1. Introduction
Assume that
$(M, \mathcal {B}, \mu )$
is a probability space and
$f: M \to M$
is an invertible map which preserves the measure
$\mu $
and is ergodic with respect to
$\mu $
. Let
$A: M \to SL(2, \mathbb {R})$
be a measurable function. The linear cocycle defined by A over the base dynamics f is the transformation:
$$ \begin{align*} (f,A):M \times \mathbb{R}^2 \to M \times \mathbb{R}^2; \quad (\theta, v) \mapsto (f(\theta), A(\theta) \cdot v). \end{align*} $$
Note that the iterates of
$(f,A)$
have the form
$(f,A)^n = (f^n, A_n)$
, where
$A_n(\theta ) = A(f^{n-1}(\theta )) \cdots A(f(\theta )) A(\theta ), n \in \mathbb {N}$
and
$A_{-n}(\theta ) = A_n(f^{-n}(\theta ))^{-1}$
. In particular, a
$C^k$
quasi-periodic linear cocycle
$(\alpha , A)$
consists of a rationally independent
$\alpha \in \mathbb {T}^d$
, which determines an ergodic torus translation on the base and
$A \in C^k(\mathbb {T}^d, SL(2,\mathbb {R}))$
which is a k times differentiable matrix-valued function with continuous kth derivatives.
In this paper, we present a purified quantitative reducibility theorem for finitely differentiable quasi-periodic cocycles. In particular, it applies to the
$C^k$
quasi-periodic Schrödinger cocycles
$(\alpha , A)$
, where
$$ \begin{align*} A(\theta)=S_E^{V}(\theta)=\begin{pmatrix} E- V(\theta) & -1 \\ 1 & 0 \end{pmatrix}\!. \end{align*} $$
This effectively extends the spectral applications in the one-dimensional discrete quasi-periodic Schrödinger operator with a
$C^k$
potential:
$$ \begin{align} (H_{V,\alpha,\theta}x)_n=x_{n+1}+x_{n-1}+V(\theta+n\alpha)x_{n},\quad n\in \mathbb{Z}, \end{align} $$
as any formal solution (not necessarily in
$\ell ^2$
) of
$H_{V,\alpha ,\theta }x=Ex$
satisfies
$$ \begin{align*} A(\theta+n\alpha)\begin{pmatrix} x_{n} \\ x_{n-1} \end{pmatrix}=\begin{pmatrix} x_{n+1} \\ x_{n} \end{pmatrix}\!. \end{align*} $$
In equation (1),
$V\in C^k(\mathbb {T}^d,\mathbb {R})$
is called the potential,
$\alpha \in \mathbb {R}^d$
is called the frequency satisfying
$\langle n,\alpha \rangle \notin \mathbb {Z}$
for any
$n\in \mathbb {Z}^d \backslash \{0\}$
and
$\theta \in \mathbb {T}^d=\mathbb {R}^d/\mathbb {Z}^d$
(or
$\mathbb {R}^d/(2\pi \mathbb {Z})^d$
if preferable) is called the initial phase,
$k,d\in \mathbb {N}^+$
. The spectrum of
$H_{V,\alpha ,\theta }$
is denoted as
$\Sigma _{V,\alpha ,\theta }$
, which is independent of the phase due to the minimality of dynamics and strong operator convergence [Reference Damanik and Fillman17]. Since V is bounded by compactness,
$H_{V,\alpha ,\theta }$
is a bounded self-adjoint operator on
$\ell ^2(\mathbb {Z})$
and
$\Sigma _{V,\alpha ,\theta } \subset \mathbb {R}$
is a compact perfect set. See the nice survey of Damanik [Reference Damanik16] for more details of the Schrödinger operator unity. Moreover, readers are invited to consult the excellent book of Damanik and Fillman [Reference Damanik and Fillman17] which is significant and timely for the community.
1.1. Quantitative reducibility
The reducibility of quasi-periodic cocycles aims to conjugate the original quasi-periodic cocycles to constant ones via transformations that are essentially coordinate changes. However, due to topological obstructions, reducibility may fail and the concept of almost reducibility naturally arises, which settles for almost conjugating to constant cocycles but has proven to be very powerful in studying the spectral theory of quasi-periodic Schrödinger operators [Reference You38]. Readers are referred to §2 for precise definitions of (almost) reducibility.
In the previous literature, almost reducibility is regarded as a prerequisite for reducibility, especially in the analytic region, due to the great flexibility of shrinking the analytic radius arbitrarily. Nevertheless, in the
$C^k$
topology, the least loss of regularity is a fixed number. To establish a
$C^k$
reducibility theorem with least initial regularity and most remainder, we perform a process parallel to the process of almost reducibility. In other words, reducibility is built not after almost reducibility, but at the same time with extra assumptions of the fibered rotation number
$\rho (\alpha , A)$
, see its definition in §2.2.
Recall that
$\alpha \in \mathbb {R}^d$
is called Diophantine if there exist
$\kappa>0$
and
$\tau>d$
such that
$\alpha \in \mathrm {DC}_d(\kappa ,\tau )$
, where
$$ \begin{align} \mathrm{DC}_d(\kappa,\tau):=\bigg\{\alpha \in\mathbb{R}^d: \inf_{j \in \mathbb{Z}}|\langle m,\alpha \rangle - j |> \frac{\kappa}{|m|^{\tau}} \text{ for all } m\in\mathbb{Z}^d\backslash\{0\} \bigg\}. \end{align} $$
Here, we denote
$$ \begin{align*} \lvert m\rvert=\lvert m_1\rvert+\lvert m_2\rvert+ \cdots + \lvert m_d\rvert \end{align*} $$
and
$$ \begin{align*} \langle m,\alpha \rangle =m_1\alpha_1+m_2\alpha_2+\cdots+m_d\alpha_d. \end{align*} $$
Denote
$\mathrm {DC}_d=\bigcup _{\kappa ,\tau } \mathrm {DC}_d(\kappa ,\tau )$
, which is of full Lebesgue measure.
Here,
$\phi \in \mathbb {R}$
is called Diophantine with respect to
$\alpha $
if it satisfies the condition
$\phi \in \mathrm {DC}^{\alpha }_d(\gamma ,\tau )$
, where
$\gamma> 0$
,
$\tau> d$
, and
$$ \begin{align} \mathrm{DC}^{\alpha}_d(\gamma,\tau):=\bigg\{\phi \in\mathbb{R}: \inf_{j \in \mathbb{Z}}|2\phi - \langle m,\alpha \rangle - j |> \frac{\gamma}{(|m|+1)^{\tau}} \text{ for all } m\in\mathbb{Z}^d \bigg\}. \end{align} $$
Additionally,
$\phi \in \mathbb {R}$
is called rational with respect to
$\alpha $
if
$2\phi = \langle m_0,\alpha \rangle $
mod
$\mathbb {Z}$
for some
$m_0 \in \mathbb {Z}^d$
.
Our main theorem is as follows.
