# Dynamical classification of analytic one-frequency quasi-periodic
$\mathrm{SO}(3,\mathbb{R})$-cocycles

Xuanji Hou111Hubei Key Laboratory of Mathematical Sciences, School of
Mathematics and Statics, Huazhong Normal University, Wuhan 430079, China, Yi
Pan222Institute of Science and Technology Austria, Klosterneuburg 3400,
Austria, Qi Zhou333Chern Institute of Mathematics, NanKai University, Tianjin
300071, China

###### Abstract

We establish a close connection between acceleration and dynamical degree for
one-frequency quasi-periodic compact cocycles, by showing that two vectors
derived separately from each coincide. Based on this, we provide a dynamical
classification of one-frequency quasi-periodic
$\mathrm{SO}(3,\mathbb{R})$-cocycles.

##  1 Introduction

The classification of dynamics is an important topic in dynamical systems. One
well-known example is the classification of circle diffeomorphisms. Let
$\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ and $f:\mathbb{T}\rightarrow\mathbb{T}$ be
an orientation-preserving $C^{r}$ homeomorphism, where
$r\in\mathbb{N}\cup\\{\infty,\omega\\}$. Poincaré laid the foundations of the
theory of circle diffeomorphisms and introduced the concept of rotation number
in the classification of these diffeomorphisms. The rotation number
$\rho(f)\in\mathbb{R}$ of $f$ is defined as the uniform limit

$\lim_{n\to\infty}\frac{{f}^{n}(x)-x}{n}.$

The rotation number is independent of the choice of $x\in\mathbb{T}$ and is
invariant under conjugations, making it a useful dynamical invariant.
Poincaré’s classification theorem states that if the rotation number $\rho(f)$
is rational $\frac{p}{q}$, then $f$ has a periodic orbit of period $q$. If the
rotation number is irrational, then $f$ is semi-conjugate to the irrational
(rigid) rotation. Denjoy [17] further showed that if the rotation number is
irrational and $\ln Df$ has bounded variation, then the semi-conjugacy is a
homeomorphism. This result was later extended by Arnold [15], as part of the
KAM theory, who showed that if $f$ is $C^{\omega}$ close to a rotation and the
rotation number $\rho(f)$ satisfies the Diophantine condition, then the
conjugacy is analytic. Subsequent work by Herman, Yoccoz, Katznelson,
Ornstein, Sinai, and others [28, 29, 51, 52, 41, 42, 43, 49] extended this
result to the global version of analytic, smooth, or finitely differentiable
circle diffeomorphisms $f$. They showed that when $f$ is analytic (resp.
$C^{\infty}$, $C^{k}$ with $k>2$) and $\rho(f)$ is Diophantine, the conjugacy
is analytic (resp. $C^{\infty}$, $C^{k-r}$ for some $0<r<k$). Moreover, the
Diophantine condition can be weakened to the Brjuno condition, which turns to
be necessary [52]. Therefore, the dynamics of the circle diffeomorphism $f$
can be completely determined by its rotation number $\rho(f)$.

A one-frequency cocycle is a natural generalization of a circle diffeomorphism
in high dimensions. Let ${\rm G}$ be a Lie-subgroup of ${\rm
GL}(s,\mathbb{C})$. A $C^{r}$ one-frequency ${\rm G}$-cocycle $(\alpha,A)$ is
a skew-product defined on $\mathbb{T}\times\mathbb{C}^{s}$ such that

$(\alpha,A):\,\mathbb{T}\times\mathbb{C}^{s}\to\mathbb{T}\times\mathbb{C}^{s},\quad(x,v)\mapsto(x+\alpha,A(x)\cdot
v),$

where $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, $A\in C^{r}(\mathbb{T},{\rm
GL}(s,\mathbb{C}))$, $r\in\mathbb{N}\cup\\{\infty\\}$. The iterates of the
cocycle $(\alpha,A)^{n}=(n\alpha,A_{n})$ are defined as $A_{0}(x)=I$,

$A_{n}(\cdot)=A(\cdot+(n-1)\alpha)\cdots A(\cdot),\quad n\geq 1,$

and $A_{n}(\cdot)=A_{-n}(\cdot-n\alpha)^{-1}$ for $n\leq-1$.

When $G=\mathrm{SO}(2,\mathbb{R})$, taking projective action, it naturally
induces the cocycle

$T_{\phi}:\mathbb{T}\times\mathbb{T}\to\mathbb{T}\times\mathbb{T},\quad(x,g)\mapsto(x+\alpha,\phi(x)\cdot
g).$

Moreover, if $\phi$ is smooth, then dynamical properties of $T_{\phi}$ depend
on the topological degree $d(\phi)$ of $\phi$. For example, if $\phi$ is
$C^{2}$ with $d(\phi)\neq 0$, then $T_{\phi}$ is ergodic and it has a
countable Lebesgue spectrum on the orthocomplement of the space of functions
depending only on the first variable [33]. If $\phi$ is absolutely continuous
with $d(\phi)=0$, then $T_{\phi}$ has a singular spectrum [26].

However, when $G$ is non-compact, hyperbolicity appears. Therefore, another
important dynamical invariant is introduced. For a cocycle
$(\alpha,A)\in\mathbb{R}\backslash\mathbb{Q}\times C^{0}(\mathbb{T},{\rm
GL}(s,\mathbb{C}))$, we define the Lyapunov exponents $L_{1}(\alpha,A)\geq
L_{2}(\alpha,A)\geq...\geq L_{s}(\alpha,A)$ as

$L_{k}(\alpha,A):=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\int_{\mathbb{T}}\ln(\sigma_{k}(A_{n}(x)))dx,\quad
k=1,\cdots,s,$

(repeatedly according to multiplicities), where for a matrix $B$ we denote by
$\sigma_{1}(B)\geq...\geq\sigma_{s}(B)$ its singular values (eigenvalues of
$\sqrt{B^{*}B}$). As the k-th exterior product $\Lambda^{k}B$ of $B$ satisfies
$\|\Lambda^{k}B\|=\sum_{j=1}^{k}\sigma_{j}(B)$, one can define

$L^{k}(\alpha,A):=\sum\limits_{j=1}^{k}L_{j}(\alpha,A)=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\int_{\mathbb{T}}\ln(\|\Lambda^{k}A_{n}(x)\|)dx,\quad
k=1,\cdots,s.$ (1.0.1)

Unfortunately, these Lyapunov exponents are far from enough to classify the
cocycles. In fact, even for cocycles with all Lyapunov exponents vanishing,
the dynamics still have a variety of possibilities [19, 40].

###  1.1 Accelerations

A breakthrough came recently, with the establishment of the global theory of
one-frequency analytic Schrödinger operators. Avila [4] classified one-
frequency analytic $\mathrm{SL}(2,\mathbb{R})$ cocycles and proposed another
dynamical invariant called "accelerations". Indeed, for any analytic cocycle
$(\alpha,A)\in\mathbb{R}\backslash\mathbb{Q}\times C^{\omega}(\mathbb{T},{\rm
GL}(s,\mathbb{C}))$, there exists $\delta>0$ such that $A$ can be
holomorphically extended to $\\{|\Im z|\leq\delta\\}$. Let
$A_{\epsilon}(x):=A(x+i\epsilon)$ for $|\epsilon|<\delta$. We can then define
the accelerations of $(\alpha,A)$ as follows:

$\omega^{k}:=\lim\limits_{\epsilon\rightarrow
0^{+}}\frac{1}{2\pi\epsilon}(L^{k}(\alpha,A_{\epsilon})-L^{k}(\alpha,A)),\qquad\omega_{k}:=\omega^{k}-\omega^{k-1}.$
(1.1.1)

Furthermore, we define $(\omega_{1},\cdots,\omega_{s})$ as the acceleration
vector of $(\alpha,A)$. The most important observation is that the
accelerations are quantized [4, 7], namely that there exists
$l\in\\{1,\cdots,s\\}$ such that both $l\omega_{k}$ and $l\omega^{k}$ are
integers. Actually the accelerations are dynamical invariants as they remain
invariant under conjugations. We recall that
$(\alpha,A_{i})\in\mathbb{R}\backslash\mathbb{Q}\times
C^{\omega}(\mathbb{T},G)$, $i=1,2$, are conjugate to each other, if there
exists $B\in C^{\omega}(\chi_{G}\mathbb{T},G)$444$\chi_{G}\in\\{1,2\\}$. In
particular, for $G=\mathrm{SO}(3,\mathbb{R})$, $\chi_{G}=1$. such that

$A_{1}(\cdot)=B(\cdot+\alpha)A_{2}(\cdot)B(\cdot)^{-1}.$

With these notations, Avila’s classification of one-frequency analytic
$\mathrm{SL}(2,\mathbb{R})$-cocycles is introduced [4]. If $A$ is homotopic to
the identity, or in other words, if the topological degree $d(A)=0$, then the
dynamical behavior can be determined by the Lyapunov exponent and
acceleration. To be precise, if $\omega_{1}=0$, then the cocycle is
subcritical (resp. uniformly hyperbolic) when $L_{1}(\alpha,A)=0$ (resp.
$L_{1}(\alpha,A)>0$). On the contrary, if $\omega_{1}\neq 0$, $(\alpha,A)$ is
critical (resp. supercritical or non-uniformly hyperbolic) when
$L_{1}(\alpha,A)=0$ (resp. $L_{1}(\alpha,A)>0$). On the other hand, if the
topological degree $d(A)\neq 0$, i.e., $A\colon$ $\mathbb{T}\to{\rm
SL}(2,\mathbb{R})$ is homotopic to $\theta\mapsto R_{d(A)\theta}$, where

$R_{\theta}:=\begin{pmatrix}\cos 2\pi\theta&-\sin 2\pi\theta\\\ \sin
2\pi\theta&\cos 2\pi\theta\end{pmatrix},$

Avila-Krikorian [11] proved that if $(\alpha,A)$ is $L^{2}$-conjugate to
rotations, then it is $C^{\omega}$-conjugate to rotations. In conclusion, the
classification of analytic $\mathrm{SL}(2,\mathbb{R})$-cocycles depends on
three dynamical invariants: the topological degree, Lyapunov exponent, and
acceleration.

The natural inquiry is whether this kind of classification can be generalized
to high-dimensional cocycles. As an initial attempt, we consider this question
for cocycles valued in ${\rm SO}(s,\mathbb{R})$, $s\in\mathbb{N}^{*}$. Note
that if $A$ takes values in $\mathrm{SO}(s,\mathbb{R})$, then
$L_{k}(\alpha,A)=0$, $k=1,\cdots,s$. Hence the monotonicity on singular values
therefore Lyapunov exponents provides that

$\omega_{1}\geq\cdots\geq\omega_{s}.$

In particular, for any $\mathrm{SO}(3,\mathbb{R})$-cocycle, as determinant is
identically $1$, we have $\omega_{1}+\omega_{2}+\omega_{3}=0.$ As singular
values $\lambda,\lambda^{-1}$ come in pairs, we deduce that nonzero
accelerations are in pairs $\omega,-\omega$. Hence the acceleration vector
must be in the form

$(\omega_{1},0,-\omega_{1}),$ (1.1.2)

i.e., essentially has one variable. We find that, contrary to the
$\mathrm{SL}(2,\mathbb{R})$ case, the acceleration determines the dynamics of
analytic one-frequency ${\rm SO}(3,\mathbb{R})$ cocycles.

######  Theorem A.

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ and $A\in
C^{\omega}(\mathbb{T},{\rm SO}(3,\mathbb{R}))$, then we have the following:

  1. 1.

$\omega_{1}=0$ if and only if $(\alpha,A)$ is $C^{\omega}$-almost reducible,
i.e., the closure of the analytic conjugate class of $(\alpha,A)$ contains a
constant cocycle.

  2. 2.

Assuming additionally that $\alpha\in\mathrm{RDC}$, if $\omega_{1}\neq 0$,
then $(\alpha,A)$ is $C^{\omega}$-conjugate to

$(\alpha,\exp\begin{pmatrix}0&\omega_{1}x+c&0\\\ -\omega_{1}x-c&0&0\\\
0&0&0\end{pmatrix}),$

for some $c\in\mathbb{R}$.

We say $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ satisfies a Diophantine
condition DC($\kappa,\tau$) if $\kappa>0$ and $\tau>0$, where

$||q\alpha-p||_{\mathbb{Z}}>\frac{\kappa}{|q|^{\tau}},~{}(p,q)\in\mathbb{Z}^{2},~{}q\neq
0,$

where $||\cdot||_{\mathbb{Z}}$ represents the distance to the nearest integer
for a real number. Furthermore, we say that
$\alpha\in\mathbb{R}\backslash\mathbb{Q}$ is recurrent Diophantine if there
exist some $\gamma>0,\tau>1$ such that
$G^{n}(\alpha)\in\mathrm{DC}(\gamma,\tau)$ for infinitely many $n$, where
$G:(0,1)\rightarrow(0,1)$ is the Gauss map defined by $G(x)=\\{\frac{1}{x}\\}$
and $\\{x\\}$ denotes the fractional part of $x$. Let $\mathrm{RDC}$ be the
set of recurrent Diophantine numbers, which has a full Lebesgue measure. We
point out that the condition $\alpha\in\mathrm{RDC}$ is used for technical
reason. It would be interesting to generalize the result to all
$\alpha\in\mathbb{R}\backslash\mathbb{Q}$.

Theorem A provides a comprehensive understanding of analytic one-frequency
$\mathrm{SO}(3,\mathbb{R})$-cocycles. This theorem is particularly interesting
for several reasons. Firstly, Theorem A (1), generalizes Avila’s well-known
Almost Reducibility Conjecture (ARC) to high-dimensional compact groups. The
ARC states that if $(\alpha,A)\in\mathbb{R}\backslash\mathbb{Q}\times
C^{\omega}$ $(\mathbb{T},{\rm SL}(2,\mathbb{R}))$ is subcritical, then it is
analytically almost reducible. This conjecture has important implications for
both dynamical and spectral aspects, as highlighted in various works [2, 4, 8,
9, 13, 27, 45]. Notably, our proof of the generalized ARC in Theorem A is
completely different from the proof in the case of $\mathrm{SL}(2,\mathbb{R})$
[2, 3]. Our method is applicable to compact group, demonstrating that ARC
holds for general compact groups. For the sake of conciseness, we present the
result in the case of $\mathrm{SO}(3,\mathbb{R})$.

Moreover, Theorem A provides classification for one-frequency analytic
$\mathrm{SO}(3,\mathbb{R})$-cocycle: all cocycles can be classified into
almost reducible ones and non-almost-reducible ones, since the normal form
cocycle in Theorem A (2) is obviously not almost-reducible. Furthermore,
Theorem A (1) states that almost reducibility holds for all irrational
frequency (not merely $\mathrm{RDC}$ as in [25, 39]), which is a completely
new result.

Finally, we would like to mention that for the case $\omega_{1}=0$, the result
should hold for general compact groups. However, when $\omega_{1}\neq 0$, the
rigidity result, as demonstrated in Theorem A (2), can only be expected in a
weaken version (achieving by a sequence of conjugations) for general compact
groups. This is due to extra difficulty solving cohomological equation,
arising in the possibility of the matrix form of dynamical degree being
degenerate and having a large centralizer. One can consult [47] for more
discussions.

###  1.2 Dynamical Degree

As mentioned earlier, for $G=\mathrm{SO}(2,\mathbb{R})$ or
$\mathrm{SL}(2,\mathbb{R})$, where the fundamental group of $G$ is nontrivial,
the topological degree plays a significant role in dynamical classification.
However, for $G=\mathrm{SO}(s,\mathbb{R})$, $s\geq 3$, the fundamental group
is $\mathbb{Z}/2\mathbb{Z}$, rendering the topological degree less applicable.
In contrast, another degree called the "dynamical degree" can be defined and
utilized. The concept of dynamical degree was initially proposed by Krikorian
[39] and formally introduced by Frączek [25] for $\mathrm{SU}(2)$-cocycles. It
can be extended to any compact cocycles and general cocycles $L^{2}$-conjugate
to rotations [12, 34, 47].

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ and $A\in
C^{1}(\mathbb{T},\text{SO}(s,\mathbb{R}))$, $s\geq 3$, $\mathcal{A}_{\alpha}$
be the operator defined as following:

$\mathcal{A}_{\alpha}:L^{2}(\mathbb{T},\mathfrak{so}(s,\mathbb{R}))\longleftrightarrow
L^{2}(\mathbb{T},\mathfrak{so}(s,\mathbb{R})),\quad Y\mapsto
A^{-1}(\cdot)Y(\cdot+\alpha)A(\cdot),$

and $\mathcal{A}_{\alpha}^{j}$ represent its $j$-th iteration. It is known
that $\mathcal{A}_{\alpha}$ is a unitary operator. According to von Neumann’s
ergodic theorem, there exists an $\mathcal{A}_{\alpha}$-invariant $D\in
L^{2}(\mathbb{T},\mathfrak{so}(s,\mathbb{R}))$, such that the following
convergence holds:

$\displaystyle\frac{1}{n}A_{n}(\cdot)^{-1}\partial
A_{n}(\cdot)=\frac{1}{n}\sum_{j=0}^{n-1}\mathcal{A}_{\alpha}^{j}(A(\cdot)^{-1}\partial
A(\cdot))\xrightarrow{L^{2}}D,$ (1.2.1)

and $A(x)^{-1}D(x+\alpha)A(x)=D(x)$ almost everywhere. We call any matrix $D$
in the algebraic conjugate class the dynamical degree of $(\alpha,A)$. The
dynamical degree is invariant under $C^{1}$-conjugation. Since
$D\in\mathfrak{so}(s,\mathbb{R})$, there exist $Q\in\text{SU}(s)$ and
$d_{1},\cdots,d_{s}\in\mathbb{R}$ such that $D$ is conjugate to the diagonal
form

$D=Q\cdot 2\pi i\text{diag}\left(d_{1},\cdots,d_{s}\right)\cdot Q^{-1}.$

Furthermore, notice that conjugation of $\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\in\mathrm{SO}(2,\mathbb{R})$ on diagonal matrix acts like a
transposition on diagonal elements, we can properly order the diagonal entries
of $D$ such that

$d_{1}\geq\cdots\geq d_{s}.$

We thus define the vector of dynamical degree to be
$\vec{d}=(d_{1},\cdots,d_{s}).$

We note that the dynamical degree is quantized ($d_{k}\in\mathbb{Z}$,
$k=1,\cdots,s$), as proven in [12, 47]. Since acceleration is also quantized
[4, 7], a natural question arises as to whether there is any relation between
them. In fact, for analytic $\text{SO}(s,\mathbb{R})$-cocycles, they are the
same.

######  Theorem B.

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ and $A\in
C^{\omega}(\mathbb{T},\mathrm{SO}(s,\mathbb{R}))$. The acceleration vector of
$(\alpha,A)$ is the same as the vector of dynamical degree of $(\alpha,A)$.

As the vector of dynamical degree has integer entries, we deduce the following
result.

