# Sharpness of $C^{0}$ conjugacy for the non-autonomous differential equations
with Lipschitzian perturbation 111 This paper was jointly supported from the
National Natural Science Foundation of China under Grant (No. 11671176,
11931016) and Grant Fondecyt 1170466.

Weijie Lua Manuel Pintob Y-H. Xiaa222Corresponding author. Yonghui Xia,
xiadoc<EMAIL_ADDRESS>
a College of Mathematics and Computer Science, Zhejiang Normal University,
321004, Jinhua, China
b Departamento de Matemáticas, Universidad de Chile, Santiago, Chile
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The classical $C^{0}$ linearization theorem for the non-autonomous
differential equations states the existence of a $C^{0}$ topological conjugacy
between the nonlinear system and its linear part. That is, there exists a
homeomorphism (equivalent function) $H$ sending the solutions of the nonlinear
system onto those of its linear part. It is proved in the previous literature
that the equivalent function $H$ and its inverse $G=H^{-1}$ are both Hölder
continuous if the nonlinear perturbation is Lipschitzian. Questions: is it
possible to improve the regularity? Is the regularity sharp? To answer this
question, we construct a counterexample to show that the equivalent function
$H$ is exactly Lipschitzian, but the inverse $G=H^{-1}$ is merely Hölder
continuous. Furthermore, we propose a conjecture that such regularity of the
homeomorphisms is sharp (it could not be improved anymore). We prove that  the
conjecture is true for the systems with linear contraction. Furthermore, we
present the special cases of linear perturbation, which are closely related to
the spectrum.
Keywords: topological conjugacy; exponential dichotomy; Hartman-Grobman;
regularity; Lipschitzian
MSC2020: 34C41; 34D09; 34D10

##  1 Introduction

###  1.1 History of linearization

The well-known Hartman-Grobman theorem [1, 2] pointed out that nearby the
hyperbolic equilibrium $u^{*}=0$, the dynamic behaviour of the nonlinear
system $u^{\prime}=Au+f(u)$, where $f\in C^{1}(\mathbb{R}^{n})$,
$f(x)=o(\|x\|)$, is topologically conjugated to its linear problem
$v^{\prime}=Av$. Pugh [3] initially studied global linearization and presented
a particular case of the Hartman-Grobman theorem as long as $f$ satisfies some
goodness conditions, like continuity, boundedness or being Lipschitzian.
Moreover, the Hartman-Grobman theorem has been extended to many classes of
differential equations such as retarded functional equations by Farkas [4],
scalar reaction-diffusion equations by Lu [5], Cahn-Hilliard equation and
phase field equations by Lu and Bates [6] and semilinear hyperbolic evolution
equations by Hein and Prüss [7]. Except for the $C^{0}$ linearization of the
differential equations, Sternberg ([8, 9]) initially investigated $C^{r}$
linearization for $C^{k}$ diffeomorphisms. Sell ([10]) extended the theorem of
Sternberg. More mathematicians paid particular attention on the $C^{1}$
linearization. Belitskii [11], ElBialy [12], Rodrigues and Solà-Morales [13]
studied that $C^{1}$ linearization of hyperbolic diffeomorphisms on Banach
space independently. Recently, Zhang and Zhang [14] proved that the $C^{1}$
linearization is Hölder continuous. Zhang et al. [15, 16, 17] showed the sharp
regularity of linearization for $C^{1}$ or $C^{1,1}$ hyperbolic
diffeomorphisms, and it is proved that the linearization is Hölder continuous.

Efforts were made to the linearization of the nonautonomous dynamical system
initially by Palmer [18]. To weakened the conditions of Palmer’s theorem,
Jiang [19] gave a version of linearization result by a generalized exponential
dichotomy. Recently, Barreira and Valls [20, 21, 22, 23, 24, 25] proved
several versions of Hartman-Grobman theorem provided that the linear
homogenous system admits a nonuniform contraction or nonuniform dichotomy. Xia
et al. [26] improved Palmer’s linearization by assuming that the nonlinear
perturbation would be unbounded or non-Lipschitzian. Huerta et al. [27, 28]
constructed conjugacies between linear system and an unbounded nonlinear
perturbation for a nonuniform contraction. Inspired by the work of Reinfelds
and Šteinberga [29], Backes et al. [30] extended the linearization theorem to
a class of non-hyperbolic system. Newly, Backes and Dragičević [31] further
gave some results of smooth linearization on the basis of [30]. Furthermore,
in [32], they presented the multiscale linearization of nonautonomous systems.
It is interesting that if the nonlinear perturbations are well-behaved, they
do not assume any asymptotic behaviour of the linear system. Other
generalizations of Hartman-Grobman theorem were given for the instantaneous
impulsive system [33, 34, 35], dynamic systems on time scales [36],
differential equations with piecewise constant argument [37, 38].

###  1.2  Counterexample and Motivation

In this paper, we pay particular attention to the regularity of the equivalent
function $H$ which transforms the solutions of nonlinear system onto its
linear part. It is proved in the previous works that the equivalent function
and its inverse are Hölder continuous for the Hartman-Grobman theorem in
$C^{0}$ linearization when the nonlinear perturbations are Lipschitzian (see
e.g. Hein and Prüss [7], Jiang [19], Barreira and Valls [20, 21, 22, 23,
25],Xia et al [26], Backes et al [30], Xia et al [35],Shi and Zhang [39]).
However, the following counterexample is to show that the equivalent function
$H$ is exactly Lipschitzian, but the inverse $G=H^{-1}$ is merely Hölder
continuous.

####  1.2.1 Counterexample to Hölder continuity of the transformation

######  Example 1.1.

For the sake of readable and easily verification, we consider the scalar
equation

$x^{\prime}=-x+f(x).$ (1.1)

We take the nonlinear perturbation $f$ as follows

$f(x)=\begin{cases}\epsilon,&x\geq 1,\\\ \epsilon x,&0\leq x<1,\\\ \epsilon
x^{3},&-1<x<0,\\\ -\epsilon,&x\leq-1,\end{cases}$

where $0<\epsilon<\frac{1}{3}$. We see that for all $x,x_{1},x_{2}$,

$|f(x)|\leq\epsilon,\quad|f(x_{1})-f(x_{2})|\leq 3\epsilon|x_{1}-x_{2}|.$

Moreover, the linear equation $x^{\prime}=-x$ has an uniform contraction with
constant $k=1$ and exponent $\alpha=1$. Taking $k=1,L_{f}=3\epsilon,\alpha=1$
($L_{f}K/\alpha=3\epsilon<1$), it is easy to show that there exists a
homeomorphism $H(x)$ transforming the solution $x(t)$ of (1.1) onto that of
its linear part

$x^{\prime}=-x.$ (1.2)

To show the regularity of $H$ and $G=H^{-1}$ (the inverse). We dive it into
three steps.

Step 1. We obtain the explicit expressions of $H$ and $G$. Notice that

$-x+f(x)\begin{cases}<0,&\mathrm{if}\;x>0,\\\ =0,&\mathrm{if}\;x=0,\\\
>0,&\mathrm{if}\;x<0.\end{cases}$

Thus a solution is either always $0$, always $<0$ or always $>0$. Now we
divide the discussion into three cases.

(I) zero solution. $H(0)=0$ since $0$ is a solution of (1.1) and $H(0)$ is the
unique solution of (1.2). But $0$ is such a solution.

(II) positive solution $x(t)>0$.  Clearly, $x(t)$ is strictly decreasing,
i.e., $x(t)\rightarrow 0$ as $t\rightarrow+\infty$; $x(t)\rightarrow\infty$ as
$t\rightarrow-\infty$. Therefore, there must exists a unique time $t_{0}$ such
that $x(t_{0})=1$. Without loss of generality, we take $t_{0}=0$. If $t<0$,
then $x(t)>1$ and so $x^{\prime}(t)=-x(t)+\epsilon$ with $x(0)=1$. Hence,

$x(t)=(1-\epsilon)e^{-t}+\epsilon,\quad t\leq 0.$

If $t>0$, then $0<x(t)<1$ and so $x^{\prime}(t)=-x(t)+\epsilon x(t)$ with
$x(0)=1$. Hence,

$x(t)=e^{(-1+\epsilon)t},\quad t\geq 0.$

We need to find the unique solution $y(t)$ of (1.2) such that $|y(t)-x(t)|$ is
bounded. Looking at $x(t)$ when $t\leq 0$, we see that
$y(t)=(1-\epsilon)e^{-t}$. Hence for all $t$,

$H(x(t))=(1-\epsilon)e^{-t}.$

Then

$H(1)=H(x(0))=1-\epsilon.$

If $\xi>1$, then there exists a unique time $t<0$ such that
$x(t)=(1-\epsilon)e^{-t}+\epsilon=\xi$. Then

$H(\xi)=H(x(t))=(1-\epsilon)e^{-t}=x(t)-\epsilon=\xi-\epsilon.$

If $0<\xi<1$, then there exists a unique time $t>0$ such that
$x(t)=e^{(-1+\epsilon)t}=\xi$. Then

$H(\xi)=H(x(t))=(1-\epsilon)e^{-t}=(1-\epsilon)x(t)^{\frac{1}{1-\epsilon}}=(1-\epsilon)\xi^{\frac{1}{1-\epsilon}}.$

Therefore,

$H(x)=\begin{cases}(1-\epsilon)x^{\frac{1}{1-\epsilon}},&0<x<1,\\\
x-\epsilon,&x\geq 1.\end{cases}$

(III) negative solution $x(t)<0$.  Clearly, $x(t)$ is strictly increasing,
i.e., $x(t)\rightarrow 0$ as $t\rightarrow\infty$; $x(t)\rightarrow-\infty$ as
$t\rightarrow-\infty$. So there must exists a unique time $t_{0}$ such that
$x(t_{0})=-1$. Without loss of generality, we take $t_{0}=0$. If $t<0$, then
$x(t)<-1$ and so $x^{\prime}(t)=-x(t)-\epsilon$ with $x(0)=-1$. Hence,

$x(t)=(\epsilon-1)e^{-t}-\epsilon,\quad t\leq 0.$

If $t>0$, then $-1<x(t)<0$ and so $x^{\prime}(t)=-x(t)+\epsilon x^{3}(t)$ with
$x(0)=-1$. Letting $z=x^{-2}$, then $z^{\prime}=2z-2\epsilon$ with $z(0)=1$.
Hence, $z(t)=(1-\epsilon)e^{2t}+\epsilon$, that is,

$x(t)=-\left[(1-\epsilon)e^{2t}+\epsilon\right]^{-\frac{1}{2}},\quad t\geq 0.$

We need to find the unique solution $y(t)$ of (1.2) such that $|y(t)-x(t)|$ is
bounded. Looking at $x(t)$ when $t\leq 0$, we see that
$y(t)=(\epsilon-1)e^{-t}$. Hence for all $t$,

$H(x(t))=(\epsilon-1)e^{-t}.$

Then

$H(1)=H(x(0))=\epsilon-1.$

If $\xi<-1$, then there exists a unique time $t<0$ such that
$x(t)=(\epsilon-1)e^{-t}-\epsilon=\xi$. Consequently,