Theorem 1.1. Let
$A \in SL(2, \mathbb {R}), \alpha \in \mathrm {DC}_d (\kappa , \tau ) $
and
$f \in C^k(\mathbb {T}^d, sl(2, \mathbb {R}))$
with
$k> 14\tau + 2$
. There exists
$\epsilon = \epsilon (\kappa , \tau , d, k, \|A\|)$
such that if
$\lVert f(\theta ) \rVert _k \leqslant \epsilon $
and
$\rho (\alpha , Ae^f\!)$
is Diophantine or rational with respect to
$\alpha $
, then
$(\alpha , Ae^{f(\theta )})$
is
$C^{k,k_0}$
reducible with
$k_0 < k-10\tau -3$
.
Remark 1.2. The quantitative version of Theorem 1.1 can be found in §3, specifically in Theorem 3.8. Our theorem is a significant improvement over the results presented in [Reference Cai and Ge12]. Notably, the loss of regularity
$10\tau +3$
does not depend on the parameter k.
As there are few reducibility results in the
$C^k$
topology, we only briefly review the development of analytic reducibility. Let us first discuss it for local cocycles of the form
$(\alpha , Ae^{f(\theta )})$
. An achievement was initiated by Dinaburg and Sinai [Reference Dinaburg and Sinai22], who applied the classical Kolmogorov–Arnold–Moser (KAM) scheme. They established the positive measure reducibility in terms of the rotation number for continuous quasi-periodic Schrödinger equations featuring small analytic potentials in the perturbative regime. Moser and Pöschel [Reference Moser and Pöschel32] later expanded on this achievement by using a resonance-cancellation technique, which extended the positive measure reducibility to a class of rotation numbers that are rational with respect to
$\alpha $
. Moreover, the breakthrough came from Eliasson [Reference Eliasson23], who established weak almost reducibility for all energies E and full measure reducibility for Diophantine frequencies and small analytic potentials. For further insights into strong almost reducibility results, readers can refer to the works of Chavaudret [Reference Chavaudret14] and Leguil et al [Reference Leguil, You, Zhao and Zhou31]. In the non-perturbative regime, Puig [Reference Puig33] employed the localization method to derive a non-perturbative version of Eliasson’s reducibility theorem. As for continuous linear systems, Hou and You [Reference Hou and You26] proved weak almost reducibility results for all rotation numbers and frequencies
$\omega = (1, \alpha ) \in \mathbb {T}^2$
under small analytic perturbations.
In the case of global cocycle
$(\alpha , A(\theta ))$
with
$A \in C^\omega (\mathbb {T}, SL(2, \mathbb {R}))$
and
$\alpha \in \mathbb {T}$
, Avila and Krikorian [Reference Avila and Krikorian5] applied the renormalization scheme to show that, for
$\alpha $
satisfying certain recurrent Diophantine conditions and almost every (a.e.) E, the quasi-periodic Schrödinger cocycle is either reducible or non-uniform hyperbolic. Additionally, Avila, Fayad, and Krikorian [Reference Avila, Fayad and Krikorian3] proved that for irrational
$\alpha $
and a.e. E, the quasi-periodic Schrödinger cocycle is either rotations reducible or non-uniformly hyperbolic.
In the realm of finitely differentiable topology, reducibility results are scarce and the existing ones are rough in the sense of very large initial regularity and very high loss. We note that our philosophy is that nicer reducibility implies nicer spectral applications. In this direction, Theorem 1.1, which was established via keeping all the parameters, allows us to obtain the most general applications according to the technique.
1.2. Spectral type and structure
Over the past forty years, advances have been made in studying the spectral theory of the Schrödinger operator, focusing on understanding the spectral type and the structure of the spectrum. In the 21st century, people found that quantitative dynamical estimates lead to quantitative spectral applications [Reference Avila and Jitomirskaya4]. In particular, quantitative (almost) reducibility is one of the most powerful techniques, with abundant fruitful spectral results [Reference Avila, Bochi and Damanik2, Reference Cai10–Reference Cai and Wang13, Reference Damanik, Goldstein and Lukic18, Reference Jian and Shi27, Reference Johnson and Moser28, Reference Puig33, Reference Wang and Zhang37]. One is invited to consult You’s 2018 ICM survey [Reference You38]. Generally, spectral type results refer to absolutely continuous spectrum, singular continuous spectrum, pure point spectrum, ballistic transport, Anderson localization, etc. while regarding spectral structure, common results often involve the Cantor spectrum and homogeneous spectrum. For the sake of concision, we only list one spectral type application and one spectral structure application, though our Theorem 1.1 along with its quantitative version is promisingly applicable to many other spectral results in relation.
Theorem 1.3. (Spectral type application)
Assume
$\alpha \in \mathrm {DC}_d(\kappa , \tau )$
,
$V\in C^k(\mathbb {T}^d, \mathbb {R})$
with
$k>14\tau +2$
. If there exists
$\epsilon '=\epsilon '(\kappa , \tau , k, d)$
such that
$\lVert V \rVert _k \leqslant \epsilon '$
and
$\rho (\alpha , S_E^V)$
is Diophantine or rational with respect to
$\alpha $
for
$E \in \Sigma _{V,\alpha }$
, then
$H_{V,\alpha ,\theta }$
has strong ballistic transport for all
$\theta \in \mathbb {T}^d$
.
Remark 1.4. The definition of (global) strong ballistic transport is too lengthy, so we prefer to put it in §4 rather than here. This theorem compensates for the work of Ge and Kachkovskiy [Reference Ge and Kachkovskiy25] by providing the precise requirement of k for a
$C^k$
quasi-periodic Schrödinger family to be reducible.
Indeed, there is a qualitative connection between spectral type and transport properties, which can be understood through the RAGE theorem [Reference Damanik16]. However, quantitative results also illustrate the connection between them. For example, except for purely absolutely continuous spectra, we generally do not expect to observe ballistic transport phenomena. In particular, it has been proven in [Reference Simon34] that point spectra cannot support any ballistic motion. However, when the operator is restricted to a subspace that supports purely absolutely continuous spectrum, we can still expect to observe ballistic transport phenomena. Additionally, the works of Guarneri, Combes, and Last [Reference Combes15, Reference Last30] have also made contributions to the connection. In particular, the Guarneri–Combes–Last theorem [Reference Last30] quantitatively provides insights into transport phenomena in one-dimensional systems with absolutely continuous spectrum. In subsequent studies, Asch and Knauf [Reference Asch and Knauf1], as well as Damanik [Reference Damanik, Lukic and Yessen20], demonstrated the occurrence of strong ballistic transport in periodic continuous Schrödinger operators, which are widely known to have absolutely continuous spectra. Later, Zhang and Zhao [Reference Zhao and Zhang42] explicitly established a connection between the values of transport exponents and absolutely continuous spectra in the setting of discrete single-frequency quasi-periodic operators. More recently, based on the results and techniques in the discrete setting of Fillman [Reference Fillman24], Young’s research [Reference Young39] discovered strong ballistic transport in a class of continuum limit-periodic operators known to possess absolutely continuous spectra.
Let us now state our spectral structure result. First, recall the following definition.
