# Takens Theorem for nonautonomous partially hyperbolic dynamical systems

Davor Dragičević  Faculty of Mathematics
University of Rijeka
Rijeka 51000, Croatia<EMAIL_ADDRESS>,  Xiao Tang  School of
Mathematical Sciences
Chongqing Normal University
Chongqing 401331, China<EMAIL_ADDRESS>and  Wenmeng Zhang  School of
Mathematical Sciences
Chongqing Normal University
Chongqing 401331, China<EMAIL_ADDRESS>
###### Abstract.

Takens Theorem for a partially hyperbolic dynamics provides a normal
linearization along the center manifold. In this paper, we give the
nonautonomous version of Takens Theorem under non-resonance conditions
formulated in terms of the dichotomy spectrum. In our proof, one difficulty is
to solve homological equations for the normal form theory which involve a
center variable, while another difficulty is to find the dichotomy spectrum of
a certain matrix cocycle that is block lower triangular. In order to overcome
those difficulties, in comparison with the autonomous case, we need an
additional term in the (nonautonomous) non-resonance conditions to guarantee
certain spectral gap conditions. This additional term disappears naturally in
the autonomous case.

Keywords: Nonautonomous system; dichotomy spectrum; center manifold;
homological equation; matrix cocycle

AMS (2010) subject classification: 37D10

##  1\. Introduction

Smooth linearization is one of the most important topics in the qualitative
theory of dynamical systems since it is highly related to the classification,
simplification and the structural stability of dynamical systems. Its study
goes back to the pioneering work of Poincaré ([19]), who proved that an
analytic diffeomorphism defined on $\mathbb{C}^{d}$ ($d\in\mathbb{N}$) can be
analytically conjugated to its linear part provided that its eigenvalues lie
in the Poincaré domain, i.e., all of the eigenvalues $\lambda_{i}$ for
$i=1,\ldots,d$ have modules $<1$ or $>1$ (which is a special case of the
hyperbolicity) and satisfy the non-resonance conditions of all orders:

$\displaystyle\lambda_{1}^{t_{1}}\cdots\lambda_{d}^{t_{d}}\neq\lambda_{j}\qquad\forall
j=1,\ldots,d,$ (1.1)

where $t_{i}\geq 0$ are all possible integers such that
$\sum_{i=1}^{d}t_{i}\geq 2$. On $\mathbb{R}^{d}$, Sternberg ([29, 30]) proved
that for any $k\in\mathbb{N}$ there is a sufficiently large $N\in\mathbb{N}$
such that each $C^{N}$ hyperbolic diffeomorphism is locally $C^{k}$ conjugated
to its linear part if the non-resonance condition of order $N$ holds, i.e.,
(1.1) holds for integers $t_{i}\geq 0$ such that $2\leq\sum_{i=1}^{d}t_{i}\leq
N$.

In the non-hyperbolic case, Takens ([31]) further considered diffeomorphisms
with center direction. More precisely, he proved that for any $k\in\mathbb{N}$
there is a sufficiently large $N\in\mathbb{N}$ such that if

$\displaystyle|\lambda_{1}^{t_{1}}\cdots\lambda_{d}^{t_{d}}|\neq|\lambda_{j}|,\qquad\forall
j=1,\ldots,d,$ (1.2)

for integers $t_{i}\geq 0$ such that $2\leq\sum_{i=1}^{d}t_{i}\leq N$, then a
$C^{N}$ diffeomorphism with a center direction is locally $C^{k}$ conjugated
to its Takens normal form

$\displaystyle\left(\begin{array}[]{lll}x_{s}\vspace{1ex}\\\
x_{c}\vspace{1ex}\\\
x_{u}\end{array}\right)\mapsto\left(\begin{array}[]{lll}\Lambda_{s}(x_{c})x_{s}\vspace{1ex}\\\
\Lambda_{c}x_{c}+f_{c}(x_{c})\vspace{1ex}\\\
\Lambda_{u}(x_{c})x_{u}\end{array}\right),$

where $x_{*}$ ($*:=s,c,u$) are variables corresponding to the stable, center
and unstable subspaces and $\Lambda_{*}$ ($*:=s,c,u$) are linear operators
such that the eigenvalues of $\Lambda_{s}(0)$, $\Lambda_{c}$ and
$\Lambda_{u}(0)$ have modulus $<1$, $=1$ and $>1$, respectively. Basically,
Takens Theorem can be used to obtain the normal form of a partially hyperbolic
system (i.e., a system with a center direction) via a $C^{k}$ transformation.
This is important because the Takens normal form can help one to linearize a
partially hyperbolic system along its center manifold and can also help to
obtain the existence of a smooth invariant foliation near the center manifold.
Consequently, Takens Theorem has wide applications to the studies of
bifurcations ([25, 26]), nonlinear oscillations ([6]), Hilbert’s 16th problem
([12]), $N$-body problem ([15]), boundary value problems ([14]), random tree
model ([9]), etc.

Due to the importance of Takens Theorem, providing its extensions and
generalizations has been an active area of research since the 1990s. On one
hand, $C^{\infty}$ versions of Takens Theorem were obtained when the
hyperbolic linear part is independent of the center variable or the hyperbolic
direction is purely contractive (or expansive) (see e.g. [4, 13, 16]). On the
other hand, Takens Theorem was extended to the setting of random dynamical
systems ([23]). Most of these works employed the so-called “path method” or
“homotopy method” developed by Mather ([24]). In this method, a crucial
intermediate result is to show that two systems having identical derivatives
up to the $N$-th order on their common center manifold are $C^{k}$ conjugated.
This result is known as the Belitskii-Samavol Theorem ([21]). After 2000,
$C^{\infty}$ versions of Belitskii-Samavol Theorem were obtained in [3, 20] on
$\mathbb{R}^{d}$ and on Banach spaces, respectively.

In the case of nonautonomous dynamics on $\mathbb{R}^{d}$, we are concerned
with the system

$x_{n+1}=F_{n}(x_{n}):=A_{n}x_{n}+f_{n}(x_{n}),\quad\forall n\in\mathbb{Z},$
(1.3)

where $A_{n}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ are invertible matrices and
$f_{n}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ are smooth mappings. The smooth
conjugacy of two nonautonomous systems $(F_{n})_{n\in\mathbb{Z}}$ and
$(G_{n})_{n\in\mathbb{Z}}$ is a sequence $(\Psi_{n})_{n\in\mathbb{Z}}$ of
diffeomorphisms with bounded derivatives near $0$ such that

$\Psi_{n+1}\circ F_{n}=G_{n}\circ\Psi_{n},\quad\forall n\in\mathbb{Z}.$

In the uniformly hyperbolic case, i.e., when $(A_{n})_{n\in\mathbb{Z}}$ admits
a uniform exponential dichotomy, the nonautonomous version of Poincaré Theorem
can be found in [27, 32]. In 2019, the nonautonomous version of Sternberg
Theorem was given in [7, Theorem 5]. The conditions of [7, Theorem 5] are
expressed in terms of the so-called dichotomy spectrum $\Sigma(A_{n})$ which
consists of all $\lambda>0$ with the property that the sequence
$(\lambda^{-1}A_{n})_{n\in\mathbb{Z}}$ does not admit an exponential
dichotomy. Provided that $\sup_{n\in\mathbb{Z}}\|A_{n}\|<+\infty$ and
$\sup_{n\in\mathbb{Z}}\|A_{n}^{-1}\|<+\infty$, we have that

$\Sigma(A_{n})=\bigcup_{i=1}^{r}[a_{i},b_{i}]\quad\text{for some $r\leq d$,}$

where

$0<a_{1}\leq b_{1}<\ldots<a_{r}\leq b_{r}.$

Provided that $1\notin\Sigma(A_{n})$, it is proved in [7, Theorem 5] that for
any $k\in\mathbb{N}$, there is an integer $N\in\mathbb{N}$ such that if the
non-resonance condition

$\displaystyle\left([a_{1},b_{1}]^{t_{1}}\cdots[a_{r},b_{r}]^{t_{r}}\right)\cap[a_{j},b_{j}]=\emptyset,\quad\forall
j=1,\ldots,r,$ (1.4)

holds for integers $t_{i}\geq 0$ such that $2\leq t_{1}+\ldots+t_{r}\leq N$,
where $[a,b]^{s}:=[a^{s},b^{s}]$ and $[a,b]\cdot[c,d]:=[ab,cd]$ for
$a,b,c,d,s\in\mathbb{R}$, then the $C^{N}$ system $(F_{n})_{n\in\mathbb{Z}}$
can be locally $C^{k}$ linearized. For different smooth linearizations results
dealing with nonautonomous dynamics, we refer to [10, 11]. In addition, we
note that the study of nonautonomous linearization goes back to the seminal
work of Palmer [18].

The main objective of the present paper is to extend the Takens Theorem to the
nonautonomous case when the linear part $(A_{n})_{n\in\mathbb{Z}}$ of (1.3)
admits a uniform exponential trichotomy. Provided that
$(A_{n})_{n\in\mathbb{Z}}$ does not admit an exponential dichotomy, its
dichotomy spectrum has the form
$\Sigma(A_{n})=\bigcup_{i=1}^{r}[a_{i},b_{i}]\cup[\nu_{-},\nu_{+}]$ with

$0<a_{1}\leq b_{1}<\ldots a_{l}\leq b_{l}<\nu_{-}\leq 1\leq\nu_{+}<a_{l+1}\leq
b_{l+1}<\ldots a_{r}\leq b_{r},$

for some $r<d$ and $l\in\\{1,\ldots,r-1\\}$. Then, we have the following main
result of our paper.

Main Theorem. For any $k\in\mathbb{N}$, there is $N\in\mathbb{N}$ such that if
the non-resonance conditions

$\displaystyle\begin{split}&([a_{1},b_{1}]^{q_{1}}\cdots[a_{r},b_{r}]^{q_{r}}\cdot[\nu_{-},\nu_{+}]^{N})\cap[\nu_{-},\nu_{+}]=\emptyset,\\\
&([a_{1},b_{1}]^{t_{1}}\cdots[a_{r},b_{r}]^{t_{r}}\cdot[\nu_{-},\nu_{+}]^{N})\cap[a_{j},b_{j}]=\emptyset,\quad\forall
j=1,\ldots,r,\end{split}$ (1.5)

hold for integers $q_{i},t_{i}\geq 0$ such that $1\leq q_{1}+\ldots+q_{r}\leq
N$ and $2\leq t_{1}+\ldots+t_{r}\leq N$, then the $C^{N+1}$ system
$(F_{n})_{n\in\mathbb{N}}$ is locally $C^{k}$ conjugated to

$(x_{s},x_{c},x_{u})^{T}\mapsto(A_{n}^{s}(x_{c})x_{s},A_{n}^{c}x_{c}+f_{n}^{c}(x_{c}),A_{n}^{u}(x_{c})x_{u})^{T},$

where $A_{n}^{s}(x_{c})$ and $A_{n}^{u}(x_{c})$ are linear operators, which
are $C^{N-1}$ in $x_{c}$, and $f_{n}^{c}$ is a $C^{N}$ map.

We note that the above theorem includes the hyperbolic case (when the center
direction does not exist) in which we interpret (1.5) as (1.4). Moreover,
(1.5) reduces to (1.2) in the autonomous partially hyperbolic case (when
$\nu_{-}=\nu_{+}=1$). One of the difficulties in the proof of the above
theorem is to solve homological equations for the normal form theory which now
involve a center variable (see (4.30) and (4.54) below), while another
difficulty is to find the dichotomy spectrum of a certain matrix cocycle that
is block lower triangular (see (4.42)). In order to overcome those
difficulties, in comparison with (1.2), we need an additional term
$[\nu_{-},\nu_{+}]^{N}$ in (1.5) to guarantee certain spectral gap conditions
(see (4.47) and (4.51)).

The paper is organized as follows. In Section 2, we recall the notions of
exponential dichotomy and exponential trichotomy as well as that of the
dichotomy spectrum. In Section 3, we give a center manifold result in the
nonautonomous case by lifting the nonautonomous system on $\mathbb{R}^{d}$ to
the autonomous one which acts on a certain Hilbert space. Then, we prove our
main theorem in Section 4 by first establishing the nonautonomous version of
Belitskii-Samavol Theorem by using the “lifting method”. Finally, in the
Appendix, we give the Belitskii-Samavol Theorem on Hilbert spaces for
autonomous dynamics.

##  2\. Preliminaries

Let $\mathbb{R}^{d}$ denote the $d$-dimensional Euclidean space equipped with
the Euclidean norm $\|\cdot\|$. The associated operator norm on the space of
linear operators on $\mathbb{R}^{d}$ will also be denoted by $\|\cdot\|$. Let
$(A_{n})_{n\in\mathbb{Z}}$ be a two-sided sequence of invertible linear
operators acting on $\mathbb{R}^{d}$. We define the associated _linear
cocycle_ by

$\displaystyle{\mathcal{A}}(m,n):=\begin{cases}A_{m-1}\cdots A_{n}&\text{if
$m>n$,}\\\ {\operatorname{Id}}&\text{if $m=n$,}\\\ A_{m}^{-1}\cdots
A_{n-1}^{-1}&\text{if $m<n$,}\end{cases}$ (2.1)

where $\operatorname{Id}$ denotes the identity operator on $\mathbb{R}^{d}$.
We recall the notions of exponential dichotomy and exponential trichotomy as
follows.

######  Definition 2.1.

The linear system $(A_{n})_{n\in\mathbb{Z}}$ admits an exponential dichotomy
if there exist constants $K>0$, $0<\alpha_{+}<1<\beta_{-}$ and projections
$\Pi^{s}_{n},\Pi^{u}_{n}$, $n\in\mathbb{Z}$ on $\mathbb{R}^{d}$ such that the
following conditions hold:

  * •

for every $n\in\mathbb{Z}$, $\Pi^{s}_{n}+\Pi^{u}_{n}=\operatorname{Id};$

  * •

for every $n\in\mathbb{Z}$, $\Pi^{s}_{n+1}A_{n}=A_{n}\Pi^{s}_{n}$;

  * •

for $m\geq n$,

$\displaystyle\|\mathcal{A}(m,n)\Pi^{s}_{n}\|\leq
K\alpha_{+}^{m-n},\qquad\|\mathcal{A}(n,m)\Pi^{u}_{m}\|\leq K\beta_{-}^{n-m}.$

In addition, if there exist $\alpha_{-}\leq\alpha_{+}$ and
$\beta_{+}\geq\beta_{-}$ such that

$\displaystyle\|\mathcal{A}(n,m)\Pi^{s}_{m}\|\leq
K\alpha_{-}^{n-m},\qquad\|\mathcal{A}(m,n)\Pi^{u}_{n}\|\leq K\beta_{+}^{m-n}$

hold for $m\geq n$, then $(A_{n})_{n\in\mathbb{Z}}$ is said to admit a strong
exponential dichotomy.

######  Definition 2.2.

The linear system $(A_{n})_{n\in\mathbb{Z}}$ admits an exponential trichotomy
if there exist constants

$K>0,\quad 0<\mu_{+}<\lambda_{-}\leq 1\leq\lambda_{+}<\varrho_{-},$

and projections $\Pi^{s}_{n},\Pi^{c}_{n},\Pi^{u}_{n}$, $n\in\mathbb{Z}$ on
$\mathbb{R}^{d}$ such that the following conditions hold:

  * •

for every $n\in\mathbb{Z}$,

$\Pi^{s}_{n}+\Pi^{c}_{n}+\Pi^{u}_{n}=\operatorname{Id};$ (2.2)

  * •

for every $n\in\mathbb{Z}$,

$\Pi^{s}_{n+1}A_{n}=A_{n}\Pi^{s}_{n},\quad\Pi^{c}_{n+1}A_{n}=A_{n}\Pi^{c}_{n},\quad\Pi^{u}_{n+1}A_{n}=A_{n}\Pi^{u}_{n};$
(2.3)

  * •

for $m\geq n$,

$\displaystyle\|\mathcal{A}(m,n)\Pi^{s}_{n}\|\leq K\mu_{+}^{m-n},$
$\displaystyle\|\mathcal{A}(n,m)\Pi^{u}_{m}\|\leq K\varrho_{-}^{n-m},$ (2.4)
$\displaystyle\|\mathcal{A}(m,n)\Pi^{c}_{n}\|\leq K\lambda_{+}^{m-n},$
$\displaystyle\|\mathcal{A}(n,m)\Pi^{c}_{m}\|\leq K\lambda_{-}^{n-m}.$ (2.5)

In addition, if there exist $0<\mu_{-}\leq\mu_{+}$ and
$\varrho_{+}\geq\varrho_{-}$ such that

$\displaystyle\|\mathcal{A}(n,m)\Pi^{s}_{m}\|\leq
K\mu_{-}^{n-m},\qquad\|\mathcal{A}(m,n)\Pi^{u}_{n}\|\leq K\varrho_{+}^{m-n}$
(2.6)

hold for $m\geq n$, then $(A_{n})_{n\in\mathbb{Z}}$ is said to admit a strong
exponential trichotomy.

######  Remark 2.1.

It is easy to show that a sequence $(A_{n})_{n\in\mathbb{Z}}$ which admits an
exponential dichotomy (resp. trichotomy) admits a strong exponential dichotomy
(resp. trichotomy) if any only if

$\sup_{n\in\mathbb{Z}}\|A_{n}\|<+\infty\quad\text{and}\quad\sup_{n\in\mathbb{Z}}\|A_{n}^{-1}\|<+\infty.$
(2.7)

Indeed, suppose for example that $(A_{n})_{n\in\mathbb{Z}}$ admits a strong
exponential trichotomy. Then, for $m\geq n$, (2.2), (2.4)-(2.6) give that

$\displaystyle\|\mathcal{A}(m,n)\|$
$\displaystyle=\|\mathcal{A}(m,n)(\Pi^{s}_{n}+\Pi^{c}_{n}+\Pi^{u}_{n})\|$
$\displaystyle\leq\|\mathcal{A}(m,n)\Pi^{s}_{n}\|+\|\mathcal{A}(m,n)\Pi^{c}_{n}\|+\|\mathcal{A}(m,n)\Pi^{u}_{n}\|$
$\displaystyle\leq 3K\varrho_{+}^{m-n},$ (2.8)

and

$\begin{split}\|\mathcal{A}(n,m)\|&=\|\mathcal{A}(n,m)(\Pi^{s}_{n}+\Pi^{c}_{n}+\Pi^{u}_{n})\|\\\
&\leq\|\mathcal{A}(n,m)\Pi^{s}_{n}\|+\|\mathcal{A}(n,m)\Pi^{c}_{n}\|+\|\mathcal{A}(n,m)\Pi^{u}_{n}\|\\\
&\leq 3K\mu_{-}^{n-m}.\end{split}$ (2.9)

Clearly, (2.8) and (2.9) imply (2.7). The converse implication is obvious.