$H(\xi)=H(x(t))=(\epsilon-1)e^{-t}=x(t)+\epsilon=\xi+\epsilon.$

If $-1<\xi<0$, then there exists a unique time $t>0$ such that
$x(t)=-\left[(1-\epsilon)e^{2t}+\epsilon\right]^{-\frac{1}{2}}=\xi$.
Consequently,

$H(\xi)=H(x(t))=(\epsilon-1)e^{-t}=-(1-\epsilon)^{\frac{3}{2}}\left(\frac{1}{(-x(t))^{2}}-\epsilon\right)^{-\frac{1}{2}}=-(1-\epsilon)^{\frac{3}{2}}\left(\frac{1}{(-\xi)^{2}}-\epsilon\right)^{-\frac{1}{2}}.$

Thus we obtain that

$H(x)=\begin{cases}-(1-\epsilon)^{\frac{3}{2}}\left(\frac{1}{(-x)^{2}}-\epsilon\right)^{-\frac{1}{2}},&-1<x<0,\\\
x+\epsilon,&x\leq-1.\end{cases}$

Summarizing the above three cases, we see that

$H(x)=\begin{cases}x-\epsilon,&x\geq 1,\\\
(1-\epsilon)x^{\frac{1}{1-\epsilon}},&0<x<1,\\\ 0,&x=0,\\\
-(1-\epsilon)^{\frac{3}{2}}\left(\frac{1}{(-x)^{2}}-\epsilon\right)^{-\frac{1}{2}},&-1<x<0,\\\
x+\epsilon,&x\leq-1,\end{cases}$

where $0<\epsilon<\frac{1}{3}$.

Step 2. We show that  $H$ is Lipschitzian, but it is not $C^{1}$. It suffices
to show that $H$ is continuous at $0$ and $\pm 1$, and $H^{\prime}(x)$ is
bounded, but is not $C^{1}$ at $0$. In fact,
(1) $H(x)$ is continuous at $x=1$:

$\begin{split}&\lim\limits_{x\rightarrow
1^{-}}(1-\epsilon)x^{\frac{1}{1-\epsilon}}=1-\epsilon,\\\
&\lim\limits_{x\rightarrow 1^{+}}x-\epsilon=1-\epsilon.\end{split}$

(2) $H(x)$ is continuous at $x=0$:

$\begin{split}&\lim\limits_{x\rightarrow
0^{+}}(1-\epsilon)x^{\frac{1}{1-\epsilon}}=0,\\\ &\lim\limits_{x\rightarrow
0^{-}}-(1-\epsilon)^{\frac{3}{2}}\left(\frac{1}{(-x)^{2}}-\epsilon\right)^{-\frac{1}{2}}=0.\end{split}$

(3) $H(x)$ is continuous at $x=-1$:

$\begin{split}&\lim\limits_{x\rightarrow-1^{+}}-(1-\epsilon)^{\frac{3}{2}}\left(\frac{1}{(-x)^{2}}-\epsilon\right)^{-\frac{1}{2}}=-1+\epsilon,\\\
&\lim\limits_{x\rightarrow-1^{-}}x+\epsilon=-1+\epsilon.\end{split}$

Hence, $H(x)$ is continuous, but the following fact proves that $H$ is not
$C^{1}$ at $0$. Clearly, for $0<x<1$,
$H^{\prime}(x)=x^{\frac{\epsilon}{1-\epsilon}}$, and for $-1<x<0$,
$H^{\prime}(x)=(1-\epsilon)^{\frac{3}{2}}(1-\epsilon(-x)^{2})^{-\frac{3}{2}}$.
Therefore,

$\begin{split}&\lim\limits_{x\rightarrow
0^{+}}x^{\frac{\epsilon}{1-\epsilon}}=0,\\\ &\lim\limits_{x\rightarrow
0^{-}}(1-\epsilon)^{\frac{3}{2}}(1-\epsilon(-x)^{2})^{-\frac{3}{2}}=(1-\epsilon)^{\frac{3}{2}},\end{split}$

it implies that $H(x)$ is not in $C^{1}$.
Fortunately, $H(x)$ is Lipschitz continuous, since $H^{\prime}(x)$ is
continuous at $x$ except for $x=0$, and it is bounded with $|H^{\prime}|\leq
1$. Therefore, function $H(x)$ is globally Lipschitz continuous with Lipschitz
constant $L=1$, but is not in $C^{1}$. Fig. 1 (H) illustrates the Lipschitzian
continuity of $H$.

Step 3.  The inverse homeomorphism $G=H^{-1}$ is Hölder continuous, but not
Lipschitzian. In fact, the expression of the inverse homeomorphism $G=H^{-1}$
is given as

$G(y)=\begin{cases}y+\epsilon,&y\geq 1-\epsilon,\\\
(\frac{y}{1-\epsilon})^{1-\epsilon},&0<y<1-\epsilon,\\\ 0,&y=0,\\\
-\left((1-\epsilon)^{3}(-y)^{-2}+\epsilon\right)^{-\frac{1}{2}},&-1+\epsilon<y<0,\\\
y-\epsilon,&y\leq\epsilon-1.\end{cases}$

Obviously, $G(y)$ is continuous at $0$ and $\pm 1$. Thus, $G(y)$ is a
continuous function. However, this is not Lipschitz continuous since
$y^{1-\epsilon}$ is not Lipschitz, as $0<1-\epsilon<1$.
$G^{\prime}(y)\rightarrow\infty$ as $y\rightarrow 0$. As you see in Fig. 1
(G), the curve of $G$ is tangent to vertical-axis $G(y)$ at $y=0$.

(H) | (G)
---|---

Fig. 1 The equivalent function $H$ and its inverse $G$. See Fig. 1 (G), the
curve of $G$ is tangent to vertical-axis $G(y)$ at $y=0$.

####  1.2.2 Questions and motivation

Counterexample above demonstrated that the topological equivalent function $H$
could be exactly Lipschitzian when the nonlinear perturbations are
Lipschitzian. And we find that the equivalent function $H$ and its inverse $G$
do not have the same smoothness. This new findings are different from the
existing literature that both the equivalent function and its inverse are
Hölder continuous for the Hartman-Grobman theorem in $C^{0}$ linearization
when the nonlinear perturbations are Lipschitzian (see e.g. [7, 19, 20, 21,
22, 23, 25, 26, 30, 35, 39]). Motivated by Example 1.1, two natural questions
arise: is it possible to improve the regularity of the equivalent function to
Lipschitz continuity for any systems with Lipschitzian nonlinear
perturbations? And is the regularity sharp? General speaking, these are not
true. Then, it is very important to know how many and what of systems with
Lipschitzian nonlinear perturbations have the property that the equivalent
function is Lipschitzian.

In the present paper, we give a positive answer if the linear system admits an
exponential contraction. In this case, we show that $H$ is exactly
Lipschitzian, but the inverse $G=H^{-1}$ is still Hölder continuous if the
nonlinear perturbation is Lipschitzian. Furthermore, it is a sharp and it is
impossible to be improved anymore. To the best of our knowledge, it is the
first time to observe these facts in $C^{0}$ linearization with Lipschitzian
nonlinear perturbations.

Secondly, we also discuss two special cases of the linear perturbations, which
are closely related to the spectrum. If the perturbation is linear
homogeneous, $H$ is $C^{1}$ and the inverse $G$ is Hölder continuous; If the
perturbation is constant, both $H$ and $G$ are $C^{\infty}$.

###  1.3 Outline of the paper

The rest of this paper is organized as follows: In Section 2, our main results
are stated. In Section 3, some preliminary results are presented. In Section
4, rigorous proof is given to show the regularity of conjugacy.

##  2 Statement of main results and illustrative examples

Consider the following two nonautonomous systems:

$y^{\prime}=A(t)y$ (2.1)

and

$x^{\prime}=A(t)x+f(t,x),$ (2.2)

where $A(t)$ is a bounded and continuous $n\times n$ matrix and
$f:\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is a continuous
function.

We introduce the operator $\mathcal{K}$ given by

$\mathcal{K}(\varphi)(t)=\int_{-\infty}^{t}U(t,s)\varphi(s)ds,\quad
t\in\mathbb{R},$

where $\varphi:\mathbb{R}\rightarrow\mathbb{R}^{n}$ is a function,
$\|\mathcal{K}(\varphi)\|\leq\mathcal{L}(\|\varphi\|)$ with

$\mathcal{L}(b)(t)=\int_{-\infty}^{t}\exp\\{-\alpha(t-s)\\}b(s)ds,$

for $b:\mathbb{R}\rightarrow(0,\infty)$ a continuous function.

Throughout this paper, we always suppose that
${\bf(A_{1})}$ there exist positive constants $k>0$ and $\alpha>0$ such that
the transition matrix $U(t,s)=U(t)U^{-1}(s)$ of Eq. (2.1) satisfies:

$\|U(t,s)\|\leq k\exp\\{-\alpha(t-s)\\},\quad\mathrm{for}\;t\geq s,$ (2.3)

that is, $U(t,s)$ is contractive.
${\bf(A_{2})}$ If there is a nonnegative local integrable functions $\mu(t)$
satisfying $\sup_{t\in\mathbb{R}}\int_{t}^{t+1}\mu(s)ds\leq C_{1}$ ($C_{1}$ is
a constant) such that for any $t\in\mathbb{R}$ and $x\in\mathbb{R}^{n}$,
$\|f(t,x)\|\leq\mu(t)$;
${\bf(A_{3})}$ If there is a nonnegative local integrable functions $r(t)$
satisfying small enough $C_{2}=\sup_{t\in\mathbb{R}}\int_{t}^{t+1}r(s)ds$ such
that for any $t\in\mathbb{R}$ and $x_{1},x_{2}\in\mathbb{R}^{n}$,
$\|f(t,x_{1})-f(t,x_{2})\|\leq r(t)\|x_{1}-x_{2}\|$;
${\bf(A_{4})}$ $\sup_{t\in\mathbb{R}}\mathcal{L}(r)(t)=\theta<k^{-1}$.

We remark that if $\mu(t)\equiv\mu$ and $r(s)\equiv L_{f}$,
${\bf(A_{2})}-{\bf(A_{4})}$ reduce to the following particular cases:
${\bf(\widetilde{A}_{2})}$ the perturbation $f$ is continuous and bounded,
i.e., for any $t\in\mathbb{R}$ and $x\in\mathbb{R}^{n}$,
$\|f(t,x)\|\leq\mu<+\infty$;
${\bf(\widetilde{A}_{3})}$ for any $t\in\mathbb{R}$ and
$x_{1},x_{2}\in\mathbb{R}^{n}$, $\|f(t,x_{1})-f(t,x_{2})\|\leq
L_{f}\|x_{1}-x_{2}\|$;
${\bf(\widetilde{A}_{4})}$ the Lipschitz constant $L_{f}$ satisfies:
$L_{f}\leq\alpha/k$.

