
The area of metric fixed point theory applied to relational metric spaces has received significant attention since the appearance of the relation-theoretic contraction principle. In recent times, a number of fixed point theorems addressing the various contractivity conditions in the relational metric space has been investigated. Such results are extremely advantageous in solving a variety of boundary value problems, matrix equations, and integral equations. This article offerred some fixed point results for a functional contractive mapping depending on a control function due to Boyd and Wong in a metric space endued with a local class of transitive relations. Our findings improved, developed, enhanced, combined and strengthened several fixed point theorems found in the literature. Several illustrative examples were delivered to argue for the reliability of our findings. To verify the relevance of our findings, we conveyed an existence and uniqueness theorem regarding the solution of a first-order boundary value problem.
Citation: Ahmed Alamer, Faizan Ahmad Khan. Boyd-Wong type functional contractions under locally transitive binary relation with applications to boundary value problems[J]. AIMS Mathematics, 2024, 9(3): 6266-6280. doi: 10.3934/math.2024305
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The area of metric fixed point theory applied to relational metric spaces has received significant attention since the appearance of the relation-theoretic contraction principle. In recent times, a number of fixed point theorems addressing the various contractivity conditions in the relational metric space has been investigated. Such results are extremely advantageous in solving a variety of boundary value problems, matrix equations, and integral equations. This article offerred some fixed point results for a functional contractive mapping depending on a control function due to Boyd and Wong in a metric space endued with a local class of transitive relations. Our findings improved, developed, enhanced, combined and strengthened several fixed point theorems found in the literature. Several illustrative examples were delivered to argue for the reliability of our findings. To verify the relevance of our findings, we conveyed an existence and uniqueness theorem regarding the solution of a first-order boundary value problem.
It is well known that the problem of reducibility and stability of Sobolev norms for quasi-periodically forced linear operators on the torus is a classical one in the theory of infinite dimensional dynamical systems, which has received new attention in the past few years. Roughly speaking, given a linear operator with coefficients that depend on time in a quasi-periodic way, we say that it is reducible if there exists a bounded change of variables depending quasi-periodically on time (say mapping Hs→Hs for all times), which makes its coefficients constant. Actually, the notion of reducibility has been first introduced for ordinary differential equations (ODEs), going back to Bogolyubov [11] and Moser [32]. Also, there is a large literature around the reducibility of ODEs by means of the KAM tools. Regarding recent developments in this direction, we invite the reader to consult [4,16,22,25,26] and the references therein. Such kind of reducibility results for PDEs using KAM machinery have been well developed, see [7,8,9,10,13,18,21,30,31]. Due to the large amount of work in reducible KAM theory for non linear partial differential equations (PDEs), we do not list them, and refer to the introduction in [20] for more details.
Investigating the reducibility of linear operators is important when reducibility problems of nonlinear PDEs are considered. Very recently, Feola et al. [20] considered the reducibility of linear equations of the form
∂tu+(ζ+a(ω,ζ)(ωt,x))⋅∂xu=0, | (1.1) |
on the torus x∈Td where ω∈Rν and ζ∈Rd are parameters (and the dependence on these parameters is denoted by the corresponding subscripts) while a(ω,ζ)∈C∞(Tν+d,Rd). They proved that if for any s1∈Z+ large enough and ‖a‖s1=‖a(ω,ζ)‖Hs1(Tν+d,Rd) small enough, then there exists a Borl set O∞⊂O0 (a bounded domain in Rν+d) with positive Lebesgue measure such that for any (ω,ζ)∈O∞, there is a quasi-periodic family of bounded and invertible linear operators Ψ(ωt)=Ψω,ζ(ωt) and the equation (1.1) is reduced by v=Ψ(ωt)−1[u] into constant coefficients
∂tv+mω,ζ⋅∂xv=0, |
where |mω,ζ−ζ|=O(‖a‖s1) uniformly in (ω,ζ)∈O0, see Theorem 1 in [20]. As a direct consequence, the Sobolev norms of the solutions of the Cauchy problem associated to (1.1) are controlled uniformly in time. The main strategies in [20] are divided into two steps: Firstly, the problem of reducing equations (1.1) to constant coefficients, by the identification between first order operators and vector fields, can be formulated as the problem of finding a family of diffeomorphism Ψξ:θ↦θ+hξ(θ) that conjugates a weakly perturbed constant vector field
(ξ+fξ(θ))⋅∂θ |
on Tν+d to constant Diophantine vector fields αξ⋅∂θ. Here, ξ is a parameter ranging in a bounded domain O⊂Rν+d and fξ,hξ∈C∞(Tν+d,Rν+d). In other words, the authors obtained a new version of J. Moser's theorem on straightening vector fields on tori (called the tame Moser theorem), see Theorem 2 in [20]. Second, the tame Moser theorem is proven via an iterative KAM-type scheme in the Sobolev spaces Hs. Approximations by analytic functions are not used. Instead, the authors approached the problem in the spirit of the Nash-Moser theory, where one employs interpolation and smoothing estimates in order to control the loss of regularity due to the presence of small divisors.
Consider two classes of first order, linear quasi-periodically forced PDEs as the following:
{∂tu+(ξ+f(x,ωt,ξ))⋅∂xu=0,u(x,0)=u0(x), | (1.2) |
where x∈Tm,ω∈Rd,ξ∈O⊂Rm, f:Tm+d→Rm and u0:Tm→R are both Cω functions, and
{∂tu+(ρ+f(x,ωt))⋅∂xu=0,u(x,0)=u0(x), | (1.3) |
where x∈T,ω∈Rd,ρ∈R, f:T1+d→R and u0:T→R are both Cω functions. We say (1.2) and (1.3) satisfy the Cauchy problem with initial condition u(x,0)=u0(x) if u(x,t) is the solution of (1.2) and (1.3) and satisfies u(x,0)=u0(x). Furthermore, we say this Cauchy problem is stable if the norm of u(x,⋅) can be controlled by the norm of u0(x) for each t∈R. Inspired by the works in [20], we study the stability of the Cauchy problem of (1.2) and (1.3) in analytic Sobolev norm under multi-Liouvillean forced frequency. To this end, we have to discuss the rotations reducibility* of PDEs
*Equation (1.4) is said to be rotations reducible, if there exists a quasi-periodic transformation u=Φ(ωt)[v] such that system (1.4) is transformed into ∂tv+m(ωt,ξ))⋅∂xv=0 and m is close to constant.
∂tu+(ξ+f(x,ωt,ξ))⋅∂xu=0, | (1.4) |
and
∂tu+(ρ+f(x,ωt))⋅∂xu=0 | (1.5) |
since it is quite difficult to study the reducibility of PEDs in the case of Liouvillean forced frequency, and such rotations reducibility is able to ensure the stability of solutions of Cauchy problems (1.2) and (1.3). See Theorems 3.1 and 3.2 in Section 3 for more detail. As done in [20], we use the identification between derivation operators and vector fields in order to change rotations reducible for PDEs (1.4) and (1.5) into a corresponding rotations reducible for analytic vector field on Tm+d
ω⋅∂∂φ+(ξ+f(θ,φ,ξ))⋅∂∂θ | (1.6) |
and for analytic vector field on T1+d
ω⋅∂∂φ+(ρ+f(θ,φ))⋅∂∂θ. | (1.7) |
Under appropriate hypotheses on the size of f, rotations reducibility of vector field (1.6) or (1.7) follows a KAM-type iterative scheme with multidimensional Liouvillean forced frequency in the analytic Sobolev spaces Hs. See Propositions 1 and 2 in Section 3 for more detail; these results can be regarded as the generalization of V.I. Arnold's [1] Cω reducible result on analytic vector fields in the case of Diophantine frequency to C∞ rotations reducible in the case of multidimensional Liouvillean forced frequency.
Let us briefly review recent developments in KAM theory with Liouvillean forced frequency. In 2011, Avila et al. [4] considered the following linear skew-products
(α,A):T×R2×T×R2(x,ω)→(x+α,A(x)⋅ω) | (1.8) |
where α∈R and A:T→SL(2;R) is analytic, and gave a KAM scheme for the rotations reducibility of SL(2;R) cocycles with Liouvillean frequency by using the technique of continued fractions. Subsequently, X. Hou and J. You [24] further proved that a quasi-periodic linear differential equation (1.8) in sl(2,R) with two frequencies (1,α) is almost reducible provided that the coefficients are analytic and close to a constant. In the case that α is Diophantine, they got the non-perturbative reducibility. Reducibility and the rotations reducibility for an arbitrary irrational α under some assumption on the rotation number were also obtained. For more references, see [38,39,40,41,42,43,44]. Starting from the works [4] and [24], the question of reducibility on quasi-periodically forced systems beyond Diophantine or Brjuno condition † has been one of the central themes of the subject---when can the dynamics of a given system be related to those of a linear model, as for example periodic or quasi-periodic motion on a torus? However, very little is known about reducibility (or linearization) of quasi-periodically forced nonlinear flows on a m-dimensional torus under a non-resonance condition on forced frequency weaker than the Diophantine or Brjuno condition.
†If there exist γ>0 and τ>d such that
|⟨k,ω⟩|≥γ|k|τ,k∈Zd∖{0}, |
then we say the frequency ω satisfies Diophantine conditions. Slightly weaker than Diophantine conditions can be often required as Brjuno conditions, which are defined by
∑n≥02−nmax0<|k|≤2n,k∈Zdln1|⟨k,ω⟩|<∞. |
If the frequency ω does not satisfy Brjuno condition, we call it Liouvillean.
An analytic quasi-periodically forced (qpf) m-torus flow is the flow
˙θ=f(θ,φ),˙φ=ω |
defined by an analytic vector field of the form
ω⋅∂∂φ+f(θ,φ)⋅∂∂θ, | (1.9) |
where f:Tm×Td→Rm is analytic, and ω∈Rd is rationally independent. We denote by (ω,f) the flow of the vector field (1.9) for simplicity. Recently, in the case of m=1, Krikorian et al. [27] proved that the flow (ω,ρ+f) with ω=(1,α), where f:T×T2→R is analytic, α satisfies not super-Liouvillean condition
β(α):=limsupn>0lnlnqn+1lnqn<∞ | (1.10) |
with {pnqn} being the continued fraction approximate to α and its rotation number ρf is Diophantine with respect to the basic frequency ω ‡, is C∞ rotations reducible provided that f is sufficiently small. Their inspiration comes from reducibility theory of quasi-periodic SL(2,R) cocycle [2,3,4,5,6,14,15,19,24] and their breakthrough is solving a homological equations of variable coefficient taking advantage of diagonally dominant operators. This work generalizes the linear result [4,24] to general nonlinear quasi-periodically forced circle flows. Subsequently, using the same Liouvillean non-resonant condition, i.e., ω=(1,α) satisfies not super-Liouvillean condition (1.10), J. Wang and J. You [35] proved the boundedness of solutions for non-linear quasi-periodic differential equations. In addition, there are many interesting works on the Stoker's problem (existence of response solutions) for nonlinear mechanical models with Liouvillean forced frequency. Since the Stoker's problem is different from that of the reducible problem, its Liouvillean forced frequency can be much weaker. Readers can consult [12,29,36,43] for related results.
‡that is
|⟨k,ω⟩+lρf|≥γ|k|τ,for all0≠(k,l)∈Zd×Z, |
with γ>0 and τ>2.
All the above reducible results about quasi-periodically forced (qpf) flows with Liouvillean forced frequency are mainly concerned with two-dimensional frequency. However, for multidimensional Liouvillean forced frequency, the theory of high-dimensional continued fractions is not completely satisfactory. Thus, we must look for more appropriate methods to deal with the case of multidimensional Liouvillean forced frequency. In this direction, there are some works on ODEs or PDEs, see [12,34,37,38]. In particular, Xu et al. [38] provided quasi-periodic solutions for Hamiltonian PDE with frequency vector ω=(¯ω1,¯ω2)∈R2×Rd−2, ¯ω1=(1,α), α∈R∖Q, which is called weak Liouvillean frequency, i.e., for γ>0,τ>d+1, there is
{β(α):=limsupn>0lnlnqn+1lnqn<∞,|⟨k1,¯ω1⟩+⟨k2,¯ω2⟩|≥γ(|k1|+|k2|)τ,fork1∈Z2,k2∈Zd−2∖{0}, | (1.11) |
where {pnqn} is the continued fraction approximate to α. We note that in (1.11) only ¯ω1 is allowed to be Liouvillean, not the entire frequency ω.
In the case of multidimensional Liouvillean forced frequency, a key step the stability of the Cauchy problem (1.2) or (1.3) is to study rotations reducibility of the qpf m-torus flow (ω,ρ+f). Even under Brjuno conditions, this problem becomes more complex in higher-dimensional tori (m≥2) because there is no conception of rotation number in higher-dimensional tori. The corresponding conception are rotation vectors, which are highly relying on the orbit of flow. In most cases, the rotation vector is not unique but a set. Thus, it is difficult to classify the quasi-periodically forced tori using the characters of rotation vector. In the case m≥1, d≥2, W. Si and J. Si[33] proved that the qpf flow (ω,ρ+f), where ρ∈Rm, f:Tm×Td→Tm is analytic and (ω,ρ) satisfies the Brjuno-Rüssmann's non-resonant condition
|⟨k,ω⟩+⟨l,ρ⟩|≥γΔ(|k|+|l|),for all0≠(k,l)∈Zd×Zm, |
is Cω reducible provided that f is sufficiently small and satisfies some non-degeneracy condition (see [33] for the definition of Δ).