######  Remark 2.2.

Assume that a sequence $(A_{n})_{n\in\mathbb{Z}}$ admits a strong exponential
trichotomy. Then, it is easy to verify the following:

  * •

for $\gamma\in(0,\mu_{-})$, the sequence $(\gamma^{-1}A_{n})_{n\in\mathbb{Z}}$
admits a strong exponential dichotomy with respect to projections
$\Pi_{n}^{\gamma,s}=0$ and $\Pi_{n}^{\gamma,u}=\operatorname{Id}$ for
$n\in\mathbb{Z}$;

  * •

for $\gamma\in(\mu_{+},\lambda_{-})$, the sequence
$(\gamma^{-1}A_{n})_{n\in\mathbb{Z}}$ admits a strong exponential dichotomy
with respect to projections $\Pi_{n}^{\gamma,s}=\Pi_{n}^{s}$ and
$\Pi_{n}^{\gamma,u}=\Pi_{n}^{c}+\Pi_{n}^{u}$ for $n\in\mathbb{Z}$;

  * •

for $\gamma\in(\lambda_{+},\varrho_{-})$, the sequence
$(\gamma^{-1}A_{n})_{n\in\mathbb{Z}}$ admits a strong exponential dichotomy
with respect to projections $\Pi_{n}^{\gamma,s}=\Pi_{n}^{s}+\Pi_{n}^{c}$ and
$\Pi_{n}^{\gamma,u}=\Pi_{n}^{u}$ for $n\in\mathbb{Z}$;

  * •

for $\gamma\in(\varrho_{+},\infty)$, the sequence
$(\gamma^{-1}A_{n})_{n\in\mathbb{Z}}$ admits a strong exponential dichotomy
with respect to projections $\Pi_{n}^{\gamma,s}=\operatorname{Id}$ and
$\Pi_{n}^{\gamma,u}=0$ for $n\in\mathbb{Z}$.

Let us fix an arbitrary sequence $(A_{n})_{n\in\mathbb{Z}}$ that admits a
strong exponential trichotomy and which does not admit an exponential
dichotomy. We define the dichotomy spectrum of $(A_{n})_{n\in\mathbb{Z}}$ by

$\displaystyle\Sigma(A_{n}):=\Big{\\{}$
$\displaystyle\gamma\in(0,\infty):(\gamma^{-1}A_{n})_{n\in\mathbb{Z}}~{}\mbox{admits
no exponential dichotomy}\Big{\\}}.$ (2.10)

According to the Spectral Theorem ([2]) and Remark 2.2, we know that there is
$r\leq d-1$ such that

$\displaystyle\Sigma=\Sigma(A_{n})=\bigcup_{i=1}^{r}[a_{i},b_{i}]\cup[\nu_{-},\nu_{+}]\subset[\mu_{-},\mu_{+}]\cup[\lambda_{-},\lambda_{+}]\cup[\varrho_{-},\varrho_{+}],$
(2.11)

where

$\mu_{-}\leq a_{1}\leq b_{1}<\ldots<a_{\ell}\leq
b_{\ell}\leq\mu_{+}<\nu_{-}\leq 1\leq\nu_{+}$

and

$\varrho_{-}\leq a_{\ell+1}\leq b_{\ell+1}<\ldots<a_{r}\leq
b_{r}\leq\varrho_{+},$

with a certain $\ell\in\\{0,\ldots,r\\}$. In the sequel we will assume that
$r\geq 1$, i.e. that $\Sigma$ does not consist only of $[\nu_{-},\nu_{+}]$.
Observe that in (2.11) we have supposed that
$\Sigma\cap[\lambda_{-},\lambda_{+}]$ consists of a single interval. It can be
easily seen that this can be assumed without any loss of generality, as it can
be achieved by redefining projections (and corresponding constants) in the
trichotomy of the sequence $(A_{n})_{n\in\mathbb{Z}}$. Moreover, remark that
when $\ell=0$ (or $\ell=r$), the system has only expansive (or contractive)
and center directions, which is a special case of the partial hyperbolicity.
In what follows, we do not discuss the case of $\ell\in\\{0,r\\}$ because it
can be reduced to the case of $\ell\in\\{1,\ldots,r-1\\}$ by adding some dummy
variables to the expansive (or contractive) part. Finally, we note that in
(2.11) the union
$\bigcup_{i=1}^{r}[a_{i},b_{i}]\subset[\mu_{-},\mu_{+}]\cup[\varrho_{-},\varrho_{+}]$
corresponds to the hyperbolic part of $\Sigma$ as
$1\notin\bigcup_{i=1}^{r}[a_{i},b_{i}]$, while the interval
$[\nu_{-},\nu_{+}]$ corresponds to the center part of $\Sigma$ as
$1\in[\nu_{-},\nu_{+}]\subset[\lambda_{-},\lambda_{+}]$.

Furthermore, according to [28], we may assume that $(A_{n})_{n\in\mathbb{Z}}$
is block-diagonalized, i.e.,

$A_{n}={\rm
diag}(A_{n}^{1},\ldots,A_{n}^{\ell},A_{n}^{c},A_{n}^{\ell+1},\ldots,A_{n}^{r})=:{\rm
diag}(A_{n}^{s},A_{n}^{c},A_{n}^{u}),$

where

$\Sigma(A_{n}^{i})=[a_{i},b_{i}]\ \text{for $i=1,\ldots,r$
and}\quad\Sigma(A_{n}^{c})=[\nu_{-},\nu_{+}].$ (2.12)

The above block-diagonal form of operators $A_{n}$ generates the corresponding
decomposition of $\mathbb{R}^{d}$ given by

$\mathbb{R}^{d}=X_{1}\oplus\cdots\oplus X_{\ell}\oplus X_{c}\oplus
X_{\ell+1}\oplus\cdots\oplus X_{r}.$

For convenience, denote

$\displaystyle X_{s}:=X_{1}\oplus\ldots\oplus X_{\ell},\quad
X_{u}:=X_{\ell+1}\oplus\ldots\oplus X_{r},\quad X_{su}:=X_{s}\oplus X_{u},$
$\displaystyle x_{s}:=x_{1}+\ldots+x_{\ell}\in X_{s},\qquad
x_{u}:=x_{\ell+1}+\ldots+x_{r}\in X_{u}$

for $x_{i}\in X_{i}$, $i=1,\ldots,r$, and denote

$\pi_{i}x:=x_{i},\quad\pi_{s}x:=x_{s},\quad\pi_{c}x:=x_{c},\quad\pi_{u}x:=x_{u},\quad\pi_{su}:=\pi_{s}+\pi_{u},$

for $x=x_{s}+x_{c}+x_{u}\in\mathbb{R}^{d}$. Taking into account (2.12), it
follows that for a sufficiently small $\varsigma>0$ there exists $C>0$ such
that

$\displaystyle\begin{split}&\|{\mathcal{A}}(m,n)\pi_{i}x\|\leq
C(b_{i}+\varsigma)^{m-n}\|\pi_{i}x\|,\quad m\geq n,\\\
&\|{\mathcal{A}}(m,n)\pi_{i}x\|\leq C(a_{i}-\varsigma)^{m-n}\|\pi_{i}x\|,\quad
m\leq n,\\\ &\|{\mathcal{A}}(m,n)\pi_{c}x\|\leq
C(\nu_{+}+\varsigma)^{m-n}\|\pi_{c}x\|,\quad m\geq n,\\\
&\|{\mathcal{A}}(m,n)\pi_{c}x\|\leq
C(\nu_{-}-\varsigma)^{m-n}\|\pi_{c}x\|,\quad m\leq n,\end{split}$ (2.13)

for $i=1,\ldots,r$.

######  Remark 2.1.

In order to simplify the notation, in the sequel we will assume that estimates
in (2.13) hold with $\varsigma=0$. This will not cause any issues since every
strict inequality (see for example (4.20)-(4.21)) for $a_{i},b_{i},\nu_{-}$
and $\nu_{+}$ holds if and only if it holds for
$a_{i}-\varsigma,b_{i}+\varsigma,\nu_{-}-\varsigma$ and $\nu_{+}+\varsigma$
with a sufficiently small $\varsigma>0$.

Next, set

$Y:=\Big{\\{}\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\subset\mathbb{R}^{d}:\sum_{n\in\mathbb{Z}}\|x_{n}\|^{2}<\infty\Big{\\}},$
(2.14)

which is a Banach space when equipped with the norm
$\|\mathbf{x}\|:=(\sum_{n\in\mathbb{Z}}\|x_{n}\|^{2})^{\frac{1}{2}}.$ In
addition, $Y$ is actually a Hilbert space with respect to the scalar product
given by

$\langle\mathbf{x},\mathbf{y}\rangle_{Y}:=\sum_{n\in\mathbb{Z}}\langle
x_{n},y_{n}\rangle\quad\mathbf{x}=(x_{n})_{n\in\mathbb{Z}},\
\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}\in Y,$ (2.15)

where $\langle\cdot,\cdot\rangle$ is the Euclidean scalar product on
$\mathbb{R}^{d}$.

This enables us to define a linear operator $\mathbb{A}\colon Y\to Y$ by

$(\mathbb{A}\mathbf{x})_{n}:=A_{n-1}x_{n-1},\quad\forall\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\in
Y,~{}\forall n\in\mathbb{Z}.$ (2.16)

It follows easily from (2.7) that $\mathbb{A}$ is bounded and invertible.
Indeed, $\mathbb{A}^{-1}$ is given by

$(\mathbb{A}^{-1}\mathbf{x})_{n}=A_{n+1}^{-1}x_{n+1},\quad\forall\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\in
Y,~{}\forall n\in\mathbb{Z}.$

We define projections $\Pi_{Y}^{s}$, $\Pi_{Y}^{u}$ and $\Pi_{Y}^{c}$ on $Y$ by

$(\Pi_{Y}^{s}\mathbf{x})_{n}:=\Pi^{s}_{n}x_{n},\quad(\Pi_{Y}^{c}\mathbf{x})_{n}:=\Pi^{c}_{n}x_{n},\quad\text{and}\quad(\Pi_{Y}^{u}\mathbf{x})_{n}:=\Pi^{u}_{n}x_{n},$

for $n\in\mathbb{Z}$ and $\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\in Y$. One can
easily verify that:

  * •

$\Pi_{Y}^{s}+\Pi_{Y}^{c}+\Pi_{Y}^{u}=\operatorname{Id};$

  * •

$\mathbb{A}\Pi_{Y}^{*}=\Pi_{Y}^{*}\mathbb{A}$ for $*=s,c,u$;

  * •

for $m\geq 0$,

$\displaystyle\begin{split}&\|\mathbb{A}^{m}\Pi_{Y}^{s}\|\leq
K\mu_{+}^{m},\qquad\|\mathbb{A}^{-m}\Pi_{Y}^{s}\|\leq K\mu_{-}^{-m},\\\
&\|\mathbb{A}^{m}\Pi_{Y}^{c}\|\leq
K\lambda_{+}^{m},\qquad\|\mathbb{A}^{-m}\Pi_{Y}^{c}\|\leq
K\lambda_{-}^{-m},\\\ &\|\mathbb{A}^{m}\Pi_{Y}^{u}\|\leq
K\varrho_{+}^{m},\qquad\|\mathbb{A}^{-m}\Pi_{Y}^{u}\|\leq
K\varrho_{-}^{-m}.\end{split}$ (2.17)

In the sequel, by $Y^{*}$ we will denote the image of the projection
$\Pi_{Y}^{*}$ for $*=s,c,u$.

##  3\. Center manifolds

This section is devoted to the existence of smooth center manifolds of a
nonautonomous system whose linear part $(A_{n})_{n\in\mathbb{Z}}$ admits a
strong exponential trichotomy (but does not admit an exponential dichotomy).
Take $N\in\mathbb{N}$ and consider a sequence $(f_{n})_{n\in\mathbb{Z}}$ of
$C^{N+1}$ maps $f_{n}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$. In what follows,
by $d_{x}^{i}f_{n}$ we denote the $i$-th derivative of $f_{n}$ at $x$.
Moreover, we will write $d_{x}f_{n}$ instead of $d_{x}^{1}f_{n}$. Suppose
that:

  * •

$(A_{n})_{n\in\mathbb{Z}}$ admits a strong exponential trichotomy (but does
not admit an exponential dichotomy). In particular, $\Sigma(A_{n})$ satisfies
(2.11);

  * •

for $n\in\mathbb{Z}$,

$f_{n}(0)=0\quad\text{and}\quad d_{0}f_{n}=0;$ (3.1)

  * •

for $n\in\mathbb{Z}$, $2\leq i\leq N+1$ and $x\in\mathbb{R}^{d}$,

$\|d_{x}f_{n}\|\leq\delta\quad\text{and}\quad\|d^{i}_{x}f_{n}\|\leq M,$ (3.2)

where $\delta,M>0$ are some constants. In the sequel, we will assume that
$\delta$ can be made sufficiently small.

Let $\operatorname{Im}\Pi^{*}_{n}$ denote the image of $\Pi^{*}_{n}$ for
$*=s,c,u$. Moreover, set

$F_{n}:=A_{n}+f_{n}\colon\mathbb{R}^{d}\to\mathbb{R}^{d},\quad
n\in\mathbb{Z}.$

Then, we have the following main result of this section.

######  Proposition 3.1.

Let $(F_{n})_{n\in\mathbb{Z}}$ be given above and suppose that the spectral
gap condition

$\displaystyle b_{\ell}<\nu_{-}^{N},\qquad\nu_{+}^{N}<a_{\ell+1},$ (3.3)

holds. Then, there exists a sequence of $C^{N}$ maps
$\phi_{n}\colon\operatorname{Im}\Pi^{c}_{n}\to\operatorname{Im}\Pi^{s}_{n}\oplus\operatorname{Im}\Pi^{u}_{n}$,
$n\in\mathbb{Z}$, such that for $n\in\mathbb{Z}$:

  * •

$\phi_{n}(0)=0$, $d_{0}\phi_{n}=0$ and $\|d_{x}^{i}\phi_{n}\|\leq\tilde{M}$
near $x=0$ for $i=2,\ldots,N$, with respect to some constant $\tilde{M}>0$
independent on $n$ and $i$;

  * •

$F_{n}(\Gamma_{n})=\Gamma_{n+1}$, where
$\Gamma_{n}:=\\{x_{c}+\phi_{n}(x_{c}):\
x_{c}\in\operatorname{Im}\Pi^{c}_{n}\\}$ is the center manifold;

  * •

for any $x_{c}\in\operatorname{Im}\Pi^{c}_{n}$ and
$\gamma_{1}\in(\mu_{+},\nu_{-}^{N}),\gamma_{2}\in(\nu_{+}^{N},\varrho_{-})$,
we have

$\sup_{m>n}\left\\{\gamma_{2}^{-(m-n)}\|(F_{m-1}\circ\ldots\circ
F_{n})(x_{c}+\phi_{n}(x_{c}))\|\right\\}<\infty,$ (3.4)

and

$\sup_{m<n}\left\\{\gamma_{1}^{-(m-n)}\|(F_{m}^{-1}\circ\ldots\circ
F_{n-1}^{-1})(x_{c}+\phi_{n}(x_{c}))\|\right\\}<\infty.$ (3.5)

Remark that (3.3) holds if

$\mu_{+}<\lambda_{-}^{N}\quad\text{and}\quad\lambda_{+}^{N}<\varrho_{-},$
(3.6)

which is expressed purely in terms of constants associated to the trichotomy
of the sequence $(A_{n})_{n}$.

###### Proof of Proposition 3.1.

We define $\mathbb{F}\colon Y\to Y$ by

$(\mathbb{F}(\mathbf{x}))_{n}:=F_{n-1}(x_{n-1}):=A_{n-1}x_{n-1}+f_{n-1}(x_{n-1}),\quad\forall
n\in\mathbb{Z},$ (3.7)

for $\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\in Y$, and recall that
$\mathbb{A}\colon Y\to Y$ is given by (2.16).

######  Lemma 3.1.

$\mathbb{F}$ is $C^{N}$, $\mathbb{F}(\boldsymbol{0})=\boldsymbol{0}$ and
$d_{\boldsymbol{0}}\mathbb{F}=\mathbb{A}$. Moreover,

$\|d_{\bf x}(\mathbb{F}-\mathbb{A})\|\leq\delta,\quad\|d^{i}_{\bf
x}\mathbb{F}\|\leq M,\quad\forall{\bf x}\in Y,$ (3.8)

for $2\leq i\leq N$, where $\delta>0$ and $M>0$ are as in (3.2).

###### Proof of Lemma 3.1.

We first claim that $\mathbb{F}$ is differentiable and that

$d_{\mathbf{x}}\mathbb{F}(\boldsymbol{\xi})=(A_{n-1}\xi_{n-1}+d_{x_{n-1}}f_{n-1}\xi_{n-1})_{n\in\mathbb{Z}},$
(3.9)

for
$\mathbf{x}=(x_{n})_{n\in\mathbb{Z}},\boldsymbol{\xi}=(\xi_{n})_{n\in\mathbb{Z}}\in
Y$. We define $L\colon Y\to Y$ by

$L\boldsymbol{\xi}=(A_{n-1}\xi_{n-1}+d_{x_{n-1}}f_{n-1}\xi_{n-1})_{n\in\mathbb{Z}}.$

It is easy to verify that $L$ is a bounded linear operator on $Y$. Take
$\mathbf{y}\in Y$. Note that

$\begin{split}(\mathbb{F}(\mathbf{x}+\mathbf{y})-\mathbb{F}(\mathbf{x})-L\mathbf{y})_{n}&=\int_{0}^{1}(d_{x_{n-1}+ty_{n-1}}f_{n-1}y_{n-1}-d_{x_{n-1}}f_{n-1}y_{n-1})\,dt,\end{split}$

which together with Mean Value Theorem and (3.2) implies that

$\begin{split}\|(\mathbb{F}(\mathbf{x}+\mathbf{y})-\mathbb{F}(\mathbf{x})-L\mathbf{y})_{n}\|&\leq\int_{0}^{1}\|d_{x_{n-1}+ty_{n-1}}f_{n-1}y_{n-1}-d_{x_{n-1}}f_{n-1}y_{n-1}\|\,dt\\\
&\leq\left(\sup_{x\in\mathbb{R}^{d}}\|d^{2}_{x}f_{n-1}\|\right)\|y_{n-1}\|^{2}\\\
&\leq M\|y_{n-1}\|^{2},\end{split}$

for $n\in\mathbb{Z}$. Hence,

$\|\mathbb{F}(\mathbf{x}+\mathbf{y})-\mathbb{F}(\mathbf{x})-L\mathbf{y}\|\leq
M\|\mathbf{y}\|^{2}.$

This readily implies that (3.9) holds. In particular, we have that
$d_{\mathbf{0}}\mathbb{F}=\mathbb{A}$.