######  Corollary 1.1.

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ and $A\in
C^{\omega}(\mathbb{T},\mathrm{SO}(s,\mathbb{R}))$. The accelerations of
$(\alpha,A)$ are all integers.

We recall [7, Theorem 1.4]: when $A\in
C^{\omega}(\mathbb{T},\mathcal{L}(\mathbb{C}^{s},\mathbb{C}^{s}))$, for
$k=1,\cdots,s$, there exists an integer $1\leq l\leq s$, such that
$lw_{k}\in\mathbb{Z}$. Our improvement is essentially due to the compactness
of $\text{SO}(s,\mathbb{R})$: eigenvalues always have algebraic multiplicity
$1$, which leads to $l=1$.

Consequently, the dynamical degree can be used to classify analytic one-
frequency quasi-periodic $\text{SO}(3,\mathbb{R})$-cocycles. In particular, we
can establish the relation between acceleration, dynamical degree, and almost
reducibility.

######  Corollary 1.2.

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ and $A\in
C^{\omega}(\mathbb{T},\mathrm{SO}(3,\mathbb{R}))$. Then the following
statements are equivalent:

  1. 1.

The dynamical degree is zero,

  2. 2.

The largest acceleration $\omega_{1}$ is zero,

  3. 3.

$(\alpha,A)$ is $C^{\omega}$ almost reducible.

###  1.3 More Histories and Comments

A quasi-periodic linear system on a Lie group $G$ (referred to as a linear
system for short) is an ordinary differential equation on
$\mathbb{T}^{d}\times G$ of the form

$\displaystyle\begin{cases}\dot{x}=\tilde{A}(\theta)x,\\\
\dot{\theta}=\omega,\end{cases}$ (1.3.1)

where $\tilde{A}:\mathbb{T}^{d}\rightarrow\mathfrak{g}$ ($\mathfrak{g}$ is the
Lie algebra of $G$), and $\omega\in\mathbb{T}^{d}$ is rationally independent.
Poincaré time-$1$ map of (1.3.1) is in fact a quasi-periodic cocycle on $G$.
Naturally, conjugation and in particular almost reducibility results of the
corresponding cocycle is closely related to the ones of the original linear
system [54], cf. section 3.1.

Let the frequency $\alpha\in\mathbb{T}^{d}$ be Diophantine and $A\in
C^{\omega}(\mathbb{T}^{d},\mathrm{SL}(2,\mathbb{R}))$. In 1975, Dinaburg-Sinai
[18] used the KAM approach to prove that for a typical one-parameter family of
cocycles close to a constant, reducible cocycles are of positive measure.
Subsequently, Eliasson [19] made a significant breakthrough by proving, under
the same assumption, that reducible cocycles are of full measure, through an
essential improvement of the KAM approach. By adapting and improving
Eliasson’s approach, Krikorian [37] extended the full-measure reducibility
result in [19] to the cases of semi-simple compact Lie groups, and was further
generalized to $\mathrm{GL}(s,\mathbb{R})$ cocycles quite recently by Wang-Xu-
You-Zhou [50]. All of these results hold for smooth or finitely differentiable
cases, regardless of whether we consider linear systems or cocycles [1, 16,
22, 38].

However, the Diophantine condition on frequencies in local problems can be
relaxed when considering one-frequency cocycles or two-frequency linear
systems on $\mathrm{SL}(2,\mathbb{R})$, as shown in [2, 6, 31]. Both of these
articles developed non-standard KAM techniques to deal with conjugation
problems. Although the method developed in [31] only applies to linear
systems, the result still holds for cocycles due to the local embedding
theorem [54].

One may inquire whether any conclusions can be derived regarding conjugation
for cocycles or linear systems that are not necessarily close to constants
(referred to as the global case). In the case of one-frequency cocycles,
global results can be anticipated. In [40], Krikorian proved a profound result
stating that for a recurrent Diophantine frequency, $C^{\infty}$-reducible
smooth cocycles valued in $\mathrm{SU}(2)$ are $C^{\infty}$-dense. Moreover,
Krikorian proved that for a recurrent Diophantine frequency, non-reducible
smooth cocycles on $\mathrm{SU}(2)$ can be divided into countable conjugation
classes. In fact, the proof in [40] indicates the existence of a quantified
conjugation-invariant quantity, which is precised later by Fraczek [25]. In
the case of $\mathrm{SL}(2,\mathbb{R})$, Avila-Krikorian [10] proved that
given recurrent Diophantine frequency, for almost every energy, the
corresponding Schrödinger cocycle is either $C^{\omega}$-reducible or non-
uniformly hyperbolic, based on Kotani theory [44, 48] and the renormalization
scheme introduced in [40].

Furthermore, several interesting results have been established in [34, 35, 36,
12], which include topics such as degree, reducibility, almost-reducibility,
non-reducibility, and spectrum of one-frequency cocycles on semi-simple
compact groups. Avila’s global theory and accelerations in [4, 7] for analytic
one-frequency cocycles provide new insights into understanding global
conjugation problems.

It is important to note that most of the conclusions about cocycles on semi-
simple compact groups, whether local or global, require Diophantine or
recurrent Diophantine conditions. In this paper, we aim to examine the
relations among accelerations, dynamical degree, and almost-reducibility for
analytic cocycles over any irrational frequency. We firstly recall the
convergence of renormalization proved in[12, 47]. Then by proving the
preservation of acceleration along renormalization and using the explicit form
of limit of renormalization, we deduce Theorem B concerning coincidence of
acceleration vector and vector of dynamical degree. As for Theorem A, if
$\omega_{1}=0$, we conclude by showing a local almost-reducibility result over
any irrational frequency. We prove such result using the KAM scheme developed
by Hou-You [31] and the local embedding theorem [54], which are applicable
only to the analytic case. On the other hand, if $\omega_{1}\neq 0$, we prove
conjugation result through an adapted KAM scheme developed by Krikorian [39]
for $\mathrm{SU}(2)$-cocycles. A technical point in the function of this KAM
scheme is the estimates on length, which were developed in [39, 34] for
compact cocycles. Since accelerations can only be defined for analytic
cocycles, Corollary 1.2 holds only for the analytic case.

Finally, let us mention that the dynamics of cocycles has received significant
attention from researchers, with a focus on the conjugation problems and their
classifications. These questions are not only important in the field of
dynamical systems, but also in mathematical physics, where they have wider
applications in the spectral theory of Schrödinger operators [2, 4, 8, 9, 13,
27, 45, 53].

## Acknowledgement

X. Hou is partially supported by NNSF of China (Grant 12071083) and Funds for
Distinguished Youths of Hubei Province of China (2019CFA680). Y. Pan is
supported by ERC Advanced Grant (#885707). Q. Zhou is partially supported by
National Key R&D Program of China (2020YFA0713300), NSFC grant (12071232) and
Nankai Zhide Foundation.

##  2 Preliminary

###  2.1 Lie group and Lie algebra

Let

$\mathrm{SO}(s,\mathbb{R}):=\\{X\in\mathrm{M}(s,\mathbb{R})~{}:~{}X^{T}X=XX^{T}=I\\},$

$\mathfrak{so}(s,\mathbb{R})$ be the Lie algebra of
$\mathrm{SO}(s,\mathbb{R})$, namely

$\mathfrak{so}(s,\mathbb{R}):=\\{X\in\mathrm{M}(s,\mathbb{R})~{}:~{}X=-X^{T}\\}.$

For a matrix $D\in\mathfrak{so}(s,\mathbb{R})$, its centralizer
$\mathfrak{z}(D)$ is defined to be

$\mathfrak{z}(D)=\\{C\in\mathfrak{so}(s,\mathbb{R})~{}|~{}DC=CD\\}.$

In this article, we mainly work on $\mathrm{SO}(3,\mathbb{R})$ and
$\mathfrak{so}(3,\mathbb{R})$. As for the Lie algebra
$\mathfrak{so}(3,\mathbb{R})$, it has a basis

$J_{1}=\begin{pmatrix}0&1&0\\\ -1&0&0\\\
0&0&0\end{pmatrix},J_{2}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
-1&0&0\end{pmatrix},J_{3}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
-1&0&0\end{pmatrix},$

satisfying

$[J_{1},J_{2}]=J_{3},\quad[J_{2},J_{3}]=J_{1},\quad[J_{3},J_{1}]=J_{2}.$

Then we have the following observation.

######  Lemma 2.1.

For $D\in\mathfrak{so}(3,\mathbb{R})\backslash\\{0\\}$, its centralizer is the
$1$-dimensional linear subspace generated by $D$.

Besides, it is easy to see that for $A\in\mathfrak{so}(3,\mathbb{R})$, it has
eigenvalue $\\{||A||i,-||A||i,0\\}$ and there exists
$B\in\mathrm{SO}(3,\mathbb{R})$ such that

$B^{T}AB=||A||J_{1}.$

###  2.2 Continued fraction expansion

Define as usual for $0<\alpha<1$,

$a_{0}=0,~{}~{}\alpha_{0}=\alpha,\quad q_{0}=1,\quad p_{0}=0,\quad
q_{-1}=0,\quad p_{-1}=1,$

and inductively for $n\geq 1$,

$a_{n}=[\alpha_{n-1}^{-1}],~{}~{}\alpha_{n}=\alpha_{n-1}^{-1}-a_{n}=G(\alpha_{n-1})=\\{\frac{1}{\alpha_{n-1}}\\},$
$p_{n}=q_{n}p_{n-1}+p_{n-2},\quad q_{n}=a_{n}q_{n-1}+q_{n-2},$ (2.2.1)

where $[\cdot]$ denotes the integer part, $\\{\cdot\\}$ denotes the fractional
part and $G$ is the Gauss map $G(x)=\\{x^{-1}\\}$. Let

$\beta_{n}=\Pi_{j=0}^{n}\alpha_{j},\quad U(x)=\left(\begin{matrix}0&1\\\
1&-[x^{-1}]\end{matrix}\right).$

Define

$Q_{0}=I,\quad Q_{n}=U(\alpha_{n-1})\cdots U(\alpha_{0}).$

Then (2.2.1) shows that

$Q_{n}=(-1)^{n}\left(\begin{matrix}q_{n}&p_{n}\\\
q_{n-1}&p_{n-1}\end{matrix}\right).$

And we have

$\beta_{n}=(-1)^{n}(q_{n}\alpha-p_{n})=\frac{1}{q_{n+1}+\alpha_{n+1}q_{n}},$
$\frac{1}{q_{n+1}+q_{n}}<\beta_{n}<\frac{1}{q_{n+1}}.$

###  2.3 Renormalization of actions and cocycles

$\mathbb{Z}^{2}$-actions and renormalization of actions will be a fundamental
tool in this paper. We recall here, following [10], $\mathbb{Z}^{2}$-action
and the scheme of renormalization introduced in [40]. In particular, we focus
on analytic case for $\mathrm{SO}(s,\mathbb{R})$, $s\geq 3$.

Denote by $\Omega^{\omega}=\mathbb{R}\times
C^{\omega}(\mathbb{R},\mathrm{SO}(s,\mathbb{R}))$ the group composed of skew-
product diffeomorphisms
$(\alpha,A):\mathbb{R}\times\mathrm{SO}(s,\mathbb{R})\to\mathbb{R}\times\mathrm{SO}(s,\mathbb{R}),$

$(\alpha,A)(x,v)=(x+\alpha,A(x)v),$ (2.3.1)

with the composition being the group operation. An analytic fibered
$\mathbb{Z}^{2}$-action is a homomorphism
$\Phi:\mathbb{Z}^{2}\to\Omega^{\omega}$. We use $\Lambda^{\omega}$ to denote
the space of analytic fibered $\mathbb{Z}^{2}$-actions, endowed with the
pointwise topology.

For two elements in $\Omega^{r}$, $(\alpha_{1},A_{1}),(\alpha_{2},A_{2})$,
they form a commuting pair, denoted by $\begin{pmatrix}(\alpha_{1},A_{1})\\\
(\alpha_{2},A_{2})\end{pmatrix}$, if they are commuting with each other, i.e.,

$A_{1}(x+\alpha_{2})A_{2}(x)=A_{2}(x+\alpha_{1})A_{1}(x).$

As $\Phi(1,0)$ and $\Phi(0,1)$ determine $\Phi$ and commute with each other by
definition, they form a commuting pair. In below, we will not distinguish an
$\mathbb{Z}^{2}$-action $\Phi$ and the corresponding commuting pair
$(\begin{smallmatrix}\Phi(1,0)\\\ \Phi(0,1)\end{smallmatrix})$.

For any $\Phi\in\Lambda^{\omega}$, we define
$\gamma^{\Phi}_{n,m}:=\Pi_{1}\circ\Phi(n,m)\in\mathbb{R}$ and
$A^{\Phi}_{n,m}:=\Pi_{2}\circ\Phi(n,m)\in
C^{\omega}(\mathbb{R},\mathrm{SO}(s,\mathbb{R}))$ for all
$(n,m)\in\mathbb{Z}^{2}$, where $\Pi_{1}:\mathbb{R}\times
C^{\omega}(\mathbb{R},\mathrm{SO}(s,\mathbb{R}))\to\mathbb{R}$ and
$\Pi_{2}:\mathbb{R}\times C^{\omega}(\mathbb{R},\mathrm{SO}(s,\mathbb{R}))\to
C^{\omega}(\mathbb{R},\mathrm{SO}(s,\mathbb{R}))$ are coordinate projections.
Let $\Lambda_{0}^{\omega}$ be the set of $\Phi\in\Lambda^{\omega}$ with
$\gamma_{1,0}^{\Phi}=1$.

Notice that $C^{\omega}(\mathbb{R},\mathrm{SO}(s,\mathbb{R}))$ acts on
$\Omega^{\omega}$ by
$\mathrm{Conj}_{B}(\alpha,A(\cdot))=(\alpha,B(\cdot+\alpha)A(\cdot)B(\cdot)^{-1}).$
This action extends to an action, still denoted $\mathrm{Conj}_{B}$, on
$\Lambda^{\omega}$. For $\Phi\in\Lambda^{\omega}$, the
$\mathrm{Conj}_{B}(\Phi)$ is given by

$\mathrm{Conj}_{B}(\Phi)(n,m)=(\gamma^{\Phi}_{n,m},B(\cdot+\gamma_{n,m}^{\Phi})A_{n,m}^{\Phi}(\cdot)B(\cdot)^{-1}),\quad
n,m\in\mathbb{Z}.$

We say that $\Phi,\Phi^{\prime}\in\Lambda^{\omega}$ are $C^{\omega}$-conjugate
via $B\in C^{\omega}(\mathbb{R},\mathrm{SO}(s,\mathbb{R}))$ if
$\Phi^{\prime}=\mathrm{Conj}_{B}(\Phi)$ holds.

We say that a $\mathbb{Z}^{2}$-action $\Phi$ is normalized if
$\Phi(1,0)=(1,I)$. In this case, $\Phi(0,1)=(\alpha,A)$ can be viewed as an
analytic cocycle, since $A$ is automatically $1$-periodic. Conversely, given
an analytic cocycle $(\alpha,A)$, we associate a normalized action
$\Phi_{\alpha,A}$ by setting
$\Phi_{\alpha,A}(1,0)=(1,I),~{}~{}\Phi_{\alpha,A}(0,1)=(\alpha,A).$

We cite the normalizing lemma from [10], whose proof remains true for
$\mathbb{Z}^{2}$-actions valued in compact groups. We will provide a
quantitative version of it in section 5.2.

######  Lemma 2.2.

For any $\Phi\in\Lambda_{0}^{\omega}$, $\Phi$ is $C^{\omega}$-conjugate to a
normalized action.

Let $\Phi\in\Lambda_{0}^{\omega}$, $B\in
C^{\omega}(\mathbb{R},\mathrm{SO}(s,\mathbb{R}))$ such that
$\mathrm{Conj}_{B}(\Phi)$ is normalized. Then we call $B$ a normalizing map of
$\Phi$. The choice of $B$ may not be unique.

To define renormalization, we start with $\mathbb{Z}^{2}$-actions. We
introduce the following maps.

  1. 1.

Let $\lambda\neq 0$. Define rescaling
$M_{\lambda}:\Lambda^{\omega}\to\Lambda^{\omega}$ by

$M_{\lambda}(\Phi)(n,m):=(\lambda^{-1}\gamma_{n,m}^{\Phi},A^{\Phi}_{n,m}(\lambda\cdot)),\quad\Phi\in\Lambda^{\omega}.$
(2.3.2)

  2. 2.

Let $U\in\mathrm{GL}(2,\mathbb{Z})$. Define base change
$U:\Lambda^{\omega}\to\Lambda^{\omega}$ by

$U(\Phi)(n,m)=\Phi(n^{\prime},m^{\prime}),~{}~{}\binom{n^{\prime}}{m^{\prime}}=U\cdot\binom{n}{m},\quad\Phi\in\Lambda^{\omega}.$
(2.3.3)

We point out that these two operations are pairwise commutative. Besides, $U$
commute with $\mathrm{Conj}_{B}$ while
$M_{\lambda}\circ\mathrm{Conj}_{B}=\mathrm{Conj}_{B(\lambda\cdot)}\circ
M_{\lambda}.$

For $\mathbb{Z}^{2}$-action $\Phi\in\Lambda^{\omega}$, $n\in\mathbb{N}$, we
define the $n$-th renormalization of $\Phi$ around $0$ to be

$\mathcal{R}^{n}(\Phi):=M_{\beta_{n-1}}\circ{Q_{n}}(\Phi).$ (2.3.4)

Based on the renormalization of $\mathbb{Z}^{2}$-action, we introduce
renormalization of cocycles. For a $C^{r}$ cocycle $(\alpha,A)$, we consider
renormlization on the related $\mathbb{Z}^{2}$-action $\Phi_{\alpha,A}$. We
can also express renormalization explicitly by commuting pair:

$\mathcal{R}^{n}(\Phi_{\alpha,A})=M_{\beta_{n-1}}\circ(-1)^{n}\begin{pmatrix}p_{n-1}&-p_{n}\\\
-q_{n-1}&q_{n}\end{pmatrix}\begin{pmatrix}(1,I)\\\
(\alpha,A)\end{pmatrix},\quad n\in\mathbb{N}.$

As $\mathcal{R}^{n}(\Phi_{\alpha,A})\in\Lambda_{0}^{\omega}$, by Lemma 2.2,
there exists $B\in C^{\omega}(\mathbb{R},\mathrm{SO}(s,\mathbb{R}))$ such that
$\mathrm{Conj}_{B}\mathcal{R}^{n}(\Phi_{\alpha,A})$ is normalized. We denote
the commuting pair of this normalized $\mathbb{Z}^{2}$ action by
$\begin{pmatrix}(1,I)\\\ (\alpha_{n},\tilde{A}^{(n)})\end{pmatrix}$. Since
normalizing map is not unique, $\tilde{A}^{(n)}$ is not unique either. We call
$(\alpha_{n},\tilde{A}^{(n)})$ a representative of the $n$-th renormalization
of $(\alpha,A)$.