Definition 1.5. [Reference Cai and Wang13]
Let
$\nu> 0$
. A closed set
$\mathfrak {B} \subset \mathbb {R}$
is called
$\nu $
-homogeneous if
$$ \begin{align*} |\mathfrak{B} \cap (E-\epsilon, E+\epsilon)|>\nu \epsilon \quad \text{for all } E\in \mathfrak{B}, \text{ for all } 0<\epsilon<\mathrm{diam} \mathfrak{B}. \end{align*} $$
As another corollary of Theorem 1.1, we have the following.
Theorem 1.6. (Spectral structure application)
Assume
$\alpha \in \mathrm {DC}_d(\kappa , \tau )$
,
$V\in C^k(\mathbb {T}^d, \mathbb {R})$
with
$k>17\tau +2$
. If there exists
$\epsilon "=\epsilon "(\kappa , \tau , k, d)$
such that
$\lVert V \rVert _k \leqslant \epsilon "$
, then
$\Sigma _{V,\alpha }$
is
$\nu $
-homogeneous for some
$\nu \in (0,1)$
.
Remark 1.7. We know that the homogeneity of the spectrum is related to polynomial decay of gap length and Hölder continuity of the integrated density of states. Therefore, we need to change the assumption to
$k> 17\tau + 2$
so that we can further ensure the
$\tfrac 12$
-Hölder continuity of IDS (see [Reference Cai10, Theorems 3.3 and 3.4]). This theorem greatly reduces the initial regularity assumption in the
$C^k$
case.
The spectrum’s homogeneity plays a crucial role in the inverse spectral theory, as demonstrated in the seminal works of Sodin and Yuditskii [Reference Sodin and Yuditskii35, Reference Sodin and Yuditskii36]. Under the assumption of a finite total gap length and a reflectionless condition on the spectrum, it has been proved that the homogeneity of the spectrum implies the almost periodicity of the associated potentials [Reference Sodin and Yuditskii35]. Note that the assumption of having a finite total gap length is trivial in the discrete case since the gap length is always bounded by the diameter of the spectrum. In particular, the homogeneity of the spectrum is closely linked to Deift’s conjecture, which investigates whether the solutions of the Korteweg–De Vries (KdV) equation exhibit quasi-periodicity when the initial data are quasi-periodic [Reference Binder, Damanik, Goldstein and Lukic9, Reference Damanik, Goldstein and Lukic19]. In the continuous setting, Binder et al [Reference Binder, Damanik, Goldstein and Lukic9] demonstrated that when considering small analytic quasi-periodic initial data with Diophantine frequency, the solution of the KdV equation exhibits almost periodicity in the temporal variable. In the discrete setting, Leguil et al [Reference Leguil, You, Zhao and Zhou31] demonstrated that for the subcritical potential
$V \in C^\omega (\mathbb {T}, \mathbb {R})$
, the Toda flow is almost periodic in the time variable when considering initial data that are also almost periodic with
$\beta (\alpha ) = 0$
. Recently, Avila et al [Reference Avila, Last, Shamis and Zhou6] also constructed an intriguing counter-example that even for the AMO, its spectrum is not homogeneous if
$e^{-2/3\beta (\alpha )} < \unicode{x3bb} < e^{2/3\beta (\alpha )}$
. In the
$C^k$
case, Cai and Wang [Reference Cai and Wang13] have recently proved the homogeneity of the spectrum. They achieved this through a not so refined quantitative
$C^k$
reducibility theorem for quasi-periodic
$SL(2, \mathbb {R})$
cocycles, as well as by employing the Moser–Pöschel argument for the associated Schrödinger cocycles.
2. Preliminaries
2.1. Conjugation and reducibility
For a bounded analytic (possibly matrix valued) function
$F(\theta )$
defined on
$\mathcal {S}_h:= \{ \theta =(\theta _1,\ldots , \theta _d)\in \mathbb {C}^d\ |\ \ | \Im \theta _i | < h \text { for all } 1\leqslant i\leqslant d\}$
, let
$|F|_h= \sup _{\theta \in \mathcal {S}_h } \| F(\theta )\| $
and denote by
$C^\omega _{h}(\mathbb {T}^d,*)$
the set of all these
$*$
-valued functions (
$*$
will usually denote
$\mathbb {R}$
,
$sl(2,\mathbb {R})$
,
$SL(2,\mathbb {R})$
). Let
$C^\omega (\mathbb {T}^d,*)=\bigcup _{h>0}C^\omega _{h}(\mathbb {T}^d,*)$
and
$C^{k}(\mathbb {T}^{d},*)$
be the space of k times differentiable with continuous kth derivatives functions. Define the norm as
$$ \begin{align*} \lVert F \rVert _{k}=\sup_{\substack{ \lvert k^{'}\rvert\leqslant k, \theta \in \mathbb{T}^{d} }}\lVert \partial^{k^{'}}F(\theta) \rVert. \end{align*} $$
For two cocycles
$(\alpha ,A_1)$
,
$(\alpha ,A_2)\in \mathbb {T}^d \times C^{\ast }(\mathbb {T}^d,SL(2,\mathbb {R}))$
, ‘
$\ast $
’ represents ‘
$\omega $
’ or ‘k’, we can say that they are
$C^{\ast }$
conjugated if there exists
$Z\in C^{\ast }(2\mathbb {T}^d, SL(2,\mathbb {R}))$
, such that
$$ \begin{align*} Z(\theta+\alpha)A_1(\theta)Z^{-1}(\theta)=A_2(\theta). \end{align*} $$
Notably, we want to define Z on the
$2\mathbb {T}^d=\mathbb {R}^d/(2\mathbb {Z})^d$
for the purpose of making it still real-valued.
An analytic cocycle
$(\alpha , A)\in \mathbb {T}^d \times C^{\omega }_h(\mathbb {T}^d, SL(2,\mathbb {R}))$
is called almost reducible if there exist a sequence of constant matrices
$A_j\in SL(2,\mathbb {R})$
, a sequence of conjugations
$Z_j\in C^{\omega }_{h_j}(2\mathbb {T}^d, SL(2,\mathbb {R}))$
, and a sequence of small perturbation
$f_j \in C^{\omega }_{h_j}(\mathbb {T}^d, sl(2,\mathbb {R}))$
such that
$$ \begin{align*} Z_j(\theta+\alpha)A(\theta)Z_j^{-1}(\theta)=A_j e^{f_j(\theta)} \end{align*} $$
with
$$ \begin{align*} \lvert f_j(\theta)\rvert_{h_j}\rightarrow 0, \quad j\rightarrow \infty. \end{align*} $$
Furthermore, it is said to be weak
$(C^{\omega })$
almost reducible if
$h_j \rightarrow 0$
and it is said to be strong
$(C^{\omega }_{h_j,h'})$
almost reducible if
$h_j\rightarrow h'>0$
. We also claim
$(\alpha , A)$
is
$C^{\omega }_{h,h'}$
reducible if there exist a constant matrix
$\widetilde {A} \in SL(2,\mathbb {R})$
and a conjugation map
$\widetilde {Z}\in C^{\omega }_{h'}(2\mathbb {T}^d, SL(2,\mathbb {R}))$
such that
$$ \begin{align*} \widetilde{Z}(\theta+\alpha)A(\theta)\widetilde{Z}^{-1}(\theta)=\widetilde{A}(\theta). \end{align*} $$
To avoid repetition, we have provided an equivalent definition of
$C^k$
(almost) reducibility as follows.