In this paper, we will prove that, in the case m≥1 and d≥2, the analytic qpf m-torus flow (ω,ξ+f) can be C∞ rotations reducible when ω possesses finite uniform Diophantine exponent (see (2.3) below), f is sufficiently small and ξ∈Oγ, where Oγ⊂O is Cantor subset of compact set O in Rm with positive Lebesgue measure. In the case m=1 and d≥2, the analytic qpf circle flow (ω,ρ+f) can be C∞ rotations reducible when ω possesses finite uniform Diophantine exponent (see (2.3) below), its rotation number ρf is Diophantine with respect to the forced frequency ω, and f is sufficiently small.
Essentially, the Liouvillean property in multidimensional Liouvillean frequency (1.11) comes from 2-dimensional frequency ˉω1. In this paper, we introduce multidimensional Liouvillean frequency by using finite uniform Diophantine exponent, which can holistically describe the Liouvillean property of a frequency. We can also prove that our Liouvillean frequency set contains all the Liouvillean frequencies in (1.11). This is the main novelty of our paper.
The rest of this paper is organized as follows. In Section 2, we introduce some notations, definitions, and relevant concepts, which will be used subsequently. In Section 3, we first introduce the results of rotations reducibility for qpf m-torus flow (ω,ξ+f) in the case m≥1 and d≥2 and qpf circle flow (ω,ρ+f) in the case m=1 and d≥2, i.e., Proposition 3.1 and Proposition 3.2. Then we give the proof of the main result, Theorem 1. In Section 4, we give the proof of Propositions 3.1 and 3.2. In Section 5, we give an Appendix that introduces the definition of CD-bridge.
In this section, we first give some notations, definitions, and relevant concepts, which will be used subsequently..
Denote by Z and Z+ the sets of integers and positive integers, respectively. Let Tl=Rl/2πZl be the standard l-dimensional real torus. Given m∈Z+, we consider a real valued function f∈L2(Td,Rm)
f(φ)=∑k∈Zdfkei⟨k,φ⟩. |
We define the sets
Wr(Td):={φ∈(Td⊕iRd):|Imφ|<r},Ws,r(Tm×Td):={(θ,φ)∈(Tm⊕iRm)×(Td⊕iRd):|Imθ|<s,|Imφ|<r}. |
For an analytic function f(φ)=∑k∈Zdfkei⟨k,φ⟩ defined in Wr(Td), we define the spaces of analytic functions
Hr:=Hr(Wr(Td),Rm)={f:‖f‖2r=∑k∈Zd|fk|2e2|k|r<+∞}, |
where r>0. For any N>0, we define the truncation and projection operators TN, RN on Hr,a
TNf(φ)=∑|k|<Nfkei⟨k,φ⟩,RNf(φ)=∑|k|≥Nfkei⟨k,φ⟩. |
If we consider two variables (θ,φ)∈Tm+d, we may consider a real valued function f(θ,φ)∈L2(Tm+d,Rm) as a φ-dependent family of functions f(θ,⋅)∈L2(Tm,Rm) with the Fourier series expansion
f(θ,φ)=∑l∈Zmfl(φ)ei⟨l,θ⟩=∑(l,k)∈Zm+dfklei(⟨l,θ⟩+⟨k,φ⟩) |
In this case, we describe the spaces of analytic functions defined in Ws,r(Tm×Td)
Hs,r:=Hs,r(Ws,r(Tm×Td),Rm)={f:‖f‖2s,r=∑l∈Zm‖fl(φ)‖2re2|l|s=∑l∈Zm,k∈Zd|fkl|2e2|l|s+2|k|r<+∞,}, |
where s>0, r>0. It is clear that the spaces Hr and Hs,r are Banach algebra under their norms, see [28]. Fix m∈Z+ and let O be a compact subset of Rm. For a function f:O→Hr, we define the norm of f as
‖f‖2r,O=∑k∈Zd|fk|2Oe2|k|r, |
where
|fk|O=supξ∈O|fk(ξ)|+supξ∈O|dfk(ξ)dξ|. |
For a function f:O→Hs,r, we define the norm of f as
‖f‖2s,r,O=∑l∈Zm,k∈Zd|fkl|2Oe2|l|s+2|k|r, |
where
|fkl|O=supξ∈O|fkl(ξ)|+supξ∈O|dfkl(ξ)dξ|. |
Here, the derivative with respect to ξ is in the sense of Whitney.
For an analytic function f(θ,ϕ) defined in Ws,r(Tm×Td), its supremum norm is defined by
|f(θ,ϕ)|s,r=supImθ<s,Imφ<r|f(θ,ϕ)|. |
By the attenuation of Fourier coefficients fkl, i.e., |fkl|≤|f|s,re|l|s+|k|r, one can check that
‖f‖2s,r=∑l∈Zm,k∈Zd|fkl|2e2|l|s+2|k|r≤|f|2s,r. |
In addition, one can easily check that |f|2s,r≤‖f‖2s,r, which implies ‖f‖s,r=|f|s,r.
Consider a diffeomorphisms of the m-dimensional torus
Φ:Tm→Tm,θ↦θ+h(θ)=ˆθ, | (2.1) |
where h:Tm→Rm is an analytic function with ‖h‖s≤1/2. We denote the inverse of Φ by
Φ−1:Tm→Tm,Φ−1:θ↦θ+˜h(θ), |
with ˜h an analytic function. Using the same notation, we denote transformations like (2.1) with the corresponding linear operators acting on Hr as
Φ:Hr→Hr,f(θ)↦Φf(θ):=f(θ+h(θ)). |
Similarly, we consider the action of Φ on the vector fields on Tm by the pushforward. Explicitly, we denote by T(Tm) the tangent space of Tm.
Now given a vector field X:Tm→T(Tm)
X(θ)=m∑j=1Xj(θ)∂∂θj,X1,⋯,Xm∈Cω(Tm,R), |
its pushforward is
(Φ∗X)(θ)=dΦ(Φ−1(θ))[X(Φ−1(θ))]=m∑i=1Φ−1(Xi+m∑j=1∂Xi∂θjXj)∂∂ˆθi. |
A Cω vector field X(θ)=∑mj=1Xj(θ)∂∂θj induces a linear operator acting on the space of functions f:Tm→R, that we denote by X(θ)⋅∂θ=∑mj=1Xj(θ)∂∂θj. More precisely, the action of such a linear operator is given by
Hr→Hr,u(θ)↦X(θ)⋅∂θu(θ). |
We consider the case of m=1. Suppose (ω,f) is a qpf circle flow defined by
ω⋅∂φ+f(θ,φ)⋅∂θ. |
Let
˜ρ(ω,f)=limt→∞ˆΦtφ(ˆθ)t |
be the fibred rotation number associated with (ω,f), where ˆΦtφ(ˆθ):R1+×R1×Td→R1, via (t,ˆθ,φ)↦ˆΦtφ(ˆθ) denotes the lift of the flow (ω,f) of the first variable θ. The limit exists and is independent of (ˆθ,φ) [23]. As a direct consequence of the definition, we have the following well-known results:
Lemma 2.1. ([23]) Let ω∈Rd,ρ∈R and ‖f(θ,φ)‖C0≤ϵ, then
|˜ρ(ω,ρ+f)−˜ρ(ω,ρ)|≤ϵ. |
Lemma 2.2. ([23]) Suppose (ω,f) is a qpf circle flow and H∈C0(T×Td,T×Td) is a homeomorphism homotopic to the identity that projects to the identity on the second factor. Then the fibred rotation numbers of (ω,f) and H∘(ω,f)∘H−1 are the same.
In this subsection, we recall the definition of uniform Diophantine exponent of forced frequency ω0∈Rd introduced in [17].
• Recall the Diophantine exponent denoted by ω(ω0) that the supremum of all positive real numbers κ such that there exists k∈Zd∖{0} and |⟨k,ω0⟩|≤|k|−κ.
• Recall the uniform Diophantine exponent denoted by ˆω(ω0) that the supremum of all positive real numbers κ such that for any sufficiently large N, there exists k∈Zd∖{0} such that |k|≤N and |⟨k,ω0⟩|≤N−κ. In other words, if we define the set
S:={κ|forallNlargeenough,thereexistsk∈Rd∖{0},0<|k|≤Nsuchthat|⟨k,ω0⟩|≤N−κ}, | (2.2) |
then ˆω(ω0)=supS.
By the definitions of ˆω(ω0) and ω(ω0), one obtains that ˆω(ω0)≤ω(ω0), and while ω(ω0) measures how small linear forms with integer coefficients of a given size can become when evaluated at ω0. It is clear that if ω(ω0)<∞ then ω0 is Diophantine vector and if ω(ω0)=∞ then ω0 is not Diophantine vector. For uniform Diophantine exponent ˆω(ω0), if ˆω(ω0)=∞, then ω0 is not Diophantine vector. But if ˆω(ω0)<∞, then ω0 also contains Liouvillean frequency. For example, ˆω(ω0) is always finite if ω0=(1,α) with α∈R∖Q because |⟨k,ω0⟩|>1/2qn, for 0<|k|≤qn, where pnqn is the continued fraction approximated to α.
In this paper, we strengthen the condition ˆω(ω0)<∞ to exist a τ1 and an increasing sequence Kn such that
|⟨k,ω0⟩|≥1Kτ1n,∀0<|k|≤Kn, | (2.3) |
where the sequence {Kn}n≥0 satisfies
(1) For the given constant τ1>0, there exists a positive constant M>d/2 such that
Kn+1>KMn; | (2.4) |
(2) There exists a positive constant N such that
KNn≥lnKn+1. | (2.5) |
Remark 2.1. Assumptions (2.4) and (2.5) are really technical conditions summarized from not super-Liouvillean condition in [27]. This property can also be reflected in high dimensional Liouvillean frequencies. One of the purposes of this paper is an observation of this property in higher dimensions.
Definition 2.1 (Diophantine condition). Given a frequency ω∈Rd, we call ω Diophantine if there exists a sequence {Kn}n≥0 with Kn=Kˆβn−1 and 1<ˆβ<2 such that
|⟨k,ω⟩|≥γKτ1n,∀0<|k|<Kn, |
where γ∈(0,1), τ1>d−1.
Definition 2.2 (MNτ1-admissibility). For a given sequence {Kn}n≥0, N,τ1>0 and M>d/2, we call {Kn}n≥0 is (MNτ1)-admissible if it satisfies the inequalities (2.4) and (2.5). The set of all such sequences is denoted by
K(τ1,M,N)={K|K={Kn}n≥0andKis(MNτ1)−admissible}. |
Definition 2.3 (Liouvillean vectors set). For given K={Kn}n≥0∈K(τ1,M,N), we define the set
OK={ω∈D||⟨k,ω⟩|≥CKτ1n,∀0<|k|≤Kn}, | (2.6) |
where D∈Rd is compact and C is a positive number. Setting
L:=⋃τ1,M,N>0⋃K∈K(τ1,M,N)OK, |
we call L Liouvillean vectors set.
We will show that the set L allows many Liouvillean vectors. In the following discussions, we always assume C≡1 occurring in the frequency set (2.6) without loss of generality.
Remark 2.2. If frequency ω0 belongs to Liouvillean vectors set L, then convergence speed of small divisor tending to 0 can be much faster than Diophantine case that is because if ω0∈L, Kn+1=O(eKn) increases much faster than the Diophantine case Kn+1=O(Knˆβ).
Remark 2.3. We can prove that all the vectors in L have finite uniform Diophantine exponent. In fact, by definition of ω0, we have that if ω0∈L, then there exists K, which is (MNτ1)-admissible, such that ω0∈OK. For the set S defined in (2.2), it is clear that S is no-empty because 0∈S, and it has the following two characters:
1. If κ1∈S, then κ∈S for all κ≤κ1. That is because if κ1∈S, then for all N large enough, there exists k∈Rd∖{0}, 0<|k|≤N, such that |⟨k,ω0⟩|≤N−κ1, which implies |⟨k,ω0⟩|≤N−κ for all κ≤κ1. Thus, κ∈S.
2. If κ1∉S, then κ∉S for all κ≥κ1. That is because if κ1∉S, then there exists N large enough, for all k∈Rd∖{0}, 0<|k|≤N, such that |⟨k,ω0⟩|>N−κ1, which implies |⟨k,ω0⟩|>N−κ for all κ≥κ1. Thus, κ∉S.
Moreover, ω0∈OK, then τ1∉S. Thus 0≤ˆω(ω0)=supS≤τ1.
Remark 2.4. Now, we illustrate that the Liouvillean vectors set L allows many Liouvillean vectors.
In two-dimensional case, we consider ω=(1,α) in R2. Let A=8 and (Qn)n∈N be the selected subsequence in Lemma A.1 of Appendix. By Corollary 2.1 in [27], if β(α)<∞ we have the following claims:
(1) Qn>QAn−1. That is because for n=1, it is clear that Q1>QA0 since Q0=1. If n≥2 and ¯Qn−1≤QAn−1, then one has Qn≥¯Qn−1≥QAn−1. If n≥2 and ¯Qn−1>QAn−1, then (¯Qn−1,Qn) and (Qn,Qn+1) are both CD(A,A,A3) bridges. Thus, we get Qn>QAn−1.
(2) lnQn+1<QUn, where U=β(α)+4lnAln2. That is because
lnlnQn+1lnQn≤4lnA+lnln¯QnlnQn≤β(α)+4lnAln2, |
which implies lnQn+1<QUn.