A fundamental observation is that the conjugation relation is invariant under
renormalizations. More precisely, we have the following result. The proof is
relatively direct by definition, we refer to [47, Lemma 4.4.1, Lemma 4.4.2]
for example.

######  Lemma 2.3.

1\. Representative of renormalization is unique up to conjugation.
2\. Two cocycles are conjugate to each other if and only if they have
conjugate representatives of renormalization.

Now we talk about the convergence of renormalization. As compact cocycle is a
special case of cocycles $L^{2}$-conjugation to rotation, by [12, 47], we have
the following results.

######  Theorem 2.1.

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, $A\in
C^{\omega}_{h}(\mathbb{T},\mathrm{SO}(s,\mathbb{R}))$. There exist
$D\in\mathfrak{so}(s,\mathbb{R})$ satisfying $\exp(D)=I$, a sequence of
representatives of renormalization $(\alpha_{n},\tilde{A}^{(n)})$,
$C_{n}\in\mathfrak{z}(D)$, $\varphi_{n}\in
C_{h}^{\omega}(\mathbb{T},\mathfrak{so}(s,\mathbb{R}))$ such that

$\tilde{A}^{(n)}(x)=\exp(\varphi_{n}(x))\exp((-1)^{n}Dx+C_{n}),$ (2.3.5)

satisfying the estimate

$\lim_{n\to\infty}||\varphi_{n}||_{h}=0.$

This means representatives of renormalization $(\alpha_{n},\tilde{A}^{(n)})$
converges to a precise normal form. This kind of normal form result was
firstly stated for $\mathrm{SU}(2)$-cocycles in the $C^{\infty}$-topology [39,
25], and then for $\mathrm{SL}(2,\mathbb{R})$-cocycles in the $C^{\infty}$ and
$C^{\omega}$-topology [10, 11]. It is also generalized to quasi-periodic
coycles valued in semi-compact Lie groups in the $C^{\infty}$-topology [34].

##  3 Local results I. close to constant

To prove the conjugation result in Theorem A, by the preservation of
conjugation along renormalization, it can be achieved by conjugating
renormalizations. Moreover, by the convergence of renormalization given in
Theorem 2.1, we will fall into a neighborhood of some normal form after deep
enough renormalization. Therefore we only need to study conjugation in local
case, namely close to some normal form $(\alpha,\exp(Dx+C))$, where
$\exp(D)=I$, $c\in\mathfrak{z}(D)$. We will deal with $D=0$ in this section
and $D\neq 0$ in the next section.

In this section, we will prove that any one-frequency
$\mathrm{SO}(3,\mathbb{R})$-cocycle close to constant is $C^{\omega}$-almost-
reducible.

We start with introducing some notations. Let $h>0$, $d\in\mathbb{N}^{*}$. We
define the strip

$\mathbb{T}_{h}^{d}:=\\{\theta=(\theta_{1},\cdots,\theta_{d})\in\mathbb{C}^{d}\,\big{|}\,|\Im\theta_{1}|+\cdots+|\Im\theta_{d}|\leq
h\\}/\mathbb{Z}^{d}.$

We denote by $C_{h}^{\omega}(\mathbb{T}^{d},*)$ the space of all maps
$F:\mathbb{T}^{d}\to*$ admitting an analytic extension on
$\mathbb{T}_{h}^{d}$, equipped with the norm

$\|F\|_{h}:=\sup_{\theta\in\mathbb{T}^{d}_{h}}\|F(\theta)\|,$

where $*$ could be $\mathbb{R}$, $\mathbb{C}$, $\mathrm{SO}(3,\mathbb{R})$ or
$\mathfrak{so}(3,\mathbb{R})$. Then
$C^{\omega}(\mathbb{T}^{d},*):=\bigcup_{h>0}C_{h}^{\omega}(\mathbb{T}^{d},*)$
becomes the set of all analytic $*$-valued function on $\mathbb{T}^{d}$.

Our main result in this section is the following.

######  Theorem 3.1.

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, $h>0$, $L\geq 1$,
$C\in\mathfrak{so}(3,\mathbb{R})$. There exists $\delta=\delta(h,L)$, such
that for any $A\in C_{h}^{\omega}(\mathbb{T},\mathrm{SO}(3,\mathbb{R}))$, if
$||A-\exp(C)||_{h}<\delta$, then $(\alpha,A)$ is $C^{\omega}$-almost-
reducible, in the sense that, there exist $h_{n}>0$, $B_{n}\in
C_{h_{n}}^{\omega}(\mathbb{T},\mathrm{SO}(3,\mathbb{R}))$,
$n\in\mathbb{N}^{*}$, such that

$\lim_{n\rightarrow\infty}\|A_{n}-\exp(C_{n})\|_{h_{n}}\|B_{n}\|_{h_{n}}^{L}=0,$
(3.0.1)

for $A_{n}:=B_{n}(\cdot+\alpha)AB_{n}^{-1}$ and some
$C_{n}\in\mathfrak{so}(3,\mathbb{R})$, $n\in\mathbb{N}^{*}$.

######  Remark 3.1.

The perturbation $\delta=\delta(h)$ does not depend on the frequency $\alpha$.
Thus the result is of semi-local nature [23].

For technical reasons, we will not prove Theorem 3.1 directly. Instead we
consider almost reducibility of linear systems. We will firstly formulate a
parallel theorem to Theorem 3.1 for linear systems and show that one can
deduce Theorem 3.1 from the parallel result in section 3.1. Then we will work
on almost reducibility of linear systems in the rest of this section.

###  3.1 Change to almost reducibility of linear systems

Let $\omega\in\mathbb{R}^{2}$, $A\in
C^{\omega}(\mathbb{T}^{2},\mathrm{SO}(3,\mathbb{R}))$. Consider the linear
system

$\begin{cases}\dot{x}=A(\theta)x\\\ \dot{\theta}=\omega.\end{cases}$ (3.1.1)

We say that $B\in C^{\omega}(\mathbb{T}^{2},\mathrm{SO}(3,\mathbb{R}))$
conjugates (3.1.1) to another linear system

$\begin{cases}\dot{x}=A_{+}(\theta)x\\\ \dot{\theta}=\omega,\end{cases}$
(3.1.2)

for some $A_{+}\in C^{\omega}(\mathbb{T}^{2},\mathrm{SO}(3,\mathbb{R}))$, if

$\partial_{\omega}B(\theta)=B(\theta)A(\theta)-A_{+}(\theta)B(\theta),$

where

$\partial_{\omega}B(\theta_{1},\theta_{2}):=\omega_{1}\frac{\partial}{\partial\theta_{1}}B(\theta_{1},\theta_{2})+\omega_{2}\frac{\partial}{\partial\theta_{2}}B(\theta_{1},\theta_{2}),\quad\text{for
}\omega=(\omega_{1},\omega_{2}).$

Denote by $\Phi\in
C^{\omega}(\mathbb{R}\times\mathbb{T}^{2},\mathrm{SO}(3,\mathbb{R}))$ the
basic solution (which is in fact unique) of (3.1.1) that satisfies

$\displaystyle\frac{d}{dt}\Phi(t,\theta)=A(\theta+t\omega)\Phi(t,\theta),\quad\Phi(0,\theta)=I,\quad
t\in\mathbb{R},\,\theta\in\mathbb{T}^{2}.$

Let $\Phi_{+}$ be the basic solution of (3.1.2). Then $B$ conjugating (3.1.1)
to (3.1.2) is equivalent to

$B(\theta+t\omega)\Phi(t,\theta)B(0,\theta)^{-1}=\Phi_{+}(t,\theta).$ (3.1.3)

We claim the following almost reducibility result for linear system, whose
proof will be given later.

######  Theorem 3.2.

Let $\alpha\in(0,1)\backslash\mathbb{Q}$,
$\omega=(\alpha,1)\in\mathbb{R}^{2}$, $C\in\mathfrak{so}(3,\mathbb{R})$. For
$h>0$ and $L\geq 1$, there exists $\delta=\delta(h,L)$ such that for any $F\in
C^{\omega}_{h}(\mathbb{T}^{2},\mathfrak{so}(3,\mathbb{R}))$ satisfying
$||F||_{h}<\delta$, the system

$\begin{cases}\dot{x}=(C+F(\theta))x\\\ \dot{\theta}=\omega\end{cases}$
(3.1.4)

is $C^{\omega}$-almost-reducible, in the sense that, there exist $h_{n}>0$,
$B_{n}\in C_{h_{n}}^{\omega}(\mathbb{T}^{2},\mathrm{SO}(3,\mathbb{R}))$,
$n\in\mathbb{N}^{*}$, such that $B_{n}$ conjugates (3.1.4) to

$\begin{cases}\dot{x}=(C_{n}+F_{n}(\theta))x\\\
\dot{\theta}=\omega,\end{cases}$ (3.1.5)

for some $C_{n}\in\mathfrak{so}(3,\mathbb{R}),F_{n}\in
C^{\omega}_{h_{n}}(\mathbb{T}^{2},\mathfrak{so}(3,\mathbb{R}))$,
$n\in\mathbb{N}^{*}$, and the following estimate holds

$\lim_{n\rightarrow\infty}\|F_{n}\|_{h_{n}}\|B_{n}\|_{h_{n}}^{L}=0.$ (3.1.6)

To show that we can deduce Theorem 3.1 from Theorem 3.2 directly, we need to
use in addition the following local embedding theorem proved in [54]. It was
proved for any Lie group $G$, and we will restrict on the case
$\mathrm{SO}(3,\mathbb{R})$.

######  Theorem 3.3 ([54], Local embedding theorem).

Let $h>0$, $d\in\mathbb{N}^{*}$, $\mu\in\mathbb{T}^{d-1}$ with $(\mu,1)$
rationally independent, $C\in\mathfrak{so}(3,\mathbb{R})$, $H\in
C^{\omega}_{h}(\mathbb{T}^{d-1},\mathfrak{so}(3,\mathbb{R}))$. There exist
$\epsilon=\epsilon(C,h,|\mu|)>0$, $c=c(C,h,|\mu|)>0$ such that the quasi-
periodic cocycle $(\mu,e^{C}e^{H(\cdot)})$ can be analytically embedded into a
quasi-periodic linear system provided that $\|H\|_{h}<\epsilon$. More
precisely, there exists $F\in
C^{\omega}_{h/(1+|\mu|)}(\mathbb{T}^{d},\mathfrak{so}(3,\mathbb{R}))$ such
that $(\mu,e^{C}e^{H(\cdot)})$ is the Poincaré map of

$\displaystyle\left\\{\begin{array}[]{l}\dot{x}=(C+F(\theta))x\\\
\dot{\theta}=(\mu,1),\end{array}\right.$

and the estimate holds

$\|F\|_{h/(1+|\mu|)}\leq c\|H\|_{h}.$

###### Proof of Theorem 3.1.

Assume that $\alpha\in(0,1)$ without lose of generality. For $A\in
C_{h}^{\omega}(\mathbb{T},\mathrm{SO}(3,\mathbb{R}))$ with $||A-\exp(C)||_{h}$
small for some constant $C\in\mathfrak{so}(3,\mathbb{R})$, by Theorem 3.3,
there exists $F\in
C^{\omega}_{h/(1+\alpha)}(\mathbb{T}^{2},\mathfrak{so}(3,\mathbb{R}))$ such
that $(\alpha,A)$ is the Poincaré map of the linear system

$\displaystyle\left\\{\begin{array}[]{l}\dot{x}=(C+F(\theta))x\\\
\dot{\theta}=(\alpha,1),\end{array}\right.$ (3.1.10)

and the estimate holds

$\|F\|_{h/(1+\alpha)}\leq c||A-\exp(C)||_{h}.$

As long as $||A-\exp(C)||_{h}$ is small enough, by Theorem 3.2, (3.1.10) is
$C^{\omega}$-almost-reducible, i.e., there exist $h_{n}>0$, $B_{n}\in
C_{h_{n}}^{\omega}(\mathbb{T}^{2},\mathrm{SO}(3,\mathbb{R}))$,
$n\in\mathbb{N}^{*}$, such that $B_{n}$ conjugates (3.1.10) to

$\begin{cases}\dot{x}=(C_{n}+F_{n}(\theta))x\\\
\dot{\theta}=\omega,\end{cases}$ (3.1.11)

for some $C_{n}\in\mathfrak{so}(3,\mathbb{R}),F_{n}\in\mathcal{B}_{h_{n}}$,
$n\in\mathbb{N}^{*}$, and the following estimate holds

$\lim_{n\rightarrow\infty}\|F_{n}\|_{h_{n}}\|B_{n}\|_{h_{n}}^{L}=0.$ (3.1.12)

Concerning the quantitative estimates, suppose that $\Phi(t,\theta)$,
$\Phi_{n}(t,\theta)$ are the corresponding basic solution of
$\eqref{change.to.lin.s}$ and $(\ref{loc.cont.1-1.pf})$, then we have

$\Phi_{n}(t,\theta)=e^{C_{n}t}\Big{(}I+\int_{0}^{t}e^{-C_{n}s}F_{n}(\theta+s(\alpha,1))\tilde{\Phi}(s,\theta)ds\Big{)}.$
(3.1.13)

Let $g_{n}(t)=||e^{-Ct}\Phi_{n}(t,\theta)||_{h_{n}}$. We deduce from (3.1.13)
that

$g(t)\leq 1+\int_{0}^{t}\|F_{n}\|_{h_{n}}g(s)ds.$

Gronwall’s inequality therefore provides

$g(t)\leq e^{\|F_{n}\|_{h_{n}}t}.$ (3.1.14)

Back to (3.1.13), let $t=1$, $\theta=(\theta_{1},0)$, we have

$\Phi_{n}(1,\theta_{1},0)=e^{C_{n}}+\widetilde{F}_{n}(\theta_{1}),$ (3.1.15)

where $\widetilde{F}_{n}\in
C_{h_{n}}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$ is given by

$\tilde{F}_{n}(\theta_{1})=e^{C_{n}}\int_{0}^{1}e^{-C_{n}s}F_{n}(\theta_{1}+s\alpha,s)\tilde{\Phi}(s,\theta_{1},0)ds.$

Then by (3.1.14) we deduce that

$\|\widetilde{F}_{n}\|_{h_{n}}\leq\int_{0}^{1}\|F_{n}\|_{h_{n}}g(s)ds\leq
e^{\|F_{n}\|_{h_{n}}}-1\leq 2\|F_{n}\|_{h_{n}},$ (3.1.16)

where the last inequation is provided by smallness of $\|F_{n}\|_{h_{n}}$.

Since $B_{n}$ conjugates (3.1.10) to (3.1.11), we have the relation of basic
solutions

$B_{n}(\theta_{1}+t\alpha,t)\Phi(t,\theta_{1},0)B_{n}(\theta_{1},0)^{-1}=\Phi_{n}(t,\theta_{1},0).$

Let $t=1$. As $(\alpha,A)$ is the Poincaré map of (3.1.10), we deduce that

$A_{n}(\theta_{1})=B_{n}(\theta_{1}+\alpha,0)A(\theta_{1})B_{n}(\theta_{1},0)^{-1}=e^{C_{n}}+\widetilde{F}_{n}(\theta_{1}),$

where we use the fact that $B$ is $1$-periodic in both coordinates. Besides,
(3.1.16) and (3.1.12) give that

$||A_{n}-e^{C_{n}}||_{h_{n}}||B_{n}||_{h_{n}}^{L}=0.$

Hence we have completed the proof of Theorem 3.1. ∎

###  3.2 Proof of Theorem 3.2

In this subsection, we will introduce an alternative analytic space and norm
to work with. We will also introduce an iteration scheme, by applying which we
will be able to prove Theorem 3.2. The proof of the iteration scheme will be
given in the next subsection.

####  3.2.1 Spaces and Norms

To make the future estimates simpler, we introduce an alternative analytic
space and norm. For $h>0$, we define $\mathcal{B}_{h}(*)$ as the space of
functions $F\in C^{\omega}(\mathbb{T}^{2},*)$ ($*$ can be
$\mathfrak{so}(3,\mathbb{R})$, $\mathrm{SO}(3,\mathbb{R})$, $\mathbb{R}$ or
$\mathbb{C}$) satisfying
$\sum_{k\in\mathbb{Z}^{2}}||\hat{F}(k)||e^{2\pi|k|h}<\infty$, where
$|k|:=|k_{1}|+|k_{2}|$ for $k=(k_{1},k_{2})$, equipped with the norm

$||F||^{\\#}_{h}:=\sum_{k\in\mathbb{Z}^{2}}||\hat{F}(k)||e^{2\pi|k|h}<\infty.$

It follows immediately that for any
$F,G\in\mathcal{B}_{h}(\mathrm{SO}(3,\mathbb{R}))$, we have
$FG\in\mathcal{B}_{h}(\mathrm{SO}(3,\mathbb{R}))$, and

$||FH||^{\\#}_{h}\leq||F||^{\\#}_{h}||H||^{\\#}_{h}.$

Sometimes, for an analytic matrix valued function, we need to consider the
relations between its analytic norm and the analytic norms of its entry
functions. As for $\mathfrak{so}(3,\mathbb{R})$ case, we obviously have the
conclusion

$\|\sum_{s=1}^{3}f_{s}J_{s}\|^{\\#}_{h}=\left(\sum_{s=1}^{3}(\|f_{s}\|^{\\#}_{h})^{2}\right)^{\frac{1}{2}},\quad\forall
f_{1},f_{2},f_{3}\in\mathcal{B}_{h}(\mathbb{R}),\,h>0.$ (3.2.1)

For any $N>0$, we define the operators $\mathcal{T}_{N}$, $\mathcal{R}_{N}$ on
$\mathcal{B}_{h}(*)$ as

$(\mathcal{T}_{N}F)(\theta):=\sum_{k\in\mathbb{Z}^{d},|k|<N}\hat{F}(k)e^{2\pi
i<k,\theta>},\quad(\mathcal{R}_{N}F)(\theta):=\sum_{k\in\mathbb{Z}^{d},|k|\geq
N}\hat{F}(k)e^{2\pi i<k,\theta>}.$

Then $\mathcal{T}_{N}+\mathcal{R}_{N}=I$. And for any $0<h^{\prime}<h$, since
$||\hat{F}(k)||\leq||F||_{h}e^{-2\pi|k|h}$, we compute that

$\displaystyle||\mathcal{R}_{N}F||_{h^{\prime}}^{\\#}$
$\displaystyle=\sum_{k\in\mathbb{Z}^{d},|k|\geq
N}||\hat{F}(k)||e^{2\pi|k|h^{\prime}}\leq\frac{2||F||_{h}}{\min\\{1,(h-h^{\prime})^{2}\\}}e^{-2\pi
N(h-h^{\prime})}.$ (3.2.2)

The spaces $\mathcal{B}_{h}(*)$ and $C^{\omega}(\mathbb{T}^{2},*)$ are closely
related. For $h>0$, $F\in\mathcal{B}_{h}(*)$, we have

$\displaystyle\sup_{x\in\mathbb{T}^{2}_{h}}||F(x)||\leq||F||^{\\#}_{h}.$
(3.2.3)

Hence $F\in C_{h}^{\omega}(\mathbb{T}^{2},*)$. It follows that

$\mathcal{B}_{h}(*)\subseteq C_{h}^{\omega}(\mathbb{T}^{2},*).$

And for any $h>h^{\prime}>0$, $F\in C_{h}^{\omega}(\mathbb{T}^{2},*)$, we have
the estimate

$\sum_{k\in\mathbb{Z}^{2}}||\hat{F}(k)||e^{2\pi|k|h^{\prime}}\leq\sum_{k\in\mathbb{Z}^{2}}||F||_{h}e^{-2\pi|k|h}e^{2\pi|k|h^{\prime}}\leq\frac{2||F||_{h}}{\min\\{1,(h-h^{\prime})^{2}\\}}.$
(3.2.4)

Therefore $F\in\mathcal{B}_{h^{\prime}}(*)$. We deduce that

$C_{h}^{\omega}(\mathbb{T}^{2},*)\subseteq\mathcal{B}_{h^{\prime}}(*).$

####  3.2.2 Proof of local almost reducibility of linear systems

We will prove Theorem 3.2 via a nonstandard KAM iterations. The key iteration
lemma is the following.