One can say that a finitely differentiable cocycle
$(\alpha ,A)$
is
$C^{k,k_1}$
almost reducible if
$A\in C^k(\mathbb {T}^d,SL(2,\mathbb {R}))$
and the
$C^{k_1}$
-closure of its
$C^{k_1}$
conjugacies contains a constant. In addition, we say
$(\alpha ,A)$
is
$C^{k,k_1}$
reducible if
$A\in C^k(\mathbb {T}^d,SL(2,\mathbb {R}))$
and its
$C^{k_1}$
conjugacies contain a constant.
2.2. Rotation number and degree
Suppose
$A\in C^0(\mathbb {T}^d,SL(2,\mathbb {R}))$
is homotopic to identity. Then we show the projective skew-product
$F_A:\mathbb {T}^d \times \mathbb {S}^1 \rightarrow \mathbb {T}^d \times \mathbb {S}^1$
with
$$ \begin{align*} F_A(x,\omega):=\bigg( x+\alpha, \frac{A(x)\cdot \omega}{\lvert A(x)\cdot \omega\rvert}\bigg), \end{align*} $$
which is homotopic to identity as well. Thus, we will lift
$F_A$
to a map
$\widetilde {F}_A:\mathbb {T}^d\times \mathbb {R}\rightarrow \mathbb {T}^d\times \mathbb {R}$
with
$\widetilde {F}_A(x,y)=(x+\alpha ,y+\psi (x,y))$
, where for every
$x \in \mathbb {T}^d$
,
$\psi (x,y)$
is
$\mathbb {Z}$
-periodic in y. The map
$\psi :\mathbb {T}^d \times \mathbb {R} \rightarrow \mathbb {R}$
is said to be a lift of A. Assume
$\mu $
, which is invariant by
$\widetilde {F}_A$
, is any probability measure on
$\mathbb {T}^d\times \mathbb {R}$
. Its projection on the first coordinate is provided by the Lebesgue measure. The number
$$ \begin{align} \rho_{(\alpha,A)}:=\int_{\mathbb{T}^d\times \mathbb{R}} \psi(x,y)\,d\mu(x,y)\quad\mod \mathbb{Z} \end{align} $$
has nothing to do with the choices of the lift
$\psi $
or the measure
$\mu $
. One calls it the fibered rotation number of cocycle
$(\alpha , A)$
(readers can refer to [Reference Johnson and Moser28] for more details).
Assume
$$ \begin{align*} R_{\phi}:=\begin{pmatrix} \cos 2\pi \phi & -\sin 2\pi \phi \\ \sin 2\pi \phi & \cos 2\pi \phi \end{pmatrix}\! \end{align*} $$
if
$A\in C^0(\mathbb {T}^d,SL(2,\mathbb {R}))$
is homotopic to
$\theta \rightarrow R_{\langle n,\theta \rangle }$
for some
$n\in \mathbb {Z}^d$
, then we call n the degree of A and denote it by degA. Furthermore,
$$ \begin{align} \deg(AB)=\deg A+\deg B. \end{align} $$
Please note that the fibered rotation number remains invariant under real conjugacies that are homotopic to the identity map. Generally speaking, when the cocycle
$(\alpha ,A_1)$
is conjugated to
$(\alpha ,A_2)$
by
$B\in C^0(2\mathbb {T}^d, SL(2,\mathbb {R}))$
, that is,
$B(\cdot +\alpha )A_1(\cdot )B^{-1}(\cdot )=A_2(\cdot )$
, we have
$$ \begin{align} \rho_{(\alpha,A_2)}=\rho_{(\alpha,A_1)}-\frac{\langle \deg B,\alpha\rangle}{2}. \end{align} $$
2.3. Hyperbolicity and integrated density of states
We call the cocycle
$(\alpha ,A)$
uniformly hyperbolic if for every
$\theta \in \mathbb {T}^d$
, there exists a continuous decomposition
$\mathbb {C}^2 = E^s(\theta ) \oplus E^u(\theta )$
such that for some constants
$C> 0$
,
$ c> 0$
, and every
$n \geqslant 0$
,
$$ \begin{align*} |A_n(\theta)v| \leqslant Ce^{-cn}|v|,\quad v \in E^s(\theta), \end{align*} $$
$$ \begin{align*} |A_{-n}(\theta)v| \leqslant Ce^{-cn}|v|,\quad v \in E^u(\theta). \end{align*} $$
This decomposition is invariant by the dynamics, which means that for any
$\theta \in \mathbb {T}^d$
,
$A(\theta )E^*(\theta ) = E^*(\theta + \alpha )$
for
$*= s, u$
. In the
$C^0$
topology, the set of uniformly hyperbolic cocycles is an open set. Specifically, in the case of quasi-periodic Schrödinger operators, the cocycle
$(\alpha , S_E^V)$
is uniformly hyperbolic if and only if
$E \notin \Sigma _{V,\alpha }$
, or in other words, if the energy lies within a spectral gap [Reference Johnson29].
Let us consider the Schrödinger operators
$H_{V,\alpha ,\theta }$
, where an important concept is the integrated density of states (IDS). The IDS is a function
$N_{V,\alpha }: \mathbb {R}\rightarrow [0,1]$
that can be defined by
$$ \begin{align*} N_{V,\alpha}(E) = \int_{\mathbb{T}^{d}}\mu_{V,\alpha,\theta}(-\infty,E]\,d\theta, \end{align*} $$
where
$\mu _{V,\alpha ,\theta } = \mu _{V,\alpha ,\theta }^{e_{-1}}+\mu _{V,\alpha ,\theta }^{e_{0}}$
is the universal spectral measure of
$H_{V,\alpha ,\theta }$
and
$\{e_{i}\}_{i\in \mathbb {Z}}$
is the canonical basis of
$\ell ^{2}(\mathbb {Z})$
. We say
$\{e_{-1}, e_{0}\}$
is the pair of cyclic vectors of
$H_{V,\alpha ,\theta }$
here.
There exist alternative approaches to defining the IDS by counting eigenvalues of the truncated Schrödinger operator. For further details, readers may consult [Reference Avron and Simon7]. Furthermore, there is a connection between
$\rho (\alpha , S^V_E)$
and the IDS, which can be expressed as follows:
$$ \begin{align} N_{V,\alpha}(E)=1-2\rho(\alpha,S^{V}_{E}) \ \mod \mathbb{Z}. \end{align} $$
To gain a better understanding of the various types of spectral measures, I recommend referring to the book [Reference Damanik16].