We also have ⟨k,ω⟩≥12Qn for all 0<|k|<Qn, which implies ω∈∪K∈K(1,A,U)OK. Thus L includes some frequencies beyond Brjuno in two-dimensional case.
In multidimensional case, L includes all the frequency satisfying condition (1.11). That is because if ω=(¯ω1,¯ω2)∈R2×Rd−2 satisfies condition (1.11), then
|⟨k1,¯ω1⟩+⟨k2,¯ω2⟩|≥γQτn,forall|k1|+|k2|≤Qn,|k2|≠0 |
and
|⟨k1,¯ω1⟩+⟨k2,¯ω2⟩|≥12Qn,forall|k1|≤Qn,|k2|=0, |
where (Qn)n∈N is the selected subsequence in Lemma A.1 of Appendix. If τ1=max{1,τ}, then ω∈∪K∈K(τ1,A,U)OK, and so ω∈L.
For any integer m≥1 and d≥2, we have the following Proposition.
Proposition 3.1. Let O be a bounded closed set in Rm not containing zero, r>0, s>0, τ>m+d−1, γ>0, ω∈L and ξ∈O. Consider a C1-smooth family of vector fields on Tm+d
X:=ω⋅∂∂φ+(ξ+f(θ,φ,ξ))⋅∂∂θ |
where f(⋅;ξ)∈Hs,r(Ws,r(Tm×Td),Rm). Then, there exists a sufficiently small constant ϵ=ϵ(τ,τ1,γ,s,r, M,N,O) such that if
‖f(θ,φ)‖s,r,O≤ϵ, | (3.1) |
then there exists a set
O∞={ξ∈O:|⟨k,ω⟩|+⟨l,ρ∞(ξ)⟩≥γ∞(|k|+|l|)τ,∀(k,l)∈Zd+m,|l|≠0,}, | (3.2) |
where ρ∞(ξ):O∞→Rm is a C1 function and there exists a map
h(θ,φ,ξ):Tm+d×O∞→Rm,supφ∈Td‖h(θ,⋅)‖s2,O∞≤Cϵ, | (3.3) |
which is C∞-smooth in φ, analytic in θ and C1-smooth in ξ, so that for all ξ∈O∞ the pushforward of vector fields X by diffeomorphism Ψ:(φ,θ)↦(φ,θ+h(θ,φ))=(φ,ˆθ) is
Ψ∗X=ω⋅∂∂φ+Ψ−1(ω⋅∂φh+(ξ+f)+(ξ+f)⋅∂φh)⋅∂∂ˆθ=ω⋅∂∂φ+(ρ∞(ξ)+m∞(φ,ξ))⋅∂∂ˆθ. |
where m∞:Td×O∞→Rm, (φ,ξ)↦m∞(φ,ξ) is C1-smooth in ξ, analytic in φ and for each ξ∈O∞ satisfies
supφ∈Td|m∞(φ,ξ)|≤Cϵ. |
For m=1 and any integer d≥2, we have the following Proposition.
Proposition 3.2. Let ρ∈R, ω∈L, r>0, s>0, τ>d and γ>0. We consider a vector field on Td+1
X:=ω⋅∂∂φ+(ρ+f(θ,φ))⋅∂∂θ |
where f(θ,φ)∈Hs,r(Ws,r(T×Td),Rm). Assume that ˜ρ(ω,ρ+f(θ,φ))=ρf∈DCω(γ,τ) in the sense
|⟨k,ω⟩+lρf|≥γ(|k|+|l|)τ,∀(k,l)∈Zd×Z,l≠0. |
Then there exists a sufficiently small constant ϵ=ϵ(τ,τ1,γ,s,r,M,N) such that if ‖f(θ,φ)‖s,r≤ϵ, and there exists a map
h(θ,φ):T1+d→R,supφ∈Tm‖h(θ,⋅)‖s2≤Cϵ, | (3.4) |
which is C∞-smooth in φ, analytic in θ, so that the pushforward of vector fields X by diffeomorphism Ψ:(φ,θ)↦(φ,θ+h(θ,φ))=(φ,ˆθ) is
Ψ∗X=ω⋅∂∂φ+(ρf+m∞(φ))⋅∂∂ˆθ, |
where m∞(φ) is C∞-smooth in φ and satisfies
supφ∈Td|m∞(φ)|≤Cϵ. |
Remark 3.1. The proof of Proposition 3.1 or Proposition 3.2 is based on a KAM iterative scheme under the assumption that forced frequency has finite uniform Diophantine exponent. As we know, the results existing in the literature deal with two-dimensional frequency, and exploit the theory of continued fractions to control the small divisor problem. The results in this paper extend the analysis to higher dimensional frequency with finite uniform Diophantine exponent, allowing a class of Liouvillean frequencies. More specifically, Proposition 1 can be regarded as a generalization of the work in [20] from Diophantine frequency to Liouvillean frequency and work in [33] from Brjuno frequency to Liouvillean frequency, while Theorem 3.1 can be regarded as a generalization of the work in [27] from two-dimensional not super-Liouvillean forced frequency to high-dimensional forced frequency with finite uniform Diophantine exponent. The overall strategy of this paper comes from the literature [27,33], but the method is still in the spirit of [4,17,24], which, however, has to overcome essential obstructions in techniques for dealing with the questions considered here.
Remark 3.2. Compared with the results in [20], which reduce the variables (θ,φ), our results only reduce variable θ because of the difficulty from Liouvillean frequency. Even so, the analytic norms of the solutions of the Cauchy problem can also be controlled uniformly in time. This is the following Theorem 1.
Theorem 1. Let ω∈L and consider the transport equation
∂tu+(ξ+f(x,ωt,ξ))⋅∂xu=0. | (3.5) |
Then if f(⋅;ξ)∈Hs,r(Ws,r(Tm×Td),Rm) is C1-smooth with respect to ξ and (3.1) is fulfilled, for ξ∈O∞ (see (3.2)), under the change of variable u=Ψ(ωt)[v]=v(x+h(x,ωt)) defined in (3.3), the PDE (3.5) transforms into the equation with coefficients independent of spatial variable x
∂tv+(ξ+m∞(ωt,ξ))⋅∂xv=0. |
As a consequence, for u0∈Hs2, the only solution of the Cauchy problem
{∂tu+(ξ+f(x,ωt,ξ))⋅∂xu=0,u(x,0)=u0(x), |
satisfies ‖u(x,t)‖s4≤‖u0(x)‖s2 for any t∈R, i.e., this Cauchy problem is stable.
Proof. Let ω∈L and ξ∈O∞. Under the change of coordinates u=Ψ(ωt)[v]=v(x+h(x,ωt)), the equation transforms (3.5) into the PDE
∂tv+Ψ(ωt)−1(ω⋅∂φh+ξ+f+(ξ+f)⋅∂xh)⋅∂xv=0. |
In view of Proposition 3.1, one obtains that
∂tv+(ξ+m∞(ωt,ξ))⋅∂xv=0, |
where
m∞(ωt,ξ)=Ψ(ωt)−1(ω⋅∂φh+f+(ξ+f)⋅∂xh). |
Let v=∑l∈Zmvl(t)ei⟨l,x⟩. Such a PDE with coefficients independent of a spatial variable can be integrated explicitly, implying that for any l∈Zm, vl(t)=vl(0)ei⟨ξ+∫t0m∞(ωs)ds,l⟩ and
‖v(x,t)‖2s=∑l∈Zm|vl(0)ei⟨ξt+∫t0m∞(ωs)ds,l⟩|2e2s|l|=∑l∈Zm|vl(0)|2e2s|l|=‖v(0)‖2s. |
By Proposition 3.1 and inverse mapping theorem, we get Ψ(ωt)−1[u]=u(x+˜h(x,ωt)), where
supφ∈Td‖˜h(x,⋅)‖s2−σ≤Cϵ |
with σ=s/12. Then, given u0(x)∈Hs(Tm,R), one gets that for each t∈R
‖u(x,t)‖s4=‖Ψ(ωt)[v]‖s4=‖v(x+h(x,ωt),t)‖s4≤‖v(x,t)‖s4+σ=‖v(0)‖2s4+σ=‖Ψ−1(ωt)[u0]‖s4+σ=‖u0(x+˜h(x,ωt))‖s4+σ≤‖u0(x)‖s4+2σ≤‖u0(x)‖s2. |
This completes the proof.
Similarly, we can prove the following theorem by using Proposition 3.2.
Theorem 2. Let ρ∈R and ω∈L and consider the transport equation
∂tu+(ρ+f(x,ωt))⋅∂xu=0. | (3.6) |
Then, if ˜ρ(ω,ρ+f(θ,φ))=ρf∈DCω(γ,τ), f∈Hs,r(Ws,r(T×Td),R) and ‖f(θ,φ)‖s,r≤ϵ, under the change of variable u=Ψ(ωt)[v]=v(x+h(x,ωt)) defined in (3.4), the PDE (3.6) transforms into the equation with coefficients independent of spatial variable x
∂tv+(ρf+m∞(ωt))⋅∂xv=0. |
As a consequence, for u0∈Hs2, the only solution of the Cauchy problem
{∂tu+(ρ+f(x,ωt))⋅∂xu=0,u(x,0)=u0(x), |
satisfies ‖u(x,t)‖s4≤‖u0(x)‖s2 for any t∈R, i.e., this Cauchy problem is stable.
In this section, we give the proofs of Propositions 3.1 and 3.2 by a modified KAM iterative scheme. To better understand this iterative scheme, we draw a block diagram as follows:
Since ω∈L, we assume that M,N,τ1 are defined as in (2.3). And we take c and α satisfying c>16α(3τ+2+d2)τN and max{M(2τ1−d−2)2M−d,1}<α<M. For r0, s0, γ>0, τ>m+d−1, we suppose K0>0 large enough such that
KM−α16M0lnK0>cτN, |
and ϵ0 is sufficiently small. We further define the following sequences for n≥1:
Δn=s010⋅2n−1,γn=γ0(1−n+1∑j=22−j),rn=r04Kαn,sn=sn−1−Δn,ϵn=ϵn−1K2n+1cτNn+1,˜ϵn=∑n−1i=0ϵi. | (4.1) |
Let
On−1={ξ∈On−2:|⟨k,ω⟩|+⟨l,ρn−1(ξ)⟩≥γn−1(|k|+|l|)τ,∀|l|≠0,|k|+|l|≤T(n−1)}, | (4.2) |
where T(n−1)=(γ332ϵ12n−1)13τ+2+d2 and O−1=O.
In (n−1)-th step, the vector fields Xn−1 can be written as
Xn−1=ω⋅∂φ+(ρn−1(ξ)+gn−1(φ,ξ)+fn−1(θ,φ,ξ))⋅∂θ, | (4.3) |
where ξ∈On−1, [gn−1]φ=0 and
‖gn−1(φ,ξ)‖rn−1,On−1≤4˜ϵn−1,‖fn−1(θ,φ,ξ)‖sn−1,rn−1,On−1≤ϵn−1. |
In what follows, we divide KAM step into three parts. In the proof, we do not write ξ explicit for simplicity if there is no confusion. First, we give a Lemma that is used to eliminate the lower-frequency terms.
