_{n\in\mathbb{N}}$
three sequences as in (3.5), $A(n)$ as in (3.6) and the system (2.1)
associated to this sequence of matrices. Let $(r_{n})_{n\in\mathbb{N}}$ be a
summable non negative sequence and define $(\gamma_{n})_{n\in\mathbb{N}}$ the
sequence given by:

$\gamma_{n}=\frac{r_{n}}{r_{n-1}+\dots+r_{1}+1}.$

Let $g\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{3}\to\mathbb{R}^{3}$ be
a bounded, differentiable Lipschitz (with constant $\leq 1$) map, define
$f\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\times\mathbb{R}^{3}\to\mathbb{R}^{3}$ given by
$f(k,x)=\gamma_{k}g(x)$ and consider the the system (2.2) associated to this
perturbation and the previous linear system. Then, if
$\mathinner{\\!\left\lVert r\right\rVert}_{1}<1$, systems (2.1) and (2.2) are
$C^{1}$-topologically equivalent on $\mathbb{Z}^{+}$.

Indeed, conditions (d0), (d1), (d2), (d5) and (d6) follow easily in the same
fashion as in the previous Example, with $P(\cdot)$ and $Q(\cdot)$ as in
(3.7). It is easy to inductively prove

$\gamma_{j}=r_{j}\left(\prod_{p=1}^{j-1}1+\gamma_{p}\right)^{-1}.$

Denote

$\Psi_{m}(j)={\prod_{p=m}^{j-1}}\mathinner{\\!\left\lVert
A(p)\right\rVert}+\gamma_{p}.$

It is easy to see $m\geq n$ implies $\Psi_{m}(j)\leq\Psi_{n}(j)$, hence for a
fixed $m\in\mathbb{Z}^{+}$

$\Psi_{m}(j)\leq\Psi_{1}(j)\leq\prod_{p=1}^{j-1}c_{p}+\gamma_{p}.$

So we have

$\displaystyle\sum_{j=m}^{\infty}D(j+1)h(j+1)\gamma(j)\Psi_{m}(j)$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{\infty}h(j+1)\gamma(j)\Psi_{1}(j)$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{\infty}r_{j}\left(\prod_{p=1}^{j-1}1+\gamma_{p}\right)^{-1}\left(\prod_{p=1}^{j}c_{p}^{-1}\right)\left(\prod_{p=1}^{j-1}c_{p}+\gamma_{p}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{\infty}r_{j}c_{j}^{-1}\left(\prod_{p=1}^{j-1}\frac{1+\gamma_{p}c_{p}^{-1}}{1+\gamma_{p}}\right)$
$\displaystyle\leq$ $\displaystyle\mathinner{\\!\left\lVert
r\right\rVert}_{1}<\infty.$

Hence (d7) is fulfilled. Finally note that for a fixed $k\in\mathbb{Z}^{+}$

$\displaystyle\sum_{j=0}^{\infty}\mathinner{\\!\left\lVert\mathcal{G}(k,j+1)\right\rVert}\gamma_{j}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{k-1}\frac{h(k)}{h(j+1)}\gamma_{j}+\sum_{j=k}^{\infty}\frac{h(j+1)}{h(k)}\gamma_{j}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{k-1}\left(\prod_{p=j+1}^{k-1}c_{p}^{-1}\right)\frac{r_{j}}{r_{j-1}+\cdots
r_{1}+1}$
$\displaystyle+\sum_{j=k}^{\infty}\left(\prod_{p=k}^{j}c_{p}^{-1}\right)\frac{r_{j}}{r_{j-1}+\cdots
r_{1}+1}$ $\displaystyle<$ $\displaystyle\mathinner{\\!\left\lVert
r\right\rVert}_{1}<1,$

hence proving (d4) and (d3) with $\mu_{j}=\gamma_{j}\mathinner{\\!\left\lVert
g\right\rVert}_{\infty}$. Applying theorem 3.2 our claim is proved.

######  Example 3.11.

Let
$(a_{n})_{n\in\mathbb{N}},(b_{n})_{n\in\mathbb{N}},(c_{n})_{n\in\mathbb{N}}$
three sequences as in (3.5) and $A(n)$ as in (3.6). Let
$(E(n))_{n\in\mathbb{N}}$ be a sequence of uniformly bounded invertible
matrices with uniformly bounded inverses. Set $\mathscr{E}^{+}$ to be a
uniform bound of $E(n)$ and $\mathscr{E}^{-}$ to be a uniform bound for their
inverses. Set $E(0)=I$ and define $B(n)=E^{-1}(n)A(n)E(n-1)$. Let
$(\gamma_{n})_{n\in\mathbb{N}}$ be a summable non negative sequence.

