Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 22, pp. 1–17.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

LINEARIZATION OF MULTI-FREQUENCY

QUASI-PERIODICALLY FORCED CIRCLE FLOWS BEYOND

BRJUNO CONDITION

ZIYANG LIANG, TAIAN JIN, JIAYI WANG, YUAN SHAN

Abstract. In this article, we considered the linearization of analytic quasi-

periodically forced circle flows. We generalized the rotational linearization

of systems with two-dimensional base frequency to systems with any finite
dimensional base frequency case. Meanwhile, we relaxed the arithmetical lim-

itations on the base frequencies. Our proof is based on a generalized Kol-

mogorov–Arnold–Moser (KAM) scheme.

1. Introduction and statement of main results

In this article, we consider the quasi-periodically forced (qpf) circle flow

ẋ = ρ+ f(φ, x),

φ̇ = Ω,
(1.1)

where f : Td ×T1 → R is a real analytic function with Td := Rd/Zd. Here, Ω ∈ Td
is rationally independent. We denote the system (1.1) by (Ω, ρ+ f) for simplicity.

The time discrete counterparts of the qpf circle flows are the qpf circle maps.
The research of qpf circle maps is an important topic in mathematical physics and
dynamical systems: The qpf circle maps are related to Arnold tongue, more specifi-
cally, Arnold circle map, which attempts to capture the motion of the spinning disks
at discrete time intervals. The qpf circle maps also provide a simple model of the
mode-locked loop in electronics, of mechanically musical instruments and of heart
issue. And the qpf circle maps appeared in the study of quasi-periodic crystals and
damped pendulum motions too. Moreover, the qpf circle maps can also be used
to investigate the quasi-periodic schrödinger operator [4], which is an important
mathematical model of the quantum Hall effect and many other quantum physics
problems.

In this paper, we mainly focus on the C∞ rotational linearization of (1.1) with
weak Liouvillean frequency, provided the analytic norm of f is small enough. We
say the system (Ω, ρ + f) is Cr (r = ∞, ω) linearizable or Cr reducible, if there
exists a Cr map H : Td × T1 → Td × T1 and ρ̃ ∈ R such that H conjugates
the system (Ω, ρ + f) to (Ω, ρ̃). And, if the qpf circle flow (Ω, ρ + f) can be Cr

(r = ∞, ω) conjugate to (Ω, g(φ)), then we say it is Cr rotation linearizable or

2010 Mathematics Subject Classification. 37C15, 34C20.
Key words and phrases. Linearization; quasi-periodically forced circle flow;

Liouvillean frequency.
c©2020 Texas State University.

Submitted September 4, 2019. Published March 12, 2020.
1



2 Z. LIANG, T. JIN, J. WANG, Y. SHAN EJDE-2020/22

reducible. Meanwhile, using the embedding result of You-Zhou [10], when f is
small enough, the discrete system can be embedded into the continuous case, that
is the qpf circle flows. This implies the equivalence of the linearization between the
discrete case and continuous case for the perturbation f small.

In recent years, the qpf circle flows have been studied extensively by many math-
ematicians. Herman [5] investigated system (1.1) with (Ω, ρf ) satisfying the Dio-
phantine condition

|〈k,Ω〉+ lρf | ≥
γ

(|k|+ |l|)τ
, ∀(k, l) ∈ Zd × Z, |k|+ |l| 6= 0, (1.2)

where ρf is the fibred rotation number of the system and proved the system is Cω

linearized provided the analytic norm of f is sufficiently small.

For a qpf circle flow (Ω, f), we say ρf = ρ(Ω, f) = limt→∞
Φ̂tφ(x̂)

t is the fibred

rotation number associated with (Ω, f), where Φ̂tφ(x̂) : R+ × Td × R → R via

(t, φ, x̂) 7→ Φ̂tφ(x̂) denotes the lift of the flow of (Ω, f) of the valuable x, and for any

ρ̃ ∈ R we have |ρ(Ω, ρ̃ + f) − ρ(Ω, ρ̃)| ≤ ε, provided that ‖f‖C0 ≤ ε small enough
[6].,

Note that, in (1.2), l = 0 implies that Ω is a Diophantine vector (depending
on τ and γ). Without the assumption of Ω being Diophantine, the problem is
quite different and there is not much work done so far. Recently, using the almost
reducibility theory, Krikorian-Wang-You-Zhou[9] managed to relax the Diophantine
assumption to non-super Liouvillean frequencies for d = 2, and get the rotational
linearization. More precisely, let Ω = (1, α) with α ∈ R\Q and {pnqn } be the best

convergence of α. Assume that α is not super-Liouvillean, that is

sup
n>0

ln ln qn+1

ln qn
< +∞. (1.3)

Then for f with sufficiently small analytic norm, the system (Ω, ρ + f) is C∞

rotation reducible, provided (Ω, ρf ) satisfying

|〈k,Ω〉+ lρf | ≥
γ

(|k|+ |l|)τ
, ∀k ∈ Z2, 0 6= l ∈ Z. (1.4)

However, when the base is of higher dimension, there is no result on this issue.
Thus, in this paper, we consider the rotation linearization for qpf circle flows with
multiple base frequencies satisfying the weak-Liouvillean condition. Furthermore,
we managed to relax Krikorian-Wang-You-Zhou’s condition (1.3) on one variable
of the multi-frequency. More precisely, for the frequency Ω = (1, α, ω̃) ∈ R1×R1×
Rd−2, we denote

Ũ(α) = sup
n>0

ln ln ln qn+1

ln ln qn
, (1.5)

where {pn/qn} is the best convergence of α. If Ũ := Ũ(α) <∞, and

|〈k, ω〉+ 〈l, ω̃〉| ≥ γ

(|k|+ |l|+ 1)τ
, ∀k ∈ Z2, l ∈ Zd−2\{0}, (1.6)

with ω = (1, α), then we say Ω is weak-Liouvillean. We will denote by WL(γ, τ, Ũ)
the set of all such vectors and

WL = ∪γ,τ>0,0<Ũ<+∞WL(γ, τ, Ũ).

It is obvious that WL is of full Lebesgue measure. Then our main result can be
stated as follows.



EJDE-2020/22 LINEARIZATION OF MULTI-FREQUENCIES FLOWS 3

Theorem 1.1. Let ρ ∈ R, d ≥ 2, γ′, τ ′, γ′′, τ ′′, r1, r2 > 0, Ω = (1, α, ω̃) ∈ Rd
with α ∈ R\Q, Ũ := Ũ(α̃) < ∞. If Ω ∈ WL(γ′, τ ′, Ũ) and ρ(Ω, ρ + f) =: ρf ∈
DCΩ(γ′′, τ ′′) in the sense that

|lρf + 〈k,Ω〉| ≥ γ

(|k|+ |l|+ 1)τ
, ∀k ∈ Zd, l ∈ Z\{0},

then there exists ε = ε(γ′, γ′′, τ ′, τ ′′, r1, r2, Ũ) > 0 such that if ‖f‖r1,r2 ≤ ε, (see
section 2.1 for a precise definition of the norm) then the system (Ω, ρ + f) is C∞

rotation linearizable.