Lemma 4.1. If we denote ˉsn=sn−1−Δn3, ˉrn=r0Kαn, then there exists hn−1(φ) with ‖hn−1(φ)‖ˉrn,On−1≤KMτ1+(d+1)αMn, such that the transformation H_{n-1}:\theta = \overline\theta+h_{n-1}(\varphi)(\mod 2\pi) conjugates the vector fields (4.3) into
\begin{eqnarray} \overline X_{n-1} = (H_{n-1})_{*} X_{n-1} = \omega \cdot\partial_\varphi +(\rho_{n-1}+\bar g_{n-1}(\varphi)+\bar f_{n-1}(\bar\theta, \varphi))\cdot \partial_{\bar\theta}, \end{eqnarray} | (4.4) |
where
\begin{eqnarray*} &&\|\bar g_{n-1}(\varphi)\|_{\bar r_{n}, {O}_{n-1}}\leq \epsilon_{n-1}^{\frac{1}{2}}, \\ &&\|\bar f_{n-1}(\bar\theta, \varphi)\|_{\bar s_{n}, \bar r_{n}, {O}_{n-1}}\leq \epsilon_{n-1}. \end{eqnarray*} |
Proof. Using transformation \theta = \bar\theta+h_{n-1}(\varphi)(\mod 2\pi) , the vector field (4.3) becomes
\begin{eqnarray} \overline X_{n-1} = \omega\cdot \partial_\varphi+(\rho_{n-1}-\partial_\omega h_{n-1}(\varphi)+g_{n-1}(\varphi)+f_{n-1}(\bar\theta+h_{n-1}(\varphi), \varphi))\cdot\partial_{\bar\theta}. \end{eqnarray} | (4.5) |
The vector field (4.5) becomes
\begin{eqnarray*} \overline X_{n-1} = \omega\cdot \partial_\varphi+\tilde f_{n-1}(\bar\theta, \varphi)\cdot\partial_{\bar\theta}, \end{eqnarray*} |
where \tilde f_{n-1}(\bar\theta, \varphi) = \rho_{n-1}+\mathcal{R}_{K_n}g_{n-1}(\varphi)+f_{n-1}(\bar\theta+h_{n-1}(\varphi), \varphi) , if homological equation \partial_\omega h_{n-1}(\varphi) = \mathcal{T}_{K_n}g_{n-1}(\varphi) is solvable. In view of |\langle k, \omega\rangle| > \frac{1}{K_n^{\tau_1}} for all |k|\leq K_n , one can check that
\begin{eqnarray*} \|h_{n-1}(\varphi)\|^2_{\frac{r_{n-1}}{2}, O_{n-1}}&\leq& \sum\limits_{0 < |k|\leq K_n}|h_{n-1, k}|_{O_{n-1}}^2e^{2|k|\frac{r_{n-1}}{2}}\\ &\leq&\sum\limits_{0 < |k|\leq K_n}|\frac{g_{n-1, k}}{\langle k, \omega\rangle}|_{O_{n-1}}^2e^{2|k|\frac{r_{n-1}}{2}}\\ &\leq& K_n^{2 \tau_1} \|g_{n-1}(\varphi)\|^2_{r_{n-1}, O_{n-1}} \sum\limits_{0 < |k|\leq K_n} e^{-|k|\frac{r_{n-1}}{2}}\\ &\leq& K_n^{2 \tau_1} \|g_{n-1}(\varphi)\|^2_{r_{n-1}, O_{n-1}} (\frac{2}{r_{n-1}})^d\\ & < &K_n^{\frac{2M\tau_1+d\alpha}{M}}, \end{eqnarray*} |
since K_{n-1} < K_n^{\frac{1}{M}} and \epsilon_0 sufficiently small. In order to control f_{n-1}(\bar\theta+h_{n-1}(\varphi), \varphi) , we should illustrate that {\rm Im} h_{n-1}(\varphi) can be well controlled. Let \varphi = \varphi_1+{\rm i} \varphi_2 for \varphi_1\in \mathbb{T}^{d} and \varphi_2\in \mathbb{R}^d and define
h_{n-1}^1(\varphi_1) = \sum\limits_{0 < |k|\leq K_n} \frac{g_{n-1, k}}{{\rm i}\langle k, \omega\rangle} e^{{\rm i}\langle k, \varphi_1 \rangle}. |
Notice that g_{n-1}(\varphi) is real analytic, which implies {\rm Im} h_{n-1}^1(\varphi_1) = 0 . Since \frac{1}{K_j^{\frac{\alpha (2m-d)-M(2\tau_1-d-2)}{2M}}} goes to zero much faster than \Delta_j , we can assume \frac{1}{K_j^{\frac{\alpha (2m-d)-M(2\tau_1-d-2)}{2M}}} < \frac{\Delta_j}{3} for all j\geq 0 without loss of generality. Thus, applying Cauchy-Schwarz inequality, we can get
\begin{eqnarray*} \sup\limits_{\varphi\in \mathbb{T}^d_{\bar r_n}}|{\rm Im} h_{n-1}(\varphi)|& = & \sup\limits_{\varphi\in \mathbb{T}^d_{\bar r_n}}|\sum\limits_{0 < |k|\leq K_n} \frac{g_{n-1, k}}{\langle k, \omega\rangle}(e^{{\rm i}\langle k, \varphi_1+{\rm i}\varphi_2\rangle}-e^{{\rm i}\langle k, \varphi_1\rangle}) |\\ &\leq& \sum\limits_{0 < |k|\leq K_n} \left|\frac{g_{n-1, k}}{\langle k, \omega\rangle}\right|_{O_{n-1}}\sup\limits_{\varphi\in \mathbb{T}^d_{\bar r_n}}|e^{-\langle k, \varphi_2\rangle}-1|\\ &\leq & \sum\limits_{0 < |k|\leq K_n} \left|\frac{g_{n-1, k}}{\langle k, \omega\rangle}\right|_{O_{n-1}} |k|\bar{r}_{n}\\ &\leq& K_n\frac{r_0}{K_n^\alpha}\sqrt{\sum\limits_{|k| > 0}|g_{n-1, k}|^2}\sqrt{\sum\limits_{0 < |k|\leq K_n} \langle k, \omega \rangle^{-2}}\\ &\leq&CK_n^{\tau_1+\frac{d}{2}+1-\alpha}K_{n}^{\frac{\alpha d}{2}} \|g_{n-1}(\varphi)\|_{r_{n-1}, O_{n-1}}\leq K_n^{\tau_1+\frac{d}{2}+1-\alpha}K_{n}^{\frac{\alpha d}{2M}}\\ &\leq& C\frac{1}{K_n^{\frac{\alpha (2m-d)-M(2\tau_1-d-2)}{2M}}}\leq \frac{\Delta_n}{3}. \end{eqnarray*} |
Letting \bar g_{n-1}(\varphi) = \mathcal{R}_{k_n}g_{n-1}(\varphi) and \bar f_{n-1}(\bar\theta, \varphi) = f_{n-1}(\bar\theta+h_{n-1}(\varphi), \varphi) , we have
\begin{eqnarray*} &&\|\bar f_{n-1}(\bar\theta, \varphi)\|_{\bar s_n, \bar r_n, O_{n-1}} = \|f_{n-1}(\bar\theta+h_{n-1}(\varphi), \varphi)\|_{\bar s_n, \bar r_n, O_{n-1}}\leq \|f_{n-1}(\theta, \varphi)\|_{ s_{n-1}, r_{n-1}, O_{n-1}}\leq \epsilon_{n-1}, \\ &&\|\bar g_{n-1}(\varphi)\|^2_{\bar s_n, \bar r_n, O_{n-1}} = \|\mathcal{R}_{k_n}g_{n-1}(\varphi)\|^2_{\bar r_n, O_{n-1}}\leq\epsilon_0e^{-K_n(r_{n-1}-\bar r_n)}\leq \epsilon_0e^{-K_n^{\frac{M-\alpha}{4M}}}\leq \epsilon^{\frac{1}{2}}_{n-1}, \end{eqnarray*} |
if K_0 > (2c\tau N\ln K_0)^{\frac{2M}{M-\alpha}}. The proof is completed.
In what follows, we should make the perturbation smaller taking advantage of diagonally dominant operators such that the order of perturbation reach that of the next KAM step. But we can not achieve this using one transformation. So we have to make several transformations. We give the following Lemma.
Lemma 4.2. Denote r_{n+} = \frac{r_0}{2K_n^\alpha} , s_{n+} = \bar s_n-\frac{\Delta}{3} . Under the assumptions of Lemma 4.1 and \xi \in O_{n-1} , there exists a transformation \overline {H}_{n-1} with estimates
\begin{eqnarray*} &&\|\overline H_{n-1}-id\|_{s_{n+}, r_{n+}, O_{n-1}}\leq 4\epsilon_{n-1}^{\frac{3}{4}}, \\ &&\|D(\overline H_{n-1}-id)\|_{s_{n+}, r_{n+}, O_{n-1}}\leq 4\epsilon_{n-1}^{\frac{3}{4}}, \end{eqnarray*} |
such that \overline H_{n-1} conjugates the vector field (4.4) to
\begin{eqnarray} \overline X_{n-1, +} = \omega\cdot\partial_\varphi+(\rho_{n-1}+\bar g_{n-1, +}(\varphi)+\bar f_{n-1, +}(\bar\theta_+, \varphi))\cdot\partial_{\bar\theta_+} \end{eqnarray} | (4.6) |
with
\begin{eqnarray*} &&\|\bar g_{n-1, +}(\varphi)\|_{r_{n+}, O_{n-1}}\leq 2\epsilon^{\frac{1}{2}}_{n-1}, \\ &&\|\bar f_{n-1, +}(\bar\theta_+, \varphi)\|_{s_{n+}, r_{n+}, O_{n-1}}\leq \epsilon_{n}, \\ &&\|\bar g_{n-1, +}(\varphi)-\bar g_{n-1}(\varphi)\|_{r_{n+}, O_{n-1}}\leq 4\epsilon_{n-1}. \end{eqnarray*} |
Proof. In order to find \overline H_{n-1} , we divide \overline H_{n-1} into \mathcal N transformations where \mathcal{N} is determined later. Letting \tilde r = \bar r_n = \frac{r_0}{K_n^\alpha} , \tilde s = \bar s_n = s_{n-1}-\frac{\Delta}{3} , \tilde \eta = 2\epsilon_{n-1} , we define the following sequences
\begin{eqnarray*} \begin{array}{l} \tilde r_0 = \tilde r, \quad \tilde s_0 = \tilde s, \quad \tilde\eta_\nu = \tilde \eta^{(\frac{3}{2})^\nu}, \\ \sigma_1 = \frac{\tilde r}{4}, \quad \sigma_{\nu+1} = \frac{1}{2^\nu}\sigma_1, \\ \delta_1 = \frac{\Delta_n}{6}, \quad\delta_{\nu+1} = \frac{1}{2^\nu}\delta_1, \\ \tilde r_\nu = \tilde r_{\nu-1}- \sigma_{\nu}, \quad \tilde s_\nu = \tilde s_{\nu-1}- \delta_{\nu}, \\ T_\nu = \frac{1}{\sigma_\nu}\ln\frac{1}{\tilde \eta_{\nu-1}}. \end{array} \end{eqnarray*} |
In the proof, we do not write n explicitly for simplicity. At (\nu-1) -th step, we write vector field as
\begin{eqnarray} \overline X_{\nu-1} = \omega \cdot\partial_\varphi+(\rho+\bar g_{\nu-1}(\varphi)+\bar f_{\nu-1}(\bar\theta_{\nu-1}, \varphi))\cdot\partial_{\bar\theta_{\nu-1}}, \end{eqnarray} | (4.7) |
where
\begin{eqnarray*} &&\|\bar g_{\nu-1}(\varphi)\|_{\tilde r_{\nu-1}}\leq 2\epsilon_{n-1}^{1/2}, \\ &&\|\bar f_{\nu-1}(\bar\theta_{\nu-1}, \varphi)\|_{\tilde s_{\nu-1}, \tilde r_{\nu-1}}\leq \tilde \eta_{\nu-1}. \end{eqnarray*} |
Applying transformation \phi_{\nu-1}:(\bar\theta_{\nu-1}, \varphi) = (\bar\theta_\nu+h_{\nu}(\bar\theta_\nu, \varphi), \varphi) , the vector field (4.7) becomes
\begin{eqnarray} \overline X_\nu = \omega\cdot\partial_\varphi +(\rho+\bar g_{\nu}(\varphi)+(1+\frac{\partial h_\nu}{\partial \bar\theta_\nu})^{-1}(\bar f_{\nu-1}(\bar\theta_\nu+h_{\nu}(\bar\theta_\nu, \varphi), \varphi)-\bar f_{\nu-1}(\bar\theta_{\nu}, \varphi))\cdot\partial_{\bar\theta_{\nu}}, \end{eqnarray} | (4.8) |
where \bar g_{\nu}(\varphi) = \bar g_{\nu-1}(\varphi)+[\bar f_{\nu-1}(\bar\theta_{\nu}, \varphi)]_{\bar\theta_\nu}, if the homological equation
\begin{eqnarray} \partial_\omega h^i_\nu+\langle\frac{\partial h^i_\nu}{\partial \bar\theta_\nu}, \rho+\bar g_{\nu}(\varphi)\rangle = \bar f^i_{\nu-1}(\bar\theta_{\nu}, \varphi)-[\bar f^i_{\nu-1}(\bar\theta_{\nu}, \varphi)]_{\bar\theta_\nu}, \; \forall \; 1\leq i\leq m, \end{eqnarray} | (4.9) |
is solved. Since homological equation (4.9) may have no analytic solution, we solve its approximate equation for each 1\leq i\leq m .
\begin{eqnarray} \partial_\omega h^i_\nu+\langle\frac{\partial h^{i}_\nu}{\partial \bar\theta_\nu}, \rho\rangle+\mathcal{T}_{T_{\nu}}\langle\frac{\partial h^{i}_\nu}{\partial \bar\theta_\nu}, \bar g_{\nu}(\varphi)\rangle = \mathcal{T}_{T_{\nu}}(\bar f_{\nu-1}(\bar\theta_{\nu}, \varphi)-\bar[f_{\nu-1}(\bar\theta_{\nu}, \varphi)]_{\theta_\nu}). \end{eqnarray} | (4.10) |
Let
\begin{eqnarray*} \begin{array}{ll} \bar f^i_{\nu-1}(\bar\theta_{\nu}, \varphi) = \sum\limits_{l\in \mathbb{Z}^m}\bar f^i_{\nu-1, l}(\varphi)e^{{\rm i}\langle l, \bar\theta_{\nu}\rangle}, & \bar f^i_{\nu-1, l}(\varphi) = \sum\limits_{k\in\mathbb{Z}^d}\bar f_{\nu-1, l}^{ik}e^{{\rm i}\langle k, \varphi \rangle }, \\ h^i_{\nu}(\bar\theta_{\nu}, \varphi) = \sum\limits_{0 < |l|\leq T_\nu}h^i_{\nu, l}(\varphi)e^{{{\rm i}\langle l, \bar\theta_{\nu}\rangle}}, & h^i_{\nu, l}(\varphi) = \sum\limits_{|k| < T_\nu-|l|}h_{\nu, l}^{ik}e^{{\rm i}\langle k, \varphi \rangle }. \end{array} \end{eqnarray*} |
In order to solve equation (4.10), it is equivalent to solve
\begin{eqnarray} \partial_\omega h^i_{\nu, l}(\varphi)+{\rm i}\langle l, \rho\rangle h^i_{\nu, l}(\varphi)+\mathcal{T}_{T_\nu-|l|}h^i_{\nu, l}(\varphi)\langle {\rm i}l, \bar g_{\nu}(\varphi)\rangle = \mathcal{T}_{T_\nu-|l|}\bar f^i_{\nu-1, l}(\varphi), \end{eqnarray} | (4.11) |
for 0 < |l| < T_\nu .