######  Proposition 3.1.

Let $0<h\leq 1$, $C\in\mathfrak{so}(3,\mathbb{R})$,
$F\in\mathcal{B}_{h}(\mathfrak{so}(3,\mathbb{R}))$,
$\epsilon=||F||_{h}^{\\#}$, $p\leq q<q_{+}$ be positive integers. Under the
following assumptions

$e^{-\frac{1}{2}q_{+}h}<\epsilon\leq\min\\{(10L)^{-100L},\,h^{4(L+1)},\,e^{-\frac{1}{2}qh}\\},$
(3.2.5) $\frac{1}{2q_{+}}\leq|q\alpha-p|\leq\frac{1}{q_{+}},$ (3.2.6)
$\\{k\in\mathbb{Z}^{2}~{}|\,\langle
k,\,\omega\rangle|<\frac{1}{7q},\,|k|<\frac{q_{+}}{6}\\}\subseteq\\{l(q,-p)~{}|~{}l\in\mathbb{Z}\\},$
(3.2.7)

one can construct $B\in\mathcal{B}_{h_{+}}(\mathrm{SO}(3,\mathbb{R}))$,
$C_{+}\in\mathfrak{so}(3,\mathbb{R})$ and
$F_{+}\in\mathcal{B}_{h_{+}}(\mathfrak{so}(3,\mathbb{R}))$, with
$h_{+}=\frac{h}{6(L+2)}$, such that $B$ conjugates
$\begin{cases}\dot{x}=(C+F(\theta))x\\\ \dot{\theta}=\omega\end{cases}$ to
$~{}\begin{cases}\dot{x}=(C_{+}+F_{+}(\theta))x\\\
\dot{\theta}=\omega\end{cases}$ , and for
$\epsilon_{+}=||F_{+}||_{h_{+}}^{\\#}$, the estimate holds

$\displaystyle\epsilon_{+}\left(||B||^{\\#}_{h_{+}}\right)^{2L}\leq\epsilon^{2}e^{-q_{+}h_{+}}.$
(3.2.8)

We point out that (3.2.7) requires the resonant sites to be in some special
form. In fact this is satisfied as a consequence of following small divisor
lemma given in [31, Lemma 4.1].

######  Lemma 3.1 ([31]).

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, $\omega=(\alpha,1)$,
$\frac{p_{n}}{q_{n}}$ be the best approximation of $\alpha$, we have

$\\{k\in\mathbb{Z}^{2}~{}|\,\langle
k,\,\omega\rangle|<\frac{1}{7q_{n}},\,|k|<\frac{q_{n+1}}{6}\\}\subseteq\\{l(q_{n},-p_{n})~{}|~{}l\in\mathbb{Z}\\}.$
(3.2.9)

The proof of Proposition 3.1 is rather technical and will be given in the next
subsection. Now we use the proposition to prove Theorem 3.2.

###### Proof of Theorem 3.2.

We choose

$\delta=\delta(h,L)=\frac{h^{2}}{8}\min\\{(10L)^{-100L},(\frac{h}{2})^{4(L+1)},e^{-\frac{h}{4}}\\}.$
(3.2.10)

For $F\in C_{h}^{\omega}(\mathbb{T}^{2},\mathfrak{so}(3,\mathbb{R}))$ with
$||F||_{h}<\delta$, by (3.2.4), we deduce that
$F\in\mathcal{B}_{\frac{h}{2}}(\mathfrak{so}(3,\mathbb{R}))$ satisfies the
estimate

$||F||_{\frac{h}{2}}^{\\#}\leq\frac{8}{h^{2}}||F||_{h}<\frac{8}{h^{2}}\delta.$
(3.2.11)

To study almost reducibility of the linear system

$\begin{cases}\dot{x}=(C+F(\theta))x\\\ \dot{\theta}=\omega,\end{cases}$
(3.2.12)

we construct conjugation inductively.

Let $B_{0}=I$, $h_{0}=\frac{h}{2}$, $C_{0}=C$, $F_{0}=F$. Then $B_{0}$
automatically conjugates (3.2.12) to

$\begin{cases}\dot{x}=(C_{0}+F_{0}(\theta))x\\\
\dot{\theta}=\omega,\end{cases}$

and the estimate holds for $\epsilon_{0}=\|F_{0}\|_{h_{0}}^{\\#}$,

$\displaystyle\epsilon_{0}\left(||B_{0}||^{\\#}_{h_{0}}\right)^{L}=\epsilon_{0}\leq\min\\{(10L)^{-100L},h_{0}^{4(L+1)},e^{-\frac{1}{2}q_{0}h_{0}}\\},$
(3.2.13)

where we use (3.2.10) and (3.2.11).

To do induction, we assume that for some $n\in\mathbb{N}^{*}$, there exist
$h_{n}>0$, $B_{n}\in\mathcal{B}_{h_{n}}(\mathrm{SO}(3,\mathbb{R}))$,
$C_{n}\in\mathfrak{so}(3,\mathbb{R})$ and
$F_{n}\in\mathcal{B}_{h_{n}}(\mathfrak{so}(3,\mathbb{R}))$, such that
$B_{n}\in\mathcal{B}_{h_{n}}(\mathrm{SO}(3,\mathbb{R}))$ conjugates (3.2.12)
to

$\begin{cases}\dot{x}=(C_{n}+F_{n}(\theta))x\\\
\dot{\theta}=\omega,\end{cases}$ (3.2.14)

and the estimate holds for $\epsilon_{n}=\|F_{n}\|_{h_{n}}^{\\#}$,

$\displaystyle\epsilon_{n}\left(||B_{n}||^{\\#}_{h_{n}}\right)^{L}\leq\min\\{(10L)^{-100L},\,(h_{n})^{4(L+1)},\,e^{-\frac{1}{2}q_{n}h_{n}}\\}.$
(3.2.15)

To construct conjugation at step $n+1$, we separate into two different cases,
depending on the size of $\epsilon_{n}$.

Case 1. If $\epsilon_{n}\leq e^{-\frac{1}{2}q_{n+1}h_{n}}$, then we trivially
define

$h_{n+1}=h_{n},\quad B_{n+1}=B_{n},\quad C_{n+1}=C_{n},\quad F_{n+1}=F_{n}.$

Immediately, $B_{n+1}$ conjugates (3.2.12) to

$\begin{cases}\dot{x}=(C_{n+1}+F_{n+1}(\theta))x\\\
\dot{\theta}=\omega,\end{cases}$

and the estimate holds for $\epsilon_{n+1}=\|F_{n+1}\|_{h_{n+1}}^{\\#}$

$\displaystyle\epsilon_{n+1}\left(||B^{(n+1)}||^{\\#}_{h_{n+1}}\right)^{L}\leq\min\\{(10L)^{-100L},\,(h_{n+1})^{4(L+1)},\,e^{-\frac{1}{2}q_{n+1}h_{n+1}}\\}.$

Case 2. If $\epsilon_{n}>e^{-\frac{1}{2}q_{n+1}h_{n}}$, then together with
(3.2.15) we have

$\displaystyle
e^{-\frac{1}{2}q_{n+1}h_{n}}<\epsilon_{n}\leq\min\\{(10L)^{-100L},\,(h_{n})^{4(L+1)},\,e^{-\frac{1}{2}q_{n}h_{n}}\\}.$
(3.2.16)

By preceding estimate and Lemma 3.1, we are able to apply Proposition 3.1.
This provides the existence of
$B^{(n+1)}\in\mathcal{B}_{h_{n+1}}(\mathrm{SO}(3,\mathbb{R}))$,
$C_{n+1}\in\mathfrak{so}(3,\mathbb{R})$ and
$F_{n+1}\in\mathcal{B}_{h_{n+1}}(\mathfrak{so}(3,\mathbb{R}))$, such that
$B^{(n+1)}$ conjugates (3.2.14) to

$\begin{cases}\dot{x}=(C_{n+1}+F_{n+1}(\theta))x\\\
\dot{\theta}=\omega,\end{cases}$ (3.2.17)

and the estimate holds

$\epsilon_{n+1}\left(||B^{(n+1)}||^{\\#}_{h_{n+1}}\right)^{2L}\leq\epsilon_{n}^{2}e^{-q_{n+1}h_{n+1}}.$
(3.2.18)

In particular,

$\epsilon_{n+1}\leq\epsilon_{n}^{2}.$ (3.2.19)

Then $B_{n+1}:=B^{(n+1)}B_{n}$ conjugates (3.2.12) to (3.2.17). Moreover, by
(3.2.15),(3.2.18) and (3.2.19), we have the following estimate

$\displaystyle\epsilon_{n+1}\left(||B_{n+1}||^{\\#}_{h_{n+1}}\right)^{L}\leq$
$\displaystyle\epsilon_{n+1}\left(||B^{(n+1)}||^{\\#}_{h_{n+1}}\right)^{L}\left(||B_{n}||^{\\#}_{h_{n+1}}\right)^{L}$
$\displaystyle\leq$
$\displaystyle\left(\epsilon_{n+1}\left(||B^{(n+1)}||^{\\#}_{h_{n+1}}\right)^{2L}\right)^{\frac{1}{2}}\cdot\epsilon_{n}\left(||B_{n}||^{\\#}_{h_{n+1}}\right)^{L}$
$\displaystyle\leq$
$\displaystyle\left(\min\\{(10L)^{-100L},\,(h_{n})^{4(L+1)},\,e^{-\frac{1}{2}q_{n}h_{n}}\\}\right)^{2}\cdot
e^{-\frac{1}{2}q_{n+1}h_{n+1}}$ $\displaystyle\leq$
$\displaystyle\min\\{(10L)^{-100L},\,(h_{n+1})^{4(L+1)},\,e^{-\frac{1}{2}q_{n+1}h_{n+1}}\\}.$

Hence in both case, we are able to construct conjugacy
$B_{n}\in\mathcal{B}_{h_{n+1}}(\mathrm{SO}(3,\mathbb{R}))$ and satisfy
estimates in the same form.

By induction, we conclude that for any $n\in\mathbb{N}$, there exist $h_{n}$,
$B_{n}\in\mathcal{B}_{h_{n}}(\mathrm{SO}(3,\mathbb{R}))$,
$C_{n}\in\mathfrak{so}(3,\mathbb{R})$,
$F_{n}\in\mathcal{B}_{h_{n}}(\mathfrak{so}(3,\mathbb{R}))$, such that $B_{n}$
conjugates (3.2.12) to

$\begin{cases}\dot{x}=(C_{n}+F_{n}(\theta))x,\\\
\dot{\theta}=\omega,\end{cases}$

and the estimate holds

$\displaystyle\|F_{n}\|_{h_{n}}^{\\#}\left(||B_{n}||^{\\#}_{h_{n}}\right)^{L}\leq\min\\{(10L)^{-100L},\,(h_{n})^{4(L+1)},\,e^{-\frac{1}{2}q_{n}h_{n}}\\}.$
(3.2.20)

Moreover, (3.2.20) provides that

$\lim_{n\to\infty}\|F_{n}\|_{h_{n}}^{\\#}\left(||B_{n}||^{\\#}_{h_{n}}\right)^{L}=0.$

As shown in (3.2.2), $\mathcal{B}_{h_{n}}(*)\subset
C_{h}^{\omega}(\mathbb{T}^{2},*)$ with $||\cdot||_{h}\leq||\cdot||_{h}^{\\#}$,
we deduce that $B_{n}\in
C^{\omega}_{h_{n}}(\mathbb{T}^{2},\mathrm{SO}(3,\mathbb{R}))$, $F_{n}\in
C^{\omega}_{h_{n}}(\mathbb{T}^{2},\mathfrak{so}(3,\mathbb{R}))$ and

$\lim_{n\to\infty}\|F_{n}\|_{h_{n}}\left(||B_{n}||_{h_{n}}\right)^{L}=0.$

Hence we complete the proof of Theorem 3.2. ∎

###  3.3 Proof of iteration scheme

In this subsection, we construct the iteration scheme. We will firstly
introduce a nice subspace of $\mathcal{B}_{h}(\mathfrak{so}(3,\mathbb{R}))$ in
the sense of conjugation, then give the proof of Proposition 3.1.

For simplicity of notation, we denote
$\mathcal{B}_{h}(\mathfrak{so}(3,\mathbb{R}))$ briefly by $\mathcal{B}_{h}$.
Without lose of generality, we assume that

$C=2\pi\rho J_{1},\qquad\rho\in\mathbb{R}.$

####  3.3.1 A nice subspace of $\mathcal{B}_{h}$

For $F\in\mathcal{B}_{h}$, to construct
$B\in\mathcal{B}_{h}(\mathrm{SO}(3,\mathbb{R}))$ so that for some
$C_{+}\in\mathfrak{so}(3,\mathbb{R})$ and a smaller $F_{+}\in\mathcal{B}_{h}$,
$B$ conjugating $\begin{cases}\dot{x}=(C+F(\theta))x\\\
\dot{\theta}=\omega\end{cases}$ to
$\quad\begin{cases}\dot{x}=(C_{+}+F_{+}(\theta))x\\\
\dot{\theta}=\omega\end{cases}$, it is sufficient to solve

$\partial_{\omega}B=B(C+F)-(C_{+}+F_{+})B.$ (3.3.1)

If $B$ is close to the identity, namely $B=e^{Y}$ with $Y\in\mathcal{B}_{h}$
small, then (3.3.1) provides a linearized equation

$\partial_{\omega}Y-[C,Y]=F+D,$ (3.3.2)

for some constant $D\in\mathfrak{so}(3,\mathbb{R})$. However, when solving
(3.3.2), one may encounter the small divisor problem, i.e., the liner operator

$\mathcal{B}_{h}\rightarrow\mathcal{B}_{h},\quad
Y\mapsto\partial_{\omega}Y-[C,Y],$ (3.3.3)

may be not invertible or invertible but with an inverse operator of too large
norm.

As studied in [31, 32], a key observation is that there exists a subspace
$\mathcal{B}_{h}^{(nre)}$ of $\mathcal{B}_{h}$, satisfying the property that
for any $Y\in\mathcal{B}_{h}^{(nre)}$, one has

$\partial_{\omega}Y,\,[C,Y]\in\mathcal{B}_{h}^{(nre)},$ (3.3.4)

and the estimate holds

$||\partial_{\omega}Y-[C,Y]||_{h}^{\\#}\geq\eta||Y||^{\\#}_{h},$ (3.3.5)

for a uniform constant $\eta>0$. In other word, the restricted linear operator
(3.3.3) on $\mathcal{B}_{h}^{(nre)}$ behaves well. Therefore, we are able to
solve (3.3.2) in $\mathcal{B}_{h}^{(nre)}$. To formulate the result precisely,
we decompose the space $\mathcal{B}_{h}$ into a direct sum
$\mathcal{B}_{h}^{(nre)}\oplus\mathcal{B}_{h}^{(re)}$.

######  Lemma 3.2 ([31, 32]).

Let $h>0$, $\eta>0$, $\omega\in\mathbb{R}^{2}$,
$C\in\mathfrak{so}(3,\mathbb{R})$. For any $F\in\mathcal{B}_{h}$ satisfying
$||F||^{\\#}_{h}\leq 10^{-8}\eta^{2}$, there exist $Y\in\mathcal{B}_{h}$,
$H\in\mathcal{B}_{h}^{(re)}$ such that $e^{Y}$ conjugate
$\begin{cases}\dot{x}=(C+F(\theta))x\\\ \dot{\theta}=\omega\end{cases}$ to
$\begin{cases}\dot{x}=(C+H(\theta))x\\\ \dot{\theta}=\omega\end{cases}$,
satisfying the estimates

$||Y||^{\\#}_{h}\leq\frac{10}{\eta}||F||^{\\#}_{h},\qquad||H||^{\\#}_{h}\leq
2||F||^{\\#}_{h}.$ (3.3.6)

######  Remark 3.2.

Preceding lemma was firstly given in [31] and there are some minor
improvements in later articles such as [32]. Here we use the version in [32].

Given $\epsilon>0$. We define

$\Lambda_{1}:=\\{k\in\mathbb{Z}^{2}\,|\,|\langle k,\,\omega\rangle|\geq
2\epsilon^{\frac{1}{4}}\\},\quad\Lambda_{2}:=\\{k\in\mathbb{Z}^{2}\,|\,|\langle
k,\,\omega\rangle-\rho|\geq\epsilon^{\frac{1}{4}}\\}.$

Then it follows directly that

$\Lambda_{2}^{c}-k_{*}:=\\{k-k_{*}\,|\,k\in\Lambda_{2}^{c}\,\\}\subseteq\Lambda_{1}^{c},\quad\forall
k_{*}\in\Lambda_{2}^{c}.$ (3.3.7)

Furthermore, we have the following simple fact.