Next theorems are the regularity of the topological equivalent function $H$
and its inverse $G=H^{-1}$. According to the different classification of
perturbation $f$, different regularity for three cases are presented as
follows.

Case 1: the perturbation $f(t,x)$ is nonlinear (not linear) with respect to
$x$.

######  Theorem 2.1.

Suppose that the perturbation $f(t,x)$ is not linear with respect to $x$. If
${\bf(A_{1})}-{\bf(A_{4})}$ hold, then there is an equivalent function
$H(t,x)$ (homeomorphism) transforming the solution of (2.2) onto that of
linear part (2.1). Furthermore, the equivalent function $H(t,x)$ and its
inverse $G(t,y)$ fulfill:
(i) $H$ is exactly Lipschitzian, that is, for any
$x,\bar{x}\in\mathbb{R}^{n}$, there exists a positive constant $p_{1}>0$ such
that

$\|H(t,x)-H(t,\bar{x})\|\leq p_{1}\|x-\bar{x}\|;$

(ii) the inverse $G=H^{-1}$ is Hölder continuous, that is, for any
$y,\bar{y}\in\mathbb{R}^{n}$, there exist positive constants $p_{2}>0$ and
$0<q<1$ such that

$\|G(t,y)-G(t,\bar{y})\|\leq p_{2}\|y-\bar{y}\|^{q}.$

(iii) $H$ is Lipschitzian, but not $C^{1}$; $G$ is Hölder continuous, but not
Lipschitzian. The regularity of $H$ and $G$ can not be improved any more.

######  Corollary 2.2.

If the perturbation $f(t,x)$ satisfies ${\bf({A}_{1})}$,
${\bf(\widetilde{A}_{2})}$, ${\bf(\widetilde{A}_{3})}$ and
${\bf(\widetilde{A}_{4})}$ hold, then all the conclusions of Theorem 2.1 are
valid.

Instead of the uniform exponential contraction, next theorem is for the
nonuniform case, that is, the linear system admits a nonuniform exponential
contraction (see Barreira and Valls ([23, 24]).

######  Theorem 2.3.

Suppose that the linear system admits a nonuniform exponential contraction,
Theorem 2.1 is true for

$\widetilde{\mu}(t)=\mu(t)\exp\\{-\epsilon|t|\\}\quad\mathrm{and}\quad\widetilde{r}(t)=r(t)\exp\\{-\epsilon|t|\\},$

where $\mu(t)$ and $r(t)$ are given in $(A_{2})$ and $(A_{3})$, respectively.

In fact, in view of $\widetilde{\mu}(t)=\mu(t)\exp\\{-\epsilon|t|\\}$, it is
easy to see that

$\mathcal{L}\\{\exp\\{\epsilon|\cdot|\\}\widetilde{\mu}(\cdot)\\}=\int_{-\infty}^{t}\exp\\{-\alpha(t-s)\\}\exp\\{\epsilon|s|\\}\widetilde{\mu}(s)ds=\mathcal{L}\\{\mu(\cdot)\\},$

similarly,
$\mathcal{L}\\{\exp\\{\epsilon|\cdot|\\}\widetilde{r}(\cdot)\\}=\mathcal{L}\\{r(\cdot)\\}$.
Thus, all conditions of Theorem 2.1 in the nonuniform case are satisfied.

######  Remark 2.4.

Barreira and Valls [23, 24] constructed conjugacies between linear and non-
linear non-uniform contractions and proved that both the homeomorphisms are
Hölder continuous. As Theorem 2.3 shows, the homeomorphism is Lipschitzian and
its inverse is Hölder continuous for the non-uniform contractions. Thus, our
result improves the known results in [23, 24].

For the autonomous systems, we have following corollary on the regularity of
the conjugacy. That is, we consider the autonomous system

$\dot{x}=Ax+f(x).$ (2.4)

######  Corollary 2.5.

Assume that the perturbation $f(x)$ satisfies $|f(x)|\leq\mu$, and
$|f(x)-f(y)|\leq L_{f}\|x-y\|$, and $L_{f}\leq\alpha/k$. If all the real parts
of eigenvalues of $A$ are strictly less than 0, , then the equivalent function
$H(x)$ and its inverse $G(y)$ fulfill:
(i) $H$ is exactly Lipschitzian, that is, for any
$x,\bar{x}\in\mathbb{R}^{n}$, there exists a positive constant $p_{1}>0$ such
that

$\|H(x)-H(\bar{x})\|\leq p_{1}\|x-\bar{x}\|;$

(ii) the inverse $G=H^{-1}$ is Hölder continuous, that is, for any
$y,\bar{y}\in\mathbb{R}^{n}$, there exist positive constants $p_{2}>0$ and
$0<q<1$ such that

$\|G(y)-G(\bar{y})\|\leq p_{2}\|y-\bar{y}\|^{q}.$

(iii) $H$ is Lipschitzian, but not $C^{1}$; $G$ is Hölder continuous, but not
Lipschitzian. The regularity of $H$ and $G$ can not be improved any more.

Now we pay particular attention to two special cases:

Case 2: $f$ is linear homogeneous perturbation.

If $f(t,x)=B(t)x$, then equation (2.2) reduces to

$x^{\prime}=A(t)x+B(t)x,$ (2.5)

where $B(t)$ is a continuous $n\times n$ matrix.

We need the following definition.

######  Definition 2.6 ([40]).

The exponential dichotomy spectrum of equation (2.1) is the set

$\Sigma(A(t))=\\{\gamma\in\mathbb{R}|x^{\prime}=(A(t)-\gamma
I)x\;\mathrm{admits}\;\mathrm{no}\;\mathrm{exponential}\;\mathrm{dichotomy}\\},$

and the resolvent set $\rho(A(t))=\mathbb{R}\backslash\Sigma(A(t))$.

It is known that the exponential dichotomy spectrum $\Sigma(A(t))$ of (2.1) is
the disjoint union of $k$ closed intervals where $0\leq k\leq n$. i.e.,
$\Sigma(A(t))=\bigcup_{i=1}^{k}[a_{i},b_{i}]$, $a_{1}\leq b_{1}<a_{2}\leq
b_{2}<\cdots<a_{k}\leq b_{k}$, where $a_{1}$ could be $-\infty$ and $b_{k}$
could be $\infty$. If $k=0$, then $\Sigma(A(t))=\emptyset$ (see Sacker and
Sell [40], Xia et al [41, 42]). Notice that instead of $f(t,x)$ being
Lipschitz continuous with respect to $x$, it is differentiable with respect to
$x$. Thus, a better regularity will be proved. Namely, $H$ is $C^{1}$ and its
inverse $G$ is Hölder continuous.

######  Theorem 2.7.

Suppose that $f(t,x)=B(t)x$ and (2.1) satisfies condition ${\bf(A_{1})}$ with
$\Sigma(A(t))=\bigcup_{i=1}^{k}[a_{i},b_{i}]$, where $b_{k}<0$. If there exist
$\delta>0$ and appropriately small $0\leq\epsilon<-b_{k}$ such that
$\|B(t)\|\leq\delta$ and
$\Sigma(A(t)+B(t))\subseteq\bigcup_{i=1}^{k}[a_{i}+\epsilon,b_{i}+\epsilon]$,
then the equivalent function $H(t,x)$ is $C^{1}$ and its inverse $G(t,y)$ is
Hölder continuous.

######  Remark 2.8.

In addition, if (2.1) satisfies condition ${\bf(\widetilde{A}_{1})}$ with
$\Sigma(A(t))=\bigcup_{i=1}^{k}[a_{i},b_{i}]$, where $a_{1}>0$, and there
exist $\widetilde{\delta}>0$ and appropriately small $0\leq\varepsilon<a_{1}$
such that $\|B(t)\|\leq\widetilde{\delta}$ and
$\Sigma(A(t)+B(t))\subseteq\bigcup_{i=1}^{k}[a_{i}-\varepsilon,b_{i}-\varepsilon]$,
then our result is similar to Theorem 2.7.

######  Example 2.9.

For Eq. (1.1), we consider the nonlinear term $f=\epsilon x$, $0<\epsilon<1$.
In this case, $\Sigma(A+B)=-1+\epsilon<0$ and $|B|=\epsilon$. Similar to the
procedure just shown, we have that

$H(x)=\begin{cases}(1-\epsilon)x^{\frac{1}{1-\epsilon}},&x>0,\\\ 0,&x=0,\\\
(\epsilon-1)(-x)^{\frac{1}{1-\epsilon}},&x<0.\end{cases}$

This function $H(x)$ is at least in $C^{1}$, and $H(x)$ is Lipschitzian. And
the inverse function $G=H^{-1}$ is

$G(y)=\begin{cases}(\frac{y}{1-\epsilon})^{1-\epsilon},&y>0,\\\ 0,&y=0,\\\
-(\frac{-y}{1-\epsilon})^{1-\epsilon},&y<0.\end{cases}$

This function $G(y)$ is Hölder continuous.

Case 3: $f$ is constant.

If the nonlinear term $f(t,x)$ is a constant function, then equation (2.2)
reduces to a non-homogeneous linear equation. Clearly,

######  Remark 2.10.

If the nonlinear term $f(t,x)$ is a constant, then there is a $C^{\infty}$
coordinate transformation such that Eq. (2.2) is topologically conjugated to
Eq. (2.1), namely, the equivalent function $H(t,x)$ and its inverse $G(t,y)$
are both $C^{\infty}$.

######  Example 2.11.

If $f(x)$ is a constant, then Eq. (1.1) reduces to

$x^{\prime}=-x+\delta.$

In this case, linearization theorem is always satisfied. Moreover, it is easy
to obtain that

$H(x)=x-\delta,\quad G(y)=y+\delta.$

Thus, $H,G$ are $C^{\infty}$.

##  3 Preliminary results

In what follows, we always suppose that the conditions of Theorem 2.1 are
satisfied. Let $X(t,t_{0},x_{0})$ be a solution of Eq. (2.2) satisfying the
initial condition $X(t_{0})=x_{0}$ and $Y(t,t_{0},y_{0})$ is a solution of Eq.
(2.1) satisfying the initial condition $Y(t_{0})=y_{0}$.

Note that Xia et al [26] has proved that if Eq. (2.1) admits an exponential
dichotomy, and nonlinear perturbation $f$ satisfies
${\bf(A_{2})}-{\bf(A_{4})}$, then there exists a unique function
$H:\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that Eq.
(2.2) is topologically conjugated to Eq. (2.1). Since the assumption
${\bf(A_{1})}$ (contraction) is a particular case of the exponential
dichotomy, naturally, Eq. (2.2) is topologically conjugated to Eq. (2.1).

Therefore, we can construct the two homeomorphisms as follows.