For any fixed l , (4.11) can be written as a matrix equation
(A_l+G_l)\bar h_l = \bar f_l, |
where
\begin{eqnarray*} \begin{array}{ll} \bar h_l = (h_{\nu, l}^{ik})^T_{|k| < T_\nu-|l|}, &A_l = diag({\rm i}\langle k, \omega \rangle+{\rm i} \langle l, \rho\rangle :|k| < T_\nu-|l|), \\ \bar f_l = (\bar f_{\nu-1, l}^{ik})^T_{|k| < T_\nu-|l|}, & G_l = {\rm i}\langle l, \bar g_{\nu, p-q}\rangle_{|p|, |q| < T_\nu-|l|}, \end{array} \end{eqnarray*} |
with \bar g_{\nu, k} being the Fourier coefficients of Fourier expansion of \bar g_\nu(\varphi) . If we denote
\Omega_{l, r'} = {\rm diag}(\cdots, e^{|k|r'}, \cdots) |
for any r'\leq \bar r_n, then we have
(\tilde A_{l, r'}+\tilde G_{l, r'})\tilde h_{l, r'} = \tilde f_{l, r'}, |
where
\begin{eqnarray*} \begin{array}{ll} \tilde A_{l, r'} = \Omega_{l, r'}A_l\Omega_{l, r'}^{-1}, &\tilde G_{l, r'} = \Omega_{l, r'}G_l\Omega_{l, r'}^{-1}, \\ \tilde h_{l, r'} = \Omega_{l, r'}\bar h_l, & \tilde f_{l, r'} = \Omega_{l, r'}\bar f_l. \end{array} \end{eqnarray*} |
Letting \mathcal N = [2^nc_1\tau N\ln K_n]+1 with c_1 = \frac{c}{ 16(3\tau+2+\frac{d}{2})\ln3}-\frac{\alpha}{ \tau N\ln3}, we have
\begin{eqnarray} T_\nu& = &\frac{1}{\sigma_\nu}\ln\frac{1}{\tilde \eta_{\nu-1}}\leq \frac{4 K_n^\alpha 3^{\nu-1}}{r_0}\ln\frac{1}{2\epsilon_{n-1}}\\ &\leq&\frac{4K_n^\alpha3^{2^n c_1\tau N\ln K_n}}{r_0}\ln\frac{1}{2\epsilon_{n-1}}\\ &\leq& \frac{4K_n^{\alpha+2^n c_1\tau N\ln 3}}{r_0}\ln\frac{1}{2\epsilon_{n-1}}\\ &\leq& \epsilon_{n-1}^{-\frac{1}{16(3\tau+2+\frac{d}{2})}-\frac{1}{16(3\tau+2+\frac{d}{2})}}\\ &\leq& (\frac{\gamma^3}{32\epsilon_{n-1}^{\frac{1}{2}}})^{\frac{1}{3\tau+2+\frac{d}{2}}} = T^{(n-1)}, \end{eqnarray} | (4.12) |
since \epsilon_0 sufficiently small. In view of \xi\in O_{n-1} and (4.12), we get
\begin{eqnarray*} \|\tilde A_{l, r'}^{-1}\|&\leq&\max\limits_{k\leq T_\nu}\sup\limits_{\xi\in O_{n-1}}(\frac{1}{|\langle k, \omega \rangle+\langle l, \rho\rangle|}+\frac{|\langle l, \frac{d\rho}{d\xi}\rangle|}{|\langle k, \omega \rangle+\langle l, \rho\rangle|^2})\nonumber\\ &\leq& 8\frac{T_\nu^{3\tau+1}}{\gamma^3}\leq 8\frac{{(T^{n-1})}^{3\tau+1}}{\gamma^3} \end{eqnarray*} |
for all 0 < |k|+|l| < T_\nu , |l|\neq 0 . Meanwhile, since the (k_1, k_2) -th variable of \tilde G_{l, r'} is {\rm i}e^{(|k_1|-|k_2|)r'}\langle l, \bar g_{\nu, k_1-k_2}\rangle, we obtain that
\begin{eqnarray} \|\tilde G_{l, r'}\|&\leq& T_\nu\max\limits_{|k_2|\leq T_\nu}\sum\limits_{|k_1|\leq T_\nu}e^{(|k_1-k_2|)r'}|\bar g_{\nu, k_1-k_2}|\\ &\leq&T_\nu \max\limits_{|k_2|\leq T_\nu}\sqrt{\sum\limits_{|k_1| > 0}e^{(2|k_1-k_2|)r'}|\bar g_{\nu, k_1-k_2}|^2}\sqrt{\sum\limits_{|k_1|\leq T_\nu} 1}\\ &\leq & T_\nu^{1+\frac{d}{2}}\|\bar g_{\nu-1}\|_{r', O_{n-1}}, \end{eqnarray} | (4.13) |
which implies the diagonally dominant operators \tilde A_{l, r'}+\tilde G_{l, r'} have a bounded inverse and the following estimate
\|(\tilde A_{l, r'}+\tilde G_{l, r'})^{-1}\|\leq\|\tilde A_{l, r'}^{-1}\|\|(I+\tilde A_{l, r'}^{-1}\tilde G_{l, r'})\| < 2 T_\nu^{\tau} |
for r' = \tilde r_{\nu-1}-\sigma_\nu . One can check that
\begin{eqnarray*} \|h^i_\nu\|^2_{\tilde s_{\nu-1}-\frac{\delta_\nu}{2}, \tilde r_{\nu-1}-\frac{\sigma_\nu}{2}, O_{n-1}}&\leq&\sum\limits_{|l| < T_\nu}\sum\limits_{|k|\leq |l|-T\nu}|h_{\nu, l}^{ik}|_{O_{n-1}}^2e^{2|k|(\tilde r_{\nu-1}-\frac{\sigma_\nu}{2})}e^{2|l|(\tilde s_{\nu-1}-\frac{\delta_\nu}{2})}\\ &\leq&4T_\nu^{4a}\sum\limits_{|l| < T_\nu}\langle\tilde h_{l, \tilde r_{\nu-1}-\frac{\sigma_\nu}{2}}, \tilde h_{l, \tilde r_{\nu-1}-\frac{\sigma_\nu}{2}}\rangle e^{2|l|(\tilde s_{\nu-1}-\frac{\delta_\nu}{2})}\\ &\leq& 4T_\nu^{4a}\sum\limits_{|l| < T_\nu}\langle (\tilde A_{l, r'}+\tilde G_{l, r'})^{-1}\tilde f_{l, \tilde r_{\nu-1}-\frac{\sigma_\nu}{2}}, (\tilde A_{l, r'}+\tilde G_{l, r'})^{-1}\tilde f_{l, \tilde r_{\nu-1}-\frac{\sigma_\nu}{2}}\rangle \times\\ && \times e^{2|l|(\tilde s_{\nu-1}-\frac{\delta_\nu}{2})}\\ &\leq&8 T_\nu^{4a+2\tau} \sum\limits_{|l| < T_\nu}\langle\tilde f_{l, \tilde r_{\nu-1}-\frac{\sigma_\nu}{2}}, \tilde f_{l, \tilde r_{\nu-1}-\frac{\sigma_\nu}{2}}\rangle e^{2|l|(\tilde s_{\nu-1}-\frac{\delta_\nu}{2})}\\ &\leq&8 T_\nu^{4a+2\tau}\sum\limits_{|l| < T_\nu}\sum\limits_{|k| < T_\nu-|l|}|f_{\nu-1, l}^k|_{O_{n-1}}^2e^{2|k|(\tilde r_{\nu-1}-\frac{\sigma_\nu}{2})}e^{2|l|(\tilde s_{\nu-1}-\frac{\delta_\nu}{2})}\\ &\leq&\frac{T_\nu^{4a+2\tau}2^{d+4+\tau}\tilde\eta_{\nu-1}^2}{\gamma\sigma_\nu^{d+2+\tau}}\leq \frac{1}{m^2}\tilde\eta_{\nu-1}^{\frac{3}{2}} \end{eqnarray*} |
since \epsilon_0 sufficiently small. And the error
P_\nu = \mathcal{R}_{T_{\nu}}(\bar g_{\nu}(\varphi)\frac{\partial h_\nu}{\partial \bar\theta_\nu})-\mathcal{R}_{T_{\nu}}(\bar f_{\nu-1}(\bar\theta_{\nu}, \varphi)-\bar[f_{\nu-1}(\bar\theta_{\nu}, \varphi)]_{\theta_\nu}) |
satisfies
\begin{eqnarray*} \|P_\nu\|_{\tilde s_{\nu-1}-\delta_\nu, \tilde r_{\nu-1}-\sigma_\nu, O_{n-1}}\leq \frac{2}{5}\eta_{\nu-1}^{\frac{3}{2}}. \end{eqnarray*} |
Now, the system (4.8) becomes
\begin{eqnarray*} \overline X_\nu = \omega\cdot \partial_\varphi+(\rho+\bar g_{\nu}(\varphi)+\bar f_{\nu}(\bar\theta_{\nu}, \varphi))\cdot \partial_{\bar\theta_{\nu}}, \end{eqnarray*} |
where \bar g_{\nu}(\varphi) = \bar g_{\nu-1}(\varphi)+[\bar f_{\nu-1}(\bar\theta_{\nu}, \varphi)]_{\bar\theta_{\nu}} and
\bar f_{\nu}(\bar\theta_{\nu}, \varphi) = (1+\frac{\partial h_\nu}{\partial \bar\theta_\nu})^{-1}(P_\nu+\bar f_{\nu-1}(\bar\theta_\nu+h_{\nu}(\bar\theta_\nu, \varphi), \varphi)-\bar f_{\nu-1}(\bar\theta_{\nu}, \varphi)). |
By the mean value theorem and Cauchy estimate, we have for each \xi\in O_{n-1} that
\begin{eqnarray*} &&\|\bar f_{\nu-1}(\bar\theta_\nu+h_{\nu}(\bar\theta_\nu, \varphi), \varphi)-\bar f_{\nu-1}(\bar\theta_{\nu}, \varphi)\|_{\tilde s_\nu, \tilde r_\nu, O_{n-1}}\\ & = & \|\frac{\partial\bar f_{\nu-1}(\bar\theta_\nu+sh_{\nu}(\bar\theta_\nu, \varphi), \varphi)}{\partial \theta}h_{\nu}(\bar\theta_\nu, \varphi)\|_{\tilde s_\nu, \tilde r_\nu, O_{n-1}}\\ &\leq&\frac{2}{\delta_\nu} \| \bar f_{\nu-1}\|_{\tilde s_{\nu-1}-\frac{\delta_\nu}{2}, \tilde r_{\nu-1}-\frac{\sigma_\nu}{2}, O_{n-1}}\|h_\nu\|_{\tilde s_{\nu-1}-\frac{\delta_\nu}{2}, \tilde r_{\nu-1}-\frac{\sigma_\nu}{2}, O_{n-1}}\\ &\leq& \frac{2\tilde\eta_{\nu-1}}{\delta_\nu}\|h_\nu\|_{\tilde s_{\nu-1}-\frac{\delta_\nu}{2}, \tilde r_{\nu-1}-\frac{\sigma_\nu}{2}, O_{n-1}}\leq \frac{\eta_{\nu-1}^{\frac{3}{2}}}{5} \end{eqnarray*} |
where s\in (0, 1) . This implies
\|\bar f_{\nu}(\bar\theta_{\nu}, \varphi)\|_{\tilde s_{\nu}, \tilde r_{\nu}, O_{n-1}}\leq (1+\eta_{\nu-1}^{\frac{3}{4}})(\frac{\eta_{\nu-1}^{\frac{3}{2}}}{5}+\frac{2\eta_{\nu-1}^{\frac{3}{2}}}{5}+\frac{\eta_{\nu-1}^{\frac{3}{2}}}{5})\leq \tilde\eta_\nu. |
Since 1\leq \nu\leq \mathcal{N} , we should estimate \bar f_{\mathcal{N}}(\bar\theta_{\mathcal{N}}, \varphi) . First, we have the following estimate
(\frac{3}{2})^{\mathcal{N}}-1\geq (\frac{3}{2})^{2^nc_1\tau N\ln K_n}-1\geq K_n^{2^{n-1}c_1\tau N\ln (3/2)}. |
By (2.5), we have
\begin{eqnarray*} \|\bar f_{\mathcal{N}}(\bar\theta_{\mathcal{N}}, \varphi)\|_{\tilde s_{\mathcal{N}}, \tilde r_{\mathcal{N}}, O_{n-1}}&\leq&\tilde\eta^{\frac{3}{2}^{\mathcal{N}}} = \tilde\eta e^{-(\frac{3}{2}^{\mathcal{N}}-1)\ln \frac{ 1}{\tilde\eta}}\\ &\leq& \tilde\eta e^{-(K_n^{2^{n-1}c_1\tau N\ln (3/2)})2^{n+1}c\tau N} \\ &\leq & \tilde \eta\frac{1}{K_{n+1}^{2^{n+1}c\tau N}} < \epsilon_n, \end{eqnarray*} |
in view of \epsilon_0 < e^{-2c\tau N}. Let \bar\theta_{\mathcal{N}} = \bar\theta_+ and
\overline H_\nu(\bar\theta_{\nu}, \varphi) = \phi_1\circ\cdots\circ \phi_{\nu-1}\circ \phi_\nu(\bar\theta_\nu, \varphi) |
for 1\leq \nu\leq \mathcal{N} . Then, \overline H_\nu(\bar\theta_{\nu}, \varphi) is analytic in W_{\tilde s_\nu, \tilde r_\nu}(\mathbb{T}^m\times \mathbb{T}^d) and
\|\partial_{\bar \theta_\nu}(\pi\circ\overline H_\nu(\bar\theta_{\nu}, \varphi))\|_{\tilde s_\nu, \tilde r_\nu, O_{n-1}}\leq\prod\limits_{j = 1}^\nu(1+\tilde\eta_j^{\frac{3}{4}}), |
where \pi:\mathbb{T}^m\times \mathbb{T}^d\rightarrow \mathbb{T}^m denotes the natural projection to the first variable. If we rewrite \overline H_{\mathcal{N}}(\bar\theta_{+}, \varphi) = (\bar\theta_++\tilde h(\bar\theta_+, \varphi)\mod 2\pi, \varphi) , then we have
\begin{eqnarray} \begin{array}{l} \|\tilde h(\bar\theta_+, \varphi)\|_{\tilde s_{\mathcal{N}-1}-\frac{\delta_\mathcal{N}}{2}, \tilde r_{\mathcal{N}-1}-\frac{\sigma_\mathcal{N}}{2}, O_{n-1}}\leq \sum\limits_{\nu = 1}^\mathcal{N}\prod\limits_{j = 1}^{\nu-1}\|h_\nu\|_{\tilde s_\nu, \tilde r_\nu, O_{n-1}} < 2\tilde\eta^{\frac{3}{4}} < 4\epsilon_{n-1}^{\frac{3}{4}}, \\ \|\frac{\partial\tilde h(\bar\theta_+, \varphi)}{\partial \bar\theta_+}\|_{\tilde s_{\mathcal{N}}, \tilde r_{\mathcal{N}}, O_{n-1}} < 4\epsilon_{n-1}^{\frac{3}{4}}. \end{array} \end{eqnarray} | (4.14) |
In conclusion, let \tilde s_{\mathcal{N}} = s_{n+} , \tilde r_{\mathcal{N}} = r_{n+} , \bar g_{\mathcal{N}}(\varphi) = \bar g_{n-1, +}(\varphi) and \bar f_{\mathcal{N}}(\bar\theta_{+}, \varphi) = \bar f_{n-1, +}(\bar\theta_{+}, \varphi) . Then \overline H_{n-1} = \overline H_{\mathcal{N}}(\bar\theta_{+}, \varphi) conjugates (4.4) to
\begin{eqnarray*} \overline X_{n-1, +} = \omega\cdot\partial_\varphi+ (\rho_{n-1}+\bar g_{n-1, +}(\varphi)+\bar f_{n-1, +}(\bar\theta_+, \varphi))\cdot\partial_{\bar\theta_+}, \end{eqnarray*} |
with the estimates
\begin{eqnarray*} &&\|\bar g_{n-1, +}(\varphi)\|_{r_{n+}, O_{n-1}}\leq 2\epsilon^{\frac{1}{2}}_{n-1}, \\ &&\|\bar f_{n-1, +}(\bar\theta_+, \varphi)\|_{s_{n+}, r_{n+}, O_{n-1}}\leq \epsilon_{n}, \\ &&\|\bar g_{n-1, +}(\varphi)-\bar g_{n-1}(\varphi)\|_{r_{n+}, O_{n-1}}\leq \sum\limits_\nu\|\bar f_\nu\|_{s_{n}, r_{n}, O_{n-1}}\leq\sum\limits_\nu\tilde \eta_\nu\leq4\epsilon_{n-1}. \end{eqnarray*} |
The following Lemma is the end of one KAM step.