######  Lemma 3.3.

Under the assumptions of Proposition 3.1, the estimate holds

$100(L+2)^{2}q\epsilon^{\frac{1}{4}}\ll 1.$ (3.3.8)

Consequently, we have

$\Lambda_{1}^{c}\cap\\{k\in\mathbb{Z}^{2}~{}|~{}|k|<\frac{q_{+}}{6}\\}\subseteq\\{l(q,-p)~{}|~{}l\in\mathbb{Z}\\}.$
(3.3.9)

###### Proof.

By (3.2.5), we get

$\displaystyle 100(L+2)^{2}q\epsilon^{\frac{1}{4}}$ $\displaystyle\leq$
$\displaystyle 100(L+2)^{2}q\epsilon^{\frac{1}{8}}\epsilon^{\frac{1}{8}}\leq
100(L+2)^{2}qe^{-\frac{qh}{16}}\epsilon^{\frac{1}{16}}$ $\displaystyle\leq$
$\displaystyle\frac{1600(L+2)^{2}}{eh}\epsilon^{\frac{1}{8}}\leq
10^{5}L^{2}\epsilon^{\frac{1}{8}}\ll 1,$

then (3.3.9) follows from (3.2.7) and (3.3.8). ∎

Given $h>0$, now we define $\mathcal{B}_{h}^{(nre)}$, $\mathcal{B}_{h}^{(re)}$
explicitly and verify the properties (3.3.4) and (3.3.5). Let

$\displaystyle\mathcal{B}_{h}^{(nre)}:=\\{E\in\mathcal{B}_{h}\,|\,$
$\displaystyle E=fJ_{1}+\Re wJ_{2}+\Im wJ_{3},\,\text{ for
}f\in\mathcal{B}_{h}(\mathbb{R})\text{ in the form
}\sum_{k\in\Lambda_{1}}\hat{f}(k)e^{2\pi i\langle k,\theta\rangle},$
$\displaystyle w\in\mathcal{B}_{h}(\mathbb{C})\text{ in the form
}\sum_{k\in\Lambda_{2}}\hat{w}(k)e^{2\pi i\langle k,\theta\rangle}\\},$
$\displaystyle\mathcal{B}_{h}^{(re)}:=\\{H\in\mathcal{B}_{h}\,|\,$
$\displaystyle H=fJ_{1}+\Re wJ_{2}+\Im wJ_{3},\,\text{ for
}f\in\mathcal{B}_{h}(\mathbb{R})\text{ in the form
}\sum_{k\in\Lambda_{1}^{c}}\hat{f}_{1}(k)e^{2\pi i\langle k,\theta\rangle},$
$\displaystyle w\in\mathcal{B}_{h}(\mathbb{C})\text{ in the form
}\sum_{k\in\Lambda_{2}^{c}}\hat{w}(k)e^{2\pi i\langle k,\theta\rangle}\\}.$
(3.3.10)

It follows the direct sum splitting

$\mathcal{B}_{h}=\mathcal{B}_{h}^{(nre)}\oplus\mathcal{B}_{h}^{(re)}.$

By definition, for any $E\in\mathcal{B}_{h}^{(nre)}$, there exist
$f=\sum_{k\in\Lambda_{1}}\hat{f}(k)e^{2\pi i\langle
k,\theta\rangle}\in\mathcal{B}_{h}(\mathbb{R})$,
$w=\sum_{k\in\Lambda_{2}}\hat{w}(k)e^{2\pi i\langle
k,\theta\rangle}\in\mathcal{B}_{h}(\mathbb{C})$, such that $E=fJ_{1}+\Re
wJ_{2}+\Im wJ_{3}$. We can express $E$ explicitly

$\displaystyle E=$ $\displaystyle\left(\sum_{k\in\Lambda_{1}}\hat{f}(k)e^{2\pi
i\langle k,\theta\rangle}\right)J_{1}$
$\displaystyle+\frac{1}{2}\left(\sum_{k\in\Lambda_{2}}\hat{w}(k)e^{2\pi
i\langle k,\theta\rangle}+\sum_{k\in\Lambda_{2}}\overline{\hat{w}(k)}e^{-2\pi
i\langle k,\theta\rangle}\right)J_{2}$
$\displaystyle-\frac{i}{2}\left(\sum_{k\in\Lambda_{2}}\hat{w}(k)e^{2\pi
i\langle k,\theta\rangle}-\sum_{k\in\Lambda_{2}}\overline{\hat{w}(k)}e^{-2\pi
i\langle k,\theta\rangle}\right)J_{3}.$

Then we compute that

$\displaystyle\partial_{\omega}E=$ $\displaystyle 2\pi
i\left(\sum_{k\in\Lambda_{1}}\langle k,\,\omega\rangle\hat{f}(k)e^{2\pi
i\langle k,\theta\rangle}\right)J_{1}$ $\displaystyle+\pi
i\left(\sum_{k\in\Lambda_{2}}\langle k,\,\omega\rangle\hat{w}(k)e^{2\pi
i\langle k,\theta\rangle}-\sum_{k\in\Lambda_{2}}\langle
k,\,\omega\rangle\overline{\hat{w}(k)}e^{-2\pi i\langle
k,\theta\rangle}\right)J_{2}$
$\displaystyle+\pi\left(\sum_{k\in\Lambda_{2}}\langle
k,\,\omega\rangle\hat{w}(k)e^{2\pi i\langle
k,\theta\rangle}+\sum_{k\in\Lambda_{2}}\langle
k,\,\omega\rangle\overline{\hat{w}(k)}e^{-2\pi i\langle
k,\theta\rangle}\right)J_{3},$

and

$\displaystyle[C,E]=$ $\displaystyle\pi
i\rho\left(\sum_{k\in\Lambda_{2}}\hat{w}(k)e^{2\pi i\langle
k,\theta\rangle}-\sum_{k\in\Lambda_{2}}\overline{\hat{w}(k)}e^{-2\pi i\langle
k,\theta\rangle}\right)J_{2}$
$\displaystyle+\pi\rho\left(\sum_{k\in\Lambda_{2}}\hat{w}(k)e^{2\pi i\langle
k,\theta\rangle}+\sum_{k\in\Lambda_{2}}\overline{\hat{w}(k)}e^{-2\pi i\langle
k,\theta\rangle}\right)J_{3},$

where we recall that $C=2\pi\rho J_{1}$, $\rho\in\mathbb{R}$. It follows that

$\partial_{\omega}E\in\mathcal{B}_{h}^{(nre)},\quad[C,E]\in\mathcal{B}_{h}^{(nre)}.$

Moreover, by direct computation, we have

$\|\partial_{\omega}E-[C,E]\|^{\\#}_{h}\geq\pi\epsilon^{\frac{1}{4}}\|E\|^{\\#}_{h}.$
(3.3.11)

Hence the two characterizing conditions are satisfied therefore the space
$\mathcal{B}_{h}^{(nre)}$ is well-defined.

####  3.3.2 Proof of Proposition 3.1

Based on the nice space $\mathcal{B}^{(nre)}$ defined in the preceding
section, we can now prove Proposition 3.1. We will also make use of Floquet
theory.

###### Proof of Proposition 3.1.

We start with the linear system

$\begin{cases}\dot{x}=(C+F(\theta))x\\\ \dot{\theta}=\omega,\end{cases}$
(3.3.12)

satisfying the assumptions (3.2.5), (3.2.6), (3.2.7). Motivated by the
quantitative estimate (3.3.11) of $\mathcal{B}^{(nre)}$, we work with
$\eta=\pi\epsilon^{\frac{1}{4}}.$ Then (3.2.5) provides that

$||F||_{h}^{\\#}=\epsilon<10^{-8}\eta,$

which enables us to apply Lemma 3.2. It shows the existence of
$Y\in\mathcal{B}_{h}$, $H\in\mathcal{B}_{h}^{(re)}$, such that $e^{Y}$
conjugates (3.3.12) to

$\begin{cases}\dot{x}=(C+H(\theta))x\\\ \dot{\theta}=\omega,\end{cases}$
(3.3.13)

and the estimates hold

$||Y||^{\\#}_{h}\leq\frac{10}{\eta}\epsilon<\epsilon^{\frac{1}{2}},\quad||H||^{\\#}_{h}\leq
2\epsilon,$ (3.3.14)

where (3.2.5) is used for the first inequation. By construction of
$\mathcal{B}_{h}^{(re)}$ in (3.3.1), there exist

$f=\sum_{k\in\Lambda_{1}^{c}}\hat{f}(k)e^{2\pi i\langle
k,\theta\rangle}\in\mathcal{B}_{h}(\mathbb{R}),\quad
w=\sum_{k\in\Lambda_{2}^{c}}\hat{w}(k)e^{2\pi i\langle
k,\theta\rangle}\in\mathcal{B}_{h}(\mathbb{C}),$ (3.3.15)

such that

$H=fJ_{1}+\Re wJ_{2}+\Im wJ_{3}.$

To study further conjugation of (3.3.13), we separate into two cases depending
on $\Lambda_{2}^{c}$.

Case 1. If
$\Lambda_{2}^{c}\cap\\{k\in\mathbb{Z}^{2}||k|<\frac{q_{+}}{6}\\}=\emptyset$,
then $\mathcal{T}_{\frac{q_{+}}{6}}H=\mathcal{T}_{\frac{q_{+}}{6}}fJ_{1}$. By
Lemma 3.3,

$\Lambda_{1}^{c}\cap\\{k\in\mathbb{Z}^{2}~{}|~{}|k|<\frac{q_{+}}{6}\\}\subseteq\\{l(q,-p)~{}|~{}l\in\mathbb{Z}\\}.$

Correspondingly, we define

$\displaystyle B_{1}(\theta):=$
$\displaystyle\exp\left(\sum_{k\in\Lambda_{1}^{c}\backslash\\{0\\},~{}|k|<\frac{q_{+}}{6}}\frac{\hat{f}(k)e^{2\pi
i\langle k,\theta\rangle}}{2\pi i\langle k,\,\omega\rangle}J_{1}\right)$
$\displaystyle=$
$\displaystyle\exp\left(\sum_{l(q,-p)\in\Lambda_{1}^{c}\backslash\\{0\\},|l(q,-p)|<\frac{q_{+}}{6}}\frac{\hat{f}(k)e^{2\pi
i\langle l(q,-p),\theta\rangle}}{2\pi il(q\alpha-p)}J_{1}\right).$ (3.3.16)

Immediately, $B_{1}$ conjugates (3.3.13) to

$\begin{cases}\dot{x}=(C_{+}+F_{+})x\\\ \dot{\theta}=\omega,\end{cases}$
(3.3.17)

where $C_{+}=C+\hat{H}(0)$,
$F_{+}=\mathrm{Ad}(B_{1})\mathcal{R}_{\frac{q_{+}}{6}}H$. By the assumptions
(3.2.5) and (3.2.6), for $h_{+}:=\frac{h}{6(L+2)}$, the construction of
$B_{1}$ (3.3.2) provides the bound

$\displaystyle\|B_{1}\|^{\\#}_{h_{+}}\leq e^{\frac{q_{+}\epsilon}{\pi}}\ll
e^{\frac{1}{2(L+1)}q_{+}h_{+}}.$ (3.3.18)

Let $B=B_{1}e^{Y}$. Then $B$ conjugates (3.3.12) to (3.3.17). For
$\epsilon_{+}=\|F_{+}\|^{\\#}_{h_{+}}$, by (3.3.18), (3.3.14), (3.2.5), we
compute that

$\displaystyle\epsilon_{+}\left(\|B\|^{\\#}_{h_{+}}\right)^{2L}\leq$
$\displaystyle\|\mathcal{R}_{\frac{q_{+}}{6}}H\|^{\\#}_{h_{+}}\cdot\left(\|B_{1}\|^{\\#}_{h_{+}}\right)^{2(L+1)}\cdot\left(\|e^{Y}\|^{\\#}_{h_{+}}\right)^{2L}$
$\displaystyle\leq$ $\displaystyle 2\epsilon e^{-\frac{2\pi
q_{+}\cdot(6L+11)h_{+}}{6}}\cdot e^{q_{+}h_{+}}\cdot
e^{2L\epsilon^{\frac{1}{2}}}$ $\displaystyle\ll$
$\displaystyle\epsilon^{2}e^{-q_{+}h_{+}}.$ (3.3.19)

Hence (3.2.8) holds, which completes the proof in Case 1.

Case 2. If
$\Lambda_{2}^{c}\cap\\{k\in\mathbb{Z}^{2}||k|<\frac{q_{+}}{6}\\}\neq\emptyset$,
then there exists $k_{*}\in\Lambda_{2}^{c}$, such that

$|k_{*}|=\min_{k\in\Lambda_{2}^{c}}|k|,\quad 0\leq|k_{*}|<\frac{q_{+}}{6}.$

We deal with $k_{*}$ through a rotation step as follows.

######  Lemma 3.4.

Let $Q=\exp\\{-2\pi\langle k_{*},\,\theta\rangle J_{1}\\}$,
$C_{1}=2\pi(\rho-\langle k_{*},\theta\rangle)J_{1}$,

$F_{1}=fJ_{1}+\Re\\{we^{-2\pi\langle
k_{*},\,\theta\rangle}\\}J_{2}+\Im\\{we^{-2\pi\langle
k_{*},\,\theta\rangle}\\}J_{3}.$ (3.3.20)

Then $Q$ conjugates (3.3.13) to the system

$\begin{cases}\dot{x}=(C_{1}+F_{1}(\theta))x\\\
\dot{\theta}=\omega,\end{cases}$ (3.3.21)

and the estimates hold

$\|C_{1}\|\leq
2\pi\epsilon^{\frac{1}{4}},\qquad\|F_{1}\|^{\\#}_{\frac{h}{3}}\leq 2\epsilon.$
(3.3.22)

Moreover, $\mathcal{T}_{\frac{q_{+}}{6}}F_{1}$ is in the form

$\mathcal{T}_{\frac{q_{+}}{6}}F_{1}=\sum_{k\in(q,-p)\mathbb{Z},|k|<\frac{q_{+}}{6}}\hat{F_{1}}(k)e^{2\pi
i\langle k,\theta\rangle}.$ (3.3.23)

###### Proof.

Direct computation shows that $Q$ conjugates (3.3.13) to
$\begin{cases}\dot{x}=(C_{1}+\mathrm{Ad}(Q)H(\theta))x\\\
\dot{\theta}=\omega,\end{cases}$. In addition, we compute that

$\displaystyle\mathrm{Ad}(Q)H=$ $\displaystyle\mathrm{Ad}(Q)\cdot(fJ_{1}+\Re
wJ_{2}+\Im wJ_{3})$ $\displaystyle=$ $\displaystyle fJ_{1}+\Re wAd(Q)\cdot
J_{2}+\Im wAd(Q)\cdot J_{3}$ $\displaystyle=$ $\displaystyle fJ_{1}+\\{\Re
w\cos 2\pi\langle k_{*},\,\theta\rangle+\Im w\sin 2\pi\langle
k_{*},\,\theta\rangle\\}J_{2}$ $\displaystyle+\\{\Im w\cos 2\pi\langle
k_{*},\,\theta\rangle-\Re w\sin 2\pi\langle k_{*},\,\theta\rangle\\}J_{3}$
$\displaystyle=$ $\displaystyle fJ_{1}+\Re\\{we^{-2\pi\langle
k_{*},\,\theta\rangle}\\}J_{2}+\Im\\{we^{-2\pi\langle
k_{*},\,\theta\rangle}\\}J_{3}.$ (3.3.24)

Hence we deduce (3.3.20) for $F=\mathrm{Ad}(Q)H$.

Recall (3.3.15), we can rewrite

$we^{-2\pi\langle
k_{*},\,\theta\rangle}=\sum_{k\in\Lambda_{2}^{c}}\hat{w}(k)e^{2\pi i\langle
k-k_{*},\,\theta\rangle}=\sum_{k\in\Lambda_{2}^{c}-k_{*}}\hat{w}(k+k_{*})e^{2\pi
i\langle k,\,\theta\rangle}.$ (3.3.25)

As $k_{*}\in\Lambda_{2}^{c}$, by (3.3.7) and (3.3.9), we have

$(\Lambda_{2}^{c}-k_{*})\cap\\{k\in\mathbb{Z}^{2}~{}|~{}|k|<\frac{q_{+}}{6}\\}\subseteq\Lambda_{1}^{c}\cap\\{k\in\mathbb{Z}^{2}~{}|~{}|k|<\frac{q_{+}}{6}\\}\subseteq\\{l(q,-p)~{}|~{}l\in\mathbb{Z}\\}.$

Therefore we get the truncated form of $F$ (3.3.23). ∎

To do further conjugation of (3.3.21), we firstly work with the truncated
system

$\begin{cases}\dot{x}=(C_{1}+\mathcal{T}_{\frac{q_{+}}{6}}F_{1})x\\\
\dot{\theta}=\omega.\end{cases}$ (3.3.26)

By the precise from of $\mathcal{T}_{\frac{q_{+}}{6}}F_{1}$ given in (3.3.23),
we can apply Lemma A.1. By (3.3.22),

$\|C_{1}+\mathcal{T}_{\frac{q_{+}}{6}}F_{1}\|^{\\#}_{\frac{h}{3}}\leq
10\epsilon^{\frac{1}{4}}.$ (3.3.27)

Applying Lemma A.1 to the system (3.3.26) on the domain of
$|\Im\theta|\leq\frac{h}{3}$, there exists $B_{1}\in
C^{\omega}_{\frac{h}{3}}(\mathbb{T}^{2},\mathrm{SO}(3,\mathbb{R}))$ which
conjugates (3.3.26) to $\begin{cases}\dot{x}=C_{+}x\\\
\dot{\theta}=\omega\end{cases}$ for some
$C_{+}\in\mathfrak{so}(3,\mathbb{R})$. By (3.3.27), Lemma A.1 and Lemma 3.3,
we have the estimate

$\|B_{1}\|_{\frac{h}{3}}\leq e^{15q_{+}q\epsilon^{\frac{1}{4}}h}\leq
e^{\frac{1}{(L+1)}q_{+}h_{+}}.$

Then by the assumptions (3.2.4), (3.2.5), we deduce the estimate for
$h_{+}=\frac{h}{6(L+2)}$,

$\displaystyle\|B_{1}\|^{\\#}_{h_{+}}\leq\frac{72(L+2)^{2}}{(2L+3)^{2}h^{2}}e^{\frac{1}{(L+1)}q_{+}h_{+}}\leq
200\epsilon^{\frac{1}{L+1}}e^{\frac{1}{2(L+1)}q_{+}h_{+}}\leq
e^{\frac{1}{2(L+1)}q_{+}h_{+}}.$ (3.3.28)

Consequently, $B_{1}$ conjugates (3.3.21) to

$\begin{cases}\dot{x}=(C_{+}+F_{+}(\theta))x\\\
\dot{\theta}=\omega,\end{cases}$ (3.3.29)

where $F_{+}=\mathrm{Ad}(B_{1})\mathcal{R}_{\frac{q_{+}}{6}}F_{1}$. To sum up,
let

$B=B_{1}Qe^{Y}=B_{1}\exp\\{-2\pi\langle k_{*},\,\theta\rangle J\\}e^{Y}.$

Then $B$ conjugates (3.3.12) to (3.3.29).

By (3.2.5) and (3.3.28), recalling $|k_{*}|\leq\frac{1}{6}q_{+}$, for
$\epsilon_{+}=\|F_{+}\|^{\\#}_{h_{+}}$, we compute that

$\displaystyle\epsilon_{+}\left(\|B\|^{\\#}_{h_{+}}\right)^{2L}\leq$
$\displaystyle\|\mathcal{R}_{\frac{q_{+}}{6}}F_{1}\|^{\\#}_{h_{+}}\epsilon_{+}\left(\|B_{1}\|^{\\#}_{h_{+}}\right)^{2L+2}\left(\|Q\|^{\\#}_{h_{+}}\right)^{2L}\left(\|e^{Y}\|^{\\#}_{h_{+}}\right)^{2L}$
$\displaystyle\leq$ $\displaystyle 2\epsilon e^{-2\pi\cdot\frac{q_{+}}{6}\cdot
2(L+\frac{3}{2})h_{+}}\cdot
e^{q_{+}h_{+}}\cdot\left(e^{2\pi\cdot\frac{1}{6}q_{+}h_{+}}\right)^{2L}\cdot
2^{2L}$ $\displaystyle\ll$ $\displaystyle\epsilon^{2}e^{-q_{+}h_{+}}.$

Hence (3.2.8) holds, which completes the proof in Case 2. Therefore we have
completed the proof of Proposition 3.1.