Next, we further prove that for each $2\leq i\leq N+1$

$d_{\mathbf{x}}^{i}\mathbb{F}(\boldsymbol{\xi}^{1},\ldots,\boldsymbol{\xi}^{i})=\left(d^{i}_{x_{n-1}}f_{n-1}(\xi^{1}_{n-1},\ldots,\xi^{i}_{n-1})\right)_{n\in\mathbb{Z}},$
(3.10)

for
$\mathbf{x}=(x_{n})_{n\in\mathbb{Z}},\boldsymbol{\xi}^{j}=(\xi^{j}_{n})_{n\in\mathbb{Z}}\in
Y$ with $2\leq j\leq i$. This will be proved by induction. When $i=2$, let
$\mathbb{D}^{2}\colon Y\times Y\to Y$ be a multilinear operator defined by

$\mathbb{D}^{2}(\boldsymbol{\xi}^{1},\boldsymbol{\xi}^{2}):=\left(d^{2}_{x_{n-1}}f_{n-1}(\xi^{1}_{n-1},\xi^{2}_{n-1})\right)_{n\in\mathbb{Z}}$

for $\boldsymbol{\xi}^{1},\boldsymbol{\xi}^{2}\in Y$. Observe that the second
inequality of (3.2) (with $i=2$) implies that
$\|\mathbb{D}^{2}(\boldsymbol{\xi}^{1},\boldsymbol{\xi}^{2})\|\leq
M\|\boldsymbol{\xi}^{1}\|\|\boldsymbol{\xi}^{2}\|,$ and therefore
$\mathbb{D}^{2}$ is bounded. Moreover, by the second inequality of (3.2) (with
$i=3$) again and by the Mean Value Theorem, we deduce that

$\displaystyle\|d_{\mathbf{x}+\mathbf{y}}\mathbb{F}(\boldsymbol{\xi})-d_{\mathbf{x}}\mathbb{F}(\boldsymbol{\xi})-\mathbb{D}^{2}(\boldsymbol{\xi},\mathbf{y})\|$
$\displaystyle=\Big{\|}\Big{(}d_{x_{n-1}+y_{n-1}}f_{n-1}(\xi_{n-1})-d_{x_{n-1}}f_{n-1}(\xi_{n-1})-d^{2}_{x_{n-1}}f_{n-1}(\xi_{n-1},y_{n-1})\Big{)}_{n\in\mathbb{Z}}\Big{\|}$
$\displaystyle\leq\left(\sum_{n\in\mathbb{Z}}\int_{0}^{1}\|(d^{2}_{x_{n-1}+ty_{n-1}}f_{n-1}-d^{2}_{x_{n-1}}f_{n-1})(\xi_{n-1},y_{n-1})\|^{2}\,dt\right)^{\frac{1}{2}}$
$\displaystyle\leq M\|\mathbf{y}\|^{2}\|\boldsymbol{\xi}\|,$ (3.11)

for
$\mathbf{y}=(y_{n})_{n\in\mathbb{Z}},~{}\boldsymbol{\xi}=(\xi_{n})_{n\in\mathbb{Z}}\in
Y$, which implies that

$\displaystyle\lim_{\bf y\to 0}\frac{\|d_{\bf x+\bf y}\mathbb{F}-d_{\bf
x}\mathbb{F}-\mathbb{D}^{2}(\cdot,\mathbf{y})\|}{\|\bf y\|}$
$\displaystyle=\lim_{\bf y\to 0}\,\sup_{\boldsymbol{\xi}\in
Y\backslash\\{0\\}}\frac{\|d_{\mathbf{x}+\mathbf{y}}\mathbb{F}(\boldsymbol{\xi})-d_{\mathbf{x}}\mathbb{F}(\boldsymbol{\xi})-\mathbb{D}^{2}(\boldsymbol{\xi},\mathbf{y})\|}{\|\bf
y\|\|\boldsymbol{\xi}\|}=0.$ (3.12)

This implies that $\mathbb{F}$ is twice differentiable and that the second
order derivative at ${\bf x}$ is given by $d_{{\bf x}}^{2}F=\mathbb{D}^{2}$.
Hence, (3.10) holds for $i=2$. Assume now that (3.10) holds for some $2\leq
i\leq N$, and define a multilinear operator $\mathbb{D}^{i+1}\colon Y^{i+1}\to
Y$ by

$\mathbb{D}^{i+1}(\boldsymbol{\xi}^{1},\ldots,\boldsymbol{\xi}^{i+1}):=\left(d_{x_{n-1}}^{i+1}f_{n-1}(\xi_{n-1}^{1},\ldots,\xi_{n-1}^{i+1})\right)_{n\in\mathbb{Z}},$

for $\boldsymbol{\xi}^{j}=(\xi_{n}^{j})_{n\in\mathbb{Z}}\in Y$ and $2\leq
j\leq i+1$, which is clearly bounded. Then, similarly to (3)-(3), we can show
that (3.10) holds for $i+1$. By induction on $i$ we see that (3.10) holds for
all $2\leq i\leq N+1$. This shows that $\mathbb{F}$ is differentiable $N+1$
times and therefore it is $C^{N}$.

It remains to establish (3.8). We infer from (3.9) and the first inequality in
(3.2) that

$\displaystyle\|d_{\bf x}(\mathbb{F}-\mathbb{A})\|$
$\displaystyle=\sup_{\boldsymbol{\xi}\in Y\backslash\\{0\\}}\frac{\|d_{\bf
x}(\mathbb{F}-\mathbb{A})\boldsymbol{\xi}\|}{\|\boldsymbol{\xi}\|}=\sup_{\boldsymbol{\xi}\in
Y\backslash\\{0\\}}\frac{\|(d_{x_{n-1}}f_{n-1}(\xi_{n-1}))_{n\in\mathbb{Z}}\|}{\|\boldsymbol{\xi}\|}$
$\displaystyle=\sup_{\boldsymbol{\xi}\in
Y\backslash\\{0\\}}\frac{\left(\sum_{n\in\mathbb{Z}}\|d_{x_{n-1}}f_{n-1}(\xi_{n-1})\|^{2}\right)^{\frac{1}{2}}}{\|\xi\|}$
$\displaystyle\leq\sup_{\boldsymbol{\xi}\in
Y\backslash\\{0\\}}\frac{\left(\sum_{n\in\mathbb{Z}}\delta^{2}\|\xi_{n-1}\|^{2}\right)^{\frac{1}{2}}}{\|\xi\|}$
$\displaystyle=\sup_{\boldsymbol{\xi}\in
Y\backslash\\{0\\}}\frac{\delta\|\boldsymbol{\xi}\|}{\|\boldsymbol{\xi}\|}=\delta,$
(3.13)

which proves the first inequality of (3.8). Moreover, by (3.10) and the second
inequality of (3.2), similarly to (3), we can prove the second inequality of
(3.8). This completes the proof of Lemma 3.1. ∎

We continue with the proof of Proposition 3.1. It follows from (3.6), Lemma
3.1 and the Center Manifold Theorem on Banach spaces (see e.g. [17, Theorem
4.1]) that there exists a $C^{N}$ map $\Phi\colon Y^{c}\to Y^{s}\oplus Y^{u}$
such that $\Phi(\boldsymbol{0})=\boldsymbol{0}$, $d_{\boldsymbol{0}}\Phi=0$,
$\|d_{\bf x}^{i}\Phi\|\leq\tilde{M}$ near $\bf x=0$ for $i=2,\ldots,N$ with a
constant $\tilde{M}>0$ and $\mathbb{F}(\Gamma)=\Gamma$, where

$\Gamma:=\\{{\mathbf{x}}_{c}+\Phi({\mathbf{x}}_{c}):\ {\mathbf{x}}_{c}\in
Y^{c}\\}.$

In addition, $\Gamma$ is also characterized as

$\Gamma=\bigg{\\{}{\bf x}\in Y:\sup_{n\geq
0}\\{\gamma_{2}^{-n}\|\mathbb{F}^{n}({\bf x})\|\\}<\infty,\quad\sup_{n\geq
0}\\{\gamma_{1}^{n}\|\mathbb{F}^{-n}({\bf x})\|\\}<\infty\bigg{\\}}.$ (3.14)

Next, for any given $n\in\mathbb{Z}$ and
$x_{c}\in\operatorname{Im}\Pi^{c}_{n}$, define ${\bf
v}=(v_{m})_{m\in\mathbb{Z}}\in Y^{c}$ by

$\displaystyle v_{n}:=x_{c}\quad{\rm and}\quad v_{m}:=0\quad{\rm for}~{}m\neq
n,$ (3.15)

and define
$\phi_{n}:\operatorname{Im}\Pi^{c}_{n}\to\operatorname{Im}\Pi^{s}_{n}\oplus\operatorname{Im}\Pi^{u}_{n}$
by

$\displaystyle\phi_{n}(x_{c}):=(\Phi({\bf v}))_{n}.$ (3.16)

Similarly to (3.7) and (3.10), we see that each $\phi_{n}$, $n\in\mathbb{Z}$,
is $C^{N}$ such that

$d^{i}_{x_{c}}\phi_{n}(\zeta^{1},\ldots,\zeta^{i})=(d^{i}_{\bf
v}\Phi({\boldsymbol{\zeta}}^{1},\ldots,{\boldsymbol{\zeta}}^{i}))_{n},\quad\forall
1\leq i\leq N,$

for $\zeta^{1},\ldots,\zeta^{i}\in\operatorname{Im}\Pi^{c}_{n}$, where
${\boldsymbol{\zeta}}^{i}=(\zeta_{m}^{i})_{m\in\mathbb{Z}}\in Y^{c}$ is
defined by $\zeta_{n}^{i}:=\zeta^{i}$ and $\zeta_{m}^{i}=0$ for $m\neq n$.
Thus, $\phi_{n}(0)=0$, $d_{0}\phi_{n}=0$ and
$\|d_{x_{c}}^{i}\phi_{n}\|\leq\tilde{M}$ near $x_{c}=0$ for $i=2,\ldots,N$.

For the invariance property, we first show that
$F_{n}(\Gamma_{n})\subset\Gamma_{n+1}$ for each $n\in\mathbb{Z}$. Fix
$n\in\mathbb{Z}$ and choose any $x_{c}+\phi_{n}(x_{c})\in\Gamma_{n}$.
According to the definitions of $\phi_{n}$ and $\mathbb{F}$,

$\displaystyle F_{n}(x_{c}+\phi_{n}(x_{c}))=F_{n}(({\bf v}+\Phi({\bf
v}))_{n})=(\mathbb{F}({\bf v}+\Phi({\bf v})))_{n+1},$ (3.17)

where ${\bf v}\in Y^{c}$ is defined by (3.15). Since ${\bf v}+\Phi({\bf
v})\in\Gamma$ and $\mathbb{F}(\Gamma)=\Gamma$, there exists $\tilde{\bf v}\in
Y^{c}$ such that $\mathbb{F}({\bf v}+\Phi({\bf v}))=\tilde{\bf
v}+\Phi(\tilde{\bf v})\in\Gamma$. Set $\tilde{x}_{c}:=(\tilde{\bf
v})_{n+1}\in\operatorname{Im}\Pi^{c}_{n+1}$. We claim that

$\displaystyle(\tilde{\bf v}+\Phi(\tilde{\bf
v}))_{n+1}=\tilde{x}_{c}+\phi_{n+1}(\tilde{x}_{c}).$ (3.18)

Indeed, let $\hat{\bf v}=(\hat{v}_{m})_{m\in\mathbb{Z}}\in Y^{c}$ be defined
by

$\hat{v}_{n+1}:=\tilde{x}_{c}\quad{\rm and}\quad\hat{v}_{m}:=0\quad{\rm
for}~{}m\neq n+1,$

which means that $(\tilde{\bf v})_{n+1}=(\hat{\bf v})_{n+1}$. Due to the
definition of $\phi_{n+1}$, it suffices to prove that

$\displaystyle\Phi(\tilde{\bf v})=\Phi(\hat{\bf v}).$ (3.19)

For this purpose, we need the following result.

######  Lemma 3.2.

Let ${\bf z}^{1}=(z_{n}^{1})_{n\in\mathbb{Z}}$ and ${\bf
z}^{2}=(z_{n}^{2})_{n\in\mathbb{Z}}$ belong to $Y^{c}$ and satisfy
$z_{k}^{1}=z_{k}^{2}$ for some $k\in\mathbb{Z}$. Then, $(\Phi({\bf
z}^{1}))_{k}=(\Phi({\bf z}^{2}))_{k}$.

###### Proof of Lemma 3.2.

Assume that $\Phi({\bf z}^{1})_{k}\neq\Phi({\bf z}^{2})_{k}$. Let ${\bf
y}=(y_{n})_{n\in\mathbb{Z}}\in Y$ be defined by

$y_{n}=\begin{cases}(\Phi({\bf z}^{1}))_{n}&\text{if}~{}~{}n\neq k,\\\
(\Phi({\bf z}^{2}))_{k}&\text{if}~{}~{}n=k.\end{cases}$ (3.20)

Then, we see from (3.7) that

$(\mathbb{F}^{m}({\bf z}^{1}+{\bf
y}))_{n}=\begin{cases}F_{n-1}\circ\ldots\circ F_{n-m}(z^{1}_{n-m}+(\Phi({\bf
z}^{1}))_{n-m})&\text{if}~{}~{}n\neq m+k,\\\ F_{n-1}\circ\ldots\circ
F_{n-m}(z^{2}_{n-m}+(\Phi({\bf
z}^{2}))_{n-m})&\text{if}~{}~{}n=m+k,\end{cases}$

and that

$(\mathbb{F}^{-m}({\bf z}^{1}+{\bf
y}))_{n}=\begin{cases}F_{n}^{-1}\circ\ldots\circ
F_{n+m-1}^{-1}(z^{1}_{n+m}+(\Phi({\bf z}^{1}))_{n+m})&\text{if}~{}~{}n+m\neq
k,\\\ F_{n}^{-1}\circ\ldots\circ F_{n+m-1}^{-1}(z^{2}_{n+m}+(\Phi({\bf
z}^{2}))_{n+m})&\text{if}~{}~{}n+m=k,\end{cases}$

for every $n\in\mathbb{Z}$ and $m\in\mathbb{N}$. This together with (3.14)
easily implies that

$\sup_{m\geq 0}\\{\gamma_{2}^{-m}\|\mathbb{F}^{m}({\bf z}^{1}+{\bf
y})\|\\}<\infty,\quad\sup_{m\geq 0}\\{\gamma_{1}^{m}\|\mathbb{F}^{-m}({\bf
z}^{1}+{\bf y})\|\\}<+\infty.$

By (3.14) we have that ${\bf z}^{1}+{\bf y}\in\Gamma$, and thus ${\bf
y}=\Phi({\bf z}^{1})$. This contradicts (3.20) and the proof of lemma 3.2 is
completed. ∎

By Lemma 3.2, we have that (3.19) holds, which implies (3.18). Then, combining
(3.17) and (3.18), we obtain that

$F_{n}(x_{c}+\phi_{n}(x_{c}))=(\mathbb{F}({\bf v}+\Phi({\bf
v})))_{n+1}=(\tilde{\bf v}+\Phi(\tilde{\bf
v}))_{n+1}=\tilde{x}_{c}+\phi_{n+1}(\tilde{x}_{c})\in\Gamma_{n+1},$

and therefore $F_{n}(\Gamma_{n})\subset\Gamma_{n+1}$. Conversely, similar
arguments imply that $\Gamma_{n+1}\subset F_{n}(\Gamma_{n})$ and therefore
$\Gamma_{n+1}=F_{n}(\Gamma_{n})$.

It remains to observe that (3.4) and (3.5) follow readily from (3.14). The
proof of proposition 3.1 is completed. ∎

##  4\. Nonautonomous version of Takens theorem

In this section, we prove our main theorem, as described in the introduction.

###  4.1. Nonautonomous Belitskii-Samavol Theorem

We consider two sequences $f_{n}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$g_{n}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$, $n\in\mathbb{Z}$ of $C^{N+1}$
maps such that

  * •

for $n\in\mathbb{Z}$,

$f_{n}(0)=g_{n}(0)=0\quad\text{and}\quad d_{0}f_{n}=d_{0}g_{n}=0;$ (4.1)

  * •

for $2\leq i\leq N+1$ and $x\in\mathbb{R}^{d}$

$\|d_{x}f_{n}\|\leq\delta,\quad\|d^{i}_{x}f_{n}\|\leq
M,\quad\|d_{x}g_{n}\|\leq\delta,\quad\|d^{i}_{x}g_{n}\|\leq M,$ (4.2)

where $\delta,M>0$ are constants with $\delta$ being sufficiently small.

By Proposition 3.1, there are sequences of nonautonomous center manifolds
$\Gamma_{n}$ and $\Gamma^{\prime}_{n}$, $n\in\mathbb{Z}$ that correspond to
sequences $F_{n}:=A_{n}+f_{n}$ and $G_{n}:=A_{n}+g_{n}$ for $n\in\mathbb{Z}$,
respectively. Similarly to (3.7), we define a map $\mathbb{G}\colon Y\to Y$ by

$(\mathbb{G}(\mathbf{x}))_{n}:=G_{n-1}(x_{n-1}):=A_{n-1}x_{n-1}+g_{n-1}(x_{n-1}),\quad\forall
n\in\mathbb{Z},$ (4.3)

for $\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\in Y$. Let $\Gamma$ and
$\Gamma^{\prime}$ be center manifolds of $\mathbb{F}$ and $\mathbb{G}$,
respectively. Then, we have the following lemma.