$H(t,x)=x+h(t,(t,x)),$ (3.1)

where

$h(t,(\tau,\xi))=-\int_{-\infty}^{t}U(t,s)P(s)f(s,X(s,\tau,\xi))ds,$ (3.2)

is the unique bounded solution of the auxiliary equation

$z^{\prime}=A(t)z-f(t,X(t,\tau,\xi)),\,\,\,\mbox{for any
fixed}\,\,\,\,(\tau,\xi).$

And

$G(t,y)=y+g(t,(t,y)),$ (3.3)

where

$g(t,(\tau,\xi))=\int_{-\infty}^{t}U(t,s)P(s)f(s,Y(s,\tau,\xi)+g(s,(\tau,\xi)))ds,$
(3.4)

is the unique bounded solution of the auxiliary equation

$z^{\prime}=A(t)z+f(t,Y(t,\tau,\xi)+z),\,\,\,\mbox{for any
fixed}\,\,\,\,(\tau,\xi).$

To prove the regularity of the homeomorphisms, we need the following lemma.

######  Lemma 3.1.

If $r(t)$ is a nonnegative local integrable function and
$C_{2}=\sup\limits_{t\in\mathbb{R}}\int_{t}^{t+1}r(s)ds<\infty$, then we have

$\|X(t,t_{0},x_{0})-X(t,t_{0},\bar{x}_{0})\|\leq
ke^{kC_{2}}\|x_{0}-\bar{x}_{0}\|\cdot e^{(kC_{2}-\alpha)|t-t_{0}|}.$ (3.5)

###### Proof.

We only prove the case of $t\geq t_{0}$, and the case of $t\leq t_{0}$ can be
obtained analogously. Since $r(t)$ is a local integrable function, i.e.,
$C_{2}=\sup_{t\in\mathbb{R}}\int_{t}^{t+1}r(s)ds$, we have

$\begin{split}\int_{t_{0}}^{t}r(s)ds=&\int_{t_{0}}^{1}r(s)ds+\int_{1}^{2}r(s)ds+\cdots+\int_{t-1}^{t}r(s)ds\\\
\leq&([t-t_{0}]+1)C_{2},\end{split}$

where $[t-t_{0}]$ means taking the largest integer not great than $t-t_{0}$.
Note that

$X(t,t_{0},x_{0})=U(t,t_{0})x_{0}+\int_{t_{0}}^{t}U(t,s)f(s,X(s,t_{0},x_{0}))ds.$

Then for any $x_{0},\bar{x}_{0}\in\mathbb{R}^{n}$, the above equality leads to

$\begin{split}&\|X(t,t_{0},x_{0})-X(t,t_{0},\bar{x}_{0})\|\\\
\leq&ke^{-\alpha(t-t_{0})}\|x_{0}-\bar{x}_{0}\|+\int_{t_{0}}^{t}ke^{-\alpha(t-s)}r(s)\|X(s,t_{0},x_{0})-X(s,t_{0},\bar{x}_{0})\|ds.\end{split}$

Using the Bellman inequality, we have

$\begin{split}\|X(t,t_{0},x_{0})-X(t,t_{0},\bar{x}_{0})\|\leq&ke^{\alpha
t_{0}}\|x_{0}-\bar{x}_{0}\|e^{\int_{t_{0}}^{t}kr(s)ds-\alpha t}\\\
\leq&ke^{kC_{2}}\|x_{0}-\bar{x}_{0}\|e^{(kC_{2}-\alpha)(t-t_{0})},\quad t\geq
t_{0}.\end{split}$

The proof for $t\leq t_{0}$ is similar. ∎

##  4 Proofs of main results

Now we are in a position to prove our results.

###### Proof of Theorem 2.1..

We divide the proof of Theorem 2.1 into four steps.
Step 1. We are going to prove the Lipschitz continuity of the equivalent
function $H(t,x)$ by Lemma 3.1. That is, we will show that

$\|H(t,x)-H(t,\overline{x})\|\leq p_{1}\|x-\overline{x}\|,\qquad
p=\frac{1+k^{2}C_{2}e^{kC_{2}}-e^{-kC_{2}}}{1-e^{-kC_{2}}}.$

In fact, by uniqueness $X(t,(\tau,\xi))=X(t,(t,X(t,(\tau,\xi)))$. From (3.2),
it follows that

$h(t,(t,\xi))=-\int_{-\infty}^{t}U(t,s)f(s,X(s,t,\xi))ds.$

Thus we get

$\displaystyle
h(t,(t,\xi))-h(t,(t,\overline{\xi}))=-\int_{-\infty}^{t}U(t,s)(f(s,X(s,t,\xi))-f(s,X(s,t,\overline{\xi})))ds.\qquad\textbf{$(I_{1})$}$

By condition (A2) and Lemma 3.1, we have

$\begin{split}\|I_{1}\|\leq&\int_{-\infty}^{t}ke^{-\alpha(t-s)}r(s)\cdot\|X(s,t,\xi)-X(s,t,\bar{\xi})\|ds\\\
\leq&\int_{-\infty}^{t}ke^{-\alpha(t-s)}r(s)\cdot
ke^{kC_{2}}\|\xi-\bar{\xi}\|e^{(kC_{2}-\alpha)(s-t)}ds\\\
\leq&\frac{k^{2}C_{2}e^{kC_{2}}}{1-e^{-kC_{2}}}\cdot\|\xi-\bar{\xi}\|.\end{split}$

By the definition of $H(t,x)$,

$\begin{split}\|H(t,x)-H(t,\overline{x})\|\leq&\|x-\overline{x}\|+\frac{k^{2}C_{2}e^{kC_{2}}}{1-e^{-kC_{2}}}\|x-\overline{x}\|\\\
\leq&\frac{1+k^{2}C_{2}e^{kC_{2}}-e^{-kC_{2}}}{1-e^{-kC_{2}}}\|x-\overline{x}\|\\\
\equiv&p_{1}\|x-\overline{x}\|,\end{split}$

which implies that $H$ is Lipschitzian.

Step 2. We show that $H$ is not $C^{1}$. In fact, noticing $H=x+h$ and the
expression of $h$ (see (3.2)), it is easy to see that $h$ is not $C^{1}$
because $f$ is Lipschitzian, not $C^{1}$.

Step 3. We claim that there exist positive constants $p_{2}>0$ and $0<q<1$
such that for all $t\in\mathbb{R},y,\bar{y}\in\mathbb{R}^{n}$,

$\|G(t,y)-G(t,\bar{y})\|\leq p_{2}\|y-\bar{y}\|^{q}.$

Usually this point is treated with successive approximations. From (3.4), we
know that $g(t,(\tau,\xi))$ is a fixed point of the following map
$\mathcal{T}$

$\begin{split}(\mathcal{T}z)(t)=&\int_{-\infty}^{t}U(t,s)f(s,Y(s,\tau,\xi)+z(s))ds.\end{split}$
(4.1)

Let $g_{0}(t,(\tau,\xi))\equiv 0$, and by recursion define

$\begin{split}g_{m+1}(t,(\tau,\xi))=&\int_{-\infty}^{t}U(t,s)f(s,Y(s,\tau,\xi)+g_{m}(s,(\tau,\xi)))ds.\end{split}$

It is not difficult to show that

$g_{m}(t,(\tau,\xi))\rightarrow g(t,(\tau,\xi)),\quad\text{as}\ \
m\rightarrow+\infty,$

uniformly with respect to $t,\tau,\eta$.
Note that $g_{0}(t,(\tau,\xi))=g_{0}(t,(t,Y(t,\tau,\xi)))$. By induction, we
see that $g_{m}(t,(\tau,\xi))=g_{m}(t,(t,Y(t,\tau,\xi)))$, for all
$m(m\in\mathbb{N})$. Choose a sufficiently large positive constant $\lambda 0$
and a sufficiently small positive constant $0<q<1$ such that

$\begin{cases}\lambda>\frac{3kC_{1}}{1-e^{-\alpha}}+\frac{3}{2}k^{2}C_{2},\\\
q<\frac{\alpha}{\alpha+1}<1,\\\ 0<\frac{C_{2}k^{q+1}}{e^{\alpha-\alpha
q}-1}<\frac{1}{3}.\end{cases}$ (4.2)

In what follows, we prove that

$\|g_{k}(t,(t,\xi))-g_{k}(t,(t,\bar{\xi}))\|<\lambda\|\xi-\bar{\xi}\|^{q},\,0<\|\xi-\bar{\xi}\|<1,\,k\in\mathbb{N}.$
(4.3)

Clearly, (4.3) holds for $k=0$. Suppose that (4.3) holds for $k=m$. It
suffices to prove that (4.3) holds for $k=m+1$. Note

$\begin{split}&g_{m+1}(t,(t,\xi))-g_{m+1}(t,(t,\bar{\xi}))\\\
=&\int_{-\infty}^{t}U(t,s)[f(s,Y(s,t,\xi)+g_{m}(s,(t,\xi)))-f(s,Y(s,t,\bar{\xi})+g_{m}(s,(t,\bar{\xi})))]ds.\qquad\textbf{$(J_{1})$}\end{split}$

We divide $J_{1}$ into two parts:

$J_{1}=\int_{-\infty}^{t-\tau}+\int_{t-\tau}^{t}\triangleq J_{11}+J_{12},$

where $\tau=\frac{1}{\alpha+1}\ln\frac{1}{\|\xi-\bar{\xi}\|}$. Then by
conditions ${\bf(A_{1})}$ and ${\bf(A_{2})}$, we deduce that

$\displaystyle\|J_{11}\|\leq$
$\displaystyle\int_{-\infty}^{t-\tau}ke^{-\alpha(t-s)}2\mu(s)ds$
$\displaystyle\leq$
$\displaystyle\frac{2kC_{1}}{1-e^{-\alpha}}\|\xi-\bar{\xi}\|^{\frac{\alpha}{\alpha+1}}.$

Since Eq. (2.1) satisfies condition ${\bf(A_{1})}$, we have

$\|Y(s,t,\xi)-Y(s,t,\bar{\xi})\|\leq ke^{-\alpha(s-t)}\|\xi-\bar{\xi}\|.$
(4.4)