∎

##  4 Local results II. close to $\exp(Dx+C)$ with $D\neq 0$

When restricted in $3$ dimension, since the dynamical degree
$D\in\mathfrak{so}(3,\mathbb{R})$, we could always conjugate it to $2\pi
dJ_{1}$ for some $d\in\mathbb{R}$. By Theorem 2.1, $\exp D=I$. It forces
$d\in\mathbb{Z}$.

In this section, we study the normal form in the neighborhood of
$(\alpha,\exp(Dx+C))$ for $D,C\in\mathfrak{so}(3,\mathbb{R})$, where $D=2\pi
dJ_{1}$, $d\in\mathbb{Z}$, $C\in\mathfrak{z}(D)$. By Lemma 2.1, $C$ must be in
the form $cJ_{1}$ for some $c\in\mathbb{R}$. We will prove the following
theorem.

######  Theorem 4.1.

Let $h>0$, $\gamma>0,\sigma>0$, $d\in\mathbb{Z}$. There exists $\epsilon>0$
such that for $\alpha\in\mathrm{DC}(\gamma,\sigma)$, $A\in
C_{h}^{\omega}(\mathbb{T},\mathrm{SO}(3,\mathbb{R}))$, if
$||A-\exp(2\pi(dx+c)J_{1})||_{h}\leq\epsilon$ for some $c\in\mathbb{R}$ and
$(\alpha,A)$ has dynamical degree $dJ_{1}$, then $(\alpha,A)$ is
$C^{\omega}$-conjugate to $(\alpha,\exp(2\pi(dx+c^{\prime})J_{1}))$ for some
$c^{\prime}\in\mathbb{R}$.

###  4.1 Cohomological equation

Let $\alpha\in\mathrm{DC}(\gamma,\sigma)$. $A$ can be written in the form

$A(x)=\exp(2\pi(dx+c)J_{1})\exp\varphi(x),$

where $\varphi\in C^{\omega}_{h}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$
satisfies $||\varphi||_{h}\leq\epsilon$. We try to find the simplest function
$\varphi^{\prime}\in
C^{\omega}_{h^{\prime}}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$ and a
$\psi\in C^{\omega}_{h^{\prime}}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$ for
some $0<h^{\prime}<h$ such that

$\exp\psi(x+\alpha)\exp(2\pi(dx+c)J_{1})\exp\varphi(x)\exp(-\psi(x))=\exp(2\pi(dx+c)J_{1})\exp\varphi(x).$
(4.1.1)

If $\varphi$ is small enough and if we seek a small $\psi$, the linearized
equation becomes

$\mathrm{Ad}(\exp(-2\pi(dx+c)J_{1}))\psi(x+\alpha)+\varphi(x)-\psi(x)=\varphi^{\prime}(x).$
(4.1.2)

Let $\\{w_{1},w_{2},w_{3}\\}$ be a basis of $\mathfrak{so}(3,\mathbb{C})$
given by the adjoint action of $\exp(2\pi(dx+c)J_{1})$, i.e.,

$\exp(-2\pi(dx+c)J_{1})w_{k}\exp(2\pi(dx+c)J_{1})=e^{2\pi
i(l_{k}x+\lambda_{k})}w_{k},\quad k=1,2,3,$

for some $l_{k},\lambda_{k}\in\mathbb{C}$. As $\exp(2\pi dJ_{1})=I$ we have
$l_{k}\in\mathbb{Z}$. We express $\varphi$ with respect to the basis
$\varphi(x)=\sum_{k=1}^{3}\varphi_{k}(x)w_{k}.$ Then (4.1.2) becomes

$e^{2\pi
i(l_{k}x+\lambda_{k})}\psi_{k}(x+\alpha)+\varphi_{k}(x)-\psi_{k}(x)=0,\quad
k=1,2,3.$ (4.1.3)

By Lemma 2.1, there exits only one $k\in\\{1,2,3\\}$ such that $l_{k}=0$. We
assume that $l_{3}=0$. Then Lemma 2.1 gives that $w_{3}=J_{3}$. Therefore
$\lambda_{3}=0$. In this case, we have the following classical lemma:

######  Lemma 4.1.

Let $\alpha\in\mathrm{DC}(\gamma,\sigma)$, $h>0$, $\varphi\in
C_{h}^{\omega}(\mathbb{T},\mathbb{C})$. For $0<h^{\prime}<h$, there exists
$\psi\in C^{\omega}_{h^{\prime}}(\mathbb{T},\mathbb{C})$, such that

$-\varphi(x)+\hat{\varphi}(0)=\psi(x+\alpha)-\psi(x),$

and

$||\psi||_{h^{\prime}}\leq\mathrm{cst}.||\varphi||_{h},$

where the constant depends only on $\gamma$.

For $k\in\\{1,2\\}$, $l_{k}\neq 0$. Similar to Proposition 8.1 in [39], we
have the following lemma.

######  Lemma 4.2.

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q},h>0,\varphi\in
C_{h}^{\omega}(\mathbb{T},\mathbb{C}),l\in\mathbb{Z}\backslash\\{0\\},\lambda\in\mathbb{R}$.
For $0<h^{\prime}<h$, there exists $\psi\in
C^{\omega}_{h^{\prime}}(\mathbb{T},\mathbb{C})$ such that

$e^{2\pi
i(lx+\lambda)}\psi(x+\alpha)-\psi(x)=-\varphi(x)+P_{\alpha,l,\lambda}(\varphi)(x),$
(4.1.4)

where

$P_{\alpha,l,\lambda}(\varphi)(x)=\begin{cases}\sum_{m=0}^{-l-1}e^{2\pi
imx}P_{\alpha,l,\lambda}(\varphi)(m),\quad\text{if }l<0;\\\
\sum_{m=-1+1}^{0}e^{2\pi imx}P_{\alpha,l,\lambda}(\varphi)(m),\quad\text{if
}l>0,\\\ \end{cases}$ $\displaystyle P_{\alpha,l,\lambda}(\varphi)(m)$
$\displaystyle=$ $\displaystyle\sum_{j=0}^{\infty}\hat{\varphi}(m-jl)e^{2\pi
i\\{[(j+1)m-\frac{j(j-1)}{2}l]\alpha+(j+1)\lambda\\}}$ $\displaystyle+$
$\displaystyle\sum_{j=1}^{\infty}\hat{\varphi}(m+jl)e^{-2\pi
i\\{[jm+\frac{j(j+1)}{2}l]\alpha+j\lambda\\}},$

and the following estimates hold:

$||P_{\alpha,l,\lambda}(\varphi)||_{h^{\prime}}\leq||\varphi||_{h}(1+\frac{2e^{-2\pi
lh}}{1-e^{-2\pi lh}})\sum_{j=1}^{|l|}e^{-2\pi
k(h-h^{\prime})}\leq\mathrm{cst}.\frac{||\varphi||_{h}}{h},$
$||\psi||_{h^{\prime}}\leq\frac{||\varphi||_{h}}{1-e^{-2\pi
h}}(1+\frac{1}{1-e^{-2\pi(h-h^{\prime})}})\leq\mathrm{cst}.\frac{||\varphi||_{h}}{h(h-h^{\prime})}.$

For $\varphi\in C_{h}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$, we
decompose it with respect to the basis $\\{w_{1},w_{2},w_{3}\\}$ to be
$\varphi=\sum_{k=1}^{3}\varphi_{k}w_{k}$. Let
$\alpha\in\mathbb{R}\backslash\mathbb{Q},D\in\mathfrak{so}(3,\mathbb{R})\backslash\\{0\\}$,
$C\in\mathfrak{z}(D)$. We define

$\mathcal{P}_{0}\varphi=\hat{\varphi}_{3}(0)w_{3},\quad\mathcal{P}_{\alpha,d,c}\varphi=P_{\alpha,l_{1},\lambda_{1}}\varphi_{1}w_{1}+P_{\alpha,l_{2},\lambda_{2}}\varphi_{2}w_{2}.$

Combining Lemma 4.1 and 4.2 in each direction, and using preceding lemma, we
deduce the following result.

######  Corollary 4.1.

Let $\epsilon>0$ small, $\alpha\in\mathrm{DC}(\gamma,\sigma)$, $h>0$,
$\varphi\in C_{h}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$ with
$||\varphi||_{h}\leq\epsilon$, $d\in\mathbb{Z},c\in\mathbb{R}$. Then there
exists $\psi,F\in C_{h^{\prime}}^{\omega}(\mathbb{T},\mathbb{C})$,
$h^{\prime}<h$, such that

$\displaystyle\exp(\psi(x+\alpha))\exp(2\pi(dx+c)J_{1})\exp(\varphi(x))\exp(-\psi(x))$
$\displaystyle=$
$\displaystyle\exp(2\pi(dx+c)J_{1}+\mathcal{P}_{0}\varphi)\exp(\mathcal{P}_{\alpha,d,c}\varphi(x)+F(x)),$

where $F=O_{2}(\varphi,\psi,\mathcal{P}_{\alpha,d,c}\varphi)$ and the
following estimates hold:

$||P_{\alpha,d,c}(\varphi)||_{h}\leq\mathrm{cst}.\frac{||\varphi||_{h}}{h},\quad||\psi||_{h^{\prime}}\leq\mathrm{cst}.\frac{||\varphi||_{h}}{h(h-h^{\prime})}.$

###  4.2 Proof of Theorem 4.1

Let $D=2\pi dJ_{1}$ with $d\in\mathbb{Z}$, $\\{w_{1},w_{2},w_{3}\\}$ be the
basis given by adjoint action of $d$, namely

$\exp(-Dx)w_{k}\exp(Dx)=e^{2\pi il_{k}x}w_{k},$

for some $l_{k}\in\mathbb{Z}$, $\varphi\in
C_{h}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$. We assume that
$l_{3}=0$. Then we decompose $\varphi$ with respect to
$\\{w_{1},w_{2},w_{3}\\}$, namely $\varphi=\sum_{k=1}^{3}\varphi_{k}w_{k}$. We
define $\Pr\varphi=\sum_{k=1}^{2}T_{l_{k}}\varphi_{k}w_{k},$ where

$T_{l}\psi=\begin{cases}\sum_{m=0}^{-l-1}e^{2\pi
imx}\hat{\psi}(m),\quad\text{if }l<0;\\\ \sum_{m=-l+1}^{0}e^{2\pi
imx}\hat{\psi}(m),\quad\text{if }l>0.\end{cases}$

We cite [34, Lemma 6.1] and reformulate it as following. It is proved through
the estimates on length.

######  Lemma 4.3.

There exists a positive constant $\mathrm{cst}$ such that for
$c\in\mathbb{R}$, $\varphi\in
C_{h}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$ small enough, if
$(\alpha,\exp(2\pi(dx+c)J_{1})\exp\varphi)$ is of dynamical degree $2\pi
dJ_{1}$, then it satisfies

$||\Pr\varphi||_{L^{2}}\leq c||(I-\Pr)\partial\varphi||_{L^{2}}.$ (4.2.1)

###### Proof of Theorem 4.1.

By our assumption, we can write $A(x)=\exp(2\pi(dx+c)J_{1})\exp\varphi(x)$ for
some $\varphi\in C_{h}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$ with
$||\varphi||_{h}\leq\epsilon$. By Corollary 4.1, there exists
$c_{1}\in\mathbb{R},\psi_{1},F_{1}\in
C_{h_{1}}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$,
$h_{1}=\frac{h}{2}+\frac{h}{4}$, such that for
$\varphi_{1}=\mathcal{P}_{\alpha,d,c}\varphi,$

$\displaystyle\exp(\psi_{1}(x+\alpha))\exp(2\pi(dx+c)J_{1})\exp(\varphi(x))\exp(-\psi_{1}(x))$
$\displaystyle=$
$\displaystyle\exp(2\pi(dx+c_{1})J_{1})\exp(\varphi_{1}(x)+F_{1}(x)),$

with estimates

$||\varphi_{1}||_{h_{1}}\leq\mathrm{cst}.\frac{||\varphi||_{h}}{h},\quad||F_{1}||_{h_{1}}\leq\mathrm{cst}.\frac{||\varphi||_{h}^{2}}{h^{2}(h-h_{1})^{2}}.$

Then Lemma 4.3 gives that

$||\Pr(\varphi_{1}+F_{1})||_{L^{2}}\leq
c||(I-\Pr)\partial(\varphi_{1}+F_{1})||_{L^{2}}.$ (4.2.2)

As $\varphi_{1}=\mathcal{P}_{\alpha,d,c}\varphi$, we have
$\varphi_{1}=\Pr\varphi_{1}$. Then (4.2.2) gives that

$||\varphi_{1}+\Pr F_{1}||_{L^{2}}\leq c||(I-\Pr)\partial F_{1}||_{L^{2}}\leq
c||F_{1}||_{h}\leq\mathrm{cst}.\frac{||\varphi||_{h}^{2}}{h^{2}(h-h_{1})^{2}}.$

Hence we deduce that

$||\varphi_{1}+F_{1}||_{h}\leq||\varphi_{1}+\Pr
F_{1}||_{h}+||(I-\Pr)F_{1}||_{h}\leq c||\varphi_{1}+\Pr
F_{1}||_{L^{2}}+\mathrm{cst}.||F_{1}||_{h}\leq\mathrm{cst}.\frac{||\varphi||_{h}^{2}}{h^{2}(h-h_{1})^{2}}.$

Let $\varphi^{1}=\varphi_{1}+F_{1},\varphi^{0}=\varphi,h_{0}=h$. Inductively,
we can find $h_{j}=\frac{h}{2}+\frac{h}{2^{j+1}}$, $c_{j}\in\mathbb{R}$,
$\varphi^{j},\psi_{j}\in
C_{h_{n}}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$, $j=1,\cdots,n$
such that

$\displaystyle\exp(\psi_{j}(x+\alpha))\exp(2\pi(dx+c_{j-1})J_{1})\exp(\varphi^{j-1}(x))\exp(-\psi_{j}(x))$
$\displaystyle=$ $\displaystyle\exp(2\pi(dx+c_{j})J_{1})\exp(\varphi^{j}(x)),$

with estimates

$||\varphi^{j}||_{h_{j}}\leq\mathrm{cst}.\frac{||\varphi^{j-1}||_{h_{j-1}}^{2}}{h_{j-1}^{2}(h_{j-1}-h_{j})^{2}},\quad
j=1,\cdots,n.$

Now we deal with $\varphi^{n}$. By Corollary 4.1, there exists
$c_{n+1}\in\mathbb{R}$, $\psi_{n+1},F_{n+1}\in
C_{h_{n+1}}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$,
$h_{n+1}=\frac{h}{2}+\frac{h}{2^{n+1}}<h_{n}$, such that for
$\varphi_{n+1}=\mathcal{P}_{\alpha,d,c_{n}}\varphi^{n},$

$\displaystyle\exp(\psi_{n+1}(x+\alpha))\exp(2\pi(dx+c_{n})J_{1})\exp(\varphi^{n}(x))\exp(-\psi_{n+1}(x))$
$\displaystyle=$
$\displaystyle\exp(2\pi(dx+c_{n+1})J_{1})\exp(\varphi_{n+1}(x)+F_{n+1}(x)),$

with estimates

$||\varphi_{n+1}||_{h_{n+1}}\leq\mathrm{cst}.\frac{||\varphi_{n}||_{h_{n}}}{h_{n}},\quad||F_{n+1}||_{h_{n+1}}\leq\mathrm{cst}.\frac{||\varphi^{n}||_{h_{n}}^{2}}{h_{n}^{2}(h_{n}-h_{n+1})^{2}}.$

Then Lemma 4.3 gives that

$||\Pr(\varphi_{n+1}+F_{n+1})||_{L^{2}}\leq
C||(I-\Pr)\partial(\varphi_{n+1}+F_{n+1})||_{L^{2}}.$ (4.2.4)

As $\varphi_{n+1}=\mathcal{P}_{\alpha,d,c}\varphi_{n}$, we have
$\varphi_{n+1}=\Pr\varphi_{n+1}$. Then (4.2.4) gives that

$||\varphi_{n+1}+\Pr F_{n+1}||_{L^{2}}\leq C||(I-\Pr)\partial
F_{n+1}||_{L^{2}}\leq
C||F_{n+1}||_{h}\leq\mathrm{cst}.\frac{||\varphi^{n}||_{h_{n}}^{2}}{h_{n}^{2}(h_{n}-h_{n+1})^{2}}.$

Hence we deduce that

$\displaystyle||\varphi_{n+1}+F_{n+1}||_{h}$ $\displaystyle\leq$
$\displaystyle||\varphi_{n+1}+\Pr F_{n+1}||_{h}+||(I-\Pr)F_{n+1}||_{h}$
$\displaystyle\leq$ $\displaystyle C||\varphi_{n+1}+\Pr
F_{n+1}||_{L^{2}}+\mathrm{cst}.||F_{n+1}||_{h_{n+1}}\leq\mathrm{cst}.\frac{||\varphi||_{h_{n}}^{2}}{h_{n}^{2}(h_{n}-h_{n+1})^{2}}.$

Let $\varphi^{n+1}=\varphi_{n+1}+F_{n+1}$. Then we have

$||\varphi^{n+1}||_{h_{n+1}}\leq\mathrm{cst}.\frac{||\varphi^{n}||_{h_{n}}^{2}}{h_{n}^{2}(h_{n}-h_{n+1})^{2}}=\mathrm{cst}.2^{2n}||\varphi^{n}||^{2}_{h_{n}}.$
(4.2.5)

Let $\epsilon_{n}=||\varphi^{n}||_{h_{n}}$. We have the following result for
the convergence of the sequence.