We want to point out that, using the method in [9] by Krikorian-Wang-You-
Zhou, we can obtain the same result as [9, Corollary 1.1]. That is (Ω, ρ+ f) is C∞

accumulated by analytic flows {(Ω, f̃n)}, where the qpf flow (Ω, f̃n) is mode-locked.
In fact, our condition on the base frequency was partially inspired by recent

progress of almost reducibility in linear quasi-periodic SL(2,R) cocycles

(α,A) : Td−1 × R2 → Td−1 × R2

(θ, v) 7→ (θ + α,A(θ)v),
(1.7)

When d = 2, as for the reducibility of quasi-periodic SL(2,R) cocycles, which is
one-dimensional base frequency case, there are fruitful results. For the local case,
meaning the cocycle is close to a constant system, Dinaburg and Sinai [2] first
proved the positive measure reducibility with Diophantine frequency α, and it was
deepened by Eliasson to full measure reducibility [3]. In these papers, the Diophan-
tine condition on α allows the authors to use a Kolmogorov–Arnold–Moser (KAM)
argument. Recently, using generalized KAM schemes, Avila-Fayad-Krikorian [1]
and Hou-You [7] proved the local Cω rotation reducible result for any base forcing
frequency α ∈ R\Q in discrete case and continuous case respectively.

Compared the above one frequency case, very little is known for multifrequency
case. Recently, Hou-Wang-Zhou [8] considered the reducibility of multi-frequency
analytic quasi-periodic SL(2,R)-cocycles (1.7) with the frequency α = (α̃, α) ∈
T1 × Td−2 satisfying the conditions

ũ(α̃) := sup
n>0

ln ln q̃n+1

ln q̃n
<∞, (1.8)

and

‖kα̃+ 〈l, α′〉‖R/Z ≥
γ

(|k|+ |l|+ 1)τ
, ∀k ∈ Z, l ∈ Zd−2\{0},

for some γ > 0, τ > d− 1, where {p̃n/q̃n} is the best convergence of α̃, and

‖a‖R/Z = inf
p∈Z
|a− p|.

They proved the local positive measure rotation reducibility for the SL(2,R) cocy-
cles provided that the cocycle is Cω close enough to constant ones.

Note that our system is continuous non-linear system, which is quite different and
much more complicated than the above SL(2,R)-cocycles (discrete linear system).
Moreover, comparing the frequency condition of Theorem 1.1 with condition (1.8),
we handled more frequencies Ω including more Liouvillean ones.



4 Z. LIANG, T. JIN, J. WANG, Y. SHAN EJDE-2020/22

2. Preliminaries

2.1. Norm and Basic definitions. Denote by Cωr1,r2(Td+1,R) the set of all R-
valued functions admitting an analytic extension on

Td+1
r1,r2 := {(φ, x) ∈ Td+1 : |=φ1| ≤ r1, . . . , |=φd| ≤ r1, |=x| ≤ r2},

where r1, r2 > 0. For any f ∈ Cωr1,r2(Td+1,R), let

‖f‖r1,r2 := sup
(φ,x)∈Td+1

r1,r2

|f(φ, x)|.

In this article, we also frequently consider real-valued functions admitting an ana-
lytic extension on

Td+1
r1,r2,r3 :=

{
(φ, x) ∈ Td+1 : |=φ1| ≤ r1, |=φ2| ≤ r1, |=φ3| ≤ r2, . . . ,

|=φd| ≤ r2, |=x| ≤ r3

}
,

where r1, r2, r3 > 0, and use ‖f‖r1,r2,r3 to denote the norm

‖f‖r1,r2,r3 := sup
(φ,x)∈Td+1

r1,r2,r3

|f(φ, x)|.

We use Cωr1,r2,r3(Td+1,R) to denote the set of all such functions.

An integrable real-valued function f on Td has the Fourier expansion f =∑
k∈Zd f̂(k)e2πi〈k,φ〉 with f̂(k) =

∫
Td f(φ)e−2πi〈k,φ〉dφ. For any N > 0, TN and

RN are used to denote the truncation operators:

TN (f) =
∑
|k|<N

f̂(k)e2πi〈k,φ〉, RN (f) =
∑
|k|≥N

f̂(k)e2πi〈k,φ〉. (2.1)

2.2. Continued fraction expansion. Let α ∈ R be irrational. Define a0 =
[α], α0 = α− a0, and for k ≥ 1,

ak = [α−1
k−1], αk = G(αk−1) = α−1

k−1 − ak.

We define p0 = 0, p1 = 1, q0 = 1, q1 = a1, and inductively,

pk = akpk−1 + pk−2, qk = akqk−1 + qk−2.

So the sequence {pn/qn}n∈N is the best rational approximation for α ∈ R\Q, and
it satisfies

‖kα‖R/Z ≥ ‖qn−1α‖R/Z, ∀ 1 ≤ k < qn,

and
1

qn + qn+1
≤ ‖qnα‖R/Z ≤

1

qn+1
.

2.3. CD bridge. For any α ∈ R\Q, we will fix in the sequel a particular sub-
sequence (qnk)k∈N of the denominators of the continued fraction expansion for
α, which is denoted by (Qk)k∈N, and the subsequence (qnk+1)k∈N is denoted by
(Q̄k)k∈N.

Definition 2.1 ([1]). Let 0 < A ≤ B ≤ C. We say that the pair of denominators
(ql, qn) (l < n) forms a CD(A,B, C) bridge if

qi+1 ≤ qAi , ∀i = l, . . . , n− 1

qCl ≥ qn ≥ qBl .



EJDE-2020/22 LINEARIZATION OF MULTI-FREQUENCIES FLOWS 5

Lemma 2.2 ([1]). For any A > 0, there exists a subsequence (Qk)k∈N such that

Q0 = 1 and for each k ≥ 0, Qk+1 ≤ Q̄A
4

k . Furthermore, either Q̄k ≥ QAk , or the

pairs (Q̄k−1, Qk) and (Qk, Qk+1) are both CD(A,A,A3) bridges.

In the sequel, we let A ≥ 3 and assume (Qn)n∈N is the selected subsequence in
above lemma accordingly. As an immediate corollary of the above lemma, we have
a corollary.

Corollary 2.3. If Ũ(α) <∞, then Qn ≥ QAn−1 for every n ≥ 1. Furthermore,

sup
n>0

ln ln lnQn+1

ln lnQn
≤ U(α),

where U(α) := Ũ(α) + ln lnA4

ln ln 3 + 36 <∞.