######  Lemma 4.1.

Let $(F_{n})_{n\in\mathbb{Z}}$ and $(G_{n})_{n\in\mathbb{Z}}$ be given above
and suppose that $\Gamma_{n}=\Gamma_{n}^{\prime}$ for $n\in\mathbb{Z}$. In
addition, assume that $d^{i}_{x_{c}}(F_{n}-G_{n})=0$ for $1\leq i\leq N$ and
$x_{c}\in\Gamma_{n}$, $n\in\mathbb{Z}$. Then, $\Gamma=\Gamma^{\prime}$ and
$d^{i}_{{\bf x}_{c}}(\mathbb{F}-\mathbb{G})={\bf 0}$ for $1\leq i\leq N$ and
${\bf x}_{c}\in\Gamma$.

###### Proof.

We recall that

$\displaystyle\Gamma_{n}=\\{x_{c}+\phi_{n}(x_{c}):\
x_{c}\in\operatorname{Im}\Pi_{n}^{c}\\},\qquad\Gamma^{\prime}_{n}=\\{x_{c}+\phi^{\prime}_{n}(x_{c}):\
x_{c}\in\operatorname{Im}\Pi_{n}^{c}\\}$

and

$\displaystyle\Gamma=\\{{\bf x}_{c}+\Phi({\bf x}_{c}):\ {\bf x}_{c}\in
Y^{c}\\},\qquad\Gamma^{\prime}=\\{{\bf x}_{c}+\Phi^{\prime}({\bf x}_{c}):\
{\bf x}_{c}\in Y^{c}\\},$

with $C^{N}$ maps $\Phi,\Phi^{\prime}\colon Y^{c}\to Y^{s}\oplus Y^{u}$.
Choose an arbitrary $\mathbf{x}_{c}=(x^{c}_{n})_{n\in\mathbb{Z}}\in Y^{c}$.
For any $n\in\mathbb{Z}$, we denote
$x^{c}_{n}=:x_{c}\in\operatorname{Im}\Pi_{n}^{c}$. Then, we have (see (3.16))
that

$\phi_{n}(x_{c})=(\Phi({\bf
v}))_{n},\qquad\phi^{\prime}_{n}(x_{c})=(\Phi^{\prime}({\bf v}))_{n},$

where ${\bf v}\in Y^{c}$ is given by (3.15). Then, Lemma 3.2 tells that

$(\Phi({\bf x}_{c}))_{n}=(\Phi({\bf
v}))_{n}=\phi_{n}(x_{c})=\phi^{\prime}_{n}(x_{c})=(\Phi^{\prime}({\bf
v}))_{n}=(\Phi^{\prime}({\bf x}_{c}))_{n},$

where the equality $\phi_{n}(x_{c})=\phi^{\prime}_{n}(x_{c})$ follows from
$\Gamma_{n}=\Gamma_{n}^{\prime}$. Therefore, $\Gamma=\Gamma^{\prime}$.

Next, we establish the second assertion of the lemma. By (3.10), we have that

$\begin{split}&d_{{\bf
x}_{c}}^{i}\mathbb{F}(\boldsymbol{\xi}^{1},\ldots,\boldsymbol{\xi}^{i})-d_{{\bf
x}_{c}}^{i}\mathbb{G}(\boldsymbol{\xi}^{1},\ldots,\boldsymbol{\xi}^{i})\\\
&=\Big{(}d^{i}_{x^{c}_{n-1}}f_{n-1}(\xi^{1}_{n-1},\ldots,\xi^{i}_{n-1})-d^{i}_{x^{c}_{n-1}}g_{n-1}(\xi^{1}_{n-1},\ldots,\xi^{i}_{n-1})\Big{)}_{n\in\mathbb{Z}}\\\
&=\Big{(}d_{x^{c}_{n-1}}^{i}(F_{n-1}-G_{n-1})(\xi^{1}_{n-1},\ldots,\xi^{i}_{n-1})\Big{)}_{n\in\mathbb{Z}}=(0)_{n\in\mathbb{Z}}=\boldsymbol{0},\end{split}$

for $1\leq i\leq N$, $\boldsymbol{\xi}^{1},\ldots,\boldsymbol{\xi}^{i}\in
Y^{c}$ and ${\bf x}_{c}\in\Gamma=\Gamma^{\prime}$. The proof of the lemma is
completed. ∎

We have the following theorem, which can be regarded as the nonautonomous
version of Belitskii-Samavol Theorem.

######  Theorem 4.1.

Take an arbitrary $k\in\mathbb{N}$. Then, there exists $N_{0}\in\mathbb{N}$
with the property that for each $N\geq N_{0}$ such that (3.3) holds and for
sequences $(f_{n})_{n\in\mathbb{Z}}$, $(g_{n})_{n\in\mathbb{Z}}$ of $C^{N+1}$
maps on $\mathbb{R}^{d}$ satisfying (4.1), (4.2) and

  * •

$\Gamma_{n}=\Gamma_{n}^{\prime}$ and $d_{x_{c}}^{i}(f_{n}-g_{n})=0$ for all
$1\leq i\leq N$, $x_{c}\in\Gamma_{n}$ and $n\in\mathbb{Z}$,

there exists a sequence $(H_{n})_{n\in\mathbb{Z}}$ of $C^{k}$ local
diffeomorphisms $H_{n}$ near $0$ such that the following holds:

  * •

for $n\in\mathbb{Z}$,

$H_{n}(0)=0;$ (4.4)

  * •

there exists a constant $r>0$ such that for $n\in\mathbb{Z}$,

$H_{n+1}\circ F_{n}(x)=G_{n}\circ H_{n}(x)\qquad\text{for}\quad\|x\|\leq r,$
(4.5)

where $F_{n}=A_{n}+f_{n}$ and $G_{n}=A_{n}+g_{n}$;

  * •

there exist $M,\rho>0$ such that

$\|d_{x}^{i}H_{n}\|\leq M\quad\text{and}\quad\|d_{x}^{i}H_{n}^{-1}\|\leq M,$
(4.6)

for each $n\in\mathbb{Z}$, each $1\leq i\leq k$, and $x\in\mathbb{R}^{d}$ with
$\|x\|\leq\rho$.

###### Proof.

It follows from Lemmas 3.1, 4.1 and Theorem 5.1 (which can be applied as $Y$
is a Hilbert space) that there exists a local $C^{k}$ diffeomorphism
$\mathbb{H}\colon Y\to Y$ and a constant $r>0$ such that

$\mathbb{H}(\boldsymbol{0})=\boldsymbol{0},$ (4.7)

and

$\mathbb{H}\circ\mathbb{F}(\mathbf{x})=\mathbb{G}\circ\mathbb{H}(\mathbf{x})\qquad\text{for}\qquad\|\mathbf{x}\|\leq
r,$ (4.8)

where $\mathbb{F}$ and $\mathbb{G}$ are given by (3.7) and (4.3),
respectively. Similarly to (3.16), the desired local conjugacy $H_{n}$ can be
constructed by

$H_{n}(x):=(\mathbb{H}({\bf w}))_{n},\quad\text{for $\|x\|\leq r$ and
$n\in\mathbb{Z}$},$

where ${\bf w}:=(w_{m})_{m\in\mathbb{Z}}\in Y$ is defined by $w_{n}:=x$ and
$w_{m}:=0$ for $m\neq n$. Then, similarly to (3.7) and (3.10), we see that
$H_{n}$ is $C^{k}$ and that

$d_{x}^{i}H_{n}(\zeta^{1},\ldots,\zeta^{i})=(d_{\bf
w}^{i}\mathbb{H}(\boldsymbol{\zeta}^{1},\ldots,\boldsymbol{\zeta}^{i}))_{n},$
(4.9)

for $1\leq i\leq k$ and $\zeta^{1},\ldots,\zeta^{i}\in\mathbb{R}^{d}$, where
$\boldsymbol{\zeta}^{j}=(\zeta_{m}^{j})_{m\in\mathbb{Z}}\in Y$ is defined by
$\zeta_{n}^{j}:=\zeta^{j}$ and $\zeta_{m}^{j}:=0$ for $m\neq n$. Moreover,
each $H_{n}$ is a local diffeomorphisms and
$H_{n}^{-1}(x)=(\mathbb{H}^{-1}(\mathbf{w}))_{n}$.

Observe that (4.4) follows readily from (4.7) and (4.9) with ${\bf w}={\bf 0}$
and $i=1$ and that (4.5) follows directly from (4.8). On the other hand, since
$\mathbb{H}$ and $\mathbb{H}^{-1}$ are $C^{k}$, there exist $M,\rho>0$ such
that for $1\leq i\leq k$,

$\|d_{{\bf w}}^{i}\mathbb{H}\|\leq M\quad\text{and}\quad\|d_{{\bf
w}}^{i}\mathbb{H}^{-1}\|\leq M,\quad\text{for ${\bf w}\in Y$ with $\|{\bf
w}\|\leq\rho$.}$

Hence, by (4.9) we obtain that

$\begin{split}\|d_{x}^{i}H_{n}(\xi^{1},\ldots,\xi^{i})\|&=\|(d_{{\bf
w}}^{i}\mathbb{H}(\boldsymbol{\zeta}^{1},\ldots,\boldsymbol{\zeta}^{i}))_{n}\|\leq\|d_{\bf
w}^{i}\mathbb{H}(\boldsymbol{\zeta}^{1},\ldots,\boldsymbol{\zeta}^{i})\|\\\
&\leq
M\prod_{j=1}^{i}\|\boldsymbol{\zeta}^{j}\|=M\prod_{j=1}^{i}\|\zeta^{j}\|,\end{split}$

for each $n\in\mathbb{Z}$ and $x\in\mathbb{R}^{d}$ such that $\|x\|\leq\rho$
(so that $\|{\bf w}\|\leq\rho$) and $1\leq i\leq k$. This establishes the
first estimate in (4.6). Similarly, one can establish the second one and the
proof of the theorem is completed. ∎

###  4.2. Nonautonomous Takens Theorem

For $q:=(q_{1},\ldots,q_{r}),~{}t:=(t_{1},\ldots,t_{r})\in\mathbb{N}_{0}^{r}$
with $\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$,
$a:=(a_{1},\ldots,a_{r})\in\mathbb{R}^{r}$,
$b:=(b_{1},\ldots,b_{r})\in\mathbb{R}^{r}$, we denote

$\displaystyle|q|:=q_{1}+\ldots+q_{r},\quad[a,b]^{q}:=[a_{1}^{q_{1}}\cdots
a_{r}^{q_{r}},~{}b_{1}^{q_{1}}\cdots b_{r}^{q_{r}}],$
$\displaystyle|t|:=t_{1}+\ldots+t_{r},\quad[a,b]^{t}:=[a_{1}^{t_{1}}\cdots
a_{r}^{t_{r}},~{}b_{1}^{t_{1}}\cdots b_{r}^{t_{r}}],$

where $a_{i}$ and $b_{i}$ for $1\leq i\leq r$ are as in (2.11). In addition,
for $N\in\mathbb{N}$ and intervals $[c_{1},c_{2}]$, $[c_{3},c_{4}]$ with
$c_{i}\in\mathbb{R}^{+}$ for $i\in\\{1,2,3,4\\}$, we denote

$[c_{1},c_{2}]^{N}:=[c_{1}^{N},c_{2}^{N}],\quad[c_{1},c_{2}]\cdot[c_{3},c_{4}]:=[c_{1}c_{3},~{}c_{2}c_{4}].$

Then, based on Theorem 4.1, we reformulate and prove the main theorem of this
paper as follows.

######  Theorem 4.2.

For any $k\in\mathbb{N}$, let $N:=3N_{0}+1$ where $N_{0}\in\mathbb{N}$
(depending on $k$) is given in Theorem 4.1. Suppose that the non-resonance
conditions

$\displaystyle\left([a,b]^{q}\cdot[\nu_{-},\nu_{+}]^{N}\right)\cap[\nu_{-},\nu_{+}]=\emptyset,$
(4.10)
$\displaystyle\left([a,b]^{t}\cdot[\nu_{-},\nu_{+}]^{N}\right)\cap[a_{i},b_{i}]=\emptyset,\quad
i=1,\ldots,r,$ (4.11)

hold for all $|q|\in\\{1,\ldots,N\\}$ and $|t|\in\\{2,\ldots,N\\}$. Then, any
sequence $(F_{n})_{n\in\mathbb{Z}}$ of $C^{N+1}$ maps of the form
$F_{n}:=A_{n}+f_{n}$ satisfying (3.1) and (3.2) is locally $C^{k}$ conjugated
to the sequence of maps

$\displaystyle\left(\begin{array}[]{lll}x_{s}\vspace{1ex}\\\
x_{c}\vspace{1ex}\\\
x_{u}\end{array}\right)\mapsto\left(\begin{array}[]{lll}A_{n}^{s}(x_{c})x_{s}\vspace{1ex}\\\
A_{n}^{c}x_{c}+f_{n}^{c}(x_{c})\vspace{1ex}\\\
A_{n}^{u}(x_{c})x_{u}\end{array}\right),\quad\forall n\in\mathbb{Z},$ (4.18)

by a conjugacy $(\Psi_{n})_{n\in\mathbb{Z}}$, where $A_{n}^{s}(x_{c}):X_{s}\to
X_{s}$, $A_{n}^{u}(x_{c}):X_{u}\to X_{u}$ are linear operators which are
$C^{N-1}$ in $x_{c}$ such that $A_{n}^{s}(0)=A_{n}^{s}$,
$A_{n}^{u}(0)=A_{n}^{u}$, and $f_{n}^{c}:X_{c}\to X_{c}$ is a $C^{N}$ map.
Moreover, for each $n\in\mathbb{Z}$,

$\displaystyle\Psi_{n}^{\pm}(0)=0,\qquad\|d_{x}^{i}\Psi_{n}^{\pm}\|\leq\tilde{M},\quad
i=2,\ldots,k,\quad{\rm for}~{}x\in\mathbb{R}^{d}~{}{\rm near}~{}0,$ (4.19)

where $\tilde{M}>0$ is a constant. Here, $\Psi_{n}^{+}=\Psi_{n}$ and
$\Psi_{n}^{-}=\Psi_{n}^{-1}$.

Remark that (4.10) implies

$\displaystyle(a_{1}^{q_{1}}\cdots a_{r}^{q_{r}})\nu_{-}^{N}>\nu_{+}\quad{\rm
or}\quad(b_{1}^{q_{1}}\cdots b_{r}^{q_{r}})\nu_{+}^{N}<\nu_{-}$ (4.20)

and (4.11) implies

$\displaystyle(a_{1}^{t_{1}}\cdots a_{r}^{t_{r}})\nu_{-}^{N}>b_{i}\quad{\rm
or}\quad(b_{1}^{t_{1}}\cdots b_{r}^{t_{r}})\nu_{+}^{N}<a_{i},\quad
i=1,\ldots,r,$ (4.21)

by the notations introduced before the statement of the theorem.

Proof of Theorem 4.2. By Proposition 3.1, the sequence
$(F_{n})_{n\in\mathbb{Z}}$ admits a nonautonomous $C^{N}$ center manifold,
which is the sequence of graphs of $C^{N}$ maps $\phi_{n}:X_{c}\to X_{s}\oplus
X_{u}.$ This enables us to define a sequence of $C^{N}$ transformations
$\varphi_{n}:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $n\in\mathbb{Z}$ by

$\begin{cases}\tilde{x}_{c}:=x_{c},\\\
(\tilde{x}_{s},\tilde{x}_{u}):=(x_{s},x_{u})-\phi_{n}(x_{c}),\end{cases}$
(4.22)

which satisfies that

$\displaystyle\begin{split}&\varphi_{n}^{\pm}(0)=0,\quad
d_{0}\varphi_{n}^{\pm}=\operatorname{Id},\quad\|d_{x}\varphi_{n}^{\pm}(x)-\operatorname{Id}\|\leq\tilde{\delta},\\\
&\|d_{x}^{i}\varphi_{n}^{\pm}(x)\|\leq\tilde{M},\quad i=2,\ldots,N,\quad{\rm
for}~{}x~{}{\rm near}~{}0,\end{split}$ (4.23)

where $\varphi_{n}^{+}=\varphi_{n}$, $\varphi_{n}^{-}=\varphi_{n}^{-1}$ and
$\tilde{\delta},\tilde{M}>0$ with the possibility of choosing $\tilde{\delta}$
arbitrarily small. Using $(\varphi_{n})_{n\in\mathbb{Z}}$ one can conjugate
$(F_{n})_{n\in\mathbb{Z}}$ to the sequence $(\tilde{F}_{n})_{n\in\mathbb{Z}}$,
i.e,

$\tilde{F}_{n}:=\varphi_{n+1}\circ
F_{n}\circ\varphi_{n}^{-1}=A_{n}+\tilde{f}_{n},$

whose center manifold is straightened up, i.e., $X_{c}$ is the (nonautonomous)
center manifold of $(\tilde{F}_{n})_{n\in\mathbb{Z}}$. Notice that, by (3.2),
(4.23) and using smooth cut-off functions, we can assume that the following
conditions hold:

  * •

$\tilde{f}_{n}(0)=0$ and $d_{0}\tilde{f}_{n}=0$;

  * •

for $2\leq i\leq N$,

$\|d_{x}\tilde{f}_{n}\|\leq\tilde{\delta},\quad\|d^{i}_{x}\tilde{f}_{n}\|\leq\tilde{M},\quad\forall
x\in\mathbb{R}^{d}.$ (4.24)

Put $v:=x_{s}+x_{u}\in X_{su}$ and write $\tilde{F}_{n}(x_{s},x_{c},x_{u})$ as
$\tilde{F}_{n}(x_{c},v)$. For an integer $p\in\\{1,\ldots,N_{0}\\}$, assume
that

$\partial_{v}^{i}(\pi_{c}\tilde{F}_{n})(x_{c},0)=0\quad{\rm
for}~{}i=1,\ldots,p-1.$

Then, the Taylor expansion of $\tilde{F}_{n}$ at $(x_{c},0)$ is given by

$\tilde{F}_{n}(x_{c},v)=\Big{(}w_{n}(x_{c})+f_{n}^{p}(x_{c},v)+o(\|v\|^{p}),~{}A_{n}^{su}(x_{c})v+O(\|v\|^{2})\Big{)}$
(4.25)

where $w_{n}(x_{c}):=\pi_{c}\tilde{F}_{n}(x_{c},0)$ is $C^{N-p}$ (actually
$C^{N}$) and $A_{n}^{su}(x_{c}):=\partial_{v}(\pi_{su}\tilde{F}_{n})(x_{c},0)$
is $C^{N-p}$ (actually $C^{N-1}$) such that

$Dw_{n}(0)=A_{n}^{c},\qquad A_{n}^{su}(0)={\rm diag}(A_{n}^{s},A_{n}^{u}).$

Moreover,
$f_{n}^{p}(x_{c},v):=\partial_{v}^{p}(\pi_{c}\tilde{F}_{n})(x_{c},0)v^{p}$ is
a homogeneous polynomial in variable $v$ of order $p$ with coefficients being
$C^{N-p}$ maps in $x_{c}$. In order to eliminate the term
$f_{n}^{p}(x_{c},v)$, we need the following lemma. We note that below and
throughout the rest of the proof, $\tilde{\delta},\tilde{M}>0$ will denote
generic constants which can change from one occurrence to the next one and
$\tilde{\delta}$ can be made arbitrarily small.