Lemma 4.3. Under the assumptions of Lemma 4.1 and Lemma 4.2, there exists \widetilde{H}_{n-1} with estimates
\begin{eqnarray*} &&\|\widetilde H_{n-1}-id\|_{s_{n}, r_{n}, O_{n-1}}\leq 4\epsilon_{n-1}^{\frac{3}{4}}, \\ &&\|D( \widetilde H_{n-1}-id)\|_{s_{n}, r_{n}, O_{n-1}}\leq 4\epsilon_{n-1}^{\frac{3}{4}}, \end{eqnarray*} |
such that \widetilde H_{n-1} conjugates vector field (4.3) to
\begin{eqnarray*} X_{n} = \omega\cdot \partial_\varphi+(\rho_{n}+ g_{n}(\varphi)+ f_{n}(\theta_+, \varphi))\cdot\partial_{\theta_+}, \end{eqnarray*} |
with the following estimates
\begin{eqnarray*} &&\|g_{n}(\varphi)\|_{r_{n}, O_{n}}\leq 4\tilde\epsilon_{n}, \\ &&\|f_{n}(\theta_+, \varphi)\|_{s_{n}, r_{n}, O_{n}}\leq \epsilon_{n}. \end{eqnarray*} |
Proof. In the Lemma 4.1, we eliminate the non-resonant terms of g_{n-1}(\varphi) and, as a result, the transformation we obtain is not close to the identity. In order to get rotations reducibility results, we need to inverse the first step, which means conjugating back by the transformation of the first step. Applying the inverse transformation H_{n-1}^{-1} :
\bar\theta_+ = \theta_+-h_{n-1}(\varphi), |
the vector field (4.6) can be conjugated to
\begin{eqnarray*} X_{n} = \omega\cdot \partial_\varphi+ (\rho_n+g_{n}(\varphi)+f_{n}(\theta_+, \varphi))\cdot\partial_{\theta_+}, \end{eqnarray*} |
where \rho_n = \rho_{n-1}+[\bar g_{n-1, +}(\varphi)]_{\varphi} and
g_n(\varphi) = \mathcal{T}_{K_n}g_{n-1}(\varphi)+\bar g_{n-1, +}(\varphi)-[\bar g_{n-1, +}(\varphi)]_{\varphi}, \; f_{n}(\theta_+, \varphi) = \bar f_{n-1, +}(\theta_+-h_{n-1}(\varphi), \varphi). |
We have
\|g_n(\varphi)-g_{n-1}(\varphi)\|_{r_n, O_{n-1}}\leq \|\bar g_{n-1, +}(\varphi)-\bar g_{n-1}(\varphi)\|_{r_{n+}, O_{n-1}}\leq 4\epsilon_{n-1}, |
which implies \|g_n(\varphi)\|_{r_n, O_{n-1}}\leq 4\tilde\epsilon_n . And it is obvious that
\|f_{n}(\theta_+, \varphi)\|_{s_{n}, r_{n}, O_{n}}\leq \epsilon_{n}. |
Let \widetilde H_{n-1} = H_{n-1}\circ\overline H_{n-1}\circ H^{-1}_{n-1} , then \widetilde H_{n-1} = \theta_++\tilde h(\theta_+-h_{n-1}(\varphi), \varphi) . By (4.14), we have
\begin{eqnarray*} &&\| \widetilde H_{n-1}-id\|_{s_{n}, r_{n}, O_{n-1}}\leq \|\tilde h\|_{s_{n+}, r_{n+}, O_{n-1}}\leq 4\epsilon_{n-1}^{\frac{3}{4}}, \\ &&\|D( \widetilde H_{n-1}-id)\|_{s_{n}, r_{n}, O_{n-1}}\leq \|D\tilde h\|_{s_{n+}, r_{n+}, O_{n-1}}\leq 4\epsilon_{n-1}^{\frac{3}{4}}. \end{eqnarray*} |
Lemmas 4.1, 4.2, and 4.3 can be summarized as the following iterative lemma.
Lemma 4.4. Suppose \omega\in\mathcal{L} and \gamma_n , r_n , s_n , \epsilon_n are defined in (4.1) for n\geq 0 . Let \epsilon_0 sufficiently small. Then the following holds for all n\geq 1 : If the vector field
\begin{eqnarray} X_n = \omega\cdot \partial_\varphi+(\rho_{n}(\xi)+ g_{n}(\varphi, \xi)+ f_{n}(\theta, \varphi, \xi))\cdot\partial_{\theta} \end{eqnarray} | (4.15) |
satisfies
\begin{eqnarray*} &&\|g_{n}(\varphi, \xi)\|_{r_{n}, O_{n}}\leq 4\tilde\epsilon_{n}, \\ &&\| f_{n}(\theta, \varphi, \xi)\|_{s_{n}, r_{n}, O_{n}}\leq \epsilon_{n}, \\ &&|\rho_{n}-\rho_{n-1}|_{O_{n}}\leq 4\epsilon_{n-1}, \end{eqnarray*} |
where [g_{n}(\varphi, \xi)]_{\varphi} = 0 and
{O}_{n} = \{\xi\in {O}_{n-1}: |\langle k, \omega \rangle| +\langle l, \rho_{n}(\xi) \rangle \geq \frac{\gamma_{n}}{(|k|+|l|)^\tau} , \; \forall |l|\neq 0, \; |k|+|l|\leq T^{(n)} \}. |
Then there exists a subset O_{n+1}\subset O_n, where
\begin{eqnarray} O_{n+1} = O_{n}\setminus\bigcup\limits_{T^{(n)} < |k|+|l|\leq T^{(n+1)}\atop |l|\neq 0}\Gamma_{kl}^{n+1}(\gamma_{n+1}) \end{eqnarray} | (4.16) |
with
\Gamma_{kl}^{n+1}(\gamma_{n+1}) = \{\xi\in {O}_n: |\langle k, \omega \rangle| +\langle l, \rho_{n+1}(\xi) \rangle < \frac{\gamma_{n+1}}{(|k|+|l|)^\tau}\}, |
and a change of variables \widetilde H_n:\mathbb{T}^m\times\mathbb{T}^d\rightarrow \mathbb{T}^m\times\mathbb{T}^d with estimates
\begin{eqnarray*} &&\| \widetilde H_{n}-id\|_{s_{n+1}, r_{n+1}, O_{n+1}}\leq 4\epsilon_{n}^{\frac{3}{4}}, \\ &&\|D( \widetilde H_{n}-id)\|_{s_{n+1}, r_{n+1}, O_{n+1}}\leq 4\epsilon_{n}^{\frac{3}{4}}, \end{eqnarray*} |
such that it transforms the vector field (4.15) to
\begin{eqnarray*} \omega\cdot \partial_\varphi+(\rho_{n+1}(\xi)+ g_{n+1}(\varphi, \xi)+ f_{n+1}(\theta, \varphi, \xi))\cdot \partial_{\theta}, \end{eqnarray*} |
with
\begin{eqnarray*} &&\|g_{n+1}(\varphi, \xi)\|_{r_{n+1}, O_{n+1}}\leq 4\tilde\epsilon_{n+1}, \\ &&\| f_{n+1}(\theta, \varphi, \xi)\|_{s_{n+1}, r_{n+1}, O_{n+1}}\leq \epsilon_{n+1}, \\ &&|\rho_{n+1}-\rho_{n}|_{O_{n+1}}\leq 4\epsilon_{n}, \end{eqnarray*} |
and [g_{n+1}(\varphi, \xi)]_{\varphi} = 0 .
Proof. Actually, Lemma 4.4 is an immediate corollary of Lemmas 4.1-4.3. The only point we need to illustrate is the definition of O_{n+1} . If \xi\in O_{n} and 0 < |k|+|l|\leq T^{(n)} , then
\begin{eqnarray} \begin{array}{l} \quad|\langle k, \omega \rangle +\langle l, {\rho}_{n+1}(\xi) \rangle|\nonumber\\ \geq|\langle k, \omega \rangle +\langle l, {\rho}_{n}(\xi)\rangle|-|l||{\rho}_{n+1}(\xi)-{\rho}_{n}(\xi)|\nonumber\\ \geq\frac{\gamma_{n}}{(|k|+|l|)^\tau}-4T^{(n)}\epsilon_{n}\nonumber\\ \geq\frac{\gamma_{n}-\epsilon_{n}^{\frac{1}{2}}}{(|k|+|l|)^\tau} > \frac{\gamma_{n+1}}{(|k|+|l|)^\tau}. \end{array} \end{eqnarray} |
Thus, by the definition of (4.16), we have
{O}_{n+1} = \{\xi\in \mathcal{O}_{n}: |\langle k, \omega \rangle| +\langle l, \rho_{n+1}(\xi) \rangle \geq \frac{\gamma_{n+1}}{(|k|+|l|)^\tau} , \; \forall |l|\neq 0, \; 0 < |k|+|l|\leq T^{{n+1}} \}, |
which satisfies the definition (4.2).
By the definition of O_n , we get \mathcal{O}_\gamma = \bigcap_{n\geq 0}O_n .