Proof. For n = 1, according to Q0 = 1, we obviously get Q1 ≥ QA0 .
For n ≥ 2, there are two cases below. If Q̄n−1 ≥ QAn−1, then Qn ≥ Q̄n−1 ≥ QAn−1.

Otherwise, since (Qn−1, Qn) is a CD(A,A,A3) bridge, and then, Qn ≥ QAn−1.

Furthermore, owing to Qn+1 ≤ Q̄A
4

n , for n ≥ 1 we obtain

ln ln lnQn+1

ln lnQn
≤ ln(lnA4 + ln ln Q̄n)

ln lnQn

≤ ln ln ln Q̄n
ln lnQn

+
ln(27 lnA4)

ln ln 3

≤ Ũ +
ln lnA4

ln ln 3
+ 36 = U. �

3. The inductive step

For convenience, in the sequel, we will rewrite the system (1.1) as

ẋ = ρ+ f(ϕ, θ, x)

θ̇ = ω̃

ϕ̇ = ω = (1, α),

(3.1)

where φ = (ϕ, θ) ∈ T2 × Td−2, Ω = (ω, ω̃). Before the linearization steps, we give
a notation for simplicity: For any r1, r2, r3, εg, εf > 0, ρ ∈ R, we denote

Fr1,r2,r3(ρ, εg, εf )

:=
{
ρ̃+ g(ϕ) + f(ϕ, θ, x) ∈ Cωr1,r2,r3(Td+1,R) : ρ(Ω, ρ̃+ g + f) = ρ,

‖g‖r1 ≤ εg, ‖f‖r1,r2,r3 ≤ εf
}
.

Let α ∈ R\Q with Ũ(α) < ∞, and (Qn)n∈N be the selected sequence of α in

Lemma 2.2 with A = 9. Then U := Ũ(α)+ ln lnA4

ln ln 3 +36 <∞. For r1,0, r2,0, r3,0, γ, τ
positive, let Qmin ≥ 3 be the smallest Q ∈ N such that for any Q ≥ Qmin we have

6(ln(2Q))U+c(τ+d) < r1,0Q
2
3 , (3.2)

where c > 1 is a constant with (ln 4)c(τ+d) · ln 3
2

64(τ+d+2) ln 3 > U+c(τ+d). Meanwhile,

let ε0 small enough such that

ε0 < min
{ (r1,0r2,0r3,0γ)12(τ+d+2)

2τ !e(ln 2Q1)U+c(τ+d)+1
, e−36(τ+d+3)U+c(τ+d)

,

Q
−6(ln 2Qmin)U+c(τ+d)−3
min

}
,

(3.3)



6 Z. LIANG, T. JIN, J. WANG, Y. SHAN EJDE-2020/22

and

ln
1

ε0
< (

1

ε0
)

1
12(τ+d+2) . (3.4)

For any given r1,0, r2,0, r3,0, ε0 > 0, we define the following sequences inductively
for j ≥ 1:

Λ1 =
r2,0

10
, Λj =

Λ1

2j−1
, ∆1 =

r3,0

10
, ∆j =

∆1

2j−1
,

r1,j =
r1,0

4Q3
j

, r2,j = r2,j−1 − Λj , r3,j = r3,j−1 −∆j ,

εj =
εj−1

e(ln 2Qj+1)U+c(d+τ)
, ε̃j =

j−1∑
m=0

εm,K
(j) =

[( γ2

2ε
1/2
j−1

) 1
τ+d+2

]
.

(3.5)

3.1. Eliminate the lower-frequency terms. To minimize the norm of the per-
turbation f , we have to solve a homological equation involving a function g(ϕ).
For solving such an equation, we will use diagonal domination, which demands the
norms of g and f are in the same order. Therefore, first, we will do the transfor-
mation in the form x+ = x− h(ϕ) to achieve this.

Lemma 3.1. For n ≥ 2, given a qpf circle flow

ẋ = ρ+ g(ϕ) + f(ϕ, θ, x)

θ̇ = ω̃

ϕ̇ = ω = (1, α)

(3.6)

with ρ+g+f ∈ Fr1,n−1,r2,n−1,r3,n−1(ρ, 4ε̃n−1, εn−1), there exists h ∈ Cωr1,n−1
(T2,R),

where ∂ωh(ϕ) = TQng(ϕ)−ĝ(0), such that the transformation x̄ = x−h(ϕ) (mod 1),
conjugates system (3.6) into

˙̄x = ρ+ ḡ(ϕ) + f̄(ϕ, θ, x̄)

θ̇ = ω̃

ϕ̇ = ω = (1, α)

(3.7)

with ρ+ḡ+f̄ ∈ Fr̄1,n,r2,n−1,r̄3,n(ρ, ε
1/2
n−1, εn−1), where r̄1,n =

r1,0
Q3
n
, r̄3,n = r3,n−1−∆n

3 .

Proof. Let x̄ = x−h(ϕ) (mod 1), where ∂ωh(ϕ) = TQng(ϕ)− ĝ(0). Then the fibred
equation becomes

˙̄x = ρ+ ĝ(0) +RQng(ϕ) + f(ϕ, θ, x̄+ h(ϕ)). (3.8)

Because the norm of h is unknown, which will affect the norm of f , we need to
estimate ‖h(ϕ)‖. Since

h(ϕ) =
∑

0<|k|<Qn

ĝ(k)

2πi〈k, ω〉
e2πi〈k,ϕ〉, (3.9)

together with the fact that |〈k, ω〉| > 1
2Qn

for 0 < |k| < Qn, we have

‖h(ϕ)‖ r1,n−1
2
≤ Qn

π

∑
0<|k|<Qn

‖g‖r1,n−1e
−2π|k|

r1,n−1
2

≤ CQnε0

r2
1,n−1

≤ Q
5
3
nε

1/2
0 .



EJDE-2020/22 LINEARIZATION OF MULTI-FREQUENCIES FLOWS 7

It is clear that, ‖h(ϕ)‖ r1,n−1
2

has an influence on f(ϕ, θ, x̄+ h(ϕ)), and in order to

indicate f within control, we should estimate the norm of f in a smaller region,
which also means to estimate |=h(ϕ)|. To establish this, we consider the region
|=ϕ| ≤ r̄1,n.

Let ϕ = ϕ1 + iϕ2, ϕ1 ∈ T2, ϕ2 ∈ R2, and

h1(ϕ1) =
∑

0<|k|<Qn

ĝ(k)

2πi〈k, ω〉
e2πi〈k,ϕ1〉,

h2(ϕ) = h(ϕ)− h1(ϕ1).