######  Lemma 4.2.

The sequence $(\tilde{F}_{n})_{n\in\mathbb{Z}}$ of $C^{N-p}$ maps is locally
conjugated to a sequence $(\hat{F}_{n})_{n\in\mathbb{Z}}$ of $C^{N-p-1}$ maps
defined by

$\hat{F}_{n}(x_{c},v):=\Big{(}w_{n}(x_{c})+o(\|v\|^{p}),~{}A_{n}^{su}(x_{c})v+O(\|v\|^{2})\Big{)}$

via a sequence $(H_{n}^{p}(x_{c},v))_{n\in\mathbb{Z}}$, where each
$H_{n}^{p}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a homogeneous polynomial in
variable $v$ of order $p$ with coefficients being $C^{N-p-1}$ maps in $x_{c}$.
Moreover,

$\displaystyle\begin{split}&(H_{n}^{p})^{\pm}(0)=0,\quad
d_{0}(H_{n}^{p})^{\pm}=\operatorname{Id},\quad\|d_{x}(H_{n}^{p})^{\pm}-\operatorname{Id}\|\leq\tilde{\delta},\\\
&\|d_{x}^{i}(H_{n}^{p})^{\pm}\|\leq\tilde{M},\quad i=2,\ldots,N-p,\quad{\rm
for}~{}x=(x_{c},v)~{}{\rm near}~{}0.\end{split}$ (4.26)

Proof of Lemma 4.2. Let $h_{n}^{p}:X_{c}\times X_{su}\to X_{c}$,
$n\in\mathbb{Z}$ be a sequence of homogeneous polynomials in variable $v$ of
order $p$ with coefficients being of a certain smoothness in $x_{c}$ (which
will be proved to be $C^{N-p}$), and consider the transformations

$H_{n}^{p}:(x_{c},v)\mapsto(x_{c}+h_{n}^{p}(x_{c},v),v)$ (4.27)

together with their inverses

$(H_{n}^{p})^{-1}:(x_{c},v)\mapsto(x_{c}-h_{n}^{p}(x_{c},v)+o(\|v\|^{p}),v).$
(4.28)

Combining (4.25) with (4.27), we calculate that

$\displaystyle H^{p}_{n+1}\circ\tilde{F}_{n}(x_{c},v)$
$\displaystyle=\Big{(}w_{n}(x_{c})+f_{n}^{p}(x_{c},v)+h_{n+1}^{p}(w_{n}(x_{c}),A_{n}^{su}(x_{c})v)+o(\|v\|^{p}),$
$\displaystyle\qquad A_{n}^{su}(x_{c})v+O(\|v\|^{2})\Big{)}.$

Employing (4.28), we further obtain that

$\displaystyle H^{p}_{n+1}\circ\tilde{F}_{n}\circ(H_{n}^{p})^{-1}(x_{c},v)$
$\displaystyle=\Big{(}w_{n}(x_{c})-Dw_{n}(x_{c})h_{n}^{p}(x_{c},v)+f_{n}^{p}(x_{c},v)+h_{n+1}^{p}(w_{n}(x_{c}),A_{n}^{su}(x_{c})v)+o(\|v\|^{p}),$
$\displaystyle\qquad A_{n}^{su}(x_{c})v+O(\|v\|^{2})\Big{)},$

where the term $O(\|v\|^{2})$ has the same smoothness as $h_{n}^{p}(\cdot,v)$.
In order to eliminate the term of order $p$, it suffices to solve the
following nonautonomous equation

$-Dw_{n}(x_{c})h_{n}^{p}(x_{c},v)+f_{n}^{p}(x_{c},v)+h_{n+1}^{p}(w_{n}(x_{c}),A_{n}^{su}(x_{c})v)=0,\quad\forall
n\in\mathbb{Z},$ (4.29)

which is equivalent to

$-Dw_{n}(x_{c})h_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v)+f_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v)+h_{n+1}^{p}(w_{n}(x_{c}),v)=0,$
(4.30)

by replacing $v$ with $A_{n}^{su}(x_{c})^{-1}v$. We recall that all the known
maps in the above equations are $C^{N-p}$ in $x_{c}$. Equation (4.30) can be
seen as a nonautonomous homological equation (cf. [1, p.182]), which depends
the center variable $x_{c}$. This is the essential difference from the
hyperbolic case and we will use the center manifold theory to solve it.

Let $\Xi^{p}$ be the linear space of all homogeneous vector polynomials of
order $p$ in the variable $v$ whose elements are of the form

$\sum_{q}C^{q}x_{1}^{q_{1}}\cdots x_{r}^{q_{r}},$

where $C^{q}:X_{1}^{q_{1}}\times\cdots\times X_{r}^{q_{r}}\to X_{c}$ are
multilinear operators and $|q|=p$. Let
$\Upsilon_{n}:\tilde{\Xi}^{p}_{c}:=X_{c}\times\Xi^{p}\to\tilde{\Xi}^{p}_{c}$,
$n\in\mathbb{Z}$ be a sequence of $C^{N-p}$ maps defined by

$\Upsilon_{n}\colon(x_{c},\psi^{p}(v))\mapsto\Big{(}w_{n}(x_{c}),~{}L_{n}(x_{c})\psi^{p}(v)-f_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v)\Big{)},$
(4.31)

where $L_{n}(x_{c}):\Xi^{p}\to\Xi^{p}$ is a linear operator defined by

$L_{n}(x_{c})\psi^{p}(v):=Dw_{n}(x_{c})\psi^{p}(A_{n}^{su}(x_{c})^{-1}v),\quad\forall\psi^{p}(v)\in\Xi^{p}.$
(4.32)

We observe that if the sets
${\mathcal{M}}_{n}:=\\{(x_{c},h_{n}^{p}(x_{c},v))\in\tilde{\Xi}^{p}_{c}:x_{c}\in
X_{c}\\}$, $n\in\mathbb{Z}$ are invariant under
$(\Upsilon_{n})_{n\in\mathbb{Z}}$, i.e.,
$\Upsilon_{n}({\mathcal{M}}_{n})\subset{\mathcal{M}}_{n+1}$, then

$h_{n+1}^{p}(w_{n}(x_{c}),v)=L_{n}(x_{c})h_{n}^{p}(x_{c},v)-f_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v),$

which is exactly the equation (4.30). Thus, we convert the problem of solving
equation (4.30) to the problem of finding an invariant manifold
$({\mathcal{M}}_{n})_{n\in\mathbb{Z}}$ of $(\Upsilon_{n})_{n\in\mathbb{Z}}$.

In order to find an invariant manifold of $(\Upsilon_{n})_{n\in\mathbb{Z}}$,
we consider the shift $\tilde{\Upsilon}_{n}$ of $\Upsilon_{n}$ defined by

$\displaystyle\\!\\!(x_{c},\psi^{p}(v))$
$\displaystyle\\!\\!\mapsto\Big{(}w_{n}(x_{c}),~{}L_{n}(x_{c})\\{\psi^{p}(v)+\kappa_{n}^{p}(x_{c},v)\\}-f_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v)-\kappa_{n+1}^{p}(x_{c},v)\Big{)},$
(4.33)

where $(\kappa_{n}^{p})_{n\in\mathbb{Z}}$ is a solution of the equation

$\displaystyle
L_{n}(0)\kappa_{n}^{p}(x_{c},v)-\kappa_{n+1}^{p}(x_{c},v)=f_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v).$
(4.34)

Set

$\displaystyle
S^{p}_{+}:=\\{q:=(q_{1},\ldots,q_{r})\in\mathbb{N}_{0}^{r}:a_{1}^{q_{1}}\cdots
a_{r}^{q_{r}}>\nu_{+},~{}|q|=p\\},$ $\displaystyle
S^{p}_{-}:=\\{q:=(q_{1},\ldots,q_{r})\in\mathbb{N}_{0}^{r}:b_{1}^{q_{1}}\cdots
b_{r}^{q_{r}}<\nu_{-},~{}|q|=p\\},$

and let $\Xi^{p,+},\Xi^{p,-}$ be two subspaces of $\Xi^{p}$ with elements

$\psi^{p,+}(v):=\sum_{q\in S_{+}^{p}}C_{+}^{q}x_{1}^{q_{1}}\cdots
x_{r}^{q_{r}},\qquad\psi^{p,-}(v):=\sum_{q\in
S_{-}^{p}}C_{-}^{q}x_{1}^{q_{1}}\cdots x_{r}^{q_{r}},$

respectively, where $C_{\pm}^{q}:X_{1}^{q_{1}}\times\cdots\times
X_{r}^{q_{r}}\to X_{c}$ are multilinear operators. By (4.20) we know that
either $a_{1}^{q_{1}}\cdots a_{r}^{q_{r}}\geq a_{1}^{q_{1}}\cdots
a_{r}^{q_{r}}\nu_{-}^{N}>\nu_{+}$ or $b_{1}^{q_{1}}\cdots b_{r}^{q_{r}}\leq
b_{1}^{q_{1}}\cdots b_{r}^{q_{r}}\nu_{+}^{N}<\nu_{-}$, which implies that

$\displaystyle\Xi^{p}=\Xi^{p,+}\oplus\Xi^{p,-},$ (4.35)

and by (4.32) we know that $\Xi^{p,\pm}$ are invariant under
$(L_{n}(0))_{n\in\mathbb{Z}}$. Let ${\mathcal{L}}(m,n)$ be the linear cocycle
generated by $(L_{n}(0))_{n\in\mathbb{Z}}$. By iterating (4.34), we obtain
that

$\kappa_{n}^{p}(x_{c},v)={\mathcal{L}}(n,m)\kappa_{m}^{p}(x_{c},v)-\sum_{i=m}^{n-1}{\mathcal{L}}(n,i+1)f_{i}^{p}(x_{c},A_{i}^{su}(x_{c})^{-1}v)$
(4.36)

for $n\geq m$ and that

$\kappa_{n}^{p}(x_{c},v)={\mathcal{L}}(n,m)\kappa_{m}^{p}(x_{c},v)+\sum_{i=n}^{m-1}{\mathcal{L}}(n,i+1)f_{i}^{p}(x_{c},A_{i}^{su}(x_{c})^{-1}v)$
(4.37)

for $n\leq m$.

In order to further compute the above $\kappa_{m}^{p}(x_{c},v)$, we define the
norm

$\|\psi^{p,\pm}(v)\|:=\max_{q\in S_{\pm}^{p}}\sup_{\|x_{1}\|\neq
0,\ldots,\|x_{r}\|\neq 0}\frac{\|C_{\pm}^{q}x_{1}^{q_{1}}\cdots
x_{r}^{q_{r}}\|}{\|x_{1}\|^{q_{1}}\cdots\|x_{r}\|^{q_{r}}},\qquad\forall\psi^{p,\pm}(v)\in\Xi^{p,\pm},$

and let

${\mathcal{A}}_{c}(n,m):={\mathcal{A}}(n,m)|_{X_{c}},\qquad{\mathcal{A}}_{i}(n,m):={\mathcal{A}}(n,m)|_{X_{i}},\quad
i=1,\ldots,r.$

Then, for $n\geq m$ we obtain from (4.32) and (2.13) (see also Remark 2.1)
that

$\displaystyle\|{\mathcal{L}}(n,m)\psi^{p,+}(v)\|$
$\displaystyle=\|Dw_{n-1}(0)\cdots
Dw_{m}(0)\psi^{p,+}(A_{m}^{su}(0)^{-1}\cdots A_{n-1}^{su}(0)^{-1}v)\|$
$\displaystyle=\max_{q\in S_{+}^{p}}\sup_{\|x_{1}\|\neq 0,\ldots,\|x_{r}\|\neq
0}\frac{\|{\mathcal{A}}_{c}(n,m)C_{+}^{q}\\{{\mathcal{A}}_{1}(m,n)x_{1}\\}^{q_{1}}\cdots\\{{\mathcal{A}}_{r}(m,n)x_{r}\\}^{q_{r}}\|}{\|x_{1}\|^{q_{1}}\cdots\|x_{r}\|^{q_{r}}}$
$\displaystyle\leq C\nu_{+}^{n-m}\max_{q\in S_{+}^{p}}\sup_{\|x_{1}\|\neq
0,\ldots,\|x_{r}\|\neq
0}\frac{\|C_{+}^{q}\\{{\mathcal{A}}_{1}(m,n)x_{1}\\}^{q_{1}}\cdots\\{{\mathcal{A}}_{r}(m,n)x_{r}\\}^{q_{r}}\|}{\|{\mathcal{A}}_{1}(m,n)x_{1}\|^{q_{1}}\cdots\|{\mathcal{A}}_{r}(m,n)x_{r}\|^{q_{r}}}$
$\displaystyle\quad\cdot\frac{\|{\mathcal{A}}_{1}(m,n)x_{1}\|^{q_{1}}\cdots\|{\mathcal{A}}_{r}(m,n)x_{r}\|^{q_{r}}}{\|x_{1}\|^{q_{1}}\cdots\|x_{r}\|^{q_{r}}}$
$\displaystyle\leq C\nu_{+}^{n-m}\max_{q\in S_{+}^{p}}(a_{1}^{q_{1}}\cdots
a_{r}^{q_{r}})^{-(n-m)}\|\psi^{p,+}(v)\|$
$\displaystyle=C\mu_{1}^{n-m}\|\psi^{p,+}(v)\|,\qquad\forall\psi^{p,+}(v)\in\Xi^{p,+},$
(4.38)

where $\mu_{1}:=\max_{q\in S_{+}^{p}}\\{\nu_{+}/(a_{1}^{q_{1}}\cdots
a_{r}^{q_{r}})\\}\in(0,1)$. Similarly, for $n\leq m$ we also have that

$\displaystyle\|{\mathcal{L}}(n,m)\psi^{p,-}(v)\|\leq
C\mu_{2}^{m-n}\|\psi^{p,-}(v)\|,\qquad\forall\psi^{p,-}(v)\in\Xi^{p,-},$
(4.39)

where $\mu_{2}:=\max_{q\in S_{-}^{p}}\\{(b_{1}^{q_{1}}\cdots
b_{r}^{q_{r}})/\nu_{-}\\}\in(0,1)$. Now, projecting (4.36) onto $\Xi^{p,+}$
(letting $m\to-\infty$) and (4.37) onto $\Xi^{p,-}$ (letting $m\to+\infty$),
respectively, we see from (4.2)-(4.39) that

$\displaystyle\kappa_{n}^{p}(x_{c},v)$
$\displaystyle=-\sum_{i=-\infty}^{n-1}{\mathcal{L}}(n,i+1)\Pi^{+}f_{i}^{p}(x_{c},A_{i}^{su}(x_{c})^{-1}v)$
$\displaystyle\quad+\sum_{i=n}^{\infty}{\mathcal{L}}(n,i+1)\Pi^{-}f_{i}^{p}(x_{c},A_{i}^{su}(x_{c})^{-1}v),$

where $\Pi^{\pm}:\Xi^{p}\to\Xi^{p,\pm}$ are projections corresponding to the
decomposition (4.35). It implies that for any $n\in\mathbb{Z}$,

$\displaystyle\|\kappa_{n}^{p}(x_{c},v)\|$ $\displaystyle\leq
C\left(\sum_{i=-\infty}^{n-1}\mu_{1}^{n-i-1}+\sum_{i=n}^{+\infty}\mu_{2}^{i+1-n}\right)\|f_{i}^{p}(x_{c},A_{i}^{su}(x_{c})^{-1}v)\|$
$\displaystyle\leq
C\left(\sum_{i=-\infty}^{n-1}\mu_{1}^{n-i-1}+\sum_{i=n}^{+\infty}\mu_{2}^{i+1-n}\right)\tilde{M}=:M_{\kappa}$
(4.40)

by (4.24) since
$f_{n}^{p}(x_{c},v):=\partial_{v}^{p}(\pi_{c}\tilde{F}_{n})(x_{c},0)v^{p}$
(see (4.25)), where we set

$\|\psi^{p}(v)\|:=\max\\{\|\Pi^{+}\psi^{p}(v)\|,\|\Pi^{-}\psi^{p}(v)\|\\},\quad\psi^{p}(v)\in\Xi^{p}.$

Moreover, noting that

$\displaystyle\\{\partial(\Pi^{\pm}f_{i}^{p})(x_{c},v)/\partial
x_{c}\\}x_{c}\in\Xi^{p,\pm},\quad\\{\partial(\Pi^{\pm}f_{i}^{p})(x_{c},v)/\partial
v\\}v\in\Xi^{p,\pm}$ $\displaystyle
A_{n}^{su}(x_{c}):=\partial_{v}(\pi_{su}\tilde{F}_{n})(x_{c},0),$

and by using similar arguments as above, we obtain that

$\displaystyle\left\|\frac{\partial\kappa_{n}^{p}(x_{c},v)}{\partial
x_{c}}\right\|\leq\tilde{M}_{\kappa},$ (4.41)

where $\tilde{M}_{\kappa}>0$ is a constant. Thus, we find the solution
$\kappa_{n}^{p}(x_{c},v)$ of equation (4.34) such that (4.2) and (4.41) hold.