Let $g\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{3}\to\mathbb{R}^{3}$ be
a bounded, differentiable Lipschitz (with constant $\leq 1$) map, define
$f\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\times\mathbb{R}^{3}\to\mathbb{R}^{3}$ given by
$f(k,x)=\gamma_{k}g(x)$. Consider the systems:

(3.9) $x(k+1)=B(k)x(k)$ (3.10) $y(k+1)=B(k)y(k)+f(k,y(k)).$

Finally, suppose there is a $\delta>0$ such that:

(3.11)
$\sum_{j=1}^{\infty}\gamma_{j}\left(\mathscr{E}^{-}\mathscr{E}^{+}+\delta\right)^{j}<\infty.$

Then, if
$\mathscr{E}^{-}\mathscr{E}^{+}\mathinner{\\!\left\lVert\gamma\right\rVert}_{1}<1$,
systems (3.9) and (3.10) are $C^{1}$-topologically equivalent on
$\mathbb{Z}^{+}$.

In fact, whitout lost of generality, $\mathscr{E}^{-},\mathscr{E}^{+}\geq 1$.
Since $0\leq\alpha\leq a_{n},b_{n}\leq c_{n}\leq M$, then
$\mathinner{\\!\left\lVert
B(n)\right\rVert}\leq\mathscr{E}^{-}M\mathscr{E}^{+}$ and
$\mathinner{\\!\left\lVert
B^{-1}(n)\right\rVert}\leq\mathscr{E}^{-}\alpha^{-1}\mathscr{E}^{+}$, hence
(d0) is fulfilled.

Define $\tilde{P}(\cdot)$ and $\tilde{Q}(\cdot)$ as

$\tilde{P}(n)=E^{-1}(n-1)P(n)E(n-1)\text{ , and
}\tilde{Q}(n)=E^{-1}(n-1)Q(n)E(n-1),$

where $P(\cdot)$ and $Q(\cdot)$ are as in (3.7). Then, if we consider
$\Phi(m,n)$ to be as in (3.8), it is easy to see that

$\tilde{\Phi}(k,n)=E^{-1}(k-1)\Phi(k,n)E(n-1),$

where $\tilde{\Phi}(k,n)$ is the transition matrix for (3.9). As in the
previous example it is immediate that

$\left\\{\begin{array}[]{lc}\mathinner{\\!\left\lVert\tilde{\Phi}(k,n)\tilde{P}(n)\right\rVert}\leq\mathscr{E}^{-}\mathinner{\\!\left\lVert
E(n-1)\right\rVert}\prod_{j=n}^{k-1}\max\\{a_{j},b_{j}\\},&\forall k\geq n\geq
0\\\ \\\
\mathinner{\\!\left\lVert\tilde{\Phi}(k,n)\tilde{Q}(n)\right\rVert}\leq\mathscr{E}^{-}\mathinner{\\!\left\lVert
E(n-1)\right\rVert}\prod_{j=k}^{n-1}c_{j}^{-1},&\forall 0\leq k\leq
n,\end{array}\right.$

hence, setting $h(n)=\prod_{p=1}^{n-1}c_{p}^{-1}$ and
$D(n)=\mathscr{E}^{-}\mathinner{\\!\left\lVert E(n-1)\right\rVert}$, (d1)
follows. Note that (d2) and (d6) follow immediately from our hypothesis, while
(d5) is satisfied in the same fashion as in Example 3.8.