######  Lemma 4.4.

Let the positive real sequence $\\{\epsilon_{j}\\}$ satisfy

$\epsilon_{j+1}\leq c4^{j}\epsilon_{j}^{2},j\in\mathbb{N}$ (4.2.6)

for some constant $c>0$. If $\epsilon_{0}$ small enough, then there exists
$\delta>0$ such that

$\epsilon_{j}\leq e^{-2^{j}\delta},\quad j\in\mathbb{N}.$

###### Proof.

Let $v_{j}=\ln\epsilon_{j}$, $\eta_{j}=\ln(c4^{j})=c+j\ln 4$. Then (4.2.6)
gives

$v_{j+1}\leq 2v_{j}+\eta_{j}.$

Inductively we get

$v_{j+1}\leq
2^{j+1}(\frac{\eta_{j}}{2^{j+1}}+\cdots+\frac{\eta_{1}}{2^{2}}+\frac{\eta_{0}}{2}+v_{0}).$

Notice that $\sum_{j=0}^{\infty}\frac{\eta_{j}}{2^{j+1}}$ converges. Denote
the limit by $\eta_{\infty}$. Choose $v_{0}<-\eta_{\infty}$, then

$v_{j}<2^{j}(v_{0}+\eta_{\infty})<0,\quad j\in\mathbb{N}.$

Let $\delta=|v_{0}+\eta_{\infty}|>0$. Therefore, $\epsilon_{j}=e^{v_{j}}\leq
e^{-2^{j}\delta}$,which completes the proof. ∎

We choose $\epsilon$ be small enough. Then we could inductively define
$\varphi^{n}\in\mathcal{B}_{h_{n}}(\epsilon)$ satisfying

$||\varphi^{n}||_{h_{n}}\leq e^{-2^{n}\delta},\quad n\in\mathbb{Z}.$

Therefore,

$||\psi_{n+1}||_{h_{n+1}}\leq\mathrm{cst}.\frac{||\varphi^{n}||_{h_{n}}}{h_{n}(h_{n}-h_{n+1})}\leq\mathrm{cst}.2^{n}e^{-2^{n}\delta},\quad
n\in\mathbb{Z}.$ (4.2.7)

Let $\tilde{\psi}_{n}\in
C_{h_{n}}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$ such that
$\exp(\tilde{\psi}_{n})=\exp(\psi_{1})\cdots\exp(\psi_{n})$. Then (4.2.7)
gives that $\tilde{\psi}_{n}$ converges in
$C_{\frac{h}{2}}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$. We denote
the limit by $\tilde{\psi}$. As
$||c_{n+1}-c_{n}||=||\mathcal{P}_{0}\varphi^{n}||\leq||\varphi^{n}||_{h_{n}}$,
$c_{n}$ converges. We denote the limite by $\tilde{c}$. By (4.2),

$\exp(-\tilde{\psi}(x+\alpha))\exp(2\pi(dx+c)J_{1})\exp(\varphi(x))\exp(-\tilde{\psi}(x))=\exp(2\pi(dx+\tilde{c})J_{1}),$

i.e., $(\alpha,A)$ is $C_{\frac{h}{2}}^{\omega}$-conjugate to
$(\alpha,\exp(2\pi(dx+\tilde{c})J_{1}))$. ∎

##  5 Proof of main results

###  5.1 Proof of Theorem B

In this subsection, we establish the equivalence of acceleration and degree.
We firstly study the accelerations along renormalization, which can be seen as
generalization of Corollary 14 of [4].

######  Lemma 5.1.

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, $G$ be a subgroup of
$\mathrm{GL}(s,\mathbb{R})$, $s\in\mathbb{N}^{*}$, $A\in
C_{h}^{\omega}(\mathbb{T},G)$, $(\alpha_{n},\tilde{A}^{(n)})$ be a
representative of $n^{th}$ renormalization, $n\in\mathbb{N}$. Then

$L^{i}(\alpha_{n},\tilde{A}^{(n)}_{\epsilon})=\frac{1}{\beta_{n-1}}L^{i}(\alpha,A_{\beta_{n-1}\epsilon}),\quad
i=1,2,\cdots,s.$

Consequently, we have

$w^{i}(\alpha,A)=w^{i}(\alpha_{n},\tilde{A}^{(n)}),\quad i=1,2,\cdots,s.$

###### Proof.

Let $N\in C^{\omega}_{h}(\mathbb{R},\mathrm{GL}(s,\mathbb{R}))$ be a
normalizing map of $n$th renormalization of $(\alpha,A)$, i.e.,

$N(x+1)A_{(-1)^{n-1}q_{n-1}}(x_{0}+\beta_{n-1}x)N(x)^{-1}=I,$
$N(x+\alpha_{n})A_{(-1)^{n}q_{n}}(x_{0}+\beta_{n-1}x)N(x)^{-1}=\tilde{A}^{(n)}.$

Then for any $k,l\in\mathbb{Z}$.

$A_{k(-1)^{n}q_{n}+l(-1)^{n-1}q_{n-1}}(x_{0}+\beta_{n-1}x)=N(x+k\alpha_{n}+l)^{-1}\tilde{A}^{(n)}_{k}N(x),$
(5.1.1)

As there exists a canonical ring endomorphism from
$\mathrm{End}(\mathbb{R}^{s})$ to $\mathrm{End}(\Lambda^{k}\mathbb{R}^{s})$,
we have for $P,Q\in\mathrm{GL}(s,\mathbb{R})$,

$\Lambda^{i}(PQ)=\Lambda^{i}P\cdot\Lambda^{i}Q.$

Hence by (5.1.1), we have

$(\Lambda^{i}A)_{k(-1)^{n}q_{n}+l(-1)^{n-1}q_{n-1}}(x_{0}+\beta_{n-1}x)=(\Lambda^{i}N(x+k\alpha_{n}+l))^{-1}(\Lambda^{i}\tilde{A}^{(n)})_{k}\Lambda^{i}N(x).$

Let $\epsilon_{0}>0$ be such that $\tilde{A}^{(n)}\in
C^{\omega}_{\epsilon_{0}}(\mathbb{T},G)$ and $N$ admits an analytic extension
to a open neighborhood of $\mathbb{R}$ containing
$V=[0,2]\times[-\epsilon_{0},\epsilon_{0}]$. Let $\Lambda_{0}=\sup_{z\in
V}||\Lambda^{i}N(z)||^{2}.$ For $k\in\mathbb{Z}$, let $l=l(k)$ be the unique
integer such that $0\leq k\alpha^{\prime}+l<1$ and
$t=t(k)=k(-1)^{n}q+l(-1)^{n-1}q_{n-1}$, then we have

$\Lambda_{0}^{-1}\leq\frac{\|(\Lambda^{i}A)_{t}(y+\beta_{n-1}\epsilon
i)\|}{\|(\Lambda^{i}\tilde{A}^{(n)})_{k}(x+\epsilon i)\|}\leq\Lambda_{0},$

where $x,y\in\mathbb{C}/\mathbb{Z}$ are related by $y=x_{0}+\beta_{n-1}x$ and
we assume that $|\Im(x+\epsilon i)|<\epsilon_{0}.$ It follows that

$\displaystyle|\int_{\mathbb{T}}\ln||(\Lambda^{i}\tilde{A}^{(n)})_{k}(x+\epsilon
i)||dx-\int_{\mathbb{T}}\ln||(\Lambda^{i}A)_{(-1)^{n}t}(x+\beta_{n-1}\epsilon
i)||dx|$ $\displaystyle=$
$\displaystyle|\int_{\mathbb{T}}\ln||(\Lambda^{i}\tilde{A}^{(n)})_{k}(x+\epsilon
i)||dx-\int_{\mathbb{T}}\ln||(\Lambda^{i}A)_{t}(x+\beta_{n-1}\epsilon
i)||dx|\leq\ln\Lambda_{0}.$

Notice that when $k$ is large, $t$ satisfies

$(-1)^{n}\frac{t}{k}=q_{n}-\frac{l}{k}q_{n-1}=q_{n}+\alpha_{n}q_{n-1}+o(1)=\frac{1}{\beta_{n-1}}+o(1).$

It follows that for large $k$,

$\frac{1}{k}\int_{\mathbb{T}}\ln||(\Lambda^{i}\tilde{A}^{(n)})_{k}(x+\epsilon
i)||dx=\frac{1+o(1)}{\beta_{n-1}}\frac{1}{(-1)^{n}}\int_{\mathbb{T}}\ln||(\Lambda^{i}A)_{(-1)^{n}t}(x+\beta_{n-1}\epsilon
i)||dx.$

Taking the limit, we get

$L^{i}(\alpha_{n},\tilde{A}^{(n)}_{\epsilon})=\frac{1}{\beta_{n-1}}L^{i}(\alpha,A_{\beta_{n-1}\epsilon}),\quad
i=1,2,\cdots,s,$

which immediately implies $w^{i}(\alpha,A)=w^{i}(\alpha_{n},\tilde{A}^{(n)})$.
∎

With preceding lemma, now we can prove Theorem B by renormalization.

###### Proof of Theorem B.

Let $\tilde{A}^{(n)}_{t}=\tilde{A}^{(n)}(\cdot+it)$, $t\in\mathbb{R}$. We
rewrite the convergence of renormalization from Theorem 2.1 to be

$\tilde{A}^{(n_{k})}_{t}\xrightarrow{C^{\omega}}\exp(C+D(x+it))\quad\text{ on
}\mathbb{R},\quad t\in[-h,h],$ (5.1.2)

for some subsequence of even integers $\\{n_{k}\\}$ and $C\in\mathfrak{z}(D)$.
We assume that $L^{i}(\alpha,A)$ is affine on $(0,h_{0})$ for some $h_{0}>0$.
By Lemma 5.1, $L^{i}(\alpha_{n},\tilde{A}^{(n)})$ is also affine on
$(0,h_{0})$ and

$\omega^{i}(\alpha_{n},\tilde{A}^{(n)})=\omega^{i}(\alpha,A)=:\omega^{i},\quad
i=1,\cdots,s,\quad n\in\mathbb{N}.$

By compactness of $\mathrm{SO}(s,\mathbb{R})$,
$L_{i}(\alpha_{n},\tilde{A}^{(n)})=0$. We deduce that for $t\in(0,h_{0})$,

$L_{i}(\alpha_{n},\tilde{A}^{(n)}_{t})=2\pi t\omega_{i},\quad
i=1,\cdots,s,\quad n\in\mathbb{N}.$

By continuity of Lyapunov exponent [7], (5.1.2) gives that

$L_{i}(\alpha,\exp(C+D(x+it)))=\lim_{n\to\infty}L_{i}(\alpha_{n},\tilde{A}^{(n)}_{t})=2\pi
t\omega_{i},\quad i=1,\cdots,s,\quad t\in(0,h_{0}).$ (5.1.3)

Meanwhile, we recall the definition of Lyapunov exponents through singular
value. As $C,D\in\mathfrak{so}(s,\mathbb{R})$ commute with each other, $C$
will not contribute in $L_{i}(\alpha,\exp(C+D(x+it)))$. Moreover, compactness
shows that there exist $Q\in\mathrm{SU}(s)$ and integers $d_{1}\geq\cdots\geq
d_{s}$ such that $D=Q\cdot 2\pi i\mathrm{diag}(d_{1},\cdots,d_{s})\cdot
Q^{-1}$. We then compute explicitly that

$L_{i}(\alpha,\exp(C+D(x+it)))=2\pi td_{i},\quad i=1,\cdots,s.$

Comparing with (5.1.3), we deduce that

$(\omega_{1},\cdots,\omega_{s})=(d_{1},\cdots,d_{s}),$

which completes the proof of Theorem B. ∎

###  5.2 Proof of Theorem A (1) $\omega_{1}=0$

Now we start to prove Theorem A. Since the case when $\omega_{1}$ zero or not
has different dynamical nature, we separate the proof of Theorem A into two
subsections. We deal with $\omega_{1}=0$ here. To study almost reducibility
along renormalization, we need the following lemma.

######  Lemma 5.2.

Let $h>0$, $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, $A\in
C_{h}^{\omega}(\mathbb{T},\mathrm{SO}(3,\mathbb{R}))$, $n\in\mathbb{N}$. If a
representative of $n$th deep renormalization $(\alpha_{n},\tilde{A}^{(n)})$
satisfies $||\tilde{A}^{(n)}-A_{0}||_{h}<\delta(h)$ for some constant
$A_{0}\in\mathrm{SO}(3,\mathbb{R})$, then $(\alpha,A)$ is $C^{\omega}$-almost
reducible and accumulated by reducible cocycles.

###### Proof.

We apply Theorem 3.1 for $h>0$ and $L=4$, denoting $\delta(h,4)$ briefly by
$\delta(h)$. when $||\tilde{A}^{(n)}-A_{0}||_{h}<\delta(h)$, Theorem 3.1 gives
that $(\alpha_{n},\tilde{A}^{(n)})$ is $C^{\omega}$-almost reducible. More
precisely, for $j\in\mathbb{N}$, there exist $h_{j}>0$, $B_{j}\in
C_{h_{j}}^{\omega}(\mathbb{T},\mathrm{SO}(3,\mathbb{R}))$,
$C_{j}\in\mathrm{SO}(3,\mathbb{R})$, such that for
$\tilde{A}_{j}=B_{j}(\cdot+\alpha_{n})\tilde{A}^{(n)}(\cdot)B_{j}(\cdot)^{-1}$,
$\epsilon_{j}=\|B_{j}\|_{h_{j}}^{4}\cdot||\tilde{A}_{j}-C_{j}||_{h_{j}}$, we
have

$\epsilon_{j}\rightarrow 0,\qquad\hbox{as $j\rightarrow\infty$.}$ (5.2.1)

We will work with large $j$ so that $\epsilon_{j}$ small.

Let
$(\alpha_{n},\tilde{A}_{j})^{m}=(m\alpha_{n},\tilde{A}_{j,m}),m\in\mathbb{N}$.
As

$\displaystyle||\tilde{A}_{j,m+1}(x)-C_{j}^{m+1}||_{h_{j}}$
$\displaystyle=||\tilde{A}_{j}(x+m\alpha)(\tilde{A}_{j,m}(x)-C_{j}^{m})+(\tilde{A}_{j}(x+m\alpha)-C_{j})C_{j}^{m}||_{h_{j}}$
$\displaystyle\leq(1+\epsilon_{j})||\tilde{A}_{j,m}(x)-C_{j}^{m}||_{h_{j}}+\epsilon_{j},$

inductively we deduce that
$||\tilde{A}_{j,m}(x)-C_{j}^{m}||\leq(1+\epsilon_{j})^{m}-1$. We choose $j$
large enough so that $\epsilon_{j}<<\frac{1}{q_{n}}$. Then we have

$||\tilde{A}_{j,q_{n}}-C_{j}^{q_{n}}||_{h}\leq q_{n}\epsilon_{j}.$ (5.2.2)

Now we define the distance for commuting pairs. We start with a notation of
maximal norm. Let $I\subset\mathbb{R}$ be an interval, $h>0$, $f:I\times
i[-h,h]\to\mathrm{SO}(3,\mathbb{R})$ be a continuous map. We define

$||f||_{h,I}=\sup_{x\in I\times i[-h,h]}||f(x)||.$

Let $\Lambda^{\omega}_{h,I}$ be the set of all $\Phi\in\Lambda^{\omega}$ with
$A^{\Phi}_{1,0},A^{\Phi}_{0,1}$ analytically defined on a set containing
$I\times i[-h,h]$. For $\Phi\in\Lambda^{\omega}_{h,I}$, we denote by
$||\Phi||_{h,I}$ the quantities

$||\Phi||_{h,I}:=\sup(||A^{\Phi}_{1,0}||_{h,I},||A^{\Phi}_{0,1}||_{h,I}),$

and then define a distance on $\Lambda^{\omega}_{h,I}$ as

$d_{h,I}(\Phi_{1},\Phi_{2}):=\sup(||A_{1,0}^{\Phi_{1}}-A_{1,0}^{\Phi_{2}}||_{h,I},||A_{0,1}^{\Phi_{1}}-A_{0,1}^{\Phi_{2}}||_{h,I}),\quad\Phi_{1},\Phi_{2}\in\Lambda^{\omega}_{h,I}.$

If $\Phi,\Phi^{\prime}\in\Lambda_{h,I}^{\omega}$ are $C^{\omega}$-conjugated
via $B\in C^{\omega}(\mathbb{R},\mathrm{SO}(s,\mathbb{R}))$ analytically
defined on a set containing $I\times i(-h,h)$, then

$||\Phi^{\prime}||_{h,I}\leq||B^{-1}||_{h,I}\max(||B||_{h,I+\gamma^{\Phi}_{1,0}},||B||_{I+\gamma^{\Phi}_{0,1}})||\Phi||_{h,I}.$

Let $\Phi_{1},\Phi_{2}\in\Lambda_{h,I}^{\omega}$, $B\in
C^{\omega}(\mathbb{R},\mathrm{SO}(3,\mathbb{R}))$ analytically defined on a
set containing $I\times i(-h,h)$,
$\Phi_{i}^{\prime}=\mathrm{Conj}_{B}(\Phi_{i})$, $i=1,2$, then

$d_{h,I}(\Phi_{1}^{\prime},\Phi_{2}^{\prime})\leq||B^{-1}||_{h,I}\max(||B||_{h,I+\gamma^{\Phi}_{1,0}},||B||_{h,I+\gamma^{\Phi}_{0,1}})d_{h,I}(\Phi_{1},\Phi_{2}).$
(5.2.3)

Let $\Phi_{1}=\mathrm{Conj}_{B_{j}}\begin{pmatrix}1,I\\\
\alpha_{n},\tilde{A}^{(n)}\end{pmatrix}$, $\Phi_{2}=\begin{pmatrix}1,I\\\
\alpha_{n},C_{j}\end{pmatrix}$. Then
$d_{h_{j},I}^{\omega}(\Phi_{1},\Phi_{2})\leq\epsilon_{j}$ for any interval
$I\subset\mathbb{R}$. It is easy to see that

$d_{h_{j},\beta_{n-1}I}^{\omega}(M_{\beta_{n-1}^{-1}}(\Phi_{1}),M_{\beta_{n-1}^{-1}}(\Phi_{2}))=\beta_{n-1}^{-1}d_{h_{j},I}^{\omega}(\Phi_{1},\Phi_{2}).$
(5.2.4)

By (5.2.2),

$d_{h_{j},\beta_{n-1}I}^{\omega}(N_{Q_{n}^{-1}}\circ
M_{\beta_{n-1}^{-1}}(\Phi_{1}),N_{Q_{n}^{-1}}\circ
M_{\beta_{n-1}^{-1}}(\Phi_{2}))\leq q_{n}\beta_{n-1}^{-1}\epsilon_{j}.$
(5.2.5)

Assume that $N\in C^{\omega}(\mathbb{R},\mathrm{SO}(3,\mathbb{R}))$ is the
normalizing map such that

$\begin{pmatrix}1,I\\\
\alpha_{n},\tilde{A}^{(n)}\end{pmatrix}=\mathrm{Conj}_{N}\circ
M_{\beta_{n-1}}\circ N_{Q_{n}}\begin{pmatrix}1,I\\\ \alpha,A\end{pmatrix}.$

Let
$\Phi_{3}:=\mathrm{Conj}_{N^{-1}(\beta_{n-1}^{-1}\cdot)}\circ\mathrm{Conj}_{B_{j}^{-1}(\beta_{n-1}^{-1}\cdot)}\circ
N_{Q_{n}^{-1}}\circ M_{\beta_{n-1}^{-1}}(\Phi_{2})$,
$I=\beta_{n-1}^{-1}[-\frac{1}{2},1]$. Then by (5.2.3) and (5.2.5),

$\displaystyle d_{h_{j},[-\frac{1}{2},1]}^{\omega}(\begin{pmatrix}1,I\\\
\alpha,A\end{pmatrix},\Phi_{3})\leq||N|_{[0,2\beta_{n-1}^{-1}]}||_{h_{j}}^{2}q_{n}\beta_{n-1}^{-1}||B_{j}||_{h_{j}}^{2}\epsilon_{j}=:\bar{\epsilon}_{j},$
(5.2.6)

By (5.2.1), $\bar{\epsilon}_{j}\to 0$ as $j\to\infty$. We work with $j$ large
so that $\bar{\epsilon}_{j}$ small enough.