It is clear that h(ϕ) is real analytic, since g(ϕ) is real analytic. Therefore, =h1(ϕ1) =
0. Then, owing to Qn ≥ Q9

n−1, we have

‖=h(ϕ)‖r̄1,n = ‖=h2(ϕ)‖r̄1,n ≤ ‖h2(ϕ)‖r̄1,n
≤ Qn

∑
0<|k|<Qn

|ĝ(k)||e−2π〈k,ϕ2〉 − 1|

≤ 2πQn
∑

0<|k|<Qn

‖g‖r1,n−1
e−2π|k|(r1,n−1−r̄1,n)|k|r̄1,n

≤ Cr1,0ε0

Q2
n(r1,n−1 − r̄1,n)3

≤ ∆n

3
,

by the selection of ε0 and ∆n. So, we have that

‖f(ϕ, θ, x̄+ h(ϕ))‖r̄1,n,r2,n−1,r̄3,n ≤ ‖f(ϕ, θ, x)‖r1,n−1,r2,n−1,r3,n−1 ≤ εn−1,

and

‖RQng(ϕ)‖r̄1,n = ‖
∑
|k|≥Qn

ĝ(k)e2πi〈k,ϕ〉‖r̄1,n

≤
∑
|k|≥Qn

‖g‖r1,n−1
e−2π|k|(r1,n−1−r̄1,n)

≤
Cε0Q

6
n−1

r2
1,0

e
−2πQn

r1,0

Q3
n−1 ≤

ε
1/2
n−1

4
,

by (3.2) and (3.3), and

|ĝ(0)| ≤ ‖f(ϕ, θ, x̄+ h(ϕ))‖r̄1,n,r2,n−1,r̄3,n + ‖RQng(ϕ)‖r̄1,n ≤
ε

1/2
n−1

2
.

Let

ḡ(ϕ) = ĝ(0) +RQng(ϕ)

f̄(ϕ, θ, x̄) = f(ϕ, θ, x̄+ h(ϕ)).

Then the result follows, since the transformation we did is homotopic to the identity,
which assures the fibred rotation number remains unchanged. �



8 Z. LIANG, T. JIN, J. WANG, Y. SHAN EJDE-2020/22

3.2. Solve the homological equation and reduce the perturbation. Under
the assumptions of Lemma 3.1, after the transformation, the norms of g(ϕ) and
f(ϕ, θ, x) are in the same order of magnitude. Then we want to reduce the pertur-
bation. However, we have no Diophantine assumption on ω, which means we need
to solve a different and more complicated homological equation in the process of
reducing ‖f‖. Moreover, it also means there might be no analytic solution of the
homological equation. Fortunately, it turns out that we can get the approximate
solution by using the method of diagonally dominant operators.

Considering the transformation: (ϕ, θ, x) = (ϕ, θ, x+ + h(ϕ, θ, x+) (mod 1)), we
obtain

ẋ = ẋ+ + ∂ωh+ ∂ω̃h+
∂h

∂x+
ẋ+.

Suppose that the system

ẋ = ρ+ g(ϕ) + f(ϕ, θ, x)

θ̇ = ω̃

ϕ̇ = ω = (1, α)

is conjugate to

ẋ+ = ρ+ g+(ϕ) + f+(ϕ, θ, x+)

θ̇ = ω̃

ϕ̇ = ω = (1, α).

In this case, we have

ẋ+ = ρ+ g + f(ϕ, θ, x+)− ∂ωh− ∂ω̃h− (ρ+ g(ϕ))
∂h

∂x+
+ h.o.t.

where h.o.t. represents the high order terms of h and f , and for simplicity, we
consider it as an error term and neglect it at this moment. Then we have the
homological equation

f(ϕ, θ, x) = ∂ωh+ ∂ω̃h+ (ρ+ g(ϕ))
∂h

∂x
. (3.10)

The following lemma gives the method to solve equation (3.10) by diagonally dom-
inant operators. We should point out the solution we obtained in the following
lemma is not an exact but approximate solution.

Lemma 3.2. Let γ′, γ′′, τ ′, τ ′′ > 0, γ := min{γ′, γ′′}, τ := max{τ ′, τ ′′}, 0 <

Ũ < ∞, r1 > δ1 > 0, r2 > δ2 > 0, r3 > δ3 > 0 with δ1 < δ2, δ1 < δ3/2,

(ω, ω̃) ∈WL(γ′, τ ′, Ũ), ρ ∈ DC(ω,ω̃)(γ
′′, τ ′′), g ∈ Cωr1(T2,R), f ∈ Cωr1,r2,r3(Td+1,R)

and
∫
T
∫
Td−2 f(ϕ, θ, x)dθdx = 0, 0 < εf ≤ εg ≤ ε

1/2
0 � 1, where ε0 satisfies (3.3)

and (3.4). If
‖g(ϕ)‖r1 ≤ εg, ‖f(ϕ, θ, x)‖r1,r2,r3 ≤ εf ,

and

K =
[ 1

πδ1
ln

1

εf

]
+ 1 <

(γ2

εg

) 1
τ+d+2

,

then the homological equation (3.10) has an approximate solution h(ϕ, θ, x) with the
estimate

‖h‖r1−δ1,r2−δ2,r3−δ3 ≤
C

γδτ+d+1
1

εf ,



EJDE-2020/22 LINEARIZATION OF MULTI-FREQUENCIES FLOWS 9

where C is a constant, and the error term P = RK(f − (ρ+ g)∂h∂x ) with

‖P‖r1−δ1,r2−δ2,r3−δ3 ≤ 2(2K)d+1ε2f .

Proof. First, we consider the truncated equation

∂ωh+ ∂ω̃h+ ρ
∂h

∂x
+ TK

(
g(ϕ)

∂h

∂x

)
= TKf(ϕ, θ, x).

Let

f(ϕ, θ, x) =
∑
l

fl(ϕ, θ)e
2πilx, h(ϕ, θ, x) =

∑
|l|<K

hl(ϕ, θ)e
2πilx,

fl(ϕ, θ) =
∑
ν

fl,ν(ϕ)e2πi〈ν,θ〉, hl(ϕ, θ) =
∑

0<|l|+|ν|<K

hl,ν(ϕ)e2πi〈ν,θ〉,

fl,ν(ϕ) =
∑
k

f̂l,ν(k)e2πi〈k,ϕ〉, hl,ν(ϕ) =
∑

|k|<K−|l|−|ν|
|l|+|ν|6=0

ĥl,ν(k)e2πi〈k,ϕ〉.

Then plugging the Fourier expansions into the truncated equation, and simplifying
it, for |l|+ |ν|+ |k| < K, |l|+ |ν| > 0, we have

(〈k, ω〉+ 〈ν, ω̃〉+ lρ)ĥl,ν(k) + l
∑

|k1|<K−|l|−|ν|

ĝ(k − k1)ĥl,ν(k1) =
f̂l,ν(k)

2πi
.