Denote

$\tilde{\Psi}_{m}(j)={\prod_{p=m}^{j-1}}\mathinner{\\!\left\lVert
B(p)\right\rVert}+\gamma_{p}.$

It is easy to see $m\geq n$ implies
$\tilde{\Psi}_{m}(j)\leq\tilde{\Psi}_{n}(j)$, hence for a fixed
$m\in\mathbb{Z}^{+}$:

$\tilde{\Psi}_{m}(j)\leq\tilde{\Psi}_{1}(j)\leq\prod_{p=1}^{j-1}\mathscr{E}^{-}\mathscr{E}^{+}c_{p}+\gamma_{p}.$

So we have

$\displaystyle\sum_{j=m}^{\infty}D(j+1)h(j+1)\gamma(j)\tilde{\Psi}_{m}(j)$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{\infty}D(j+1)h(j+1)\gamma(j)\tilde{\Psi}_{1}(j)$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{\infty}\mathscr{E}^{-}\mathinner{\\!\left\lVert
E(j)\right\rVert}\left(\prod_{p=1}^{j}c_{p}^{-1}\right)\gamma_{j}\left(\prod_{p=1}^{j-1}\mathscr{E}^{-}\mathscr{E}^{+}c_{p}+\gamma_{p}\right)$
$\displaystyle\leq$
$\displaystyle\mathscr{E}^{-}\mathscr{E}^{+}\sum_{j=1}^{\infty}\gamma_{j}c_{j}^{-1}\left(\prod_{p=1}^{j-1}\mathscr{E}^{-}\mathscr{E}^{+}+\gamma_{p}c_{p}^{-1}\right)$
$\displaystyle\leq$
$\displaystyle\mathscr{E}^{-}\mathscr{E}^{+}\sum_{j=1}^{\infty}\gamma_{j}\left(\prod_{p=1}^{j-1}\mathscr{E}^{-}\mathscr{E}^{+}+\gamma_{p}\right).$

As $\gamma_{p}\xrightarrow[p\to\infty]{}0$, there exist $p_{0}$ such that
$\gamma_{p}\leq\delta$ for every $p\geq p_{0}$. Hence, applying condition
(3.11), it is easy to see (d7) is fulfilled. Finally

$\displaystyle\sum_{j=0}^{\infty}\mathinner{\\!\left\lVert\mathcal{G}(k,j+1)\right\rVert}\gamma_{j}$
$\displaystyle\leq$
$\displaystyle\mathscr{E}^{-}\sum_{j=1}^{k-1}\frac{h(k)}{h(j+1)}\mathinner{\\!\left\lVert
E(j-1)\right\rVert}\gamma_{j}+\mathscr{E}^{-}\sum_{j=k}^{\infty}\frac{h(j+1)}{h(k)}\mathinner{\\!\left\lVert
E(j-1)\right\rVert}\gamma_{j}$ $\displaystyle\leq$
$\displaystyle\mathscr{E}^{-}\mathscr{E}^{+}\sum_{j=1}^{k-1}\left(\prod_{p=j+1}^{k-1}c_{p}^{-1}\right)\gamma_{j}$
$\displaystyle+\mathscr{E}^{-}\mathscr{E}^{+}\sum_{j=k}^{\infty}\left(\prod_{p=k}^{j}c_{p}^{-1}\right)\gamma_{j}$
$\displaystyle<$
$\displaystyle\mathscr{E}^{-}\mathscr{E}^{+}\mathinner{\\!\left\lVert\gamma\right\rVert}_{1}<1.$

Hence proving (d4) and (d3) with $\mu_{j}=\gamma_{j}\mathinner{\\!\left\lVert
g\right\rVert}_{\infty}$. By using Theorem 3.2 our claim follows.

######  Remark 3.12.

As we mentioned in the introduction, conditions (d0)-(d6) have been adopted
several times by many authors in order to study the topological equivalence
between systems (2.1) and (2.2). Hence, the novelty of this work is the
condition (d7) and its implications.

Namely, in Corollary 3.3 the fact that this condition is satisfied relies
mainly (as explained on Remark 3.4) on the properties of the exponential
dichotomy, which gives a evolution ratio for the solutions that allows (d7) to
be fulfilled. Meanwhile, in Examples 3.8 and 3.10, this condition is achieved
by a perturbation that is small enough and has a fast enough decay ratio, and
actually we completely ignore the help that the dichotomy may have presented.
Finally, in Corollary 3.5 and Example 3.11, the condition (d7) is achieved
through a combination of the contributions of both the perturbation and the
dichotomy.

This shows that there are many possibilities to achieve this condition and
hence our main result, Theorem 3.2, is applicable in many different
situations.

##  4\. Higher order derivatives

In order to study higher order derivatives of the homeomorphism of topological
equivalence, we once again consider the expression (2). Taking in account
Remark 2.10, the map $\eta\mapsto y(0,m,\eta)$ is of class $C^{r}$ ($r\geq 1$)
for every fixed $m\in\mathbb{Z}^{+}$ if conditions (d0)-(d6) are satisfied;
hence the class of differentiability of the homeomorphism relies on the
differentiability of the map $\eta\mapsto w^{*}(0;(m,\eta))$.