To continue analysis, we recall the quantitative normalizing lemma, based on
normalizing lemma in [10], with a detailed proof in [47].

######  Lemma 5.3.

For any $\Phi\in\Lambda_{0}^{\omega}$, $\Phi$ is $C^{\omega}$-conjugated to a
normalized action. Moreover, there exist an $\epsilon>0$ and a constant
$\mathrm{cst}$ such that for any $\Phi\in\Lambda_{0}^{\omega}$ satisfying
$||A_{1,0}^{\Phi}-I||_{h,{[-\frac{1}{2},1]}}\leq\epsilon$, for some $h>0$, we
can construct a normalizing map $B\in
C^{\omega}(\mathbb{R},\mathrm{SO}(3,\mathbb{R}))$ such that

$B(\cdot+1)A_{1,0}^{\Phi}(\cdot)B(\cdot)^{-1}=I,\quad||B-I||_{h,[0,2]}\leq\mathrm{cst}.||A_{1,0}^{\Phi}-I||_{h,[-\frac{1}{2},1]}.$

Notice that $\Phi_{3}$ is not necessarily normal. Since (5.2.6) gives that

$||A^{\Phi_{3}}_{1,0}-I||_{h_{j}}\leq\bar{\epsilon}_{j},$

we can construct from Lemma 5.3 a conjugacy $\tilde{B}\in
C^{\omega}_{h_{j}}(\mathbb{R},\mathrm{SO}(3,\mathbb{R}))$ such that

$\tilde{B}(\cdot+1)A^{\Phi_{3}}_{1,0}\tilde{B}(\cdot)^{-1}=I$

and satisfying

$||\tilde{B}-I||_{h_{j},[0,2]}\leq\mathrm{cst}.\bar{\epsilon}_{j}.$

As we can express $\tilde{B}^{-1}(x)$ by entries of $\tilde{B}(x)$, we get

$||\tilde{B}^{-1}-I||_{h_{j},[0,2]}\leq\mathrm{cst}.\bar{\epsilon}_{j}.$

Then

$\displaystyle||\tilde{B}(x+\alpha)A^{\Phi_{3}}_{0,1}(x)\tilde{B}(x)^{-1}-A^{\Phi_{3}}_{0,1}||_{h_{j},[0,1]}$
$\displaystyle=$
$\displaystyle||\tilde{B}(x+\alpha)A^{\Phi_{3}}_{0,1}(\tilde{B}(x)^{-1}-I)+(\tilde{B}(x+\alpha)-I)A_{0,1}^{\Phi_{3}}(x)||_{h_{j},[0,1]}\leq\mathrm{cst}.\bar{\epsilon}_{j},$

as $\bar{\epsilon}_{j}$ is small enough. Together with (5.2.6), we deduce that

$d_{h_{j},[0,1]}(\begin{pmatrix}1,I\\\
\alpha,A\end{pmatrix},\mathrm{Conj}_{\tilde{B}}\Phi_{3})\leq\mathrm{cst}.\bar{\epsilon}_{j}.$
(5.2.7)

This means that there exist $\breve{A}_{j}\in
C^{\omega}_{h_{j}}(\mathbb{T},\mathrm{SO}(s,\mathbb{R})),\breve{B}_{j}\in
C^{\omega}_{h_{j}}(\mathbb{T},\mathrm{SO}(s,\mathbb{R}))$, where
$\breve{B}_{j}(\cdot)=\tilde{B}_{j}(\cdot)B_{j}(\beta_{n-1}^{-1}\cdot)N(\beta_{n-1}^{-1}\cdot)$
satisfies

$||\breve{B}_{j}||_{h_{j}}\leq\mathrm{cst}.||B_{j}||_{h_{j}},$ (5.2.8)

such that

$||A-\breve{A}_{j}||_{h_{j}}\leq\mathrm{cst}.\bar{\epsilon}_{j}$ (5.2.9)

and

$(\alpha,\breve{A}_{j})=(0,\breve{B}_{j})\circ(\alpha,C_{j})\circ(0,\breve{B}_{j})^{-1}.$
(5.2.10)

Thus $A$ is approximated by reducible cocycles. By (5.2.8), (5.2.9), (5.2.10)
and the convexity inequality, we have

$||\breve{B}_{j}(x+\alpha)^{-1}A(x)\breve{B}_{j}(x)-C_{j}||_{h_{j}}\leq\mathrm{cst}.||B_{j}||_{h_{j}}^{2}\bar{\epsilon}_{j}.$

By (5.2.1),
$||B_{j}||_{h_{j}}^{2}\bar{\epsilon}_{j}=||B_{j}||_{h_{j}}^{4}\epsilon_{j}\to
0$ as $j\to\infty$. Hence $(\alpha,A)$ is $C^{\omega}$-almost reducible. ∎

###### Proof of Theorem A (1).

We prove the equivalence of $\omega_{1}=0$ and almost reducibility for
analytic $\mathrm{SO}(3,\mathbb{R})$-cocycles.

If $(\alpha,A)$ is $C^{\omega}$-almost-reducible, i.e., there exists $B_{n}\in
C^{\omega}(\mathbb{T},\mathrm{SO}(3,\mathbb{R}))$,
$A_{0}\in\mathrm{SO}(3,\mathbb{R})$ such that

$B_{n}(x+\alpha)^{-1}A(x)B_{n}(x)\to A_{0}.$

As acceleration is invariant by conjugation and upper semi-continuous in
$C^{\omega}$-space, we deduce that $\omega_{1}=0.$

Conversely, if $\omega_{1}=0$, by Theorem B, the dynamical degree of
$(\alpha,A)$ is zero. Then Theorem 2.1 provides the convergence of
renormalization to constant. In other word, for $\delta(h)>0$, there exists
$n_{0}\in\mathbb{N}$, such that $\tilde{A}^{(n)}$ is $\delta(h)$ close to
constant. Lemma 5.2 therefore shows that $(\alpha,A)$ is $C^{\omega}$-almost
reducible.

∎

###  5.3 Proof of Theorem A (2) $\omega_{1}\neq 0$

To prove the main theorem, we will find a conjugation representative when
$\omega_{1}\neq 0$. As conjugation is preserved along renormalization, we
firstly compute how the normal form change along renormalization.

######  Lemma 5.4.

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$,
$D\in\mathfrak{so}(3,\mathbb{R})$ satisfying $\exp D=I$,
$C\in\mathfrak{z}(D)$. Then the cocycle $(\alpha,\exp(Dx+C))$ has
representative of renormalization
$(\\{\frac{1}{\alpha}\\},\exp(-Dx-D(\frac{1}{2\alpha}+\frac{1}{2})-\frac{C}{\alpha}))$.

###### Proof.

Let $a=a_{1}=\frac{1}{\alpha}-\\{\frac{1}{\alpha}\\},\Upsilon(x)=\exp(Dx+C).$
We start with the commuting pair

$M_{\alpha}\circ\begin{pmatrix}0&1\\\ 1&-a\end{pmatrix}\begin{pmatrix}(1,I)\\\
(\alpha,H)\end{pmatrix}=\begin{pmatrix}(1,\Upsilon(\alpha x))\\\
(\frac{1}{\alpha},I)\circ(1,\Upsilon(\alpha x))^{-a}\end{pmatrix}.$ (5.3.1)

We define $J(x)=\exp(-\frac{\alpha}{2}D(x^{2}-x)-Cx)$. Then

$\Upsilon(\alpha
x)=J(x+1)^{-1}J(x),\quad\mathrm{Conj}_{J}(1,\Upsilon(\alpha\cdot))=(1,I).$
(5.3.2)

Hence $J$ is a normalizing map of the one-step renormalization (5.3.1).

Now, inversing (5.3.2) and computing its $a$ power, we get

$\mathrm{Conj}_{J}(1,\Upsilon(\alpha\cdot))^{-a}=(1,I)^{-a}=(-a,I).$

Then we compute

$\displaystyle\mathrm{Conj}_{J}((\frac{1}{\alpha},I)\circ(1,\Upsilon(\alpha\cdot))^{-a})=$
$\displaystyle\mathrm{Conj}_{J}(\frac{1}{\alpha},I)\circ\mathrm{Conj}_{J}(1,\Upsilon(\alpha\cdot))^{-a}$
$\displaystyle=$
$\displaystyle(0,J)\circ(\frac{1}{\alpha},I)\circ(0,J)^{-1}\circ(-a,I)$
$\displaystyle=$
$\displaystyle(\\{\frac{1}{\alpha}\\},J(x+\\{\frac{1}{\alpha}\\})J(x-a)^{-1}).$
(5.3.3)

Let

$\displaystyle\tilde{\Upsilon}(x)=$ $\displaystyle
J(x+\\{\frac{1}{\alpha}\\})J(x-a)^{-1}=\exp(-D(x+\frac{1}{2}+\frac{1}{2\alpha})-\frac{C}{\alpha}).$
(5.3.4)

We then get a representative $(\\{\frac{1}{\alpha}\\},\tilde{\Upsilon})$ of
one-step renormalization of $(\alpha,\Upsilon)$,

$\begin{pmatrix}(1,I)\\\
(\\{\frac{1}{\alpha}\\},\tilde{\Upsilon}(x))\end{pmatrix}=\mathrm{Conj}_{J(\alpha,d,c)}\circ
M_{\alpha}\circ\begin{pmatrix}0&-1\\\ -1&a\end{pmatrix}\begin{pmatrix}(1,I)\\\
(\alpha,\Upsilon)\end{pmatrix}.$

∎

###### Proof of Theorem A (2).

When $\omega_{1}\neq 0$, by Theorem B, we only need to prove that if dynamical
degree $D$ is nonzero, then the cocycle $(\alpha,A)$ is $C^{\omega}$-conjugate
to $(\alpha,\exp(Dx+C))$ for some $C\in\mathfrak{z}(D)$.

By Theorem 2.1, we have convergence of renormalization, namely for a sequence
of $C_{n}\in\mathfrak{z}(D)$, $\varphi_{n}\in
C_{h}^{\omega}(\mathbb{T},\mathfrak{so}(3,\mathbb{R}))$,

$\tilde{A}^{(n)}(x)=\exp(\varphi_{n}(x))\exp((-1)^{n}Dx+C_{n}),$
$\lim_{n\to\infty}||\varphi_{n}||_{h}=0.$ (5.3.5)

As $\alpha\in\mathrm{RDC}$, there exist $\kappa>0,\tau>0$ such that
$\alpha_{n}\in\mathrm{DC}(\kappa,\tau)$ for infinitely many $n\in\mathbb{N}$.
For such $n$ large enough, by (5.3.5), Theorem 4.1 gives that
$(\alpha_{n},\tilde{A}^{(n)})$ is $C^{\omega}$-conjugate to
$(\alpha_{n},\exp((-1)^{n}Dx+C^{\prime}))$ for some
$C^{\prime}\in\mathfrak{z}(D)$.

By Lemma 5.4, the form $(\alpha,\exp(\pm Dx+\cdot))$ with
$\cdot\in\mathfrak{z}(D)$ is preserved by renormalization. Hence there exists
$C\in\mathfrak{z}(D)$ such that $(\alpha,\exp(Dx+C))$ has a representative of
$n$th renormalization as $(\alpha_{n},\exp((-1)^{n}Dx+C^{\prime}))$.

In other word, $(\alpha,A)$ has a representative of $n$th renormalization
$(\alpha_{n},\tilde{A}^{(n)})$ conjugate to
$(\alpha_{n},\exp((-1)^{n}Dx+C^{\prime}))$, a representative of $n$th
renormalization of $(\alpha,\exp(Dx+C))$. Then Lemma 2.3 gives that
$(\alpha,A)$ is $C^{\omega}$-conjugate to $(\alpha,\exp(Dx+C))$. Hence we
complete the proof of Theorem A. ∎

##  Appendix A Floquet Theory

We introduce the following quantitative Floquet Theory. In [31], we have
introduced a quantitative Floquet Theory for quasi-periodic linear system on
$\mathrm{SL}(2,\mathbb{R})$. Here we consider the case of
$\mathrm{SO}(3,\mathbb{R})$, but the conclusion and the proof also hold for
general compact matrix group.

######  Lemma A.1.

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, suppose that
$F=\sum_{l\in\mathbb{Z}}\hat{F}(lq,-lp)e^{2\pi
il(q\theta_{1}-p\theta_{2})}\in\mathcal{B}_{h}$. Then there exists $B\in
C^{\omega}_{h}(\mathbb{T}^{2},\mathrm{SO}(3,\mathbb{R}))$ which conjugates the
system

$\begin{cases}\dot{x}=F(\theta)x\\\ \dot{\theta}=\omega=(\alpha,1)\end{cases}$

to the constant system

$\begin{cases}\dot{x}=Cx\\\ \dot{\theta}=\omega=(\alpha,1)\end{cases}$

with estimates

$||B||_{h}\leq\exp\left\\{\frac{(|q|+|p|)h}{|\tau|}||F||^{\\#}_{h}\right\\},\qquad\tau:=q\alpha-p.$
(A.0.1)

###### Proof.

Let $x=q\theta_{1}-p\theta_{2},\tilde{h}=(|q|+|p|)h$,
$G(\phi)=\sum_{l\in\mathbb{Z}}\hat{F}(lq,-lp)e^{2\pi il\phi}$. Then we have
$F(\theta_{1},\theta_{2})=G(q\theta_{1}-p\theta_{2})$, and

$||G||_{\tilde{h}}\leq\sum_{l\in\mathbb{Z}}||\hat{F}(lq,-lp)||e^{2\pi|l|\tilde{h}}=||F||^{\\#}_{h}.$
(A.0.2)

Let $\Phi(t)$ be its basic matrix solution of the $\frac{1}{|\tau|}$-periodic
ODE

$\frac{dx}{dt}=G(\tau t)x$ (A.0.3)

with $\Phi(0)=I$. Let
$C:=|\tau|\log\Phi(\frac{1}{|\tau|})\in\mathfrak{so}(3,\mathbb{R})$, i.e., $C$
satisfies

$\Phi(\frac{1}{|\tau|})=\exp(\frac{1}{|\tau|}\cdot
C),\qquad\|C\|\leq||G||_{\tilde{h}}\leq||F||^{\\#}_{h}.$ (A.0.4)

We consider the complexification of (A.0.3), and consider the case of $t+is$
with some fixed $t\in\mathbb{R}$ and $|s|<\frac{\tilde{h}}{|\tau|}$,

$\frac{dx}{d(is)}=G(t+\tau si)x.$ (A.0.5)

Then, by Grownwall inequality and (A.0.2), as long as
$|s|<\frac{\tilde{h}}{|\tau|}$,

$\|\Phi(t+is)\|\leq\|\Phi(t)\|e^{|s|||G||_{\tilde{h}}}\leq\|\Phi(t)\|e^{|s|||F||^{\\#}_{h}}\leq
e^{|s|||F||^{\\#}_{h}},$ (A.0.6)

($\|\Phi(t)\|=1$ for $t\in\mathbb{R}$ and $\Phi(t)$ is then
$\mathrm{SO}(3,\mathbb{R})-$valued). Then,

$\sup_{|\Im
t|<\frac{\tilde{h}}{|\tau|}}||\Phi(t)||\leq\exp\left\\{\frac{||F||^{\\#}_{h}\tilde{h}}{|\tau|}\right\\}\leq\exp\left\\{\frac{(|q|+|p|)h}{|\tau|}||F||^{\\#}_{h}\right\\}.$
(A.0.7)

And, for fixed $t\in\mathbb{R}$ and $|s|<\frac{\tilde{h}}{|\tau|}$, by (A.0.4)

$\|e^{(t+is)C}\|\leq\|e^{tC}\|\|e^{isC}\|\leq|\exp\\{|s|\|C\|\\}\leq\|e^{|s|||F||^{\\#}_{h}}$
(A.0.8)

($C\in\mathfrak{so}(3,\mathbb{R})$ and $t\in\mathbb{R}$ means that
$e^{tC}\in\mathrm{SO}(3,\mathbb{R})$), which leas to

$\sup_{|\Im
t|<\frac{\tilde{h}}{|\tau|}}\|e^{tC}\|\leq\exp\left\\{\frac{||F||^{\\#}_{h}\tilde{h}}{|\tau|}\right\\}\leq\exp\left\\{\frac{(|q|+|p|)h}{|\tau|}||F||^{\\#}_{h}\right\\}.$
(A.0.9)

Let $B_{1}(t)=\Phi(t)\exp(-Ct)$. By the definition of $C$,
$B_{1}(t)=B_{1}(t+\frac{1}{|\tau|})$, and $B_{1}$ conjugate (A.0.3) to
$\frac{dx}{dt}=Cx$. It means

$B_{1}^{\prime}(t)=B_{1}(t)G(\tau t)-CB_{1}(t).$ (A.0.10)

Define
$B(\theta_{1},\theta_{2}):=B_{1}(\frac{1}{\tau}(q\theta_{1}-p\theta_{2}))\in
C_{h}^{\omega}(\mathbb{T}^{2},\mathfrak{so}(3,\mathbb{R}))$. By (A.0.10), it
satisfies

$\displaystyle\partial_{\omega}B$ $\displaystyle=$
$\displaystyle\alpha\frac{\partial}{\partial\theta_{1}}B(\theta_{1},\theta_{2})+\frac{\partial}{\partial\theta_{2}}B(\theta_{1},\theta_{2})$
(A.0.11) $\displaystyle=$
$\displaystyle\frac{q\alpha-p}{\tau}\frac{d}{dt}B_{1}^{\prime}(\frac{1}{\tau}(q\theta_{1}-p\theta_{2}))=B_{1}^{\prime}(\frac{1}{\tau}(q\theta_{1}-p\theta_{2}))$
$\displaystyle=$ $\displaystyle
B_{1}(\frac{1}{\tau}(q\theta_{1}-p\theta_{2}))G(q\theta_{1}-p\theta_{2})-CB_{1}(\frac{1}{\tau}(q\theta_{1}-p\theta_{2}))$
$\displaystyle=$ $\displaystyle BF-CB.$

Thus $B$ conjugate $\begin{cases}\dot{x}=F(\theta)x\\\
\dot{\theta}=\omega\end{cases}$ to $\begin{cases}\dot{x}=Cx\\\
\dot{\theta}=\omega\end{cases}$. By (A.0.7, A.0.9), (A.0.1) follows ∎

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