It equals the matrix equation, for fixed 0 < |l|+ |ν| < K,

(Al,ν +Gl,ν)h̃l,ν = f̃l,ν ,

where

Al,ν = diag{〈k, ω〉+ 〈ν, ω̃〉+ lρ : |k| < K − |l| − |ν|},
Gl,ν = (lĝ(p− q))|p|,|q|<K−|l|−|ν|,

f̃l,ν = (
f̂l,ν(k)

2πi
)T|k|<K−|l|−|ν|,

h̃l,ν = (ĥl,ν(k))T|k|<K−|l|−|ν|.

If we denote Ml,ν = diag(. . . , e2π|k|(r1− δ12 ), . . . )|k|<K−|l|−|ν|, then

Ml,ν(Al,ν +Gl,ν)M−1
l,νMl,ν h̃l,ν = Ml,ν f̃l,ν .

For simplicity, we drop the subscripts l, ν temporarily and denote

A = Al,ν , G = Ml,νGl,,νM
−1
l,ν , h̄ = Ml,ν h̃l,ν , f̄ = Ml,ν f̃l,ν .

Then, we have

A(I +A−1G)h̄ = f̄ .

Obviously, to get the norm of h̄, we need to estimate the norm of (I + A−1G)−1.
We have

‖A−1G‖ ≤ ‖A−1‖‖G‖,
where ‖ · ‖ denotes the operator norm associated to the l1 norm that satisfies
‖u‖l1 =

∑
k |uk| (indeed, if X is a matrix, ‖X‖ = maxj

∑
i |Xij |). Then by the

condition (ω, ω̃) ∈WL(γ′, τ ′, Ũ), ρ ∈ DC(ω,ω̃)(γ
′′, τ ′′), we obtain

|lρ+ 〈k, ω〉+ 〈ν, ω̃〉| ≥ γ

(|k|+ |l|+ |ν|+ 1)τ
≥ γ

Kτ



10 Z. LIANG, T. JIN, J. WANG, Y. SHAN EJDE-2020/22

for |l|+ |ν|+ |k| < K and |l|+ |ν| 6= 0. We obtain that

‖A−1G‖ ≤ max
|k|<K−|l|−|ν|

|l|max|q|<K−|l|−|ν| Σ|p|<K−|l|−|ν|‖g‖r1e−π|p−q|δ1
|lρ+ 〈k, ω〉+ 〈ν, ω̃〉|

≤ 4Kτ+3

γ
‖g‖r1 .

Since K < (γ
2

εg
)

1
τ+d+2 , we know that ‖A−1G‖ < 1

2 , and thus

‖(I +A−1G)−1‖ < 2.

From the result above, there is no doubt about the existence of h̄. And it satisfies

h̄ = (I +A−1G)−1A−1f̄ .

As for the estimate of h(ϕ, θ, x), we have

‖h(ϕ, θ, x)‖
r1− δ12 ,r2−

δ2
2 ,r3−

δ3
4

≤
∑

0<|l|+|ν|<K

‖h̄l,ν‖l1e2π|ν|(r2− δ22 )e2π|l|(r3− δ34 )

=
∑

0<|l|+|ν|<K

‖(I +A−1G)−1A−1f̄l,ν‖l1e2π|ν|(r2− δ22 )e2π|l|(r3− δ34 )

≤
∑

0<|l|+|ν|+|k|<K

2(|k|+ |l|+ |ν|+ 1)τ

γ

|f̂l,ν(k)|
2π

e2π|k|(r1− δ12 )e2π|ν|(r2− δ22 )e2π|l|(r3− δ34 )

≤ Cεf

γδτ+d+1
1

,

and the error term satisfies

‖P‖r1−δ1,r2−δ2,r3−δ3

≤ ‖RKf(ϕ, θ, x)‖r1−δ1,r2−δ2,r3−δ3 + ‖RK(g(ϕ)
∂h

∂x
)‖r1−δ1,r2−δ2,r3−δ3

≤ (2K)d+1e−2πKδ1/2(εf +
Cεg

γδτ+d+2
1

εf )

< 2(2K)d+1ε2f .

�

Using diagonal domination, we can solve the approximate homological equation
(3.10), and the following lemma will apply it to reduce the perturbation.

Lemma 3.3. Under the conditions of Lemma 3.1, if (ω, ω̃) ∈ WL(γ′, τ ′, Ũ) and
ρ ∈ DC(ω,ω̃)(γ

′′, τ ′′), there exists a transformation H̄ ∈ Cωr∗1,n,r∗2,n,r∗3,n(Td+1,Td+1)

where

(r∗1,n, r
∗
2,n, r

∗
3,n) = (

r1,0

2Q3
n

, r2,n−1 −
Λn
3
, r̄3,n −

∆n

3
),

with

‖H̄ − id‖r∗1,n,r∗2,n,r∗3,n ≤ 4ε
3/4
n−1,

‖D(H̄ − id)‖r∗1,n,r∗2,n,r∗3,n ≤ 4ε
5/8
n−1,



EJDE-2020/22 LINEARIZATION OF MULTI-FREQUENCIES FLOWS 11

such that H̄ conjugates the system (3.7) to

˙̄x+ = ρ+ g∗(ϕ) + f∗(ϕ, θ, x̄+)

θ̇ = ω̃

ϕ̇ = ω = (1, α)

(3.11)

where ρ+ g∗ + f∗ ∈ Fr∗1,n,r∗2,n,r∗3,n(ρ, 2ε
1/2
n−1, εn) and ‖g∗ − ḡ‖r∗1,n ≤ 3εn−1.

Proof. For simplicity, we drop the subscript n and denote temporarily ḡ0(ϕ) = ḡ(ϕ),

f̄0(ϕ, θ, x) = f̄(ϕ, θ, x), η̃ = 2εn−1, η = 2ε
1/2
n−1, r̃1 = r̄1,n =

r1,0
Q3
n

, r̃2 = r2,n−1

r̃3 = r̄3,n = r3,n−1 − ∆n

3 . Now we define the following sequences inductively:

r̃1,0 = r̃1, r̃2,0 = r̃2, r̃3,0 = r̃3,

δ1,1 =
r̃1,0

4
, δ1,j =

1

2j−1
δ1,1, r̃1,j = r̃1,j−1 − δ1,j ,

δ2,1 =
Λn
6
, δ2,j =

1

2j−1
δ2,1, r̃2,j = r̃2,j−1 − δ2,j ,

δ3,1 =
∆n

6
δ3,j =

1

2j−1
δ3,1, r̃3,j = r̃3,j−1 − δ3,j ,

η̃j = η̃( 3
2 )j , Kj =

[ 1

πδ1,j
ln

1

η̃j−1

]
+ 1.

From the above definitions, we can assume that δ1,j < min{δ2,j , δ3,j/2}.
Let

N = [
(ln 2Qn)U+c(τ+d)

32(τ + d+ 2) ln 3
] + 1.