By (4.16), we have
\mathcal{O}\setminus\mathcal{O}_{\gamma} = \bigcup\limits_{n = 0}^\infty\bigcup\limits_{T^{(n)} < |k|+|l|\leq T^{(n+1)}\atop |l|\neq 0}\Gamma_{kl}^{n+1}(\gamma_{n+1}). |
In what follows, we start to estimate the measure of set \Gamma_{kl}^{n+1}(\gamma_{n+1}). In view of
\begin{eqnarray*} \frac{d\rho_n(\xi)}{d\xi}& = &\frac{d\rho_0(\xi)}{d\xi}+\sum\limits_{i = 1}^n(\frac{d\rho_{i}(\xi)}{d\xi}-\frac{d\rho_{i-1}(\xi)}{d\xi})\\ & = &\frac{d\rho_0(\xi)}{d\xi}+\sum\limits_{i = 1}^n(\frac{dg_{i}(\varphi, \xi)}{d\xi}-\frac{dg_{i-1}(\varphi, \xi)}{d\xi}) \end{eqnarray*} |
and \frac{d\rho_0(\xi)}{d\xi} = I , we have
{\rm rank}\left\{\frac{d\rho_n(\xi)}{d\xi}\right\} = m. |
It follows from Lemma 1.2 in [33] that
\Gamma_{kl}^{n+1}(\gamma_{n+1})\leq C\frac{\gamma_{n}}{ |l|(|k|+|l|)^{\tau}}. |
Then, we have the following estimation
\begin{eqnarray*} \begin{array}{l} {\rm meas}(\mathcal{O}\setminus\mathcal{O}_{\gamma})\leq C \sum\limits_{n\geq0}\sum\limits_{T^{(n)} < |k|+|l|\leq T^{(n+1)}}C\frac{\gamma_{n}}{ |l|(|k|+|l|)^{\tau}}\\ \quad\quad\quad\quad\quad\quad\leq C \gamma\sum\limits_{ i = 1}^\infty\frac{\ln i}{i^{\tau-(m+d-2)}}\\ \quad\quad\quad\quad\quad\quad\leq \mathcal{O}(\gamma) \end{array} \end{eqnarray*} |
provided \tau > m+d-1 .
Select \epsilon_0 sufficiently small and r_0 = r , s_0 = s . If \rho\in \mathcal{O} = \bigcap_{n\geq 0}O_n and \|f\|_{s, r, \mathcal{O}_{\gamma}} < \epsilon_0 , then the vector field
\begin{eqnarray} X = \omega\cdot \partial_\varphi+(\xi+ f(\theta, \varphi))\cdot \partial_\theta, \end{eqnarray} | (4.17) |
is C^{\infty} rotations linearizable. We can use Lemma 4.2 to system (4.17). Thus we can get H_0\in H^{s_{1}, r_{1}, a} which conjugates (4.17) to
\begin{eqnarray*} X_1 = \omega\cdot \partial\varphi+(\rho_1(\xi)+g_1(\varphi, \xi) +f_1(\theta, \varphi, \xi))\cdot \partial\theta, \end{eqnarray*} |
with
\begin{eqnarray*} &&\|g_{1}(\varphi, \xi)\|_{r_{1}, {O}_1}\leq 2\tilde\epsilon_{1}, \\ &&\| f_{1}(\theta, \varphi, \xi)\|_{s_{1}, r_{1}, {O}_1}\leq \epsilon_{1}, \end{eqnarray*} |
Then we apply Lemma 6 inductively, we can get \widetilde H_i\in H^{s_{i+1}, r_{i+1}, a} , i = 1, \cdots, n, such that H^{(n)} = \widetilde H_0\circ \widetilde H_1\circ\cdots\circ \widetilde H_n conjugates (4.17) to
\begin{eqnarray*} X_{n+1} = \omega\cdot \partial\varphi+(\rho_{n+1}(\xi)+g_{n+1}(\varphi, \xi) +f_{n+1}(\theta, \varphi, \xi))\cdot \partial\theta, \end{eqnarray*} |
with
\begin{eqnarray*} &&\|g_{n+1}(\varphi, \xi)\|_{r_{n+1}, {O}_{n+1}}\leq 2\tilde\epsilon_{n+1}, \\ &&\| f_{n+1}(\theta, \varphi, \xi)\|_{s_{n+1}, r_{n+1}, {O}_{n+1}}\leq \epsilon_{n+1}. \end{eqnarray*} |
And we have
\begin{eqnarray*} \|D H^{(n)}\|_{s_{n+1}, r_{n+1}, O_{n+1}}\leq \|D\widetilde H_{0}\|_{s_1, r_1, O_1}\|D\widetilde H_{1}\|_{s_2, r_2, O_2}\cdots\|D\widetilde H_n\|_{s_{n+1}, r_{n+1}, O_{n+1}}\leq \prod\limits_{i = 0}^{n}(1+4\epsilon_i^{\frac{3}{4}}), \end{eqnarray*} |
which implies
\begin{eqnarray*} \|H^{(n+1)}-H^{(n)}\|_{s_{n+1}, r_{n+1}, O_{n+1}}\leq \|DH^{(n)}\|_{s_n, r_n, O_{n}}\|H_n-id\|_{s_{n+1}, r_{n+1}, O_{n+1}}\leq 8\epsilon_n^{\frac{3}{4}}. \end{eqnarray*} |
By the definition of (\epsilon_n)_{\mathbb{N}} , we know that for any j\in \mathbb{Z}_+^{m+d}, there exists N\in \mathbb{N} , such that for any n\geq N , we have 8(\frac{4K_{n+1}^\alpha}{r_0})^{|j|}\epsilon_n^{\frac{3}{4}}\leq \epsilon_n^{\frac{1}{2}} . By the Cauchy estimates, if we denote x: = (\theta, \varphi)\in \mathbb{T}^{m+d} , we have
\begin{eqnarray*} |\frac{\partial^{|j|}}{\partial x^j}(H^{(n+1)}-H^{(n)})|\leq \frac{1}{r^{|j|}_{n+1}}\|H^{(n+1)}-H^{(n)}\|_{s_{n+1}, r_{n+1}, O_{n+1}}\leq 8(\frac{4K_{n+1}^\alpha}{r_0})^{|j|}\epsilon_n^{\frac{3}{4}}\leq \epsilon_n^{\frac{1}{2}}. \end{eqnarray*} |
for any n > N-1 . This guarantees the limit \lim_{n\rightarrow \infty}H^{(n)} belongs to C^\infty .
Finally, let \rho_\infty(\xi) = \lim_{n\rightarrow \infty}\rho_n(\xi) , m_{\infty}({\varphi, \xi}) = \lim_{n\rightarrow \infty}g_{n}(\varphi, \xi) and \Psi = (\lim_{n\rightarrow \infty}H^{(n)})^{-1} , the proof of Proposition 3.1 is completed.
In this section, we give the proof of Proposition 3.2. Since the proof process is very similar to [27], we only write the relevant lemma and the key points of proof that are different from Proposition 3.1. Readers can combine Proposition 3.2 with [27] for more detailed analysis.
Let N r_0 , s_0 , \gamma > 0 , M > d/2 \tau > d . Suppose K_0 > 0 large enough such that
\frac{K_0^{\frac{M-\alpha}{16M}}}{\ln K_0} > c\tau N, |
where c > \frac{16\alpha (3\tau+2+\frac{d}{2})}{\tau N} and \max\{\frac{M(2\tau_1-d-2)}{2M-d}, 1\} < \alpha < M. \epsilon_0 is sufficiently small. For given r_0 , s_0 , \epsilon_0 , we define some sequences depending on r_0 , s_0 , \epsilon_0 for n\geq 1 :
\begin{eqnarray} \begin{array}{ll} \Delta_n = \frac{s_0}{10\cdot2^{n-1}}, & r_n = \frac{r_0}{4k_n^{\alpha}}, \\ s_n = s_{n-1}-\Delta_n, & \epsilon_{n} = \frac{\epsilon_{n-1}}{K_{n+1}^{2^{n+1}c\tau {N}}}, \\ \widetilde{ \epsilon}_n = \sum\nolimits_{i = 0}^{n-1}{\epsilon_i}, & T^{(n-1)} = ({\gamma^3}/{32\epsilon_{n-1}^{\frac{1}{2}}})^{\frac{1}{3\tau+2}}, \\ \end{array} \end{eqnarray} | (4.18) |
where \max\{\frac{M\tau_1}{M-d-1}, 1\} < \alpha < M .
In (n-1) -th step, vector field can be written as
\begin{eqnarray} X_{n-1} = \omega\cdot \partial\varphi+(\rho_{f}+g_{n-1}(\varphi)+f_{n-1}(\theta, \varphi))\cdot \partial_\theta, \end{eqnarray} | (4.19) |
where
\begin{eqnarray*} &&\|g_{n-1}(\varphi)\|_{r_{n-1}}\leq 4\tilde\epsilon_{n-1}, \\ &&\| f_{n-1}(\theta, \varphi)\|_{s_{n-1}, r_{n-1}}\leq \epsilon_{n-1}. \end{eqnarray*} |
Similar to Proposition 3.1, we can get the following Lemmas.
Lemma 4.5. Denote \bar s_n = s_{n-1}-\frac{\Delta_n}{3} , \bar r_n = \frac{r_0}{K_n^\alpha} , then there exists h_{n-1}(\varphi) with \|h_{n-1}(\varphi)\|_{\bar r_n}\leq K_n^{\frac{M\tau_1+(d+1)\alpha}{M}} , such that transformation H_{n-1}:\theta = \overline\theta+h_{n-1}(\varphi)(\mod 2\pi) conjugates the vector field (4.19) into
\begin{eqnarray} \overline X_{n-1} = \omega\cdot \partial_\varphi+(\rho_{f}+\bar g_{n-1}(\varphi)+\bar f_{n-1}(\bar\theta, \varphi))\cdot \partial_{\bar\theta}, \end{eqnarray} | (4.20) |
where
\begin{eqnarray*} &&\|g_{n-1}(\varphi)\|_{\bar r_{n}}\leq \epsilon^{\frac{1}{2}}_{n-1}, \\ &&\| f_{n-1}(\theta, \varphi)\|_{\bar s_{n}, \bar r_{n}}\leq \epsilon_{n-1}. \end{eqnarray*} |
Proof. Using transformation \theta = \bar\theta+h_{n-1}(\varphi)(\mod 2\pi) , system (4.19) becomes
\begin{eqnarray*} \overline X_{n-1} = \omega\cdot \partial\varphi+\tilde f_{n-1}(\bar\theta, \varphi)\cdot \partial_{\bar\theta}, \end{eqnarray*} |
where \tilde f_{n-1}(\bar\theta, \varphi) = \rho_{f}+[g_{n-1}(\varphi)]+\mathcal{R}_{k_n}g_{n-1}(\varphi)+f_{n-1}(\bar\theta+h_{n-1}(\varphi), \varphi) , if homological equation \partial_\omega h_{n-1}(\varphi) = \mathcal{T}_{K_n}g_{n-1}(\varphi)-[g_{n-1}(\varphi)] is solvable. Then, we have
\begin{eqnarray*} &&\|f_{n-1}(\bar\theta+h_{n-1}(\varphi), \varphi)\|_{\bar s_n, \bar r_n}\leq \|f_{n-1}(\theta, \varphi)\|_{ s_{n-1}, r_{n-1}}\leq \epsilon_{n-1}, \\ &&\|\mathcal{R}_{k_n}g_{n-1}(\varphi)\|_{\bar r_n}\leq\epsilon_0e^{-K_n(r_{n-1}-\bar r_n)}\leq \epsilon_0e^{-K_n^{\frac{M-\alpha}{4M}}}\leq \frac{1}{3} \epsilon^{\frac{1}{2}}_{n-1}. \end{eqnarray*} |
By Lemma 2.2, we know \rho(\omega, \tilde f_{n-1}(\bar\theta, \varphi)) = \rho_f . By Lemma 2.1, we know that
|[g_{n-1}(\varphi)]_{\varphi}|\leq \|\mathcal{R}_{k_n}g_{n-1}(\varphi)+f_{n-1}(\bar\theta+h_{n-1}(\varphi), \varphi)\|_{\bar s_n, \bar r_n}\leq \frac{2}{3} \epsilon^{\frac{1}{2}}_{n-1}. |
Letting \bar g_{n-1}(\varphi) = [g_{n-1}(\varphi)]_{\varphi}+\mathcal{R}_{k_n}g_{n-1}(\varphi) and \bar f_{n-1}(\bar\theta, \varphi) = f_{n-1}(\bar\theta+h_{n-1}(\varphi), \varphi) , we have
\begin{eqnarray*} &&\|\bar f_{n-1}(\bar\theta, \varphi)\|_{\bar s_n, \bar r_n} = \|f_{n-1}(\bar\theta+h_{n-1}(\varphi), \varphi)\|_{\bar s_n, \bar r_n}\leq \|f_{n-1}(\theta, \varphi)\|_{ s_{n-1}, r_{n-1}, O_{n-1}}\leq \epsilon_{n-1}, \\ &&\|\bar g_{n-1}(\varphi)\|_{\bar s_n, \bar r_n} = \|[g_{n-1}(\varphi)]+\mathcal{R}_{k_n}g_{n-1}(\varphi)\|_{\bar r_n}\leq\epsilon_0e^{-K_n(r_{n-1}-\bar r_n)}\leq \epsilon_0e^{-K_n^{\frac{M-\alpha}{4M}}}\leq \epsilon^{\frac{1}{2}}_{n-1}. \end{eqnarray*} |
Lemma 4.6. Denote r_{n+} = \frac{r_0}{2K_n^\alpha} , s_{n+} = \bar s_n-\frac{\Delta}{3} . Under the assumptions of Lemma 4.5, there exists \overline {H}_{n-1} with estimates
\begin{eqnarray} &&\|\overline H_{n-1}-id\|_{s_{n+}, r_{n+}}\leq 4\epsilon_{n-1}^{\frac{3}{4}}, \end{eqnarray} | (4.21) |
\begin{eqnarray} &&\|D(\overline H_{n-1}-id)\|_{s_{n+}, r_{n+}}\leq 4\epsilon_{n-1}^{\frac{3}{4}}, \end{eqnarray} | (4.22) |
such that \overline H_{n-1} conjugates (4.20) to
\begin{eqnarray} X_{n-1, +} = \omega\cdot \partial_\varphi+(\rho_{f}+\bar g_{n-1, +}(\varphi)+\bar f_{n-1, +}(\bar\theta_+, \varphi))\cdot \partial_{\bar\theta_+} \end{eqnarray} | (4.23) |
with
\begin{eqnarray*} &&\|\bar f_{n-1, +}(\bar\theta_+, \varphi)\|_{s_{n+}, r_{n+}} = \leq 2\epsilon^{\frac{1}{2}}_{n-1}, \\ &&\|\bar g_{n-1, +}(\varphi)\|_{r_{n+}}\leq \epsilon_{n}, \\ &&\|\bar g_{n-1, +}(\varphi)-\bar g_{n-1}(\varphi)\|_{r_{n+}}\leq 4\epsilon_{n-1}. \end{eqnarray*} |
Proof. Taking advantage of (\omega, \rho_f) being Diophantine, we can obtain this Lemma immediately using the same process of Lemma 4.2. The detail proof is omitted here.