2.4. Analytic approximation
Let
$f \in C^{k}(\mathbb {T}^{d},sl(2,\mathbb {R}))$
. By Zehnder [Reference Zehnder40], there is a sequence
$\{f_{j}\}_{j\geqslant 1}$
,
$f_{j}\in C_{{1}/{j}}^{\omega }(\mathbb {T}^{d},sl(2,\mathbb {R}))$
and a universal constant
$C'$
, such that
$$ \begin{align} \lVert f_{j}-f \rVert_{k} &\rightarrow 0 , \quad j \rightarrow +\infty, \nonumber\\ \lvert f_{j}\rvert_{{1}/{j}} &\leqslant C'\lVert f \rVert_{k}, \\ \nonumber \lvert f_{j+1}-f_{j} \rvert_{{1}/({j+1})} &\leqslant C'\bigg(\frac{1}{j}\bigg)^k\lVert f \rVert_{k}. \end{align} $$
Furthermore, if
$k\leqslant \tilde {k}$
and
$f\in C^{\tilde {k}}$
, the properties in equation (8) still hold with
$\tilde {k}$
instead of k. This implies that the sequence can be constructed from f irrespective of its regularity (since
$f_{j}$
is achieved by convolving f with a map that does not depend on k).
3. Dynamical estimates: full measure reducibility
In this section, the main emphasis is on investigating the reducibility property of the following
$C^k$
quasi-periodic
$SL(2,\mathbb {R})$
cocycle:
$$ \begin{align*} (\alpha,Ae^{f(\theta)}): \mathbb{T}^d \times \mathbb{R}^2 \to \mathbb{T}^d \times \mathbb{R}^2; (\theta, v) \mapsto (\theta + \alpha, Ae^{f(\theta)} \cdot v), \end{align*} $$
where
$\alpha \in \mathrm {DC}_d(\kappa ,\tau )$
,
$A\in SL(2,\mathbb {R}), \, f\in C^k(\mathbb {T}^{d},sl(2,\mathbb {R}))$
, and
$d\in \mathbb {N}^+$
. Our approach involves initially analyzing the approximating analytic cocycles
$\{(\alpha ,Ae^{f_{j}(\theta )})\}_{j\geqslant 1}$
and subsequently transferring the obtained estimates to the targeted
$C^k$
cocycle
$(\alpha ,Ae^{f(\theta )})$
through analytic approximation techniques.
3.1. Preparations
In the following subsections, we will consider fixed parameters
$\rho ,\epsilon ,N,\sigma $
; we refer to the situation in which there exists
$m_\ast $
satisfying
$0<\lvert m_\ast \rvert \leqslant N$
such that
$$ \begin{align*} \inf_{j \in \mathbb{Z}}\lvert 2\rho- \langle m_\ast,\alpha\rangle - j\rvert< \epsilon^{\sigma}, \end{align*} $$
as the ‘resonant case’ (for simplicity, we will use the notation ‘
$\lvert 2\rho - \langle m_\ast ,\alpha \rangle \rvert $
’ to represent the left side and similarly, ‘
$\lvert \langle m_\ast ,\alpha \rangle \rvert $
’ to represent the right side). The integer vector
$m_\ast $
will be referred to as a ‘resonant site’. This type of small divisor problem commonly arises when attempting to solve the cohomological equation at each step of the KAM procedure. Resonances are connected to a useful decomposition of the space
$\mathcal {B}_r:=C^{\omega }_{r}(\mathbb {T}^{d},su(1,1))$
. For further details and the precise definition of
$su(1,1)$
and
$SU(1,1)$
, please refer to the proof of Proposition 3.3.
Given
$\alpha \in \mathbb {R}^{d}$
,
$A\in SU(1,1)$
, and
$\eta>0$
, there exists a decomposition
$\mathcal {B}_r=\mathcal {B}_r^{nre}(\eta ) \bigoplus \mathcal {B}_r^{re}(\eta )$
satisfying that for any
$Y\in \mathcal {B}_r^{nre}(\eta )$
,
$$ \begin{align} A^{-1}Y(\theta+\alpha)A\in\mathcal{B}_r^{nre}(\eta),\,\lvert A^{-1}Y(\theta+\alpha)A-Y(\theta)\rvert_r\geqslant\eta\lvert Y(\theta)\rvert_r. \end{align} $$
Additionally, denote
$\mathbb {P}_{nre}$
,
$\mathbb {P}_{re}$
as the standard projections from
$\mathcal {B}_r$
onto
$\mathcal {B}_r^{nre}(\eta )$
and
$\mathcal {B}_r^{re}(\eta )$
, respectively.
Next, we have a crucial lemma that plays a key role in eliminating all the non-resonant terms. This lemma will be used in the proof of the resonant conditions.
Lemma 3.1. [Reference Cai, Chavaudret, You and Zhou11, Reference Hou and You26]
Assume that
$A\in SU(1,1)$
,
$\epsilon \leqslant (4\lVert A\rVert )^{-4}$
, and
$\eta \geqslant 13\lVert A\rVert ^2{\epsilon }^{1/2}$
. For any
$g\in \mathcal {B}_r$
with
$|g|_r \leqslant \epsilon $
, there exist
$Y\in \mathcal {B}_r$
and
$g^{re}\in \mathcal {B}_r^{re}(\eta )$
such that
$$ \begin{align*} e^{Y(\theta+\alpha)}(Ae^{g(\theta)})e^{-Y(\theta)}=Ae^{g^{re}(\theta)}, \end{align*} $$
with estimates
$$ \begin{align*}\lvert Y \rvert_r\leqslant \epsilon^{{1}/{2}}, \quad \lvert g^{re}\rvert_r\leqslant 2\epsilon. \end{align*} $$
Remark 3.2. For the inequality ‘
$\eta \geqslant 13\lVert A\rVert ^2{\epsilon }^{1/2}$
’, ‘
$\tfrac 12$
’ is sharp because of the quantitative implicit function theorem [Reference Berti and Biasco8, Reference Deimling21]. The proof relies solely on the fact that
$\mathcal {B}_r$
is a Banach space. Therefore, it is applicable not only to the
$C^{\omega }$
topology but also to the
$C^k$
and
$C^0$
topologies. For more detailed information, please refer to [Reference Cai, Chavaudret, You and Zhou11, appendix].
3.2. Analytic KAM theorem
Following our plan, our first objective is to establish the KAM theorem for the analytic quasi-periodic
$SL(2,\mathbb {R})$
cocycle
$(\alpha , Ae^{f(\theta )})$
, where A possesses eigenvalues
${e^{i\rho },e^{-i\rho }}$
with
$\rho \in 2\pi \mathbb {R} \cup 2\pi i\mathbb {R}$
. We present our quantitative analytic KAM theorem in the following.