Next, we continue to investigate the map $\tilde{\Upsilon}_{n}$ given by
(4.2). It is clear that $(0,0)$ is the fixed point of $\tilde{\Upsilon}_{n}$
and that the derivative of $\tilde{\Upsilon}_{n}$ at $(0,0)$ has the form

${\Delta}_{n}=\begin{pmatrix}A_{n}^{c}&0\\\ M_{n}(v)&L_{n}(0)\end{pmatrix},$
(4.42)

where $A_{n}^{c}=Dw_{n}(0)$ and $M_{n}(v):X_{c}\to\Xi^{p}$ is given by

$\displaystyle M_{n}(v)$
$\displaystyle:=\frac{\partial\\{L_{n}(x_{c})\\{\psi^{p}(v)+\kappa_{n}^{p}(x_{c},v)\\}-f_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v)-\kappa_{n+1}^{p}(x_{c},v)\\}}{\partial
x_{c}}\Big{|}_{x_{c}=0,~{}\psi^{p}(v)=0}$
$\displaystyle=\frac{\partial\\{Dw_{n}(x_{c})\kappa_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v)-f_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v)-\kappa_{n+1}^{p}(x_{c},v)\\}}{\partial
x_{c}}\Big{|}_{x_{c}=0},$

by using (4.32). Recalling that $w_{n}(x_{c})=\pi_{c}\tilde{F}_{n}(x_{c},0)$,
and by (4.24), (4.2)-(4.41), we see that

$\displaystyle\|M_{n}(v)\|\leq\tilde{M},\qquad\forall n\in\mathbb{Z}.$ (4.43)

Now, we claim that the sequence $(\Delta_{n})_{n\in\mathbb{Z}}$ admits an
exponential trichotomy and that its dichotomy spectrum satisfies

$\displaystyle\Sigma(\Delta_{n})\subset(0,\mu_{1}]\cup[\nu_{-},\nu_{+}]\cup[\mu_{2}^{-1},+\infty).$
(4.44)

In fact, notice that $\tilde{\Xi}^{p,\pm}:=\\{0\\}\times\Xi^{p,\pm}$ are two
invariant subspaces of $\tilde{\Xi}^{p}_{c}:=X_{c}\times\Xi^{p}$ under
$(\Delta_{n})_{n\in\mathbb{Z}}$ and that we can equip $\tilde{\Xi}^{p}_{c}$
with the norm

$\|(x_{c},\psi^{p}(v))\|:=\max\\{\|x_{c}\|,\|\psi^{p}(v)\|\\},\quad(x_{c},\psi^{p}(v))\in\tilde{\Xi}^{p}_{c}.$

Moreover, (4.2) and (4.39) give that

$\displaystyle\begin{split}&\|\Delta_{m-1}\cdots\Delta_{n}(0,\psi^{p,+}(v))\|\\\
&=\|{\mathcal{L}}(m,n)\psi^{p,+}(v)\|\leq
C\mu_{1}^{m-n}\|(0,\psi^{p,+}(v))\|\quad{\rm for}~{}m\geq n,\\\
&\|\Delta_{m}^{-1}\cdots\Delta_{n-1}^{-1}(0,\psi^{p,-}(v))\|\\\
&=\|{\mathcal{L}}(m,n)\psi^{p,-}(v)\|\leq
C\mu_{2}^{n-m}\|(0,\psi_{n}^{p,-}(v))\|\quad{\rm for}~{}m\leq n.\end{split}$
(4.45)

Consequently, $\tilde{\Xi}^{p,\pm}$ can be regarded as the stable and unstable
subspaces of $(\Delta_{n})_{n\in\mathbb{Z}}$, respectively.

Furthermore, in order to find the center subspace of
$(\Delta_{n})_{n\in\mathbb{Z}}$, for any given $x_{c}\in X_{c}$, let
$\epsilon_{n}(v)\in\Xi^{p}$, $n\in\mathbb{Z}$ be defined by

$\displaystyle\epsilon_{n}(v):=\sum_{i=-\infty}^{n-1}{\mathcal{L}}(n,i+1)\Pi^{+}S_{i}(v)x_{c}-\sum_{i=n}^{+\infty}{\mathcal{L}}(n,i+1)\Pi^{-}S_{i}(v)x_{c},$
(4.46)

where $S_{i}(v):=M_{i}(v){\mathcal{A}}_{c}(i,0):X_{c}\to\Xi^{p}$ for
$i\in\mathbb{Z}$. We claim that $\epsilon_{n}(v)$ is well-defined. In fact, by
(2.13) (see also Remark 2.1) and (4.43), we have

$\displaystyle\sup_{i\leq
0}\\{\nu_{-}^{-i-1}\|S_{i}(v)\|\\}\leq\tilde{M}\sup_{i\leq
0}\\{\nu_{-}^{-i-1}\|{\mathcal{A}}_{c}(i,0)\|\\}<\infty,$
$\displaystyle\sup_{i\geq
0}\\{\nu_{+}^{-i-1}\|S_{i}(v)\|\\}\leq\tilde{M}\sup_{i\geq
0}\\{\nu_{+}^{-i-1}\|{\mathcal{A}}_{c}(i,0)\|\\}<\infty.$

The above together with (4.2) and (4.39) implies that for any $n\geq 0$

$\displaystyle\|\epsilon_{n}(v)\|$
$\displaystyle\leq\nu_{-}^{n}\sum_{i=-\infty}^{0}C(\mu_{1}/\nu_{-})^{n-i-1}\nu_{-}^{-i-1}\|\Pi^{+}S_{i}(v)\|$
$\displaystyle\quad+\nu_{+}^{n}\sum_{i=1}^{n-1}C(\mu_{1}/\nu_{+})^{n-i-1}\nu_{+}^{-i-1}\|\Pi^{+}S_{i}(v)\|$
$\displaystyle\quad+\nu_{+}^{n}\sum_{i=n}^{+\infty}C(\mu_{2}\nu_{+})^{i+1-n}\nu_{+}^{-i-1}\|\Pi^{-}S_{i}(v)\|<\infty,$

and that for any $n\leq 0$

$\displaystyle\|\epsilon_{n}(v)\|$
$\displaystyle\leq\nu_{-}^{n}\sum_{i=-\infty}^{n-1}C(\mu_{1}/\nu_{-})^{n-i-1}\nu_{-}^{-i-1}\|\Pi^{+}S_{i}(v)\|$
$\displaystyle\quad+\nu_{-}^{n}\sum_{i=n}^{-1}C(\mu_{2}\nu_{-})^{i+1-n}\nu_{-}^{-i-1}\|\Pi^{-}S_{i}(v)\|$
$\displaystyle\quad+\nu_{+}^{n}\sum_{i=0}^{+\infty}C(\mu_{2}\nu_{+})^{i+1-n}\nu_{+}^{-i-1}\|\Pi^{-}S_{i}(v)\|<\infty,$

since

$\displaystyle\begin{split}&\frac{\mu_{1}}{\nu_{-}}=\max_{q\in
S_{+}^{p}}\\{\nu_{+}/(\nu_{-}a_{1}^{q_{1}}\cdots
a_{r}^{q_{r}})\\}\in(0,1),\quad\frac{\mu_{1}}{\nu_{+}}=\max_{q\in
S_{+}^{p}}\\{1/a_{1}^{q_{1}}\cdots a_{r}^{q_{r}}\\}\in(0,1),\\\
&\mu_{2}\nu_{+}=\max_{q\in S_{-}^{p}}\\{(\nu_{+}b_{1}^{q_{1}}\cdots
b_{r}^{q_{r}})/\nu_{-}\\}\in(0,1),\quad\mu_{2}\nu_{-}=\max_{q\in
S_{-}^{p}}\\{b_{1}^{q_{1}}\cdots b_{r}^{q_{r}}\\}\in(0,1),\end{split}$ (4.47)

due to (4.20). Then, $\epsilon_{n}(v)$ is well-defined for $n\in\mathbb{Z}$
and

$\displaystyle\sup_{n\geq
0}(\nu_{+}^{-n}\|\epsilon_{n}(v)\|)<\infty,\qquad\sup_{n\leq
0}(\nu_{-}^{-n}\|\epsilon_{n}(v)\|)<\infty.$ (4.48)

Moreover, we verify that

$\displaystyle\Delta_{n}({\mathcal{A}}_{c}(n,0)x_{c},\epsilon_{n}(v))$
$\displaystyle=\Big{(}{\mathcal{A}}_{c}(n+1,0)x_{c},~{}M_{n}{\mathcal{A}}_{c}(n,0)x_{c}+L_{n}(0)\epsilon_{n}(v)\Big{)}$
$\displaystyle=\Big{(}{\mathcal{A}}_{c}(n+1,0)x_{c},~{}\sum_{i=-\infty}^{n-1}{\mathcal{L}}(n+1,i+1)\Pi^{+}S_{i}(v)$
$\displaystyle\qquad-\sum_{i=n}^{+\infty}{\mathcal{L}}(n+1,i+1)\Pi^{-}S_{i}(v)+S_{n}(v)\Big{)}$
$\displaystyle=({\mathcal{A}}_{c}(n+1,0)x_{c},\epsilon_{n+1}(v)),$

and therefore the sequence
$\\{({\mathcal{A}}_{c}(n,0)x_{c},\epsilon_{n}(v))\in\tilde{\Xi}^{p}_{c}:n\in\mathbb{Z}\\}$
is an orbit of $(\Delta_{n})_{n\in\mathbb{Z}}$. Conversely, using the
Variation of Constants Formula, we can prove that for any given $x_{c}\in
X_{c}$ if $(\epsilon_{n}(v))_{n\in\mathbb{Z}}\subset\Xi^{p}$ satisfies (4.48)
and
$\\{({\mathcal{A}}_{c}(n,0)x_{c},\epsilon_{n}(v))\in\tilde{\Xi}^{p}_{c}:n\in\mathbb{Z}\\}$
is an orbit of $(\Delta_{n})_{n\in\mathbb{Z}}$, then $\epsilon_{n}(v)$ are
given by (4.46). We conclude that

  * •

for any $x_{c}\in X_{c}$, there is a unique
$(\epsilon_{n}(v))_{n\in\mathbb{Z}}$ such that
$\\{({\mathcal{A}}_{c}(n,0)x_{c},\epsilon_{n}(v))\in\tilde{\Xi}^{p}_{c}:n\in\mathbb{Z}\\}$
is an orbit of $(\Delta_{n})_{n\in\mathbb{Z}}$.

The sequence $(\epsilon_{n}(v))_{n\in\mathbb{Z}}$ with the above property
corresponding to $x_{c}\in X_{c}$ will be denoted by
$(\epsilon_{n}(x_{c},v))_{n\in\mathbb{Z}}$.

The above discussion enables us to define

$\tilde{\Xi}_{n}^{c}:=\\{(x_{c},\epsilon_{n}(x_{c},v))\in\tilde{\Xi}^{p}_{c}:x_{c}\in
X_{c}\\},\quad\forall n\in\mathbb{Z},$

which is clearly a (nonautonomous) linear subspace invariant under
$(\Delta_{n})_{n\in\mathbb{Z}}$, i.e.
$\Delta_{n}\tilde{\Xi}_{n}^{c}=\tilde{\Xi}_{n+1}^{c}$ for $n\in\mathbb{Z}$.
Moreover,

$\tilde{\Xi}^{p}_{c}=\tilde{\Xi}^{p,+}\oplus\tilde{\Xi}_{n}^{c}\oplus\tilde{\Xi}^{p,-},\quad
n\in\mathbb{Z}.$

Denote the projections corresponding to the above decomposition by
$\tilde{\Pi}_{n}^{+}$, $\tilde{\Pi}_{n}^{c}$ and $\tilde{\Pi}_{n}^{-}$,
respectively. Note that

$\displaystyle\tilde{\Pi}_{n}^{c}(x_{c},\psi^{p}):=(x_{c},\epsilon_{n}(x_{c},v)),\quad\tilde{\Pi}_{n}^{+}(x_{c},\psi^{p}):=(0,\Pi^{+}(\psi^{p}-\epsilon_{n}(x_{c},v)),$
$\displaystyle\tilde{\Pi}_{n}^{-}(x_{c},\psi^{p}):=(0,\Pi^{-}(\psi^{p}-\epsilon_{n}(x_{c},v)),\quad\forall\psi^{p}\in\Xi^{p}.$

Then, (4.45) gives

$\displaystyle\begin{split}&\|\Delta_{m-1}\cdots\Delta_{n}\tilde{\Pi}_{n}^{+}(x_{c},\psi_{n}^{p}(v))\|\leq
C\mu_{1}^{m-n}\|\tilde{\Pi}_{n}^{+}(x_{c},\psi_{n}^{p}(v))\|\quad{\rm
for}~{}m\geq n,\\\
&\|\Delta_{m}^{-1}\cdots\Delta_{n-1}^{-1}\tilde{\Pi}_{n}^{-}(x_{c},\psi_{n}^{p}(v))\|\leq
C\mu_{2}^{n-m}\|\tilde{\Pi}_{n}^{-}(x_{c},\psi_{n}^{p}(v))\|\quad{\rm
for}~{}m\leq n.\end{split}$ (4.49)

Moreover, we obtain from the last two inequalities of (2.13) (see also Remark
2.1) and (4.48) that

$\displaystyle\begin{split}&\|\Delta_{m-1}\cdots\Delta_{n}\tilde{\Pi}_{n}^{c}(x_{c},\psi_{n}^{p}(v))\|\leq
C\nu_{+}^{m-n}\|\tilde{\Pi}_{n}^{c}(x_{c},\psi_{n}^{p}(v))\|\quad{\rm
for}~{}m\geq n,\\\
&\|\Delta_{m}^{-1}\cdots\Delta_{n-1}^{-1}\tilde{\Pi}_{n}^{c}(x_{c},\psi_{n}^{p}(v))\|\leq
C\nu_{-}^{m-n}\|\tilde{\Pi}_{n}^{c}(x_{c},\psi_{n}^{p}(v))\|\quad{\rm
for}~{}m\leq n.\end{split}$ (4.50)

Combining (4.49) with (4.50), we conclude that the linear cocycle generated by
$(\Delta_{n})_{n\in\mathbb{Z}}$ admits an exponential trichotomy and its
dichotomy spectrum satisfies (4.44) due to (2.10) and Remark 2.1. Moreover,
since $\sup_{n\in\mathbb{Z}}\|\Delta_{n}\|<+\infty$ and
$\sup_{n\in\mathbb{Z}}\|\Delta_{n}^{-1}\|<+\infty$,
$(\Delta_{n})_{n\in\mathbb{Z}}$ admits a strong exponential trichotomy. This
implies that the system $(\tilde{\Upsilon}_{n})_{n\in\mathbb{Z}}$ defined by
(4.2) has a (nonautonomous) $C^{N-p-1}$ center manifold, which has bounded
derivatives up order $N-p-1$ near $(0,0)$, provided that the spectral gap
condition

$\displaystyle\nu_{+}^{N-p-1}<\mu_{2}^{-1},\qquad\nu_{-}^{N-p-1}>\mu_{1}$
(4.51)

holds (see Proposition 3.1). Thus, since (4.20) implies (4.51) (actually it
implies that $\nu_{+}^{N}<\mu_{2}^{-1}$, $\nu_{-}^{N}>\mu_{1}$), we can
conclude that the equation (4.30) has a $C^{N-p-1}$ (in $x_{c}$) solution,
which has bounded derivatives up order $N-p-1$ near $(0,\kappa_{n}^{p}(0,v))$,
due to the discussion given in the second paragraph of this proof, and
therefore we find the desired sequence $(H_{n}^{p}(x_{c},v))_{n\in\mathbb{Z}}$
given in the formulation of Lemma 4.2. Moreover, it is clear that (4.26) holds
locally since $H_{n}^{p}$ is a near-identity map. This completes the proof of
this lemma. $\Box$

Applying Lemma 4.2 $N_{0}$ times, we eliminate all terms $f_{n}^{p}(x_{c},v)$
for $p=1,\ldots,N_{0}$ via $N_{0}$ conjugacies of class $C^{N-N_{0}}$ (i.e.,
$C^{2N_{0}+1}$) and obtain another sequence of maps, also denoted by
$(\hat{F}_{n}(x_{c},v))_{n\in\mathbb{Z}}$, having the form

$\hat{F}_{n}(x_{c},v):=\Big{(}w_{n}(x_{c})+o(\|v\|^{N_{0}}),~{}A_{n}^{su}(x_{c})v+O(\|v\|^{2})\Big{)},$

where $w_{n}(x_{c})$ is $C^{N-N_{0}}$ and $A_{n}^{su}(x_{c})$ is
$C^{N-N_{0}-1}$. Moreover, for a $p\in\\{2,\ldots,N_{0}\\}$, assuming that

$\partial_{v}^{i}(\pi_{su}\hat{F}_{n})(x_{c},0)=0\quad{\rm
for}~{}i=2,\ldots,p-1,$

we can rewrite $\hat{F}_{n}$ as

$\hat{F}_{n}(x_{c},v)=\Big{(}w_{n}(x_{c})+o(\|v\|^{N_{0}}),~{}A_{n}^{su}(x_{c})v+g_{n}^{p}(x_{c},v)+o(\|v\|^{p})\Big{)},$

where
$g_{n}^{p}(x_{c},v):=\partial_{v}^{p}(\pi_{su}\hat{F}_{n})(x_{c},0)v^{p}$ is a
homogeneous polynomial in variable $v$ of order $p$ with coefficients being
$C^{N-N_{0}-p}$ maps in $x_{c}$.