Furthermore, it follows from ${\bf(A_{1})}$, ${\bf(A_{3})}$ and (4.4) that

$\begin{split}\|g_{m}(s,(t,\xi))-g_{m}(s,(t,\bar{\xi}))\|=&\|g_{m}(s,(s,Y(s,t,\xi)))-g_{m}(s,(s,Y(s,t,\bar{\xi})))\|\\\
\leq&\lambda\|Y(s,t,\xi)-Y(s,t,\bar{\xi})\|^{q}\\\ \leq&\lambda
k^{q}\|\xi-\bar{\xi}\|^{q}\cdot e^{-\alpha q(s-t)},\end{split}$

and

$\begin{split}\|J_{12}\|\leq&\int_{t-\tau}^{t}ke^{-\alpha(t-s)}r(s)[\|Y(s,t,\xi)-Y(s,t,\bar{\xi})\|+\|g_{m}(s,(t,\xi))-g_{m}(s,(t,\bar{\xi}))\|]ds\\\
\leq&\int_{t-\tau}^{t}ke^{-\alpha(t-s)}r(s)[ke^{-\alpha(s-t)}\|\xi-\bar{\xi}\|+\lambda
k^{q}e^{-\alpha q(s-t)}\|\xi-\bar{\xi}\|^{q}]ds\\\
\leq&\int_{t-\tau}^{t}k^{2}r(s)\|\xi-\bar{\xi}\|ds+\int_{t-\tau}^{t}\lambda
k^{q+1}r(s)e^{(\alpha-\alpha q)(s-t)}\|\xi-\bar{\xi}\|^{q}ds\\\
\leq&\sum_{m\in[0,[\tau]]}k^{2}\|\xi-\bar{\xi}\|\int_{t-\tau+m}^{t-\tau+m+1}r(s)ds\\\
&+\sum_{m\in[0,[\tau]]}\lambda
k^{q+1}\|\xi-\bar{\xi}\|^{q}\int_{t-\tau+m}^{t-\tau+m+1}r(s)e^{(\alpha-\alpha
q)(s-t)}ds\\\
\leq&k^{2}C_{2}\tau\|\xi-\bar{\xi}\|+\sum_{m\in[0,[\tau]]}\lambda
k^{q+1}\|\xi-\bar{\xi}\|^{q}C_{2}e^{(\alpha q-\alpha)\tau}e^{(\alpha-\alpha
q)(m+1)}\\\ \leq&k^{2}C_{2}e^{\tau}\|\xi-\bar{\xi}\|+\lambda
k^{q+1}C_{2}e^{(\alpha
q-\alpha)(\tau-1)}\|\xi-\bar{\xi}\|^{q}\cdot\frac{1-e^{(\alpha-\alpha
q)[\tau]}}{1-e^{\alpha-\alpha q}}\\\
\leq&k^{2}C_{2}\|\xi-\bar{\xi}\|^{\frac{\alpha}{\alpha+1}}+\frac{\lambda
C_{2}k^{q+1}}{e^{\alpha-\alpha q}-1}\|\xi-\bar{\xi}\|^{q}.\end{split}$

Notice that $\alpha-\alpha q>0$ implies that $\exp\\{\alpha-\alpha q\\}>1$ and
$q<\frac{\alpha}{\alpha+1}$. Then

$\|J_{12}\|\leq\left(C_{2}k^{2}+\frac{\lambda C_{2}k^{q+1}}{e^{\alpha-\alpha
q}-1}\right)\|\xi-\bar{\xi}\|^{q}.$

From (4.2), we have

$\begin{split}&\|g_{m+1}(t,(t,\xi))-g_{m+1}(t,(t,\bar{\xi}))\|\leq\|J_{11}\|+\|J_{12}\|\\\
\leq&\left(\frac{2kC_{1}}{1-e^{-\alpha}}+C_{2}k^{2}+\frac{\lambda
C_{2}k^{q+1}}{e^{\alpha-\alpha q}-1}\right)\|\xi-\bar{\xi}\|^{q}\\\
\leq&\lambda\|\xi-\bar{\xi}\|^{q}.\end{split}$

Thus inequality (4.3) holds for any $k\in\mathbb{N}$. Setting
$k\rightarrow+\infty$, we have

$\|g(t,(t,\xi))-g(t,(t,\bar{\xi}))\|\leq\lambda\|\xi-\bar{\xi}\|^{q}.$

In view of $G=x=g,$

$\|G(t,y)-G(t,\bar{y})\|\leq(1+\lambda)\|y-\bar{y}\|^{\beta}=p_{2}\|y-\bar{y}\|^{q}.$

∎

Step 4. We prove that the inverse of the equivalent function $G$ could not be
Lipschitzian. In fact, if we directly prove that $G$ is Lipschitzian by above
similar arguments, then the following contradiction will appear:

$\begin{split}J_{1}&\leq\int_{-\infty}^{t}ke^{-\alpha(t-s)}r(s)[\|Y(s,t,\xi)-Y(s,t,\bar{\xi})\|+\|g_{m}(s,(t,\xi))-g_{m}(s,(t,\bar{\xi}))\|]ds\\\
&\leq\int_{-\infty}^{t}ke^{-\alpha(t-s)}r(s)[ke^{-\alpha(s-t)}\|\xi-\bar{\xi}\|+\lambda
ke^{-\alpha(s-t)}\|\xi-\bar{\xi}\|]ds\\\
&\leq\int_{-\infty}^{t}k^{2}r(s)\|\xi-\bar{\xi}\|ds+\int_{-\infty}^{t}\lambda
k^{2}r(s)\|\xi-\bar{\xi}\|ds.\end{split}$

Obviously, the integral $\int_{-\infty}^{t}r(s)ds$ is divergent in the above
right-hand estimation, thus, $J_{1}$ is not Lipschitzian. Consequently, the
equivalent function $G$ could not be Lipschitzian. The proof of main theorem
is complete.

###### Proof of Theorem 2.7..

Step 1. proof of the map $H\in C^{1}$:
Note that

$h(t,(t,\xi))=-\int_{-\infty}^{t}U(t,s)B(s)X(s,t,\xi)ds.$

Thus we can deduce that

$\frac{\partial
h(t,(t,\xi))}{\partial\xi}=-\int_{-\infty}^{t}U(t,s)B(s)\frac{\partial
X(s,t,\xi)}{\partial\xi}ds,$

and this integral is convergent, i.e., it is well defined. In fact, we need
the following auxiliary statement:

$\frac{\partial
X(t,0,\xi)}{\partial\xi}=U(t,0)+\int_{0}^{t}U(t,\tau)B(\tau)\frac{\partial
X(\tau,0,\xi)}{\partial\xi}d\tau.$

Since $\|U(t,\tau)\|\leq ke^{-\alpha(t-\tau)}$ and $\|B(\tau)\|\leq\delta$, we
have

$\left\|\frac{\partial X(t,0,\xi)}{\partial\xi}\right\|\leq ke^{-\alpha
t}+\int_{0}^{t}ke^{-\alpha(t-\tau)}\cdot\delta\cdot\left\|\frac{\partial
X(\tau,0,\xi)}{\partial\xi}\right\|d\tau.$

Using the Bellman inequality, we obtain

$\left\|\frac{\partial X(t,0,\xi)}{\partial\xi}\right\|\leq
ke^{(k\delta-\alpha)t}.$

We now continue with the proof of $H\in C^{1}$. Since

$\begin{split}\left\|\frac{\partial
h(t,(t,\xi))}{\partial\xi}\right\|\leq&\int_{-\infty}^{t}ke^{-\alpha(t-\tau)}\cdot\delta\cdot\left\|\frac{\partial
X(s,t,\xi)}{\partial\xi}\right\|ds\\\ \leq&\int_{-\infty}^{t}k\delta
e^{-\alpha(t-\tau)}\cdot ke^{(k\delta-\alpha)(s-t)}ds\\\ \leq&k,\end{split}$

$\frac{\partial h(t,(t,\xi))}{\partial\xi}$ is well defined and $h\in C^{1}$.
Hence, it is clear that $H\in C^{1}$.
Step 2. Hölder regularity of $G$: The proof is similar to that Theorem 2.1. ∎

##  5 Conflict of Interest

The authors declare that they have no conflict of interest.

##  6 Data Availability Statement

My manuscript has no associated data. It is pure mathematics.

## Contributions

We declare that all the authors have same contributions to this paper.

## References

  * [1] P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc., 11 (1960) 610–620.
  * [2] D. Grobman, Homeomorphisms of systems of differential equations, Dokl. Akad. Nauk SSSR, 128 (1965) 880–881.
  * [3] C. Pugh, On a theorem of P. Hartman, Amer. J. Math., 91 (1969) 363–367.
  * [4] G. Farkas, A Hartman-Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria, Z. Angew. Math. Phys., 52 (2001), 421–432.
  * [5] K. Lu, A Hartman-Grobman theorem for scalar reaction diffusion equations, J. Differential Equations, 93 (1991) 364–394.
  * [6] P. Bates, K. Lu, A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations, J. Dyn. Differ. Equ., 6 (1994) 101–145.
  * [7] M. Hein, J. Prüss, The Hartman-Grobman theorem for semilinear hyperbolic evolution equations, J. Differential Equations, 261 (2016) 4709–4727.
  * [8] S. Sternberg, Local $C^{n}$ transformations of the real line, Duke Math. J., 24 (1957) 97–102.
  * [9] S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79(1957) 809–824.
  * [10] G. Sell, Smooth Linearization near a fixed point, Amer. J. Math., 107 (1985) 1035–1091.
  * [11] G. Belitskii, Functional equations and the conjugacy of diffeomorphism of finite smoothness class, Funct. Anal. Appl., 7 (1973) 268–277.
  * [12] M. ElBialy, Local contractions of Banach spaces and spectral gap conditions, J. Funct. Anal., 182 (2001) 108–150.
  * [13] H. Rodrigues, J. Solà-Morales, Smooth linearization for a saddle on Banach spaces, J. Dyn. Differ. Equ., 16 (2004) 767–793.
  * [14] W. Zhang, W. Zhang, $C^{1}$ linearization for planar contractions, J. Funct. Anal., 260 (2011) 2043-2063.
  * [15] W. Zhang, W. Zhang, W. Jarczyk, Sharp regularity of linearization for $C^{1,1}$ hyperbolic diffeomorphisms, Math. Ann., 358 (2014) 69–113.
  * [16] W. Zhang, W. Zhang, $\alpha$-Hölder linearization of hyperbolic diffeomorphisms with resonance, Ergod. Theor. Dyn. Syst., 36 (2016) 310–334.
  * [17] W. Zhang, W. Zhang, Sharpness for $C^{1}$ linearization of planar hyperbolic diffeomorphisms, J. Differential Equations, 257 (2014), 4470–4502.
  * [18] K. Palmer, A generalization of Hartman’s linearization theorem, J. Math. Anal. Appl., 41 (1973) 753–758.
  * [19] L. Jiang, Generalized exponential dichotomy and global linearization, J. Math. Anal. Appl., 315 (2006) 474–490.
  * [20] L. Barreira, C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics, J. Differential Equations, 228 (2006) 285–310.
  * [21] L. Barreira, C. Valls, Conjugacies for linear and nonlinear perturbations of nonuniform behavior, J. Funct. Anal., 253 (2007) 324-358.
  * [22] L. Barreira, C. Valls, A Grobman-Hartman theorem for general nonuniform exponential dichotomies, J. Funct. Anal., 257 (2009) 1976–1993.
  * [23] L. Barreira, C. Valls, Conjugacies between linear and nonlinear non-uniform contractions, Ergod. Theor. Dyn. Syst., 28 (2008) 1–19.
  * [24] L. Barreira, C. Valls, Conjugacies between general contractions, Linear Algebra Appl., 436 (2012) 3087–3098.
  * [25] L. Barreira, C. Valls, Hölder conjugacies for random dynamical systems, Phys. D, 223 (2006) 256–269.
  * [26] Y. Xia, R. Wang, K. Kou, O’Regan, On the linearization theorem for nonautonomous differential equations, Bull. Sci. Math., 139 (2015) 829–846.
  * [27] I. Huerta, Linearization of a nonautonomous unbounded system with nonuniform contraction: A spectral approach, Discrete Contin. Dyn. Syst., 40 (2020) 5571–5590.
  * [28] Á. Castañeda, I. Huerta, Nonuniform almost reducibility of nonautonomous linear differential equations, J. Math. Anal. Appl., 485 (2020), 123822.
  * [29] A. Reinfelds, D. Šteinberga, Dynamical equivalence of quasilinear equations, Int. J. Pure Appl. Math., 98 (2015) 355-364.
  * [30] L. Backes, D. Dragičević, K. Palmer, Linearization and Hölder continuity for nonautonomous systems, J. Differential Equations, 297 (2021) 536–574.
  * [31] L. Backes, D. Dragičević, Smooth linearization of nonautonomous coupled systems, preprint, arXiv: 2202.12367.
  * [32] L. Backes, D. Dragičević, Multiscale linearization of nonautonomous systems, preprint, arXiv: 2203.03694.
  * [33] A. Reinfelds, L. Sermone, Equivalence of nonlinear differential equations with impulse effect in Banach space, Latv. Univ. Zint. Raksti., 577 (1992) 68–73.
  * [34] J. Fenner, M. Pinto, On a Hartman linearization theorem for a class of ODE with impulse effect, Nonlinear Anal., 38 (1999) 307–325.
  * [35] Y. Xia, X. Chen, V. Romanovski, On the linearization theorem of Fenner and Pinto, J. Math. Anal. Appl., 400 (2013) 439–451.
  * [36] C. Pötzche, Topological decoupling, linearization and perturbation on inhomogeneous time scales, J. Differential Equations, 245 (2008) 1210–1242.
  * [37] C. Zou, Y. Xia, M. Pinto, Hölder regularity of topological equivalence functions of DEPCAGs with unbounded nonlinear terms, Sci. Sin. Math., 49 (2019) 1–26 (In Chinese).
  * [38] M. Pinto, G. Robledo, A Grobman-Hartman Theorem for Differential Equations with Piecewise Constant Arguments of Mixed Type, Z. Anal. Anwend., 37 (2018) 101–126.
  * [39] J. Shi, J. Zhang, The Principle of Classification for Differential Equations, Science Press, Beijing, 2003 (in Chinese).
  * [40] J. Sacker, R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978) 320–358.
  * [41] Y. Xia, Y. Bai, D. O’Regan, A new method to prove the nonunifrom dichotomy spectrum theorem in $\mathbb{R}^{n}$, Proc. Amer. Math. Soc., 147 (2019) 3905–3917.
  * [42] J. Chu, F. Liao, S. Siegmund, Y. Xia, W. Zhang, Nonuniform dichotomy spectrum and reducibility for nonautonomous equations, Bull. Sci. Math., 139 (2015) 538–557.