Supposing for ν = 1, 2, . . . , j−1 < N , there are fν , hν ∈ Cωr̃1,ν ,r̃2,ν ,r̃3,ν (Td+1,R), gν ∈
Cωr̃1,ν−1

(T2,R), such that the transformation (ϕ, θ, x̄ν−1) = (ϕ, θ, x̄ν +hν(ϕ, θ, x̄ν) (

mod 1)) conjugates the system

˙̄xν−1 = ρ+ ḡν−1(ϕ) + f̄ν−1(ϕ, θ, x̄ν−1)

θ̇ = ω̃

ϕ̇ = ω = (1, α)

to

˙̄xν = ρ+ ḡν(ϕ) + f̄ν(ϕ, θ, x̄ν)

θ̇ = ω̃

ϕ̇ = ω = (1, α)

satisfying

‖f̄ν‖r̃1,ν ,r̃2,ν ,r̃3,ν ≤ η̃ν , ‖hν‖r̃1,ν ,r̃2,ν ,r̃3,ν ≤ η̃
3/4
ν−1,

and ḡν = ḡν−1+
∫
Td−2×T1 f̄ν−1(ϕ, θ, x̄ν)dθdx̄ν . For ν = j, under the similar transfor-

mation as above, (ϕ, θ, x̄j−1) = (ϕ, θ, x̄j+hj (mod 1)), the fibred equation becomes

˙̄xj = ρ+ ḡj−1(ϕ) + f̄j−1(ϕ, θ, x̄j)− ∂ωhj − ∂ω̃hj − (ρ+ ḡj−1(ϕ))
∂hj
∂x̄j

+ h.o.t.



12 Z. LIANG, T. JIN, J. WANG, Y. SHAN EJDE-2020/22

where h.o.t. is higher order terms about f̄j−1 and hj . Then the homological equation
is

∂ωhj + ∂ω̃hj + (ρ+ ḡj−1(ϕ))
∂hj
∂x̄j

= f̄j−1(ϕ, θ, x̄j)−
∫
Td−2×T

f̄j−1(ϕ, θ, x̄j)dθdx̄j .

(3.12)

First, we know that

‖ḡj−1‖r̃1,j−2
≤ ‖ḡ‖r̃1,0 +

j−2∑
ν=0

η̃ν < 2ε
1/2
n−1.

Furthermore, from (3.2)-(3.4), for j ≤ N , we obtain

Kj =
[ 1

πδ1,j
ln

1

η̃j−1

]
+ 1 <

2 · 3j−1Q3
n

r1,0
ln

1

εn−1

≤ 2Q3
n

r1,0
ec1(ln 2Qn)U+c(τ+d)

ln
1

εn−1
<
e2c1(ln 2Qn)U+c(τ+d)

r1,0
ln

1

εn−1

<
1

r1,0

1

ε2c1
n−1

( 1

εn−1

) 1
12(τ+d+2)

<
( 1

εn−1

) 1
6(τ+d+2)

(
1

εn−1
)2c1

<
( γ2

2ε
1/2
n−1

) 1
τ+d+2

,

(3.13)

where c1 = 1/
(
32(τ + d + 2)

)
. Moreover, since (ω, ω̃) ∈ WL(γ′, τ ′, Ũ) and ρ ∈

DC(ω,ω̃)(γ
′′, τ ′′), by Lemma 3.2, there exists an approximate solution of (3.12)

satisfying

‖hj‖r̃1,j−1−δ1,j/2,r̃2,j−1−δ2,j/2,r̃3,j−1−δ3,j/2 ≤
2τ+d+1C

γδτ+d+1
1,j

η̃j−1 < η̃
3/4
j−1 <

δ3,j
2
,

and the error term satisfies

‖Pj‖r̃1,j−1−δ1,j/2,r̃2,j−1−δ2,j/2,r̃3,j−1−δ3,j/2 ≤ 2(2Kj)
d+1η̃2

j−1 < η̃
7
4
j−1.

As a consequence, the fibred equation becomes

˙̄xj = ρ+ ḡj(ϕ) + f̄j(ϕ, θ, x̄j),

where ḡj = ḡj−1 +
∫
Td−2×T1 f̄j−1(ϕ, θ, x̄j)dθdx̄j and

f̄j(ϕ, θ, x̄j) + f̄j(ϕ, θ, x̄j)
∂hj
∂x̄j

= f̄j−1(ϕ, θ, x̄j + hj(ϕ, θ, x̄j))− f̄j−1(ϕ, θ, x̄j)

+ Pj −
∂hj
∂x̄j
·
∫
Td×T

f̄j−1(ϕ, θ, x̄j)dθdx̄j .

We have the estimate

‖f̄j−1(ϕ, θ, x̄j + hj(ϕ, θ, x̄j))− f̄j−1(ϕ, θ, x̄j)‖r̃1,j ,r̃2,j ,r̃3,j

≤ ‖∂f̄j−1

∂x̄j
‖r̃1,j ,r̃2,j ,r̃3,j+δ3,j/2‖hj‖r̃1,j ,r̃2,j ,r̃3,j <

η̃
3/2
j−1

3
.



EJDE-2020/22 LINEARIZATION OF MULTI-FREQUENCIES FLOWS 13

Thus,

‖f̄j‖r̃1,j ,r̃2,j ,r̃3,j < 2(
η̃

3/2
j−1

3
+ η̃

7
4
j−1 + η̃j−1

η̃
3/4
j−1

δ3,j/2
) < η̃

3/2
j−1 = η̃j .

By estimate (3.13), we know that we can do the above iterations until j = N . Now
we give the estimation of f̄N for |=ϕ| ≤ r̃1,N , |=θ| ≤ r̃2,N , |=x̄N | ≤ r̃3,N . In fact,

under the Corollary 2.3, we have lnQn+1 ≤ e(lnQn)U . Then(3

2

)N − 1 ≥ e(ln 2Qn)U+c(τ+d) c1 ln 3
2

2 ln 3

≥ e(lnQn)U (ln 2Qn)c(τ+d)
c1 ln 3

2
2 ln 3

≥ (lnQn+1)(ln 2Qn)c(τ+d)
c1 ln 3

2
2 ln 3 ,

and therefore,

‖f̄N‖r̃1,N ,r̃2,N ,r̃3,N ≤ η̃(3/2)N ≤ η̃e−(( 3
2 )N−1) ln 1

η̃

≤ η̃e−(lnQn+1)(ln 2Qn)c(τ+d)
c1 ln 3

2
2 ln 3 ln 1

η̃

< η̃e−2(lnQn+1)U+c(τ+d)2U+c(τ+d)

<
η̃

2
· e−(ln 2Qn+1)U+c(τ+d)

= εn.