######  Lemma 4.3.

The sequence $(\hat{F}_{n})_{n\in\mathbb{Z}}$ of $C^{N-N_{0}-p}$ maps is
conjugated to a sequence $(\check{F}_{n})_{n\in\mathbb{Z}}$ of
$C^{N-N_{0}-p-1}$ maps given by

$\check{F}_{n}(x_{c},v):=\Big{(}w_{n}(x_{c})+o(\|v\|^{N_{0}}),~{}A_{n}^{su}(x_{c})v+o(\|v\|^{p})\Big{)},$

via a sequence $(\hat{H}_{n}^{p}(x_{c},v))_{n\in\mathbb{Z}}$, where
$\hat{H}_{n}^{p}:X_{c}\times X_{su}\to X_{su}$ is a homogeneous polynomial in
variable $v$ of order $p$ with coefficients being $C^{N-N_{0}-p-1}$ maps in
$x_{c}$. Moreover,

$\displaystyle\begin{split}&(\hat{H}_{n}^{p})^{\pm}(0)=0,\quad
d_{0}(\hat{H}_{n}^{p})^{\pm}=\operatorname{Id},\quad\|d_{x}(\hat{H}_{n}^{p})^{\pm}-\operatorname{Id}\|\leq\tilde{\delta},\\\
&\|d_{x}^{i}(\hat{H}_{n}^{p})^{\pm}\|\leq\tilde{M},\quad
i=2,\ldots,N-N_{0}-p,\quad{\rm for}~{}x=(x_{c},v)~{}{\rm
near}~{}0.\end{split}$ (4.52)

Proof of Lemma 4.3. Let $\hat{h}_{n}^{p}:X_{c}\times X_{su}\to X_{su}$,
$n\in\mathbb{Z}$ be a sequence of homogeneous polynomials of variable $v$ of
order $p$, and consider the transformations

$\hat{H}_{n}^{p}:(x_{c},v)\mapsto(x_{c},v+\hat{h}_{n}^{p}(x_{c},v)).$

Similarly to the proof of Lemma 4.2, we calculate that

$\displaystyle\hat{H}^{p}_{n+1}\circ\hat{F}_{n}\circ(\hat{H}_{n}^{p})^{-1}(x_{c},v)$
$\displaystyle=\bigg{(}w_{n}(x_{c})+o(\|v\|^{N_{0}}),~{}A_{n}^{su}(x_{c})v-A_{n}^{su}(x_{c})\hat{h}_{n}^{p}(x_{c},v)+g_{n}^{p}(x_{c},v)+$
$\displaystyle\qquad\hat{h}_{n+1}^{p}(w_{n}(x_{c}),A_{n}^{su}(x_{c})v)+o(\|v\|^{p})\bigg{)}.$

In order to eliminate the term of order $p$, we solve the following
nonautonomous homological equation

$-A_{n}^{su}(x_{c})\hat{h}_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v)+g_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v)+\hat{h}_{n+1}^{p}(w_{n}(x_{c}),v)=0$
(4.53)

for $n\in\mathbb{Z}$. Projecting (4.53) onto $X_{i}$, $i=1,\ldots,r$, we get

$-A_{n}^{i}(x_{c})\pi_{i}\hat{h}_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v)+\pi_{i}g_{n}^{p}(x_{c},A_{n}^{su}(x_{c})^{-1}v)+\pi_{i}\hat{h}_{n+1}^{p}(w_{n}(x_{c}),v)=0.$
(4.54)

Comparing (4.54) with (4.30), we understand that if we replace $Dw_{n}(x_{c})$
with $A_{n}^{i}(x_{c})$ and use (4.21) instead of (4.20), then, similarly to
the proof of Lemma 4.2, we can prove Lemma 4.3. $\Box$

Applying Lemma 4.3 $N_{0}-1$ times, we eliminate all terms
$g_{n}^{p}(x_{c},v)$ for $p=2,\ldots,N_{0}$ via $N_{0}-1$ conjugacies of class
$C^{N-2N_{0}}$ (i.e., $C^{N_{0}+1}$) and obtain another sequence, still
denoted by $(\check{F}_{n}(x_{c},v))_{n\in\mathbb{Z}}$, having the form

$\check{F}_{n}(x_{c},v):=\Big{(}w_{n}(x_{c})+o(\|v\|^{N_{0}}),~{}A_{n}^{su}(x_{c})v+o(\|v\|^{N_{0}})\Big{)}.$

Now, we see that $(\check{F}_{n})_{n\in\mathbb{Z}}$ and (4.18) with
$f_{n}^{c}(x_{c}):=w_{n}(x_{c})-A_{n}^{c}x_{c}$ have the same derivatives up
to order $N_{0}$ on their common center manifold $X_{c}$ and that, by using
the smooth cut-off functions, (4.1)-(4.2) hold for
$(\check{F}_{n})_{n\in\mathbb{Z}}$ and the sequence given by (4.18) (replacing
$F_{n}$ and $G_{n}$, respectively).

Finally, using Theorem 4.1, we can find a sequence $(H_{n})_{n\in\mathbb{Z}}$
of $C^{k}$ maps, which satisfies (4.4) and (4.6), locally conjugating
$(\check{F}_{n})_{n\in\mathbb{Z}}$ to its Takens normal form (4.18). Hence,
the sequence

$(\Psi_{n})_{n\in\mathbb{Z}}:=(H_{n}\circ\hat{H}_{n}^{N_{0}}\circ\cdots\circ\hat{H}_{n}^{2}\circ
H_{n}^{N_{0}}\circ\cdots\circ H_{n}^{1}\circ\varphi_{n})_{n\in\mathbb{Z}}$

is the desired $C^{k}$ conjugacy from the statement of Theorem 4.2, where
$\varphi_{n}$ is given by (4.22) and $H_{n}^{p}$, $\hat{H}_{n}^{p}$ are given
by Lemmas 4.2 and 4.3, respectively. Moreover, $(\Psi_{n})_{n\in\mathbb{Z}}$
satisfies (4.19) due to (4.23), (4.26) and (4.52). This completes the proof of
this theorem. $\Box$

##  5\. Appendix: The Belitskii-Samavol Theorem on Hilbert spaces

In what follows, $(X,\|\cdot\|)$ is a Hilbert space. Let $A:X\to X$ be an
invertible and bounded linear operator which admits a strong exponential
trichotomy. In other words, we suppose that there exist projections
$\Pi^{s},\Pi^{c}$ and $\Pi^{u}:X\to X$ as well as constants $\tilde{K}>0$,
$0<\tilde{\mu}_{-}\leq\tilde{\mu}_{+}<\tilde{\lambda}_{-}\leq
1\leq\tilde{\lambda}_{+}<\tilde{\varrho}_{-}\leq\tilde{\varrho}_{+}$ such
that:

  * •

$\Pi^{s}+\Pi^{c}+\Pi^{u}={\rm Id}$;

  * •

$A\Pi^{*}=\Pi^{*}A$ for $*=s,c,u$;

  * •

for any $n\geq 0$,

$\displaystyle\|A^{n}\Pi^{s}\|$
$\displaystyle\leq\tilde{K}\tilde{\mu}_{+}^{n},\qquad$
$\displaystyle\|A^{-n}\Pi^{s}\|$
$\displaystyle\leq\tilde{K}\tilde{\mu}_{-}^{-n},$ (5.1)
$\displaystyle\|A^{n}\Pi^{c}\|$
$\displaystyle\leq\tilde{K}\tilde{\lambda}_{+}^{n},\qquad$
$\displaystyle\|A^{-n}\Pi^{c}\|$
$\displaystyle\leq\tilde{K}\tilde{\lambda}_{-}^{-n},$ (5.2)
$\displaystyle\|A^{n}\Pi^{u}\|$
$\displaystyle\leq\tilde{K}\tilde{\varrho}_{+}^{n},\qquad$
$\displaystyle\|A^{-n}\Pi^{u}\|$
$\displaystyle\leq\tilde{K}\tilde{\varrho}_{-}^{-n}.$ (5.3)

Let $X_{s}:=\Pi^{s}X$, $X_{c}:=\Pi^{c}X$ and $X_{u}:=\Pi^{u}X$. Clearly,
$X=X_{s}\oplus X_{c}\oplus X_{u}$. Let $N\in\mathbb{N}$ and $\delta,M>0$,
where we will assume in the sequel that $\delta$ is sufficiently small. Assume
that:

  1. (H1)

$F=A+f:X\to X$ is a $C^{N}$ diffeomorphism, where $f(0)=0$, $d_{0}f=0$,
$\|d_{x}f\|\leq\delta$ for all $x\in X$ and $\|d^{i}_{x}f\|\leq M$ for all
$x\in X$ and $2\leq i\leq N$;

  2. (H2)

$G=A+g:X\to X$ is a $C^{N}$ diffeomorphism, where $g(0)=0$, $d_{0}g=0$,
$\|d_{x}g\|\leq\delta$ for all $x\in X$ and $\|d^{i}_{x}g\|\leq M$ for all
$x\in X$ and $2\leq i\leq N$;

  3. (H3)

$\tilde{\mu}_{+}<\tilde{\lambda}_{-}^{N}$ and
$\tilde{\lambda}_{+}^{N}<\tilde{\varrho}_{-}$.

######  Theorem 5.1.

Take an arbitrary $k\in\mathbb{N}$. Then, there exists $N_{0}\in\mathbb{N}$
with the property that if $A$ admits a strong exponential trichotomy,
hypotheses (H1)-(H3) hold with $N\geq N_{0}$, and $F$ and $G$ have the common
center manifold $\Gamma^{c}$ such that $d^{i}_{x}(F-G)=0$ for all $1\leq i\leq
N$ and all $x\in\Gamma^{c}$, then $F$ and $G$ are locally $C^{k}$ conjugated.

This theorem will be proved using the so-called homotopy method. Our arguments
are inspired with the proof of [23, Theorem 3.2]. In accordance with the
hypotheses (H1) and (H3), $F$ has the $C^{N}$ center-stable manifold

$\Gamma^{sc}=\\{(x_{s},x_{c})+h^{sc}(x_{s},x_{c})|(x_{s},x_{c})\in X_{s}\oplus
X_{c}\\},$ (5.4)

and the $C^{N}$ center-unstable manifold

$\Gamma^{uc}=\\{(x_{u},x_{c})+h^{uc}(x_{u},x_{c})|(x_{u},x_{c})\in X_{u}\oplus
X_{c}\\},$ (5.5)

where $h^{sc}:X_{s}\oplus X_{c}\to X_{u}$ and $h^{uc}:X_{u}\oplus X_{c}\to
X_{s}$ both are $C^{N}$. The intersection of $\Gamma^{sc}$ and $\Gamma^{uc}$
is the center manifold of $F$. We now use the center-stable manifold, center-
unstable manifold and center manifold to introduce new axes. Let

$\displaystyle\tilde{x}_{s}$ $\displaystyle:=x_{s}-h^{uc}(x_{u},x_{c}),$
$\displaystyle\tilde{x}_{c}$ $\displaystyle:=x_{c},$ (5.6)
$\displaystyle\tilde{x}_{u}$ $\displaystyle:=x_{u}-h^{sc}(x_{s},x_{c}).$

By [5, Theorem 3.1], we see that Lipschitz constants Lip$(h^{uc})$ and
Lip$(h^{sc})$ are both small if Lip$(f)\leq\delta$ is small. Thus, the
transformation (5) is invertible on $X$ with Lipschitz inverse by the
Lipschitz inverse function theorem. Furthermore, it is a $C^{N}$
diffeomorphism on $X$ since $h^{sc}$ and $h^{uc}$ are both $C^{N}$.

Using the transformation (5), we have that $F$ is $C^{N}$ conjugated to a
$C^{N}$ diffeomorphism $\widetilde{F}$ of the form

$\widetilde{F}(\tilde{x}_{s},\tilde{x}_{c},\tilde{x}_{u})=(A_{s}\tilde{x}_{s},A_{c}\tilde{x}_{c},A_{u}\tilde{x}_{u})+\tilde{f}(\tilde{x}_{s},\tilde{x}_{c},\tilde{x}_{u}),$

where $A_{*}:=A|_{X_{*}}$ for $*=s,c,u$,
$\tilde{f}(\tilde{x}_{s},\tilde{x}_{c},\tilde{x}_{u})=O(\|(\tilde{x}_{s},\tilde{x}_{c},\tilde{x}_{u})\|^{2})$,
$\Pi^{s}\tilde{f}(0,\tilde{x}_{c},\tilde{x}_{u})=0$ for
$(\tilde{x}_{c},\tilde{x}_{u})\in X_{c}\oplus X_{u}$, and
$\Pi^{u}\tilde{f}(\tilde{x}_{s},\tilde{x}_{c},0)=0$ for
$(\tilde{x}_{s},\tilde{x}_{c})\in X_{s}\oplus X_{c}$.

Similarly, $G$ is also $C^{N}$ conjugated to a $C^{N}$ diffeomorphism
$\tilde{G}$ having the form

$\widetilde{G}(\tilde{x}_{s},\tilde{x}_{c},\tilde{x}_{u})=(A_{s}\tilde{x}_{s},A_{c}\tilde{x}_{c},A_{u}\tilde{x}_{u})+\tilde{g}(\tilde{x}_{s},\tilde{x}_{c},\tilde{x}_{u}),$

where
$\tilde{g}(\tilde{x}_{s},\tilde{x}_{c},\tilde{x}_{u})=O(\|(\tilde{x}_{s},\tilde{x}_{c},\tilde{x}_{u})\|^{2})$,
$\Pi^{s}\tilde{g}(0,\tilde{x}_{c},\tilde{x}_{u})=0$ for
$(\tilde{x}_{c},\tilde{x}_{u})\in X_{c}\oplus X_{u}$, and
$\Pi^{u}\tilde{g}(\tilde{x}_{s},\tilde{x}_{c},0)=0$ for
$(\tilde{x}_{s},\tilde{x}_{c})\in X_{s}\oplus X_{c}$.

Let $\widetilde{R}:=\tilde{f}-\tilde{g}$. Since $d^{i}_{x}(F-G)=0$ for $0\leq
i\leq N$ and $x\in\Gamma^{c}$, we have that
$d^{i}_{\tilde{x}_{c}}\widetilde{R}=0$ for $0\leq i\leq N$ and
$\tilde{x}_{c}\in X_{c}$. We split $\widetilde{R}$ into two parts, one of
which is flat on $X_{s}\oplus X_{c}$ while the other is flat on $X_{c}\oplus
X_{u}$.

######  Lemma 5.1.

The function $\widetilde{R}$ admits a decomposition,
$\widetilde{R}=\widetilde{R}_{1}+\widetilde{R}_{2}$, where for $0\leq
i\leq[N/2]$ we have that

$\displaystyle
d^{i}_{(0,\tilde{x}_{c},\tilde{x}_{u})}\widetilde{R}_{1}=0\qquad\text{for
all}~{}~{}(\tilde{x}_{c},\tilde{x}_{u})\in X_{c}\oplus X_{u},$ (5.7)
$\displaystyle
d^{i}_{(\tilde{x}_{s},\tilde{x}_{c},0)}\widetilde{R}_{2}=0\qquad\text{for
all}~{}~{}(\tilde{x}_{s},\tilde{x}_{c})\in X_{s}\oplus X_{c}.$ (5.8)

###### Proof.

We consider the Taylor expansion of $\widetilde{R}$ at $0$ with respect to
$\tilde{x}_{u}$,

$\widetilde{R}(\tilde{x})=\sum_{i=0}^{[N/2]}\frac{1}{i!}\frac{\partial^{i}\widetilde{R}}{\partial\tilde{x}_{u}^{i}}(\tilde{x}_{s},\tilde{x}_{c},0)\tilde{x}_{u}^{i}+o(\|\tilde{x}_{u}\|^{[N/2]}).$
(5.9)

Take partial derivatives of
$\frac{\partial^{i}\widetilde{R}}{\partial\tilde{x}_{u}^{i}}(\cdot,\cdot,0)(\cdot)^{i}$
at $(0,\tilde{x}_{c},\tilde{x}_{u})$ for all $0\leq i\leq[N/2]$ and all
$(\tilde{x}_{c},\tilde{x}_{u})\in X_{c}\oplus X_{u}$. Observe that

$\frac{\partial^{i_{1}+i_{2}}}{\partial\tilde{x}_{s}^{i_{1}}\partial\tilde{x}_{c}^{i_{2}}}\left(\frac{\partial^{i}\widetilde{R}}{\partial\tilde{x}_{u}^{i}}(0,\tilde{x}_{c},0)\right)=\frac{\partial^{i_{1}+i_{2}+i}\widetilde{R}}{\partial\tilde{x}_{s}^{i_{1}}\partial\tilde{x}_{c}^{i_{2}}\partial\tilde{x}_{u}^{i}}(0,\tilde{x}_{c},0)=0,$

for $i_{1}+i_{2}\leq[N/2]$ since $d^{i}_{(0,\tilde{x}_{c},0)}\widetilde{R}=0$
for all $0\leq i\leq N$ and all ${\tilde{x}_{c}}\in X_{c}$. Thus,
$\widetilde{R}_{1}$ chosen to be

$\widetilde{R}_{1}(\tilde{x}):=\sum_{i=0}^{[N/2]}\frac{1}{i!}\frac{\partial^{i}\widetilde{R}}{\partial\tilde{x}_{u}^{i}}(\tilde{x}_{s},\tilde{x}_{c},0)\tilde{x}_{u}^{i}$

satisfies (5.7). Let

$\widetilde{R}_{2}(\tilde{x}):=\widetilde{R}(\tilde{x})-\sum_{i=0}^{[N/2]}\frac{1}{i!}\frac{\partial^{i}\widetilde{R}}{\partial\tilde{x}_{u}^{i}}(\tilde{x}_{s},\tilde{x}_{c},0)\tilde{x}_{u}^{i}.$

We see from (5.9) that
$\widetilde{R}_{2}(\tilde{x})=o(\|\tilde{x}_{u}\|^{[N/2]})$, which implies
that (5.8) holds. The proof is completed. ∎

For simplicity, we will write $x$ instead of $\tilde{x}$. Since $X$ is a
Hilbert space, we can modify $\tilde{f}$, $\tilde{g}$, $\widetilde{R}_{1}$ and
$\widetilde{R}_{2}$ and obtain $C^{N}$ maps $\hat{f},\hat{g},\hat{R}_{1}$ and
$\hat{R}_{2}$ defined on $X$ such that $\hat{f}(x)=\tilde{f}(x)$,
$\hat{g}(x)=\tilde{g}(x)$ and $\hat{R}_{i}(x)=\widetilde{R}_{i}(x)$ for
$i=1,2$ and $\|x\|\leq\frac{r}{2}$. Moreover,
$\hat{f}(x)=\hat{g}(x)=\hat{R}_{i}(x)=0$ for $i=1,2$ and $\|x\|\geq r$.