left to the reader: if $Z$ is a closed connected $n$-manifold such that
$H^{1}(Z;{\mathbb{Z}})\simeq{\mathbb{Z}}^{n}$, then $Z$ is hypertoral if and
only if for any primitive class $\alpha\in H^{1}(Z;{\mathbb{Z}})$ there exist
classes $\alpha_{2},\dots,\alpha_{n}$ such that
$\alpha\cup\alpha_{2}\cup\dots\cup\alpha_{n}$ is a generator of
$H^{n}(Z;{\mathbb{Z}})$. (The class $\alpha$ is primitive if it is not of the
form $\lambda\beta$ for $\beta\in H^{1}(Z;{\mathbb{Z}})$ and
$\lambda\in{\mathbb{Z}}\setminus\\{-1,0,1\\}$.)

Let $p\in\Sigma$ be one of the two fixed points of $\sigma$. Let $\theta$ be a
generator of $H^{1}(S^{1};{\mathbb{Z}})$, and let
$\iota:S^{1}={\mathbb{R}}/{\mathbb{Z}}\to X$ be the map sending
$[t]\in{\mathbb{R}}/{\mathbb{Z}}$ with $t\in[0,1]$ to the class of $(t,p)$ in
$([0,1]\times Y)/\sim$. Then $\rho\circ\iota:S^{1}\to S^{1}$ is a
homeomorphism, so $\rho^{*}\theta\in H^{1}(X;{\mathbb{Z}})$ is primitive. The
Poincaré dual of $\rho^{*}\theta$ is equal to the fundamental class of any
fiber $F$ of $\rho$. Let $j:F\to X$ be the inclusion. Given
$\alpha_{2},\alpha_{3}\in H^{1}(X;{\mathbb{Z}})$, the product
$\iota^{*}\theta\cup\alpha_{2}\cup\alpha_{3}$ is a generator of
$H^{3}(X;{\mathbb{Z}})$ if and only if $j^{*}\alpha_{2}\cup j^{*}\alpha_{3}$
is a generator of $H^{2}(F;{\mathbb{Z}})$.

Now, the image of $j^{*}:H^{1}(X;{\mathbb{Z}})\to H^{1}(F;{\mathbb{Z}})\simeq
H^{1}(Y;{\mathbb{Z}})$ is contained in the group of $\sigma$-invariant classes
$H^{1}(Y;{\mathbb{Z}})^{\sigma}$. We claim that
$H^{1}(Y;{\mathbb{Z}})^{\sigma}$ is contained in the image of
$\pi^{*}:H^{1}(T^{2};{\mathbb{Z}})\to H^{1}(Y;{\mathbb{Z}})$. This follows
from applying Lemma 2.2 to the action of $G=\\{\operatorname{Id},\sigma\\}$ on
$Y$, taking into account that the action of $\sigma$ on $Y$ has fixed points
(and hence the morphism $\eta:G\to T^{2}$ given by the lemma is necessarily
trivial). Hence, for any classes $\alpha_{2},\alpha_{3}\in
H^{1}(X;{\mathbb{Z}})$ there exist classes $\beta_{2},\beta_{3}\in
H^{1}(T^{2};{\mathbb{Z}})$ such that $j^{*}\alpha_{2}\cup
j^{*}\alpha_{3}=\pi^{*}\beta_{2}\cup\pi^{*}\beta_{3}=\pi^{*}(\beta_{2}\cup\beta_{3})$.
But $\pi^{*}(\beta_{2}\cup\beta_{3})$ will never be a generator of
$H^{2}(Y;{\mathbb{Z}})$, because $\pi:Y\to T^{2}$ has degree $2$. This
concludes the proof that $X$ is not hypertoral.

##  15\. Regular self coverings of the manifolds in Theorem 1.14

Let $d$ be an odd natural number. The manifolds constructed in [14] are
products $T(h)\times H$, where $T(h)$ is the mapping torus of a self
homeomorphism $h$ of a closed topological manifold $V$ and $H$ is a closed
hyperbolic manifold. So it suffices to prove that $T(h)$ supports a regular
self covering of degree $d$. The structure of mapping torus on $T(h)$ gives a
map $T(h)\to S^{1}$, and the regular self covering we claim to exist is the
pullback, via this map, of the covering $S^{1}\to S^{1}$ sending $\theta$ to
$d\,\theta$. This pullback can be identified with the mapping torus
$T(h^{d})$, so all we need to prove is that $T(h)$ and $T(h^{d})$ are
homeomorphic.

The manifold $V$ is $W\cup_{T}(T^{n}\times[0,1])\cup_{T^{\prime}}W^{\prime}$,
where $W$ and $W^{\prime}$ are $(n+1)$-dimensional manifolds with boundaries
$T$ and $T^{\prime}$ respectively, where $n\geq 5$. Here both $T$ and
$T^{\prime}$ denote the torus $T^{n}$ with involutions $h_{T}:T\to T$ and
$h_{T^{\prime}}:T^{\prime}\to T^{\prime}$. The involution $h_{T}$ is a linear
involution, whereas $h_{T^{\prime}}$ is exotic, i.e., not conjugate to a
linear action. Both $h_{T}$ and $h_{T^{\prime}}$ have nonempty fixed point set
(see [7, §2.1] for a concrete description of the involutions $h_{T}$ and
$h_{T^{\prime}}$ used in [14]). The maps $h_{T}$ and $h_{T^{\prime}}$ are
homotopic. In the definition of $V$ we glue $T\subset W$ with
$T^{n}\times\\{0\\}$ and $T^{\prime}\subset W^{\prime}$ with
$T^{n}\times\\{1\\}$.

The involutions $h_{T}$ and $h_{T^{\prime}}$ extend to involutions $h_{W}:W\to
W$ and $h_{W^{\prime}}:W^{\prime}\to W^{\prime}$ respectively, and there is a
self homeomorphism $h_{C}$ of $C:=T^{n}\times[0,1]$ whose restriction to
$C_{0}:=T^{n}\times\\{0\\}$ resp. $C_{1}:=T^{n}\times\\{1\\}$ coincides with
$h_{T}$ resp. $h_{T^{\prime}}$. We now briefly justify the existence of
$h_{C}$. It suffices to prove the existence of a homeomorphism $\phi:C\to C$
such that $\phi|_{C_{0}}=\operatorname{Id}_{T^{n}}$ and
$\phi|_{C_{1}}=\psi:=h_{T}^{-1}\circ h_{T^{\prime}}$, for then
$h_{C}:=(h_{T}\times\operatorname{Id}_{[0,1]})\circ\phi$ has the desired
property. Now, $\psi$ is homotopic to $\operatorname{Id}_{T^{n}}$ so the
mapping tori $T(\psi)$ and $T(\operatorname{Id}_{T^{n}})$ are homotopy
equivalent. Since $T(\operatorname{Id}_{T^{n}})=T^{n+1}$, the topological
rigidity of tori implies that $T(\psi)$ and $T(\operatorname{Id}_{T^{n}})$ are
homeomorphic. The existence of $\phi$ now follows from combining: [35, Theorem
1], the observation that invertible cobordism are $h$-cobordisms, the
$s$-cobordism theorem, and the vanishing of the Whitehead group of
$\pi_{1}(T^{n})$.

Unlike $h_{T}$ and $h_{T^{\prime}}$, $h_{C}$ is not an involution. However, we
have the following:

######  Proposition 15.1.

If we chose $h_{C}$ suitably, then $h_{C}^{2}$ and $\operatorname{Id}_{C}$ are
homotopic rel. $\partial C$.

###### Proof.

We identify $T^{n}$ with ${\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$, so the universal
covering space $C^{\sharp}$ of $C$ can be identified with
${\mathbb{R}}^{n}\times[0,1]$. Let
$C^{\sharp}_{i}={\mathbb{R}}^{n}\times\\{i\\}$ for $i=0,1$. Let $f,g:C\to C$
be continuous maps such that $f|_{\partial C}=g|_{\partial C}$. Choose lifts
$f^{\sharp},g^{\sharp}:C^{\sharp}\to C^{\sharp}$. Then
$g^{\sharp}|_{C^{\sharp}_{i}}-f^{\sharp}|_{C^{\sharp}_{i}}$ is equal to some
constant $\lambda_{i}\in{\mathbb{Z}}^{n}$, because $g|_{C_{i}}-f|_{C_{i}}$ is
identically zero. Let

$\lambda(g,f):=\lambda_{1}-\lambda_{0}.$

The vector $\lambda(g,f)\in{\mathbb{Z}}^{n}$ is independent of the chosen
lifts of $f,g$.