Lemma 4.7. Under the assumptions of Lemma 4.5 and Lemma 4.6, there exists \widetilde{H}_{n-1} with estimates
\begin{eqnarray} &&\|\widetilde H_{n-1}-id\|_{s_{n}, r_{n}}\leq 4\epsilon_{n-1}^{\frac{3}{4}}, \end{eqnarray} | (4.24) |
\begin{eqnarray} &&\|D( \widetilde H_{n-1}-id)\|_{s_{n}, r_{n}}\leq 4\epsilon_{n-1}^{\frac{3}{4}}, \end{eqnarray} | (4.25) |
such that \widetilde H_{n-1} conjugates (4.20) to
\begin{eqnarray*} X_{n} = \omega\cdot \partial_\varphi+(\rho_{f}+ g_{n}(\varphi)+ f_{n}(\theta_+, \varphi))\cdot \partial_{\theta_+} \end{eqnarray*} |
with
\begin{eqnarray*} &&\|f_{n}(\theta_+, \varphi)\|_{s_{n}, r_{n}}\leq\epsilon_{n}, \\ &&\|g_{n}(\varphi)\|_{r_{n}}\leq 4\tilde\epsilon_{n}. \end{eqnarray*} |
Proof. Applying the inverse transformation H_{n-1}^{-1} :
\bar\theta_+ = \theta_+-h_{n-1}(\varphi), |
the system (4.23) can be conjugated to
\begin{eqnarray*} X_{n} = \omega\cdot \partial_\varphi+(\rho_{f}+ g_{n}(\varphi)+ f_{n}(\theta_+, \varphi))\cdot \partial_{\theta_+} \end{eqnarray*} |
where
g_n(\varphi) = \mathcal{T}_{K_n}g_{n-1}(\varphi)+\bar g_{n-1, +}(\varphi), \; f_{n}(\theta_+, \varphi) = \bar f_{n-1, +}(\theta_+-h_{n-1}(\varphi), \varphi). |
We have
\|g_n(\varphi)-g_{n-1}(\varphi)\|_{r_n, O_{n-1}}\leq \|\bar g_{n-1, +}(\varphi)-\bar g_{n-1}(\varphi)\|_{r_{n+}, O_{n-1}}\leq 4\epsilon_{n-1}, |
which implies \|g_n(\varphi)\|_{r_n, O_{n-1}}\leq 4\tilde\epsilon_n . Thus
\begin{eqnarray*} &&\|f_{n}(\theta_+, \varphi)\|_{s_{n}, r_{n}}\leq\epsilon_{n}, \\ &&\|g_{n}(\varphi)\|_{r_{n}}\leq 4\tilde\epsilon_{n}. \end{eqnarray*} |
Let \widetilde H_{n-1} = H_{n-1}\circ\overline H_{n-1}\circ H_{n-1}^{-1}. By (4.21) and (4.22), we get (4.24) and (4.25) hold.
Lemmas 4.5, 4.6, and 4.7 can be summarized as the following iterative lemma. We do not give the proof because it can be seen in [27].
Lemma 4.8. Suppose \omega\in \mathcal{L} , \tilde\rho(\omega, \rho+f(\theta, \varphi)) = \rho_f\in DC_\omega(\gamma, \tau) and r_n , s_n , \epsilon_n are defined in (4.18) for n\geq 0 . Let \epsilon_0 be sufficiently small. Then the following holds for all n\geq 1 : If the system
\begin{eqnarray} X_{n} = \omega\cdot \partial_\varphi+(\rho_{f}+ g_{n}(\varphi)+ f_{n}(\theta, \varphi))\cdot \partial_{\theta} \end{eqnarray} | (4.26) |
satisfies
\begin{eqnarray*} &&\|f_{n}(\theta, \varphi)\|_{s_{n}, r_{n}}\leq\epsilon_{n}, \\ &&\|g_{n}(\varphi)\|_{r_{n}}\leq 4\tilde\epsilon_{n}. \end{eqnarray*} |
then there exists a change of variables \widetilde H_n:\mathbb{T}\times\mathbb{T}^d\rightarrow \mathbb{T}\times\mathbb{T}^d with estimates
\begin{eqnarray*} &&\| \widetilde H_{n}-id\|_{s_{n+1}, r_{n+1}}\leq 4\epsilon_{n}^{\frac{3}{4}}, \\ &&\|D( \widetilde H_{n}-id)\|_{s_{n+1}, r_{n+1}}\leq 4\epsilon_{n}^{\frac{3}{4}}, \end{eqnarray*} |
such that it transforms the system (4.26) to
\begin{eqnarray*} X_{n+1} = \omega\cdot \partial_\varphi+(\rho_{f}+ g_{n+1}(\varphi)+ f_{n+1}(\theta, \varphi))\cdot \partial_{\theta} \end{eqnarray*} |
with
\begin{eqnarray*} &&\|f_{n+1}(\theta, \varphi)\|_{s_{n+1}, r_{n+1}}\leq\epsilon_{n+1}, \\ &&\|g_{n+1}(\varphi)\|_{r_{n+1}}\leq 4\tilde\epsilon_{n+1}. \end{eqnarray*} |
Let \epsilon_0 be sufficiently small and r_0 = r , s_0 = s . If \tilde\rho(\omega, \rho+f) = \rho_f\in DC_\omega(\gamma, \tau) and \|f\|_{s, r, \mathcal{O}_{\gamma}} < \epsilon_0/2 , then the vector field
\begin{eqnarray*} X = \omega\cdot \partial_\varphi+(\rho+ f(\theta, \varphi))\cdot \partial_{\theta} \end{eqnarray*} |
can be written as
\begin{eqnarray} X = \omega\cdot \partial_\varphi+(\rho_f+ \tilde f(\theta, \varphi))\cdot \partial_{\theta} \end{eqnarray} | (4.27) |
where \tilde f(\theta, \varphi) = \rho-\rho_f+f(\theta, \varphi) . We can use Lemma 4.6 to system (4.27). Thus we can get \widetilde H_0\in H^{s_1, r_1, a}, which conjugates (4.27) to
\begin{eqnarray*} X_1 = \omega\cdot \partial_\varphi+(\rho+ g_1(\varphi) +f_1(\theta, \varphi))\cdot \partial_{\theta} \end{eqnarray*} |
with
\begin{eqnarray*} &&\|f_{1}(\theta, \varphi)\|_{s_{1}, r_{1}}\leq \epsilon_{1}, \\ &&\|g_{1}(\varphi)\|_{r_{1}}\leq 2\tilde \epsilon_{1}. \end{eqnarray*} |
Then we apply Lemma 4.8 inductively, we can get \widetilde H_i\in H^{s_{i+1}, r_{i+1}, a} , i = 1, \cdots, n, such that H^{(n)} = \widetilde H_0\circ \widetilde H_1\circ\cdots\circ \widetilde H_n conjugates (4.27) to
\begin{eqnarray*} X_{n+1} = \omega\cdot \partial_\varphi+(\rho_{f}+ g_{n+1}(\varphi)+ f_{n+1}(\theta, \varphi))\cdot \partial_{\theta} \end{eqnarray*} |
with
\begin{eqnarray*} &&\|f_{n+1}(\theta, \varphi)\|_{s_{n+1}, r_{n+1}}\leq\epsilon_{n+1}, \\ &&\|g_{n+1}(\varphi)\|_{r_{n+1}}\leq 4\tilde\epsilon_{n+1}. \end{eqnarray*} |
And we have
\begin{eqnarray*} \|DH^{(n)}\|_{s_{n+1}, r_{n+1}}\leq \|D\widetilde H_{0}\|_{s_1, r_1}\|D\widetilde H_{1}\|_{s_2, r_2}\cdots\|D \widetilde H_n\|_{s_{n+1}, r_{n+1}}\leq \prod\limits_{i = 0}^{n}(1+4\epsilon_i^{\frac{3}{4}}), \end{eqnarray*} |
which implies
\begin{eqnarray*} \|H^{(n+1)}-H^{(n)}\|_{s_{n+1}, r_{n+1}}\leq \|DH^{(n)}\|_{s_n, r_n}\|\widetilde H_n-id\|_{s_{n+1}, r_{n+1}}\leq 8\epsilon_n^{\frac{3}{4}}. \end{eqnarray*} |
By the definition of (\epsilon_n)_{n\in\mathbb{N}} , we know that for any j\in \mathbb{Z}_+^{1+d}, there exists N\in \mathbb{N} , such that for any n\geq N , we have 8(\frac{4K_{n+1}^\alpha}{r_0})^{|j|}\epsilon_n^{\frac{3}{4}}\leq \epsilon_n^{\frac{1}{2}} . By the Cauchy estimates, if we denote x: = (\theta, \varphi)\in \mathbb{T}^{1+d} , we have
\begin{eqnarray*} |\frac{\partial^{|j|}}{\partial x^j}(H^{(n+1)}-H^{(n)})|\leq \frac{1}{r^{|j|}_{n+1}}\|H^{(n+1)}-H^{(n)}\|_{s_{n+1}, r_{n+1}}\leq 8(\frac{4K_{n+1}^\alpha}{r_0})^{|j|}\epsilon_n^{\frac{3}{4}}\leq \epsilon_n^{\frac{1}{2}}. \end{eqnarray*} |
for any n > N-1 . This guarantees the limit \lim_{n\rightarrow \infty}H^{(n)} belongs to C^\infty .
Finally, let m({\varphi}) = \lim_{n\rightarrow \infty}g_{n}(\varphi) and \Psi = (\lim_{n\rightarrow \infty}H^{(n)})^{-1} , the proof of Proposition 3.1 completed.
Xinyu Guan: Methodology, Writing-original draft; Nan Kang: Validation(equal), Writing-review & editing (equal). All authors have read and approved the final version of the manuscript for publication.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Xinyu Guan was partially supported by the National Natural Science Foundation of China (Grant No. 12301201) and the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2023QA055).
The authors declare that there is no conflict of interest in this paper.
We give the conception of continued fraction expansion and CD -bridge. Let \alpha\in (0, 1) be irrational and {\rm int}(\bullet) denote the integer part of \bullet . Define that a_0 = 0 , \alpha_0 = \alpha , and inductively for k\geq 1 ,
a_k = {\rm int}(\alpha_k^{-1}), \; \; \alpha_k = \alpha_{k-1}^{-1}-a_k. |
We also define that p_0 = 0 , p_1 = 1 , q_0 = 1 , q_1 = a_1 , and recursively,
\begin{eqnarray*} \begin{array}{l} p_k = a_kp_{k-1}+p_{k-2}\\ q_k = a_kq_{k-1}+q_{k-2}. \end{array} \end{eqnarray*} |
Then (q_n) is the sequence of denominators of the best rational approximation for \alpha . It satisfies
\|k\alpha\|_{\mathbb{T}}\geq \|q_{n-1}\alpha\|_{\mathbb{T}}, \; \; \forall 1\leq k < q_n |
and
\frac{1}{q_n+q_{n+1}} < \|q_n\alpha\|_{\mathbb{T}}\leq \frac{1}{q_{n+1}}, |
where \|x\|_{\mathbb{T}} is defined by \|x\|_{\mathbb{T}} = \inf_{p\in\mathbb{Z}}|x-p|.
For each \alpha\in \mathbb{R}/\mathbb{Q} , in the sequel we will fix a particular subsequence (q_{n_k}) of the denominators of the best rational approximations for \alpha , which for simplicity will be denote by (Q_k) . Denote the sequences (q_{n_{k}+1}) and (p_{n_k}) by (\overline{Q}_{k}) and (P_k) respectively. Now, we introduce the concept of a CD -bridge which first appeared in [4].
Definition A.1. Let 0 < \mathcal{A}\leq \mathcal{B}\leq \mathcal{C}. We say that the pair of denominators (q_l, q_n) forms a CD(\mathcal{A}, \mathcal{B}, \mathcal{C}) bridge if
• q_{i+1}\leq q_i^{\mathcal{A}}, \; \forall i = l, \ldots, n-1 ,
• q_l^{\mathcal{C}}\geq q_n\geq q_{l}^{\mathcal{B}} .
Lemma A.1. For each \mathcal{A}\geq 1 , there exists a subsequence (Q_k) such that Q_0 = 1 , and for each k\geq 0 , Q_{k+1}\leq \overline Q_{k}^{\mathcal{A}^4} , and either \overline Q_k\geq Q_k^{\mathcal{A}}, or the pairs (\overline Q_{k-1}, Q_k) and (Q_k, Q_{k+1}) are both CD(\mathcal{A}, \mathcal{A}, \mathcal{A}^3) bridges.
The proof can be seen in [4].
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