Let $\hat{G}_{\tau}(x):=Ax+\hat{g}(x)+\tau\hat{R}_{1}(x)$ for $0\leq\tau\leq
1$ and $x\in X$. We define a map $\overline{G}:X\times[0,1]\to X\times[0,1]$
by

$\overline{G}(x,\tau):=(\hat{G}_{\tau}(x),\tau),\qquad\text{for}~{}(x,\tau)\in
X\times[0,1].$

######  Lemma 5.2.

Suppose that the following homotopy equation

$\frac{\partial\hat{G}_{\tau}}{\partial x}h-h\circ\overline{G}=-\hat{R}_{1},$
(5.10)

where $h:X\times[0,1]\to X$ is an unknown map, has a $C^{k}$ solution with the
property that there exists $\hat{L}>0$ such that

$\|h(x,\tau)\|\leq\hat{L}\|x\|^{2}\qquad\text{for every $\tau\in[0,1]$ and
$\|x\|\leq 1$.}$ (5.11)

Then, $\hat{G}_{0}$ and $\hat{G}_{1}$ are $C^{k}$ locally conjugated.

###### Proof.

By (5.10), we have

$D\overline{G}\begin{pmatrix}h\\\
1\end{pmatrix}=\begin{pmatrix}\frac{\partial\hat{G}_{\tau}}{\partial
x}&\hat{R}_{1}\\\ 0&1\end{pmatrix}\begin{pmatrix}h\\\
1\end{pmatrix}=\begin{pmatrix}\frac{\partial\hat{G}_{\tau}}{\partial
x}h+\hat{R}_{1}\\\ 1\end{pmatrix}=\begin{pmatrix}h\circ\overline{G}\\\
1\end{pmatrix}=\begin{pmatrix}h\\\ 1\end{pmatrix}\circ\overline{G}.$

Set $v:=\begin{pmatrix}h\\\ 1\end{pmatrix}$ and let ${\mathcal{F}}_{v}^{t}$ be
the flow generated by the vector field $v$. Then, for any $(x_{0},\tau_{0})\in
X\times[0,1]$, it follows that

$\displaystyle\frac{d(\overline{G}({\mathcal{F}}_{v}^{t}(x_{0},\tau_{0})))}{dt}$
$\displaystyle=D\overline{G}({\mathcal{F}}_{v}^{t}(x_{0},\tau_{0}))\frac{d({\mathcal{F}}_{v}^{t}(x_{0},\tau_{0}))}{dt}$
$\displaystyle=D\overline{G}({\mathcal{F}}_{v}^{t}(x_{0},\tau_{0}))\begin{pmatrix}h({\mathcal{F}}_{v}^{t}(x_{0},\tau_{0}))\\\
1\end{pmatrix}$ $\displaystyle=\begin{pmatrix}h\\\
1\end{pmatrix}\circ\overline{G}({\mathcal{F}}_{v}^{t}(x_{0},\tau_{0})),$
(5.12)

which means that $\overline{G}({\mathcal{F}}_{v}^{t}(x_{0},\tau_{0}))$ is a
solution of the ODE that induces the flow ${\mathcal{F}}_{v}^{t}$. This yields
that

$\overline{G}({\mathcal{F}}_{v}^{t}(x_{0},\tau_{0}))={\mathcal{F}}_{v}^{t}(\overline{G}({\mathcal{F}}_{v}^{0}(x_{0},\tau_{0})))={\mathcal{F}}_{v}^{t}(\overline{G}(x_{0},\tau_{0})).$

In other words, $\overline{G}$ commutes with ${\mathcal{F}}_{v}^{t}$. Due to
(5.11), ${\mathcal{F}}_{v}^{t}(x,0)$ exists on $[0,1]$ for
$\|x\|\leq\tilde{r}$ with $\tilde{r}>0$ being a constant. Letting
${\mathcal{F}}^{1}_{v}(x,0)=(H(x),1)$ for $\|x\|\leq\tilde{r}$, we derive that

$\displaystyle((H\circ\hat{G}_{0})(x),1)$
$\displaystyle={\mathcal{F}}^{1}_{v}(\hat{G}_{0}(x),0)={\mathcal{F}}^{1}_{v}\circ\overline{G}(x,0)$
$\displaystyle=\overline{G}\circ{\mathcal{F}}^{1}_{v}(x,0)=\overline{G}(H(x),1)=((\hat{G}_{1}\circ
H)(x),1).$

Notice that, since $h$ is of class $C^{k}$, $H$ is a local $C^{k}$
diffeomorphism near $0$. This shows that $\hat{G}_{0}$ and $\hat{G}_{1}$ are
$C^{k}$ locally conjugated. The proof of the lemma is completed. ∎

######  Lemma 5.3.

The equation (5.10) has a $C^{k}$ solution satisfying (5.11) for any fixed
$k\in\mathbb{N}$ provided that (5.7) holds for a sufficiently large $N$.

To prove this lemma, we need several auxiliary lemmas. The proof of the
following lemma is analogous to the proof of [22, Lemma 4.4] (in the
particular case when $\Omega$ is taken to be a singleton).

######  Lemma 5.4.

Given any $k\in\mathbb{N}$, there exists a constant $L>0$ such that

$\max_{1\leq i\leq k+1}\bigg{\\{}\sup_{(x,\tau)\in
X\times[0,1]}\left\|d^{i}_{(x,\tau)}\hat{G}^{n}_{\tau}\right\|,\sup_{(x,\tau)\in
X\times[0,1]}\left\|d^{i}_{(x,\tau)}\hat{G}^{-n}_{\tau}\right\|\bigg{\\}}\leq
L^{n}\qquad\text{for all}\quad n\geq 0,$ (5.13)

where $\hat{G}^{n}_{\tau}$ and $\hat{G}^{-n}_{\tau}$ refer to $n$-th iterates
of $\hat{G}_{\tau}$ and $\hat{G}^{-1}_{\tau}$, respectively.

######  Lemma 5.5.

Provided that (5.7) holds for a sufficiently large $N$, the series

$h(x,\tau):=-\sum_{n=1}^{+\infty}\frac{\partial\hat{G}_{\tau}^{-n}}{\partial
x}(\hat{G}_{\tau}^{n}(x))\hat{R}_{1}\circ\hat{G}_{\tau}^{n-1}(x)$ (5.14)

converges to a $C^{k}$ map satisfying (5.11) for any fixed $k\in\mathbb{N}$.

###### Proof.

Since $d_{0}\hat{R}_{1}=0$ and $\hat{R}_{1}=0$ for $\|x\|\geq r$, we see that
the derivative of $\hat{R}_{1}$ can be made arbitrarily small on $X$ provided
that $r$ is chosen to be small, which implies that the Lipschitz constant of
the nonlinear term of $\hat{G}_{\tau}$ on $X$ is small. According to [5,
Theorem 3.1 (vi)], we have that $\bigcup_{\xi\in X_{c}\oplus X_{u}}M_{\xi}=X$,
where

$M_{\xi}=\left\\{x\in
X:\limsup_{n\to\infty}\frac{1}{n}\log\|\hat{G}_{\tau}^{n}(x)-\hat{G}_{\tau}^{n}(\xi)\|\leq\log\gamma\right\\}$

for a certain constant $\tilde{\mu}_{+}<\gamma<\tilde{\lambda}_{-}$ and
$M_{\xi}\cap M_{\zeta}=\emptyset$ whenever $\xi\neq\zeta$. Therefore, for any
$x\in X$, there exists a $\xi\in X_{c}\oplus X_{u}$ such that $x\in M_{\xi}$.
In addition, by [5, Lemma 3.3], the sequence of differences
$(q_{n}:=\hat{G}_{\tau}^{n}(x)-\hat{G}_{\tau}^{n}(\xi))_{n\geq 0}$ satisfies
the following equation:

$\displaystyle q_{n}$
$\displaystyle=A_{s}^{n}x_{s}+\sum_{i=0}^{n-1}A_{s}^{n-1-i}\Pi^{s}\\{T(\xi_{i}+q_{i})-T(\xi_{i})\\}$
$\displaystyle\quad-\sum_{i=n}^{\infty}A_{uc}^{n-1-i}({\rm
Id}-\Pi^{s})\\{T(\xi_{i}+q_{i})-T(\xi_{i})\\},~{}~{}~{}n\geq 0,$ (5.15)

where $A_{uc}$ denotes the restriction of $diag(A_{s},A_{c},A_{u})$ on
$X_{c}\oplus X_{u}$, $T:=\hat{G}_{\tau}-diag(A_{s},A_{c},A_{u})$ and
$\xi_{n}:=\hat{G}^{n}_{\tau}(\xi)$ for $n\geq 0$. According to the definition
of $M_{\xi}$, there exists $\epsilon>0$ such that
$\gamma+\epsilon<\tilde{\lambda}_{-}$ and

$C_{*}:=\sup_{n\geq 0}(\gamma+\epsilon)^{-n}\|q_{n}\|<\infty.$

From (5), we derive the estimate

$\displaystyle\|q_{n}\|$
$\displaystyle\leq\|A_{s}^{n}\|\|x_{s}\|+\sum_{i=0}^{n-1}\|A_{s}^{n-1-i}\|{\rm
Lip}(T)\|q_{i}\|+\sum_{i=n}^{\infty}\|A_{uc}^{n-1-i}\|{\rm Lip}(T)\|q_{i}\|$
$\displaystyle\leq\|A_{s}^{n}\|\|x_{s}\|+\left(\sum_{i=0}^{n-1}\|A_{s}^{n-1-i}\|(\gamma+\epsilon)^{i}+\sum_{i=n}^{\infty}\|A_{uc}^{n-1-i}\|(\gamma+\epsilon)^{i}\right){\rm
Lip}(T)C_{*},$

which implies by (5.1) and (5.2) that

$C_{*}\leq\frac{\tilde{K}}{1-c}\|x_{s}\|,$

where

$\displaystyle c$ $\displaystyle:=\sup_{n\geq
0}\left(\sum_{i=0}^{n-1}\|A_{s}^{n-1-i}\|(\gamma+\epsilon)^{i-n}+\sum_{i=n}^{\infty}\|A_{uc}^{n-1-i}\|(\gamma+\epsilon)^{i-n}\right){\rm
Lip}(T)$ $\displaystyle\leq\sup_{n\geq
0}\left(\tilde{K}\sum_{i=0}^{n-1}\left(\frac{\tilde{\mu}_{+}}{\gamma+\epsilon}\right)^{n-i}\frac{1}{\tilde{\mu}_{+}}+\tilde{K}\sum_{i=n}^{\infty}\left(\frac{\gamma+\epsilon}{\tilde{\lambda}_{-}}\right)^{i-n}\frac{1}{\tilde{\lambda}_{-}}\right){\rm
Lip}(T)$
$\displaystyle\leq\tilde{K}\left(\frac{1}{\gamma+\epsilon-\tilde{\mu}_{+}}+\frac{1}{\tilde{\lambda}_{-}-\gamma-\epsilon}\right){\rm
Lip}(T)$ $\displaystyle<1,$

provided that ${\rm Lip}(T)$ is sufficiently small. Then for all $n\geq 0$

$\|\Pi^{s}\hat{G}_{\tau}^{n}(x)\|=\|\Pi^{s}\hat{G}_{\tau}^{n}(x)-\Pi^{s}\hat{G}_{\tau}^{n}(\xi)\|\leq\tilde{K}\|q_{n}\|\leq\frac{\tilde{K}^{2}}{1-c}(\gamma+\epsilon)^{n}\|x_{s}\|.$
(5.16)

On the other hand, by (5.7) and the cut-off argument, we have that for $0\leq
i\leq[N/2]$

$d_{(0,x_{c},x_{u})}^{i}\hat{R}_{1}=0\qquad\text{for all
}\quad(x_{c},x_{u})\in X_{c}\oplus X_{u}.$

Then it follows from the Taylor formula that

$\displaystyle d^{i}_{x}\hat{R}_{1}$
$\displaystyle=d_{(0,x_{c},x_{u})}^{i}\hat{R}_{1}+d_{(0,x_{c},x_{u})}^{i+1}\hat{R}_{1}\cdot
x_{s}+\cdots+$ $\displaystyle\qquad
d_{(0,x_{c},x_{u})}^{[N/2]}\hat{R}_{1}\cdot(x_{s})^{[N/2]-i}+\int_{0}^{1}\frac{(1-t)^{[N/2]-i}}{([N/2]-i)!}d_{(tx_{s},x_{c},x_{u})}^{[N/2]+1}\hat{R}_{1}\cdot(x_{s})^{[N/2]+1-i}dt$
$\displaystyle=\int_{0}^{1}\frac{(1-t)^{[N/2]-i}}{([N/2]-i)!}d_{(tx_{s},x_{c},x_{u})}^{[N/2]+1}\hat{R}_{1}\cdot(x_{s})^{[N/2]+1-i}dt,$

which implies that for all $x\in X$

$\|d_{x}^{i}\hat{R}_{1}\|\leq\frac{Q}{([N/2]+1-i)!}\|x_{s}\|^{[N/2]+1-i}\qquad\text{for
all}\quad 0\leq i\leq k,$ (5.17)

with

$Q=\sup_{x\in X}\left\|d_{x}^{[N/2]+1}\hat{R}_{1}\right\|<\infty.$

Consider $i$, $0\leq i\leq k$, times differentiation of every term on the
right-hand side of (5.14), which is a sum of several terms. We can see every
term in this sum has one factor

$(D^{j}\hat{R}_{1})\circ\hat{G}_{\tau}^{n-1}(x)\qquad\text{for a certain}\quad
0\leq j\leq i,$

where $D^{j}\hat{R}_{1}$ denotes the function of $j$-th derivative of
$\hat{R}_{1}$, which is dominated by

$\frac{Q}{([N/2]+1-i)!}\left(\frac{\tilde{K}^{2}}{1-c}\right)^{[N/2]+1-j}(\gamma+\epsilon)^{(n-1)([N/2]+1-j)}\|x_{s}\|^{[N/2]+1-j}$

using (5.16) and (5.17). By Lemma 5.4, the remaining factors are dominated by
$L^{n}$. Consequently, the series of the $i$-th derivative of (5.14) is
absolutely convergent on any bounded domain provided that $N$ is sufficiently
large. This shows that $h$ defined by (5.14) is a $C^{k}$ map. In particular,
(5.11) holds with a constant $\hat{L}>0$ if $N\geq 2$. The proof is completed.
∎

######  Lemma 5.6.

The map $h$ given by (5.14) satisfies (5.10).

###### Proof.

We check it directly. By (5.14), we see that

$\displaystyle\bigg{(}\frac{\partial\hat{G}_{\tau}}{\partial
x}h-h\circ\overline{G}\bigg{)}(x,\tau)$
$\displaystyle=-\sum_{n=1}^{+\infty}\frac{\partial\hat{G}_{\tau}}{\partial
x}(x)\frac{\partial\hat{G}_{\tau}^{-n}}{\partial
x}(\hat{G}_{\tau}^{n}(x))\hat{R}_{1}\circ\hat{G}_{\tau}^{n-1}(x)$
$\displaystyle\quad+\sum_{n=1}^{+\infty}\frac{\partial\hat{G}_{\tau}^{-n}}{\partial
x}(\hat{G}_{\tau}^{n+1}(x))\hat{R}_{1}\circ\hat{G}_{\tau}^{n}(x)$
$\displaystyle=-\sum_{n=1}^{+\infty}\frac{\partial\hat{G}_{\tau}^{-n+1}}{\partial
x}(\hat{G}_{\tau}^{n}(x))\hat{R}_{1}\circ\hat{G}_{\tau}^{n-1}(x)$
$\displaystyle\quad+\sum_{n=1}^{+\infty}\frac{\partial\hat{G}_{\tau}^{-n}}{\partial
x}(\hat{G}_{\tau}^{n+1}(x))\hat{R}_{1}\circ\hat{G}_{\tau}^{n}(x)$
$\displaystyle=-\hat{R}_{1}(x),$

which yields (5.10). ∎

Now we are ready to prove Lemma 5.3.

Proof of Lemma 5.3. Clearly, Lemma 5.5 together with Lemma 5.6 implies the
desired conclusion. $\square$

We are now in a position to prove Theorem 5.1.

Proof of Theorem 5.1. By Lemmas 5.1, 5.2 and 5.3, we have that $\hat{G}_{0}$
and $\hat{G}_{1}$ are locally $C^{k}$ conjugated. Let
$\hat{F}_{\tau}(x):=Ax+\hat{g}(x)+\hat{R}_{1}(x)+\tau\hat{R}_{2}(x)$ for $x\in
X$ and $\tau\in[0,1]$. Using the same arguments we used to show that
$\hat{G}_{0}$ and $\hat{G}_{1}$ are locally $C^{k}$ conjugated, one can prove
that $\hat{F}_{0}$ and $\hat{F}_{1}$ are also locally $C^{k}$ conjugated.
Since $\hat{G}_{0}=\widetilde{G}$ locally, $\hat{G}_{1}=\hat{F}_{0}$ and
$\hat{F}_{1}=\widetilde{F}$ locally, we conclude that $\widetilde{G}$ is
locally $C^{k}$ conjugated to $\widetilde{F}$. Since $F$ is conjugated to
$\widetilde{F}$ and $G$ is conjugated to $\widetilde{G}$, it is derived that
$F$ and $G$ are locally conjugate. The proof is completed. $\square$

## Acknowledgements

The authors are ranked in alphabetic order of their names and should be
treated equally. D. Dragičević was supported in part by Croatian Science
Foundation under the Project IP-2019-04-1239 and by the University of Rijeka
under the Projects uniri-prirod-18-9 and uniri-prprirod-19-16. X. Tang was
supported by NSFC #12001537, the Start-up Funding of Chongqing Normal
University (Grant No. 20XLB033), and the Science and Technology Research
Program of Chongqing Municipal Education Commission (Grant No. KJQN202300540).
W. Zhang was supported in part by NSF-CQ #CSTB2023NSCQ-JQX0020, NSFC
#11831012, and the Research Project of Chongqing Education Commission
CXQT21014.

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