Proposition 3.3. [Reference Cai10, Reference Cai, Chavaudret, You and Zhou11]
Let
$\alpha \in \mathrm {DC}_d(\kappa ,\tau ), \kappa ,r>0, \tau >d,\sigma <\tfrac 16$
. Suppose that
$A \in SL(2,\mathbb {R})$
satisfying
$\|A\|$
bounded,
$f \in C_r^{\omega }(\mathbb {T}^{d},sl(2,\mathbb {R}))$
. Then for any
$0<r'<r$
, there exist constants
$c=c(\kappa ,\tau ,d), D>{2}/{\sigma }$
and
$\tilde {D}=\tilde {D}(\sigma )$
such that if
$$ \begin{align} \lvert f \rvert_r\leqslant\epsilon \leqslant \frac{c}{\lVert A\rVert^{\tilde{D}}}(r-r')^{D\tau}, \end{align} $$
then there exist
$B \in C^{\omega }_{r'}(2\mathbb {T}^d,SL(2,\mathbb {R})), A_+ \in SL(2,\mathbb {R})$
and
$f_+ \in C^{\omega }_{r'}(\mathbb {T}^d, sl(2,\mathbb {R}))$
such that
$$ \begin{align*} B(\theta+\alpha)(Ae^{f(\theta)})B^{-1}(\theta)=A_+e^{f_+(\theta)}. \end{align*} $$
More precisely, let
$N= ({2}/({r-r'})) \lvert \ln \epsilon \rvert $
, then we can distinguish two cases:
-
• (Non-resonant case) if for any
$m \in \mathbb {Z}^d$
with
$0<|m| \leqslant N$
, we have then
$$ \begin{align*} \lvert 2 \rho-\langle m,\alpha \rangle \rvert \geqslant {\epsilon}^{\sigma}, \end{align*} $$
and
$$ \begin{align*} \lvert B(\theta)- \mathrm{Id}\rvert_{r'}\leqslant \epsilon^{1-{8}/{D}}, \lvert f_+\rvert_{r'}\leqslant \epsilon^{2-{8}/{D}} \end{align*} $$
$$ \begin{align*} \lVert A_+ - A\rVert \leqslant 2\lVert A\rVert \epsilon; \end{align*} $$
-
• (Resonant case) if there exists
$m_\ast \in \mathbb {Z}^d$
with
$0<|m_\ast | \leqslant N$
such that then
$$ \begin{align*} \lvert 2 \rho-\langle m_\ast,\alpha \rangle \rvert < {\epsilon}^{\sigma}, \end{align*} $$
Moreover,
$$ \begin{align*} \lvert B \rvert_{r'} &\leqslant 8\bigg( \frac{\lVert A\rVert}{\kappa}\bigg) ^{{1}/{2}}\bigg( \frac{2}{r-r'} \lvert \ln \epsilon \rvert\bigg) ^{{\tau}/{2}}\times\epsilon^{{-r'}/({r-r'})},\\ \lVert B\rVert_0 &\leqslant 8\bigg( \frac{\lVert A\rVert}{\kappa}\bigg) ^{{1}/{2}}\bigg(\frac{2}{r-r'} \lvert \ln \epsilon \rvert\bigg) ^{{\tau}/{2}},\\ \lvert f_{+}\rvert_{r'}&\leqslant \frac{2^{5+\tau}\lVert A\rVert\lvert \ln\epsilon\rvert^{\tau}}{\kappa(r-r')^{\tau}}\epsilon e^{-N'(r-r')}(N')^de^{Nr'}\ll \epsilon^{100}, \, N'> 2N^2. \end{align*} $$
$A_+=e^{A"}$
with
$\lVert A"\rVert \leqslant 2\epsilon ^{\sigma }$
,
$A"\in sl(2,\mathbb {R})$
. More accurately, we have with
$$ \begin{align*} MA"M^{-1}=\begin{pmatrix} it & v\\ \bar{v} & -it \end{pmatrix} \end{align*} $$
$\lvert t\rvert \leqslant \epsilon ^{\sigma }$
and
$$ \begin{align*} \lvert v \rvert\leqslant \frac{2^{4+\tau}\lVert A\rVert\lvert \ln\epsilon\rvert^{\tau}}{\kappa(r-r')^{\tau}}\epsilon e^{-\lvert m_{\ast}\rvert r}. \end{align*} $$
Proof. We will only prove estimates for the non-resonant case because it is more delicate and the proof of the resonant case is the same compared with those in [Reference Cai10].
Let us recall that
$sl(2,\mathbb {R})$
is isomorphic to
$su(1,1)$
, which is a Lie algebra consisting of matrices of the form
$$ \begin{align*} \begin{pmatrix} it & v\\ \bar{v} & -it \end{pmatrix} \end{align*} $$
with
$t \in \mathbb {R}, v\in \mathbb {C}$
. The isomorphism between them is given by the map
$A \to MAM^{-1}$
, where
$$ \begin{align*}M=\frac{1}{1+i} \begin{pmatrix} 1 & -i\\ 1 & i \end{pmatrix} \end{align*} $$
and a straightforward calculation yields
$$ \begin{align*} M\begin{pmatrix} x & y+z\\ y-z & -x \end{pmatrix}\!,\quad M^{-1}= \begin{pmatrix} iz & x-iy\\ x+iy & -iz \end{pmatrix} \end{align*} $$
where
$x,y,z \in \mathbb {R}$
. Here,
$SU(1,1)$
is the corresponding Lie group of
$su(1,1)$
. We will prove this theorem within
$SU(1,1)$
, which is isomorphic to
$SL(2,\mathbb {R})$
.
We consider the non-resonant case as follows.
For
$0<|m| \leqslant N = ({2}/({r-r'})) \lvert \ln \epsilon \rvert $
, we have
$$ \begin{align} \lvert 2 \rho-\langle m,\alpha \rangle \rvert \geqslant {\epsilon}^{\sigma}, \end{align} $$
by equation (10) and
$D>{2}/{\sigma }$
, we get
$$ \begin{align} \lvert \langle m,\alpha\rangle \rvert \geqslant \frac{\kappa}{ \lvert m \rvert ^{\tau}}\geqslant \frac{\kappa}{ \lvert N \rvert ^{\tau}}\geqslant \epsilon^{{\sigma}/{2}}. \end{align} $$
Indeed, it is well known that the conditions in equations (11) and (12) play a role in addressing the small denominator problem in KAM theory.
We now define
$g \in C^\omega _r(\mathbb {T}^d,su(1,1))$
such that
$$ \begin{align*} g(\theta)=\sum_{n\in \mathbb{Z}^{d},0<\lvert n \rvert \leqslant N}\hat{f}(n)e^{2\pi i\langle n,\theta\rangle}. \end{align*} $$
From Schur’s theorem, we might as well assume
$A=(\begin {smallmatrix} e^{i\rho } & p \\ 0 & e^{-i\rho } \\ \end {smallmatrix}) \in SU(1,1).$
The condition
$\|A\|$
satisfying bounded gives a bound for p and that is the only reason for this condition.