In this section we attempt to find conditions that generalize (d7) in order to
obtain higher order derivatives for the homeomorphism.

######  Lemma 4.1.

Suppose conditions (d0)-(d7) hold, where (d6) is fulfilled with $r=2$. Also,
there are functions $\Gamma\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\to\mathbb{R}^{+}$ and
$\pi_{m}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Z}^{+}\to\mathbb{R}^{+}$
such that:

(4.1) $\mathinner{\\!\left\lVert\frac{\partial^{2}f}{\partial
u^{2}}(j,u)\right\rVert}\leq\Gamma(j)\text{ , for every }j\in\mathbb{Z}^{+}$

and

(4.2)
$\mathinner{\\!\left\lVert\frac{\partial^{2}y}{\partial\eta^{2}}(j,m,\eta)\right\rVert}\leq\pi_{m}(j)\text{
, for every }j\geq m.$

Finally, suppose that for each fixed $m\in\mathbb{Z}^{+}$, the sequences
satisfy:

(4.3)
${\sum_{j=m}^{\infty}}\left(D(j+1)h(j+1)\left\\{\pi_{m}(j)\gamma(j)+\Gamma(j)\left[{\prod_{p=m}^{j-1}}\mathinner{\\!\left\lVert
A(p)\right\rVert}+\gamma(p)\right]^{2}\right\\}\right)<+\infty.$

Then $\eta\mapsto w^{*}(0;(m,\eta))$ is a $C^{2}$ map.

###### Proof.

From the proof of Lemma 3.1 it is easy to see that for every fixed
$m\in\mathbb{Z}^{+}$

$\frac{\partial
w^{*}(0;(m,\eta))}{\partial\eta}=-\sum_{j=0}^{\infty}\mathcal{G}(0,j+1)\frac{\partial
f}{\partial u}(j,y(j,m,\eta))\frac{\partial y}{\partial\eta}(j,m,\eta).$

Hence, in order to prove the differentiability of this map, it is enough to
find a summable function that uniformly (respect to $\eta$) dominates

$\mathcal{G}(0,j+1)\left[\frac{\partial^{2}y}{\partial\eta^{2}}(j,m,\eta)\frac{\partial
f}{\partial u}(j,y(j,m,\eta))+\frac{\partial^{2}f}{\partial
u^{2}}(j,y(j,m,\eta))\left(\frac{\partial
y}{\partial\eta}(j,m,\eta)\right)^{2}\right]$

for every $j\geq 0$.

This fact is verified by hypothesis. Therefore, we apply the same strategy as
in the proof of Lemma 3.1. ∎

######  Theorem 4.2.

If conditions (d0)-(d7) hold, (d6) is satisfied with $r=2$, and conditions of
Lemma 4.1 are satisfied, then (2.1) and (2.2) are $C^{2}$-topologically
equivalent on $\mathbb{Z}^{+}$.

Note that Theorem 4.2 follows easily in the same fashion as the proof of
Theorem 3.2.

######  Corollary 4.3.

Suppose conditions (d0) and (d5) are satisfied. Also, consider system (2.1)
admits a nonuniform exponential dichotomy, which means there are two
complementary invariant projectors $P(\cdot)$ and $Q(\cdot)$ and constants
$C,\lambda,\varepsilon>0$ such that

$\left\\{\begin{array}[]{lc}\mathinner{\\!\left\lVert\Phi(k,n)P(n)\right\rVert}\leq
Ce^{-\lambda(k-n)+\varepsilon n},&\forall k\geq n\geq 0\\\ \\\
\mathinner{\\!\left\lVert\Phi(k,n)Q(n)\right\rVert}\leq
Ce^{\lambda(k-n)+\varepsilon n},&\forall 0\leq k\leq n.\end{array}\right.$

Furthermore, suppose that for every $k\in\mathbb{Z}^{+}$ $u\mapsto f(k,u)$ is
a $C^{2}$ function such that:

$|f(k,u)|\leq\kappa e^{-\tau(k+1)},$

$\mathinner{\\!\left\lVert\frac{\partial f}{\partial
u}(k,u)\right\rVert}\leq\nu e^{-\varepsilon(k+1)}$

and

(4.4) $\mathinner{\\!\left\lVert\frac{\partial f}{\partial
u}(k,x)-\frac{\partial f}{\partial u}(k,y)\right\rVert}\leq\zeta
e^{-\varepsilon(k+1)}|x-y|,$

for given $\kappa,\nu,\zeta>0$ and $\tau>\varepsilon-\lambda$. Then, if
$M^{2}e^{-\lambda}<1$, for sufficiently small $\nu>0$, the systems (2.1) and
(2.2) are $C^{2}$-topologically equivalent on $\mathbb{Z}^{+}.$

###### Proof.

By Corollary 3.5 we know that conditions (d0)-(d7) are verified and hence they
are $C^{1}$-topologically equivalent on $\mathbb{Z}^{+}$. In particular, (d4)
is satisfied with $\gamma(j)=\nu e^{-\varepsilon(j+1)}$.

It is easy to see that (4.4) implies (4.1), with $\Gamma(j)=\zeta
e^{-\varepsilon(j+1)}$. In addition, if $M^{2}e^{-\lambda}<1$, this implies,
in a similar fashion as in the proof of 3.5, that

(4.5)
${\sum_{j=m}^{\infty}}\left(D(j+1)h(j+1)\Gamma(j)\left[{\prod_{p=m}^{j-1}}\mathinner{\\!\left\lVert
A(p)\right\rVert}+\gamma(p)\right]^{2}\right)<+\infty.$

Fix $m\in\mathbb{Z}$. We know that $\eta\mapsto\frac{\partial
y}{\partial\eta}(j,m,\eta)$ is well defined for every $j\geq m$. Consider the
matrix initial value problem:

(4.6) $\left\\{\begin{array}[]{ccl}z(j+1)&=&\left[A(j)+\frac{\partial
f}{\partial u}(j,y(j,m,\eta)\right]z(j)\\\ \\\ z(m)&=&I.\end{array}\right.$

Note that $j\mapsto z(j,m,\eta)=\frac{\partial y}{\partial\eta}(j,m,\eta)$ is
a solution of (4.6), hence

$z(m+1,m,\eta)=\left[A(m)+\frac{\partial f}{\partial
u}(m,y(m,m,\eta))\right]z(m,m,\eta)=A(m)+\frac{\partial f}{\partial
u}(m,\eta).$

Then, we have

$\mathinner{\\!\left\lVert
z(m+1,m,\eta)-z(m+1,m,\tilde{\eta})\right\rVert}=\mathinner{\\!\left\lVert\frac{\partial
f}{\partial u}(m,\eta)-\frac{\partial f}{\partial
u}(m,\tilde{\eta})\right\rVert}\leq\Gamma(m)|\eta-\tilde{\eta}|.$

So, inductively we can prove

$\mathinner{\\!\left\lVert
z(j,m,\eta)-z(j,m,\tilde{\eta})\right\rVert}\leq\left(\prod_{i=m}^{j-1}\Gamma(i)\right)|\eta-\tilde{\eta}|.$

Which implies that $\pi_{m}(j)=\prod_{i=m}^{j-1}\Gamma(i)$ satisfies condition
(4.2) from Lemma 4.1. As $\Gamma(j)\to 0$ when $j\to\infty$, it is easy to see
there are constants $a_{m},b>0$, $b<1$ such that $\pi_{m}(j)\leq a_{m}b^{j}$,
which allows us to easily see

${\sum_{j=m}^{\infty}}\left(D(j+1)h(j+1)\pi_{m}(j)\gamma(j)\right)<+\infty.$

Now, condition (4.3) from Lemma 4.1 is satisfied if we consider (4.5).
Finally, applying Theorem 4.2 we obtain the result. ∎

##  5\. Discussion

Suppose that conditions (d0)-(d6) are verified. Suppose as well there are
functions $\Gamma_{s}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\to\mathbb{R}^{+}$,
$\pi_{s,m}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\to\mathbb{R}^{+}$, $1\leq s\leq r$,
$m\in\mathbb{Z}^{+}$ such that:

(5.1) $\mathinner{\\!\left\lVert\frac{\partial^{s}f}{\partial
u^{s}}(j,u)\right\rVert}\leq\Gamma_{s}(j)\text{ , for every
}j\in\mathbb{Z}^{+}$

and

(5.2)
$\mathinner{\\!\left\lVert\frac{\partial^{s}y}{\partial\eta^{s}}(j,m,\eta)\right\rVert}\leq\pi_{s,m}(j)\text{
, for every }j\geq m.$