######  Lemma 15.2.

$f$ and $g$ are homotopic rel. $\partial C$ if and only if $\lambda(g,f)=0$.

###### Proof.

The ”only if” part of the lemma is an easy exercise. For the ”if” part, note
that there is a linear map $\rho:{\mathbb{Z}}^{n}\to{\mathbb{Z}}^{n}$ such
that $f^{\sharp}(p+\mu,s)=f^{\sharp}(p,s)+(\rho(\mu),0)$ and
$g^{\sharp}(p+\mu,s)=g^{\sharp}(p,s)+(\rho(\mu),0)$ for every
$p\in{\mathbb{R}}^{n},\,\mu\in{\mathbb{Z}}^{n},\,s\in[0,1]$. Actually $\rho$
can be identified with the morphism $H_{1}(T^{n})\to H_{1}(T^{n})$ induced by
$f$ or $g$. Then the map $C^{\sharp}\times[0,1]\to C^{\sharp}$ sending
$((p,s),t)$ to $(1-t)f^{\sharp}(p,s)+tg^{\sharp}(p,s)$ is a homotopy between
$f^{\sharp}$ and $g^{\sharp}$ that descends to a homotopy rel $\partial C$
between $f$ and $g$. ∎

Now suppose that $\zeta:C\to C$ is a homeomorphism satisfying
$\zeta|_{C_{0}}=h_{T}$ and $\zeta|_{C_{1}}=h_{T^{\prime}}$. Since both $h_{T}$
and $h_{T^{\prime}}$ have fixed points, there exist $x\in C_{0}$ and $y\in
C_{1}$ such that $\zeta(x)=x$ and $\zeta(y)=y$. Choose lifts
$x^{\sharp},y^{\sharp}\in C^{\sharp}$ of $x,y$ respectively. There is a unique
lift $\zeta^{\sharp}:C^{\sharp}\to C^{\sharp}$ satisfying
$\zeta^{\sharp}(x^{\sharp})=x^{\sharp}$. As before, there is a morphism of
groups $\rho:{\mathbb{Z}}^{n}\to{\mathbb{Z}}^{n}$ such that
$\zeta^{\sharp}(p+\mu,s)=\zeta^{\sharp}(p,s)+(\rho(\mu),0)$ for all $p,\mu,s$.
Let $o:C^{\sharp}={\mathbb{R}}^{n}\times[0,1]\to{\mathbb{R}}^{n}$ denote the
projection and let
$\lambda:=o(\zeta^{\sharp}(y^{\sharp})-y^{\sharp})\in{\mathbb{Z}}^{n}$, so
that $\zeta^{\sharp}(y^{\sharp})=y^{\sharp}+(\lambda,0)$. Then:

$\displaystyle\lambda(\zeta^{2},\operatorname{Id})$
$\displaystyle=o(\zeta^{\sharp}\zeta^{\sharp}(y^{\sharp})-y^{\sharp})=o(\zeta^{\sharp}\zeta^{\sharp}(y^{\sharp})-\zeta^{\sharp}(y^{\sharp})+\zeta^{\sharp}(y^{\sharp})-y^{\sharp})$
$\displaystyle=o(\zeta^{\sharp}(y^{\sharp}+(\lambda,0))-\zeta^{\sharp}(y^{\sharp})+(\lambda,0))=\rho(\lambda)+\lambda.$

By the previous lemma, in order for $\zeta^{2}$ to be homotopic to
$\operatorname{Id}_{C}$ rel. $\partial C$ we need $\rho(\lambda)+\lambda$ to
vanish. This need not be the case, but if we define
$\xi^{\sharp}:C^{\sharp}\to C^{\sharp}$ as
$\xi^{\sharp}(p,s)=\zeta^{\sharp}(p)-s\lambda$ then we have
$\xi^{\sharp}(p+\mu,s)=\xi^{\sharp}(p,s)+(\rho(\mu),0)$ for all $p,\mu,s$, so
$\xi^{\sharp}$ descends to a homeomorphism $\xi:C\to C$ satisfying
$\xi|_{\partial C}=\zeta|_{\partial C}$. Furthermore,
$\xi^{\sharp}(y^{\sharp})=y^{\sharp}$, so
$\lambda(\xi^{2},\operatorname{Id})=0$. Consequently, $h_{C}:=\xi$ has the
desired property. ∎

Assume from now on that $h_{C}$ has been chosen in such a way that $h_{C}^{2}$
is homotopic to $\operatorname{Id}_{C}$ rel. $\partial C$, which implies that
$h_{C}^{d}$ and $h_{C}$ are homotopic rel. $\partial C$. The involution
$h:V\to V$ is defined by the condition that its restriction to the subspaces
$W,\,T^{n}\times[0,1],\,W^{\prime}$ is given by
$h_{W},\,h_{C},\,h_{W^{\prime}}$ respectively.

Since $h_{W}$ and $h_{W^{\prime}}$ are involutions, the restrictions of the
maps $h$ and $h^{d}$ to $W$ and $W^{\prime}$ are equal. Hence, to prove that
$T(h)$ and $T(h^{d})$ are homeomorphic it suffices to prove the existence of a
homeomorphism of mapping tori $\phi:T(h_{C})\to T(h_{C}^{d})$ whose
restriction to $\partial T(h_{C})$ is the natural homeomorphism $\partial
T(h_{C})\to\partial T(h_{C}^{d})$ resulting from the equalities
$h_{T}=h_{T}^{d}$ and $h_{T^{\prime}}=h_{T^{\prime}}^{d}$. We are going to
prove the existence of $\phi$ using the topological rigidity of non-positively
curved Riemannian manifolds proved in [22, 23].

By definition $T(h_{C})$ is the quotient of $C\times[0,1]$ by the relation
that identifies $(x,1)$ with $(h_{C}(x),0)$ for every $x\in C$. For $i=0,1$,
let $T_{i}(h_{C})\subset T(h_{C})$ be the image of $C_{i}\times[0,1]$ under
the projection map $C\times[0,1]\to T(h_{C})$. Then $T_{0}(h_{C})$ resp.
$T_{1}(h_{C})$ can be identified with $T(h_{T})$ resp. $T(h_{T^{\prime}})$.
Since $h_{T}$ is a linear involution, $T(h_{T})$ supports a non-positively
curved Riemannian metric. Now, $T(h_{T^{\prime}})$ is homotopy equivalent to
$T(h_{T})$, because $h_{T}$ and $h_{T^{\prime}}$ are homotopic. Hence, by
topological rigidity [22, Theorem 14.1], there is a homeomorphism
$\psi:T(h_{T^{\prime}})\to T(h_{T})$. Choosing $\psi$ appropriately, we may
and do assume that the compositions of maps
$T(h_{T})\stackrel{{\scriptstyle\psi^{-1}}}{{\longrightarrow}}T(h_{T^{\prime}})=T_{1}(h_{C})\hookrightarrow
T(h_{C})$ and $T(h_{T})=T_{0}(h_{C})\hookrightarrow T(h_{C})$ are homotopic.

We claim the existence of a homeomorphism

$\xi:T(h_{C})\to T(h_{T}\times\operatorname{Id}_{[0,1]})=T(h_{T})\times[0,1]$

whose restriction to $T_{0}(h_{C})$ resp. $T_{1}(h_{C})$ coincides with
$\operatorname{Id}_{T(h_{T})}$ resp. $\psi$. Once we prove the claim the
existence of $\phi$ will follow immediately, since by [22, Theorem 14.1]
topological rigidity applies to $T(h_{T})\times[0,1]$ (again because
$T(h_{T})$ supports a metric of non-positive curvature) and the existence of a
homotopy $h_{C}^{d}\sim h_{C}$ rel. $\partial C$ gives a homotopy equivalence
$T(h_{C})\to T(h_{C}^{d})$ whose restriction to $\partial T(h_{C})$ is a
homeomorphism.

To prove the existence of the homeomorphism $\xi:T(h_{C})\to
T(h_{T})\times[0,1]$ we rely once again on the topological rigidity of
$T(h_{T})\times[0,1]$, so we only need to prove the existence of a continuous
map $\chi:T(h_{C})\to T(h_{T})\times[0,1]$ whose restriction to $T_{0}(h_{C})$
resp. $T_{1}(h_{C})$ coincides with $\operatorname{Id}_{T(h_{T})}$ resp.
$\psi$ (these properties imply that $\chi$ is a homotopy equivalence).

If $Y\subseteq X$ is an inclusion of topological spaces and $f,g:X\to X$ are
maps preserving $Y$, satisfying $f|_{Y}=g|_{Y}$, and $f,g$ are homotopic rel.
$Y$, then there is a continuous map $\epsilon:T(f)\to T(g)$ whose restriction
to $T(f|_{Y})$ is the natural identification between $T(f|_{Y})$ and
$T(g|_{Y})$. Indeed, suppose that $H:X\times I\to X$ satisfies $H(x,0)=g(x)$,
$H(x,1)=f(x)$ and $H(y,t)=f(y)=g(y)$ for every $x\in X$, $y\in Y$,
$t\in[0,1]$. Then $\epsilon$ is defined by the map
$\widetilde{\epsilon}:X\times[0,1]\to X\times[0,1]$ given by

$\widetilde{\epsilon}(x,t)=\left\\{\begin{array}[]{ll}(x,2t)&\qquad\text{if
$t\in[0,1/2],$}\\\ (H(x,2t-1),0)&\qquad\text{if
$t\in[1/2,1]$.}\end{array}\right.$

Using the previous principle, and the facts that $h_{C}$ and
$h_{T}\times\operatorname{Id}_{[0,1]}$ are homotopic rel. $C_{0}$ and $h_{C}$
and $h_{T^{\prime}}\times\operatorname{Id}_{[0,1]}$ are homotopic rel. $C_{1}$
(which can be proved by lifting the maps to $C^{\sharp}$ as in the proof of
Proposition 15.1 and interpolating linearly), we deduce the existence of maps
$\chi_{0}:T(h_{C})\to T(h_{T})\times[0,1]$ and $\chi_{1}:T(h_{C})\to
T(h_{T^{\prime}})\times[0,1]$ such that $\chi_{i}$ restricted to
$T_{i}(h_{C})$ is the identity for $i=0,1$. Furthermore, $\chi_{0}$ and
$(\psi\times\operatorname{Id}_{[0,1]})\circ\chi_{1}$ are homotopic.