Now we want to solve the cohomological equation
$$ \begin{align} Y(\theta + \alpha)A - AY(\theta) = -Ag(\theta). \end{align} $$
Let
$$ \begin{align*} Y=\begin{pmatrix} y_1 & y_2 \\ y_3 & y_4 \\ \end{pmatrix}\!,\quad g=\begin{pmatrix} g_1 & g_2 \\ g_3 & g_4 \\ \end{pmatrix}\!, \end{align*} $$
then we can obtain the equations
$$ \begin{align*} \begin{cases} e^{i\rho}y_3(\theta + \alpha)-e^{-i\rho}y_3(\theta) = -e^{-i\rho}g_3(\theta), \\ e^{i\rho}y_1(\theta + \alpha)-e^{i\rho}y_1(\theta)-py_3(\theta) = -e^{i\rho}g_1(\theta) - pg_3(\theta), \\ py_3(\theta + \alpha) + e^{-i\rho}y_4(\theta + \alpha)-e^{-i\rho}y_4(\theta) = -e^{-i\rho}g_4(\theta), \\ py_1(\theta + \alpha) + e^{-i\rho}y_2(\theta + \alpha)-e^{i\rho}y_2(\theta)-py_4(\theta) = -e^{i\rho}g_2(\theta)-pg_4(\theta). \end{cases} \end{align*} $$
These equations can be solved by Fourier transform. Compare the corresponding Fourier coefficients of the two sides, and this shows the existence of Y. Apply equation (11) twice to solve the off-diagonal and apply equation (12) once to solve the diagonal, we can get
$$ \begin{align} \lvert Y \rvert_{r'} \leqslant c\epsilon^{-3\sigma} \lvert g \rvert_r, \quad 0<r'<r, \end{align} $$
where the constant only depends on
$\kappa , \tau $
.
By the cohomological equation (13), we obtain
$$ \begin{align*} Y(\theta+\alpha)=AY(\theta)A^{-1}-Ag(\theta)A^{-1}. \end{align*} $$
Then, we can get
$$ \begin{align*} &e^{Y(\theta+\alpha)}(Ae^{f(\theta)})e^{-Y(\theta)}\\ &\quad= e^{AY(\theta)A^{-1}-Ag(\theta)A^{-1}}(Ae^{f(\theta)})e^{-Y(\theta)}\\ &\quad= Ae^{\hat{f}(0)-(\mathcal{T}_Nf)(\theta)+Y(\theta)}e^{f(\theta)}e^{-Y(\theta)}\\ &\quad= A [e^{\hat{f}(0)}+\mathcal{O}(f(\theta)-(\mathcal{T}_Nf)(\theta)+f(\theta)Y(\theta)) ]\\ &\quad= Ae^{\hat{f}(0)} [\mathrm{Id}+e^{-\hat{f}(0)} \mathcal{O}((\mathcal{R}_Nf)(\theta)+f(\theta)Y(\theta)) ]\\ &\quad= A_+e^{f_+(\theta)}. \end{align*} $$
Here,
$\mathcal {T}_N$
is the truncation operator such that
$$ \begin{align*} (\mathcal{T}_Nf)(\theta)=\sum_{n\in \mathbb{Z}^{d},\lvert n \rvert \leqslant N}\hat{f}(n)e^{2\pi i\langle n,\theta\rangle} \end{align*} $$
and
$$ \begin{align*} (\mathcal{R}_Nf)(\theta)=\sum_{n\in \mathbb{Z}^{d},\lvert n \rvert>N}\hat{f}(n)e^{2\pi i\langle n,\theta\rangle}. \end{align*} $$
Therefore, we define
$$ \begin{align*} \begin{cases} B(\theta)=e^{Y(\theta)}, \\ A_+=Ae^{\hat{f}(0)}, \\ f_+=\mathcal{O}((\mathcal{R}_Nf)(\theta)+f(\theta)Y(\theta)). \end{cases} \end{align*} $$
Note that although we write
$D>{2}/{\sigma }$
, that is,
$\sigma>{2}/{D}$
, we consider D to be very close to
${2}/{\sigma }$
in the actual process, such as
${7}/{D}>3\sigma >{6}/{D}$
. Now we have estimates
$$ \begin{align*} &\lvert g \rvert _r \leqslant \sum_{n\in \mathbb{Z}^{d},0<\lvert n \rvert \leqslant N} \lvert \hat{f}(n)e^{2\pi i\langle n,\theta\rangle} \rvert \leqslant c\epsilon N^{d-1} < \epsilon N^{\tau-1} < \epsilon^{1-{1}/{D}},\\&\lvert Y \rvert _{r'} \leqslant c\epsilon^{-3\sigma}\lvert g \rvert _r < \epsilon^{-{7}/{D}} \cdot \epsilon^{1-{1}/{D}} < \epsilon^{1-{8}/{D}},\\&\lvert B-\mathrm{Id} \rvert _{r'} \leqslant \lvert Y \rvert _{r'} < \epsilon^{1-{8}/{D}}. \end{align*} $$
$$ \begin{align*} \lvert (\mathcal{R}_Nf)(\theta)\rvert_{r'} & \leqslant \sum_{n\in \mathbb{Z}^{d},\lvert n \rvert>N}\lvert \hat{f}(n)e^{2\pi i\langle n,\theta\rangle}\rvert_{r'}\\& \leqslant c\lvert f \rvert _r e^{-N(r-r')}\bigg(N+\frac{1}{r-r'}\bigg)^{d}\\& \leqslant c\epsilon e^{-2\log ({1}/{\epsilon})}(N)^{d}\\& < \epsilon \cdot \epsilon^{2}\cdot \epsilon^{-{1}/{D}}\\& <\epsilon^{3-{1}/{D}}. \end{align*} $$
$$ \begin{align*} &\lvert f_+ \rvert _{r'} \leqslant \lvert fY \rvert _{r'} + \lvert (\mathcal{R}_Nf)\rvert_{r'} \leqslant c\epsilon \cdot \epsilon^{1-{8}/{D}} + \epsilon^{3-{1}/{D}} < \epsilon^{2-{8}/{D}}.\\&\lVert A_+-A\rVert\leqslant \lVert A\rVert \lVert \mathrm{Id}-e^{\hat{f}(0)} \rVert \leqslant 2\lVert A\rVert \epsilon. \end{align*} $$
This finishes the proof of Proposition 3.3.
Remark 3.4. This version of analytic KAM theorem is a perfect combination of Eliasson [Reference Eliasson23] and Hou and You [Reference Hou and You26]. While the resonant case which absorbs the essence of Hou and You stays the same, the refined non-resonant case avoids eliminating irrelevant non-resonant terms via Eliasson’s way compared with Cai [Reference Cai10]. This essentially reduces the norm of conjugation maps, which is later vital for us to ensure that the final loss of regularity is independent of the initial k for our
$C^k$
reducibility theorem. Note that this is not at all considered by Cai [Reference Cai10] as almost reducibility does not really care about the convergence of the conjugation maps, but reducibility does.
3.3.
$C^k$
reducibility
As planned, we will present the quantitative
$C^k$
reducibility via analytic approximation [Reference Zehnder40].
The crucial improvement here compared with [Reference Cai10] lies at the point where we are able to obtain reducibility results in a KAM step. In fact, when the resonant steps are well separated from each other, it allows for improved control over the conjugation maps during the inductive argument. Then, after we apply certain conditions to the rotation number of cocycle
$(\alpha , Ae^{f(\theta )})$
, we can find that there are only finite resonant steps in the whole KAM iteration process. This, in return, will give us the best possible initial regularity k through the technique.
Before we prove the main theorem, we will first cite [Reference Cai10, Proposition 3.1] to simplify the main proof. We first recall some notation given in [Reference Cai10].
Let
$\{f_{j}\}_{j\geqslant 1}$
,
$f_{j}\in C_{{1}/{j}}^{\omega }(\mathbb {T}^{d},sl(2,\mathbb {R}))$
be the analytic sequence approximating
$f\in C^k(\mathbb {T}^{d},sl(2,\mathbb {R}))$
which satisfies equation (8).