Chose a fixed $m\in\mathbb{Z}^{+}$ and consider the set of functions

$\mathfrak{S}_{m}=\left\\{\Gamma_{s}\prod_{k=1}^{r}\pi_{k,m}^{e_{k}}\mathrel{\mathop{\mathchar
58\relax}}1\leq s\leq r,e_{k}\in\mathbb{Z}^{+}_{0}\right\\}.$

Consider the module $\mathbb{Z}[\mathfrak{S}_{m}]$ and define the
$\mathbb{Z}-$linear map $\mathbb{D}_{m}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}[\mathfrak{S}_{m}]\to\mathbb{Z}[\mathfrak{S}_{m}]$ given
by

  * •

$\mathbb{D}_{m}(\Gamma_{s})=\Gamma_{s+1}\pi_{1,m}$, for every $1\leq s<r.$

  * •

$\mathbb{D}_{m}(\pi_{s,m})=\pi_{s+1,m}$, for every $1\leq s<r.$

  * •

$\mathbb{D}_{m}(fg)=\mathbb{D}_{m}(f)g+f\mathbb{D}_{m}(g)$, for every
$f,g\in\mathbb{Z}[\mathfrak{S}_{m}]$ such that
$fg\in\mathbb{Z}[\mathfrak{S}_{m}].$

In order to simplify notations, suppose the existence of a certain function
$\Gamma_{0}$ such that
$\mathbb{D}_{m}(\Gamma_{0})(j)=\Gamma_{1}(j)\pi_{1,m}(j)$. Under these
conditions it is easy to see:

  * •

$\mathbb{D}^{2}_{m}(\Gamma_{0})(j)=\Gamma_{2}(j)\pi_{1,m}(j)^{2}+\Gamma_{1}(j)\pi_{2,m}(j).$

  * •

$\mathbb{D}^{3}_{m}(\Gamma_{0})(j)=\Gamma_{3}(j)\pi_{1,m}(j)^{3}+3\Gamma_{2}(j)\pi_{2,m}(j)\pi_{1,m}(j)+\Gamma_{1}(j)\pi_{3,m}(j).$

  * •

$\mathbb{D}^{4}_{m}(\Gamma_{0})(j)=\Gamma_{4}(j)\pi_{1,m}(j)^{4}+6\Gamma_{3}(j)\pi_{2,m}(j)\pi_{1,m}(j)^{2}+4\Gamma_{2}(j)\pi_{3,m}(j)\pi_{1,m}(j)+\Gamma_{1}(j)\pi_{4,m}(j)$.

And inductively it is easy to calculate higher powers of $\mathbb{D}_{m}$
applied to $\Gamma_{0}$. With this in mind, we would like to introduce the
following condition:

(DIF,r) Suppose conditions (d0)-(d6) are verified. Suppose as well there are
functions $\Gamma_{s}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\to\mathbb{R}^{+}$,
$\pi_{s,m}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{+}\to\mathbb{R}^{+}$, $1\leq s\leq r$,
$m\in\mathbb{Z}^{+}$ that satisfy (5.1) and (5.2) respectively and are such
that for every fixed $m\in\mathbb{Z}$ it is verified

${\sum_{j=m}^{\infty}}D(j+1)h(j+1)\mathbb{D}^{s}_{m}(\Gamma_{0})(j)<+\infty\text{
, for every }1\leq s\leq r.$

######  Remark 5.1.

Note that condition (d7) is a particular case of (DIF,r), namely (DIF,1), with
$\Gamma_{1}(j)=\gamma(j)$ and
$\pi_{1,m}(j)=\prod_{p=m}^{j-1}\mathinner{\\!\left\lVert
A(p)\right\rVert}+\gamma(p)$. Similarly, the conditions on Lemma (4.1) can be
summarized as (DIF,2).

Moreover, condition (d3) is equivalent to (DIF,0), defining
$j\mapsto\Gamma_{0}(j)$ as a function that uniformly (over $u$) dominates
$j\mapsto f(j,u)$, or, in the notations of (d3), $\Gamma_{0}(j)=\mu(j)$. This
makes sense, because a $C^{0}$ class of differentiability for a diffeomorphism
corresponds to an homeomorphism, which is verified by considering (d3) (in
addition with (d0)-(d5)).

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