There is no harm in denoting Hj(ϕ, θ, x̄j) = (ϕ, θ, x̄j + hj(ϕ, θ, x̄j) (mod 1)) for
1 ≤ j ≤ N . Then let

H̄j(ϕ, θ, x̄j) = H1 ◦ · · · ◦Hj−1 ◦Hj(ϕ, θ, x̄j),

and H̄j(ϕ, θ, x̄j) is analytic for (ϕ, θ, x̄j) ∈ Td+1

r̃1,j ,r̃2,j ,r̃3,j+
δ3,j
2

with

‖∂x̄j (π3 ◦ H̄j(ϕ, θ, x̄j))‖r̃1,j ,r̃2,j ,r̃3,j ≤
j−1∏
ν=0

(1 + η̃3/4
ν ),

where π3 : T2 × Td−2 × T1 → T1 is the natural projection to the third variable.
Rewriting x̄N =: x̄+ and H̄N (ϕ, θ, x̄+) =: (ϕ, θ, x̄+ + h̃(ϕ, θ, x̄+) (mod 1)), we have

‖h̃(ϕ, θ, x̄+)‖r̃1,N ,r̃2,N ,r̃3,N = ‖π3 ◦ H̄N (ϕ, θ, x̄+)− x̄+‖r̃1,N ,r̃2,N ,r̃3,N
≤ ‖π3 ◦ (H̄N (ϕ, θ, x̄+)− H̄N−1(ϕ, θ, x̄+))‖r̃1,N ,r̃2,N ,r̃3,N

+ · · ·+ ‖π3 ◦ H̄1(ϕ, θ, x̄+)− x̄+‖r̃1,1,r̃2,1,r̃3,1

≤
N∑
j=1

j−2∏
ν=0

(1 + η̃3/4
ν )‖hj‖r̃1,j ,r̃2,j ,r̃3,j

≤ 4ε
3/4
n−1,

and

‖Dh̃‖
r̃1,N−

δ1,N
2 ,r̃2,N−

δ2,N
2 ,r̃3,N−

δ3,N
2

≤ 4ε
5
8
n−1.



14 Z. LIANG, T. JIN, J. WANG, Y. SHAN EJDE-2020/22

In conclusion, letting (g∗, f∗) = (ḡN , f̄N ), H̄(ϕ, θ, x̄+) = (ϕ, θ, x̄+ + h̃(ϕ, θ, x̄+)
(mod 1) together with

‖g∗(ϕ)− ḡ(ϕ)‖r∗1,n ≤
N−1∑
j=0

‖f̄j‖r̃1,j ,r̃2,j ,r̃3,j <
N−1∑
j=0

η̃j < 3εn−1,

we obtain the result. �

3.3. Do the inverse transformation of the first step. Via the two transform-
ing steps above, our system has been simplified to some degree. However, the
transformation we made in the first step is not close to the identity. So we have to
do another transformation in this step, in order to let the ultimate transformation
be close to the identity.

Lemma 3.4. Under the conditions of Lemma 3.3, there exists a transformation

H̃ ∈ Cω(Td+1,Td+1), with

‖H̃ − id‖r1,n,r2,n,r3,n ≤ 4ε
3/4
n−1,

‖D(H̃ − id)‖r1,n,r2,n,r3,n ≤ 4ε
1/2
n−1,

such that H̃ conjugates the system (3.6) to

ẋ+ = ρ+ g+(ϕ) + f+(ϕ, θ, x+)

θ̇ = ω̃

ϕ̇ = ω

with ρ+ g+ + f+ ∈ Fr1,n,r2,n,r3,n(ρ, 4ε̃n, εn).

Proof. By Lemma 3.1 and 3.3, there exist h ∈ Cωr̄1,n(T2,R) and H̄ ∈ Cωr∗1,n,r∗2,n,r∗3,n
(Td+1,Td+1) such that H ◦ H̄ conjugates the system (3.6) to (3.11), where ∂ωh =
TQng − ĝ(0) and H(ϕ, θ, x̄) = (ϕ, θ, x̄ + h(ϕ) (mod 1), H̄(ϕ, θ, x̄+) = (ϕ, θ, x̄+ +

h̃(ϕ, θ, x̄+) (mod 1). Now let (ϕ, θ, x̄+) = (ϕ, θ, x+−h(ϕ) (mod 1) = H−1(ϕ, θ, x+).
Then by the transformation (ϕ, θ, x) = H ◦ H̄ ◦H−1(ϕ, θ, x+), the fibred equation
becomes

ẋ+ = ρ+ g∗(ϕ) + ∂ωh+ f∗(ϕ, θ, x+ − h(ϕ))

= ρ+ g + g∗(ϕ)− ḡ(ϕ) + f∗(ϕ, θ, x+ − h(ϕ))

= ρ+ g+(ϕ) + f+(ϕ, θ, x+),

where g+(ϕ) := g(ϕ) + g∗(ϕ) − ḡ(ϕ), f+(ϕ, θ, x+) = f∗(ϕ, θ, x+ − h(ϕ)). Thus,
‖g+‖r1,n ≤ 4ε̃n−1 + 4εn−1 = 4ε̃n and ‖f+‖r1,n,r2,n,r3,n ≤ ‖f∗‖r∗1,n,r∗2,n,r∗3,n ≤ εn by

Lemmas 3.1 and 3.3. Since the transformation H−1 is homotopic to the identity,
the fibred rotation number remains the same. That is,

ρ+ g+ + f+ ∈ Fr1,n,r2,n,r3,n(ρ, 4ε̃n, εn).

Let H̃ = H ◦ H̄ ◦H−1. Then

H̃(ϕ, θ, x+) = (ϕ, θ, x+ + h̃(ϕ, θ, x+ − h(ϕ))),

and hence

‖H̃ − id‖r1,n,r2,n,r3,n ≤ ‖h̃(ϕ, θ, x̄+)‖r∗1,n,r∗2,n,r∗3,n < 4ε
3/4
n−1,



EJDE-2020/22 LINEARIZATION OF MULTI-FREQUENCIES FLOWS 15

and

‖D(H̃ − id)‖r1,n,r2,n,r3,n ≤ ‖Dh̃‖r∗1,n,r∗2,n,r∗3,n‖D(H−1)‖r1,n,r2,n,r3,n ≤ 4ε
1/2
n−1.