The universal cover of $T(h_{C})$ can be identified with
${\mathbb{R}}^{n}\times[0,1]\times{\mathbb{R}}$, and that of
$T(h_{T})\times[0,1]$ with ${\mathbb{R}}^{n}\times{\mathbb{R}}\times[0,1]$.
Fix a lift $h_{T}^{\sharp}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ of
$h_{T}:T^{n}\to T^{n}$. Crucially, $h_{T}^{\sharp}$ is an affine isomorphism.
The group
$\Gamma=\pi_{1}(T(h_{T})\times[0,1])\simeq\pi_{1}(T(h_{T}))\simeq{\mathbb{Z}}^{n}\rtimes{\mathbb{Z}}$
acts on ${\mathbb{R}}^{n}\times{\mathbb{R}}\times[0,1]$ preserving the affine
structure induced by the inclusion
${\mathbb{R}}^{n}\times{\mathbb{R}}\times[0,1]\subset{\mathbb{R}}^{n+2}$: the
factor ${\mathbb{Z}}^{n}$ adds by addition on the first factor of
${\mathbb{R}}^{n}\times{\mathbb{R}}\times[0,1]$, and the action of the second
factor is generated by the transformation
$(z,t,s)\mapsto(h_{T}^{\sharp}(z),t-1,s)$.

Choose lifts of $\chi_{0}$ and
$(\psi\times\operatorname{Id}_{[0,1]})\circ\chi_{1}$ to the universal
coverings, and call them $\theta_{0}$ and $\theta_{1}$ respectively, so that
$\theta_{i}:{\mathbb{R}}^{n}\times[0,1]\times{\mathbb{R}}\to{\mathbb{R}}^{n}\times{\mathbb{R}}\times[0,1].$
Since $\theta_{0}$ and $\theta_{1}$ are homotopic there exists a morphism of
groups $\rho:\pi_{1}(T(h_{C}))\to\pi_{1}(T(h_{T})\times[0,1])$ such that both
$\theta_{0}$ and $\theta_{1}$ are $\rho$-equivariant, meaning that
$\theta_{i}(\gamma\cdot w)=\rho(\gamma)\cdot\theta_{i}(w)$ for every
$\gamma\in\pi_{1}(T(h_{C}))$ and
$w\in{\mathbb{R}}^{n}\times[0,1]\times{\mathbb{R}}$. Define
$\theta:{\mathbb{R}}^{n}\times[0,1]\times{\mathbb{R}}\to{\mathbb{R}}^{n}\times{\mathbb{R}}\times[0,1]$
as $\theta(p,s,t)=(1-s)\theta_{0}(p,s,t)+s\theta_{1}(p,s,t)$. Then $\theta$
satisfies the same $\rho$-equivariance property as $\theta_{i}$, because the
action of $\pi_{1}(T(h_{T})\times[0,1])$ on
${\mathbb{R}}^{n}\times{\mathbb{R}}\times[0,1]$ is affine. This implies that
$\theta$ descends to a map $\chi:T(h_{C})\to T(h_{T})\times[0,1]$, and by
construction the restriction of $\chi$ to $T_{0}(h_{C})$ resp. $T_{1}(h_{C})$
coincides with $\operatorname{Id}_{T(h_{T})}$ resp. $\psi$. This finishes the
proof of the theorem.

## References

  * [1] A. Assadi, D. Burghelea, Examples of asymmetric differentiable manifolds. Math. Ann. 255 (1981), no. 3, 423–430.
  * [2] M.F. Atiyah, I.G. Macdonald, Introduction to commutative algebra, Addison–Wesley, 1969.
  * [3] O. Baues, V. Cortés, Aspherical Kähler manifolds with solvable fundamental group, Geom. Dedicata 122 (2006), 215–229.
  * [4] S. Behrens, B. Kalmar, M. H. Kim, M. Powell, A. Ray (editors), The Disc Embedding Theorem, Oxford Univ. Press, Oxford, 2021.
  * [5] A.J. Berrick, A topologist’s view of perfect and acyclic groups. Invitations to geometry and topology, 1–28, Oxf. Grad. Texts Math., 7, Oxford Univ. Press, Oxford, 2002.
  * [6] L. Bessières, G. Besson, S. Maillot, M. Boileau, J. Porti, Geometrisation of 3-manifolds. EMS Tracts in Mathematics, 13. European Mathematical Society (EMS), Zürich, 2010.
  * [7] J. Block, S. Weinberger, On the generalized Nielsen realization problem, Comment. Math. Helv. 83 (2008), no. 1, 21–33.
  * [8] S. Bochner, D. Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. (2) 47 (1946), 639–653.
  * [9] A. Borel, Seminar on transformation groups, Ann. of Math. Studies 46, Princeton University Press, N.J., 1960.
  * [10] G.E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London (1972).
  * [11] T. Bröcker, T. tom Dieck, Representations of compact Lie groups. Translated from the German manuscript. Corrected reprint of the 1985 translation. Graduate Texts in Mathematics 98. Springer-Verlag, New York, 1995. x+313 pp.
  * [12] D. Burghelea, R. Schultz, On the semisimple degree of symmetry, Bull. Soc. Math. France 103 (1975), no. 4, 433–440.
  * [13] M. Bustamante, B. Tshishiku, Symmetries of exotic smoothings of aspherical space forms, preprint arXiv:2109.0919.
  * [14] S. Cappell, S. Weinberger, M. Yan, Closed aspherical manifolds with center, J. Topol. 6 (2013), no. 4, 1009–1018.
  * [15] S. Chang, S. Weinberger, A course on surgery theory. Annals of Mathematics Studies, 211. Princeton University Press, Princeton, NJ, 2021.
  * [16] M.M. Cohen, A course in simple-homotopy theory. Graduate Texts in Mathematics, Vol. 10. Springer-Verlag, New York-Berlin, 1973.
  * [17] P.E. Conner, F. Raymond, P.J. Weinberger, Manifolds with no periodic maps. Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part II, pp. 81–108. Lecture Notes in Math., Vol. 299, Springer, Berlin, 1972.
  * [18] B. Csikós, I. Mundet i Riera, L. Pyber, E. Szabó, Number of stabilizer subgroups in a finite group acting on a manifold, preprint arXiv:2111.14450.
  * [19] G. D’Ambra, Isometry groups of Lorentz manifolds, Invent. Math. 92 (1988), 555–565.
  * [20] H. Donnelly, R. Schultz, Compact group actions and maps into aspherical manifolds, Topology 21 (1982), no. 4, 443–455.
  * [21] A.L. Edmonds, Construction of group actions on four-manifolds, Trans. Amer. Math. Soc. 299 (1987), no. 1, 155–170.
  * [22] F.T. Farrell, Lectures on surgical methods in rigidity. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1996.
  * [23] F.T. Farrell, L.E. Jones, Topological rigidity for compact non-positively curved manifolds. Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), 229–274, Proc. Sympos. Pure Math., 54, Part 3, Amer. Math. Soc., Providence, RI, 1993.
  * [24] M.H. Freedman, F. Quinn, Topology of 4-manifolds. Princeton Mathematical Series, 39. Princeton University Press, Princeton, NJ, 1990.
  * [25] A. Fujiki, On automorphism groups of compact Kaehler manifolds, Invent. Math. 44 (1978) 225–258.
  * [26] D.H. Gottlieb, K.B. Lee, M. Özaydin, Compact group actions and maps into $K(\pi,1)$-spaces. Trans. Amer. Math. Soc. 287 (1985), no. 1, 419–429.
  * [27] A. Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002.
  * [28] J. Hilgert, K.-H. Neeb, Structure and geometry of Lie groups. Springer Monographs in Mathematics. Springer, New York, 2012.
  * [29] W.-c. Hsiang, C.T.C. Wall, On homotopy tori. II, Bull. London Math. Soc. 1 (1969), 341–342.
  * [30] W.-c. Hsiang, W.-y- Hsiang, The degree of symmetry of homotopy spheres, Ann. of Math. (2) 89 (1969), 52–67.
  * [31] W. Huck, Circle actions on 4-manifolds. I, Manuscripta Math. 87 (1995), no. 1, 51–70.
  * [32] M.A. Kervaire, Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67–72.
  * [33] M. Kreck, W. Lück, Topological rigidity for non-aspherical manifolds, Pure Appl. Math. Q. 5 (2009), no. 3, Special Issue: In honor of Friedrich Hirzebruch. Part 2, 873–914.
  * [34] S. Kwasik, R. Schultz, Isolated fixed points of circle actions on 4-manifolds, Forum Math. 9 (1997), no. 4, 517–546.
  * [35] T.C. Lawson, Splitting isomorphisms of mapping tori, Trans. Amer. Math. Soc. 205 (1975), 285–294.
  * [36] K.B. Lee, F. Raymond, Topological, affine and isometric actions on flat Riemannian manifolds, J. Differential Geometry 16 (1981), no. 2, 255–269.
  * [37] W. Lück, Some open problems about aspherical closed manifolds. Trends in contemporary mathematics, 33–46, Springer INdAM Ser., 8, Springer, Cham, 2014.
  * [38] L. N. Mann, J. C. Su, Actions of elementary p-groups on manifolds, Trans. Amer. Math. Soc. 106 (1963), 115–126.
  * [39] H. Matsumura, Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1989.
  * [40] H. Minkowski, Zur Theorie der positiven quadratischen Formen, Journal fur die reine und angewandte Mathematik 101 (1887), 196-202.
  * [41] J. Morgan, G. Tian, The geometrization conjecture. Clay Mathematics Monographs, 5. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2014.
  * [42] I. Mundet i Riera, Jordan’s theorem for the diffeomorphism group of some manifolds Proc. Amer. Math. Soc. 138 (2010), 2253-2262.
  * [43] I. Mundet i Riera, Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds, Transformation Groups 22 (2017), no. 2, 487–501, DOI 10.1007/s00031-016-9374-9.
  * [44] Y. Prokhorov, C. Shramov, Jordan property for Cremona groups, Amer. J. Math. 138 (2016), no. 2, 403–418.
  * [45] V. Puppe, Do manifolds have little symmetry? J. Fixed Point Theory Appl. 2 (2007), no. 1, 85–96.
  * [46] D.J.S. Robinson, A course in the theory of groups, Second edition, Graduate Texts in Mathematics 80, Springer-Verlag, New York, 1996.
  * [47] R. Schultz, Group actions on hypertoral manifolds. I. Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), pp. 364–377, Lecture Notes in Math., 788, Springer, Berlin, 1980.
  * [48] R. Schultz, Group actions on hypertoral manifolds. II. J. Reine Angew. Math. 325 (1981), 75–86.
  * [49] J.-P. Serre, Bounds for the orders of the finite subgroups of $G(k)$, Group representation theory, 405–450, EPFL Press, Lausanne, 2007.
  * [50] T. tom Dieck, Transformation groups. De Gruyter Studies in Mathematics, 8. Walter de Gruyter & Co., Berlin, 1987. x+312 pp.
  * [51] W. van Limbeek, Symmetry gaps in Riemannian geometry and minimal orbifolds, J. Differential Geom. 105 (2017), no. 3, 487–517.
  * [52] C.T.C. Wall, Surgery on compact manifolds. Second edition. Edited and with a foreword by A. A. Ranicki. Mathematical Surveys and Monographs, 69. American Mathematical Society, Providence, RI, 1999.
  * [53] J. Xu, Finite $p$-groups of birational automorphisms and characterizations of rational varieties, preprint arXiv:1809.09506.
  * [54] S. Ye, Symmetries of flat manifolds, Jordan property and the general Zimmer program, J. Lond. Math. Soc. (2) 100 (2019), no. 3, 1065–1080.