�

3.4. Iterative Lemma. According to the above three lemmas, we combine them
into the following iterative lemma.

Lemma 3.5. For any ε0, r1,0, r2,0, r3,0, γ
′, γ′′, τ ′, τ ′′ > 0, γ := min{γ′, γ′}, τ :=

max{τ ′, τ ′′}, α ∈ R\Q with Ũ = Ũ(α) < ∞, the sequences εn, ε̃n, r1,n, r2,n, r3,n

are defined as in (3.5). If ε0 satisfies (3.3) and (3.4), and (ω, ω̃) ∈ WL(γ′, τ ′, Ũ),
ρ ∈ DC(ω,ω̃)(γ

′′, τ ′′), then for all n ≥ 1 the following holds: If the system

ẋ = ρ+ gn(ϕ) + fn(ϕ, θ, x)

θ̇ = ω̃

ϕ̇ = ω = (1, α)

(3.14)

satisfies ρ + gn + fn ∈ Fr1,n,r2,n,r3,n(ρ, 4ε̃n, εn), then there exists a transformation

Hn : Td+1 → Td+1 with estimates

‖Hn − id‖r1,n+1,r2,n+1,r3,n+1
≤ 4ε3/4

n ,

‖D(Hn − id)‖r1,n+1,r2,n+1,r3,n+1 ≤ 4ε1/2
n ,

such that it transforms system(3.14) into

ẋ = ρ+ gn+1(ϕ) + fn+1(ϕ, θ, x)

θ̇ = ω̃

ϕ̇ = ω

with ρ+ gn+1 + fn+1 ∈ Fr1,n+1,r2,n+1,r3,n+1
(ρ, 4ε̃n+1, εn+1).

4. Proof of the main theorem

Let ε0 small enough satisfying (3.3) and (3.4) with τ := max{τ ′, τ ′′}, γ :=
min{γ′, γ′′}, r1,0 := r1, r2,0 := r1, r3,0 := r2. For convenience, we rewrite the
system (Ω, ρ+ f) as

ẋ = ρf + g(ϕ) + f(ϕ, θ, x)

θ̇ = ω̃

ϕ̇ = ω = (1, α)

(4.1)

with ρf + g + f ∈ Fr1,0,r2,0,r3,0(ρf , ε0, ε0), where g := ρ − ρf , Ω = (ω, ω̃), and

φ = (ϕ, θ) ∈ T2 × Td−2. Owing to the fact that (ω, ω̃) ∈ WL(γ′, τ ′, Ũ) and
ρf ∈ DC(ω,ω̃)(γ

′′, τ ′′), we can apply Lemma 3.3 and obtain the transformation

H0 ∈ Cωr1,1,r2,1,r3,1(Td+1,Td+1), such that system (4.1) can be conjugate to

ẋ = ρf + g1(ϕ) + f1(ϕ, θ, x)

θ̇ = ω̃

ϕ̇ = ω

with ρf + g1 + f1 ∈ Fr1,1,r2,1,r3,1(ρf , 4ε̃1, ε1) and ‖H0 − id‖r1,1,r2,1,r3,1 ≤ 4ε
3/4
0 ,

‖D(H0 − id)‖r1,1,r2,1,r3,1 ≤ 4ε
1/2
0 . Now, by Lemma 3.5, we obtain the sequence of

transformations Hj ∈ Cωr1,j+1,r2,j+1,r3,j+1
(Td+1,Td+1) (j = 1, . . . , n − 1) such that



16 Z. LIANG, T. JIN, J. WANG, Y. SHAN EJDE-2020/22

H(n) := H0 ◦H1 ◦· · ·◦Hn−1 conjugates (4.1) to (ω, ω̃, ρf +gn(ϕ)+fn(ϕ, θ, x)), with

‖Hj − id‖r1,j+1,r2,j+1,r3,j+1
≤ 4ε

3/4
j , ‖D(Hj − id)‖r1,j+1,r2,j+1,r3,j+1

≤ 4ε
1/2
j . Then

for H(n), we have

‖DH(n)‖r1,n,r2,n,r3,n
≤ ‖DH0‖r1,1,r2,1,r3,1‖DH1‖r1,2,r2,2,r3,2 . . . ‖DHn−1‖r1,n,r2,n,r3,n
≤ Πn−1

j=0 (1 + 4ε
1/2
j ) < 2.

Therefore, we have

‖H(n+1) −H(n)‖r1,n+1,r2,n+1,r3,n+1

≤ ‖DH(n)‖r1,n,r2,n,r3,n‖Hn − id‖r1,n+1,r2,n+1,r3,n+1
≤ 8ε3/4

n ,

which means the limit H(n) exists in C0, and we denote H = limn→∞H(n), g∞ =
limn→∞ gn. To prove the transformation H is actually in C∞, we have to prove
∂|m|H
∂ζm exists for all m ∈ Nd+1 (denoting ζ = (ϕ, θ, x)). Let r∗ = min{r1,0, r2,0, r3,0}.

Actually, for any m ∈ Nd+1, there exists nm ∈ N, so that if n ≥ nm, then we have

(
4Q3

n

r∗
)|m| < ε

− 1
4

n−1, that is

(
4Q3

n

r∗
)|m|ε

3/4
n−1 < ε

1/2
n−1, ∀n ≥ nm.

Meanwhile, we have the following inequality, for n ≥ nm − 1,∣∣∂|m|(H(n+1) −H(n))

∂ζm
∣∣ ≤ ‖H(n+1) −H(n)‖r1,n+1,r2,n+1,r3,n+1

r
|m|
1,n+1

≤ 8
(4Q3

n+1

r∗

)|m|
ε3/4
n < 8ε1/2

n .

In conclusion, the transformation H is actually in C∞, and under this transfor-
mation, the system(4.1) is conjugate to

ẋ = ρf + g∞(ϕ),

θ̇ = ω̃,

ϕ̇ = ω,

which can be rewritten as (Ω, ρf + g∞(ϕ)). The result follows.

Acknowledgements. This work was partially supported by The Fundamental Re-
search Funds for the Central Universities, No. 30918011336. Y. Shan was supported
by the Natural Science Foundation of Jiangsu Province( Grant No. BK20161053),
and the National Science Foundation of China(Grant No. 11701285). Z. Liang, T.
Jin and J.Y. Wang were supported by the National research project at the Nanjing
University of Science and Technology. The authors would like to thank Jing Wang
for inspiring and stimulating discussions.

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Ziyang Liang
Department of Mathematics, School of Science, Nanjing University of Science and

Technology, Nanjing, 210094, China

Email address: lianggzy98@163.com

Taian Jin

Department of Mathematics, School of Science, Nanjing University of Science and
Technology, Nanjing, 210094, China

Email address: 1056418286@qq.com

Jiayi Wang

Department of Mathematics, School of Science, Nanjing University of Science and

Technology, Nanjing, 210094, China
Email address: 13236582927@163.com

Yuan Shan (corresponding author)
School of Statistics and Mathematics, Nanjing Audit University, Nanjing 210029, China

Email address: shannjnu@gmail.com


	1. Introduction and statement of main results
	2. Preliminaries
	2.1. Norm and Basic definitions
	2.2. Continued fraction expansion
	2.3. CD bridge

	3. The inductive step
	3.1. Eliminate the lower-frequency terms
	3.2. Solve the homological equation and reduce the perturbation
	3.3. Do the inverse transformation of the first step
	3.4. Iterative Lemma 

	4. Proof of the main theorem
	Acknowledgements

	References