For
$0<r'<r$
, denote
$$ \begin{align} \epsilon_0'(r,r')=\frac{c}{(2\lVert A\rVert)^{\tilde{D}}}(r-r')^{D\tau} \end{align} $$
and
$$ \begin{align} \epsilon_m=\frac{c}{(2\lVert A\rVert)^{\tilde{D}}m^{D\tau+{1}/{2}}},\quad m\in\mathbb{Z}^+, \end{align} $$
where c depends on
$\kappa , \tau , d$
, and
$D,\tilde {D} \in \mathbb {Z}$
depend on
$\sigma $
.
Then for any
$0<s\leqslant {1}/({6D\tau +3})$
fixed, there exists
$m_0$
such that for any
$m\geqslant m_0$
, we can get
$$ \begin{align} \frac{c}{(2\lVert A\rVert)^{\tilde{D}}m^{D\tau+{1}/{2}}}\leqslant \epsilon_0'\bigg( \frac{1}{m},\frac{1}{m^{1+s}}\bigg). \end{align} $$
We will begin with
$M> \max \{{(2\lVert A\rVert )^{\tilde {D}}}/{c},m_0\}$
,
$M\in \mathbb {N}^+$
. Let
$l_j=M^{(1+s)^{j-1}}$
,
$j\in \mathbb {N}^+$
. Because
$l_j$
is not an integer, we pick
$[l_j]+1$
instead of
$l_j$
.
Now, let
$\Omega =\{n_1,n_2,n_3,\ldots \}$
denote the sequence of all resonant steps. In other words, the
$(n_j)$
th step is obtained by the resonant case. By the analytic approximation in equation (8) and Proposition 3.3 in each iteration step, we can establish the following almost reducibility result concerning each
$(\alpha , Ae^{f_{l_j}(\theta )})$
by applying induction.
Proposition 3.5. [Reference Cai10, Reference Cai and Ge12]
Let
$\alpha \in \mathrm {DC}_d(\kappa ,\tau )$
,
$\sigma <\tfrac 16$
. Assume that
$A\in SL(2,\mathbb {R})$
satisfying A bounded,
$f\in C^k(\mathbb {T}^d, sl(2,\mathbb {R}))$
with
$k>(D+2)\tau +2$
and
$\{f_j\}_{j\geqslant 1}$
be as in §2.4. There exists
$\bar \epsilon =\bar \epsilon (\kappa ,\tau ,d,k, \lVert A \rVert ,\sigma )$
such that if
$$ \begin{align} \lVert f\rVert_k \leqslant \bar \epsilon \leqslant \epsilon_0'\bigg( \frac{1}{l_1},\frac{1}{l_2}\bigg), \end{align} $$
then there exist
$B_{l_j}\in C^{\omega }_{{1}/{l_{j+1}}}(2\mathbb {T}^d, SL(2,\mathbb {R}))$
,
$A_{l_j}\in SL(2,\mathbb {R})$
, and
$f^{'}_{l_j}\in C^{\omega }_{{1}/{l_{j+1}}}(\mathbb {T}^d, sl(2, \mathbb {R}))$
such that
$$ \begin{align} B_{l_j}(\theta+\alpha)(Ae^{f_{l_j}(\theta)})B^{-1}_{l_j}(\theta)=A_{l_j}e^{f_{l_j}^{'}(\theta)}, \end{align} $$
with estimates
$$ \begin{align} \lvert B_{l_j}(\theta)\rvert_{{1}/{l_{j+1}}} &\leqslant 64\bigg(\frac{\lVert A\rVert}{\kappa}\bigg) \bigg( \frac{2}{({1}/{l_j})-({1}/{l_{j+1}})} \ln {\frac{1}{\epsilon_{l_j}}}\bigg) ^{\tau}\nonumber\\&\quad\times{\epsilon_{l_j}}^{-({{2}/{l_{j+1}}})/({({1}/{l_j})-({1}/{l_{j+1}})})} \leqslant \epsilon_{l_{j}}^{-{\sigma}/{2}-s}, \end{align} $$
$$ \begin{align} \lVert B_{l_j}(\theta)\rVert_0 &\leqslant 64\bigg( \frac{\lVert A\rVert}{\kappa}\bigg) \bigg( \frac{2}{({1}/{l_j})-({1}/{l_{j+1}})} \ln {\frac{1}{\epsilon_{l_j}}}\bigg) ^{\tau} \leqslant \epsilon_{l_{j}}^{-{\sigma}/{2}}, \end{align} $$
$$ \begin{align} \lvert \deg B_{l_j}\rvert &\leqslant 4l_j \ln {\frac{1}{\epsilon_{l_j}}}, \end{align} $$
$$ \begin{align} \lvert f_{l_j}^{'}(\theta)\rvert_{{1}/{l_{j+1}}} &\leqslant \epsilon_{l_{j}}^{2-{8}/{D}},\quad \lVert A_{l_j}\rVert\leqslant 2\lVert A\rVert. \end{align} $$
Moreover, there exists
$\overline f_{l_j}\in C^{k_0}(\mathbb {T}^d, sl(2,\mathbb {R}))$
with
$k_0 \in \mathbb {N}, k_0 \leqslant ({k-10\tau -3})/({1+s})$
such that
$$ \begin{align} B_{l_j}(\theta+\alpha)(Ae^{f(\theta)})B^{-1}_{l_j}(\theta)=A_{l_j}e^{\overline f_{l_j}(\theta)}, \end{align} $$
with estimate
$$ \begin{align} \lVert \overline f_{l_j}(\theta)\rVert_{k_0}\leqslant \epsilon_{l_{j}}^{{3}/{D}}. \end{align} $$
Remark 3.6. By the new analytic KAM scheme and the original proof method in [Reference Cai10, Reference Cai and Ge12], we can also get the results of the proposition. The fact that the existing estimates in equations (23) and (25) are more delicate than the original is also due to the use of the new
$C^\omega $
KAM theorem.
Remark 3.7. The estimates of equation (22) can be done better in the non-resonant case because the non-resonant step does not change the degree, thus the estimates of the degree in the non-resonant case can be considered as the estimates of the resonant step closest to this step.
With Proposition 3.5 in hand, we are going to transfer all the estimates from
$(\alpha , Ae^{f_{l_j}(\theta )})$
to
$(\alpha , Ae^{f(\theta )})$
through analytic approximation. We establish the following quantitative
$C^k$
reducibility theorem.
Theorem 3.8. Let
$\alpha \in \mathrm {DC}_d(\kappa ,\tau )$
,
$\sigma <\tfrac 16$
,
$A\in SL(2,\mathbb {R})$
, and
$f\in C^k(\mathbb {T}^{d}, sl(2,\mathbb {R}))$
with
$k>(D+2)\tau +2$
. Then there exists
$\epsilon _0=\epsilon _0(\kappa ,\tau ,d,k,\lVert A\rVert ,\sigma )$
such that if
$$ \begin{align} \lVert f\rVert_k \leqslant \epsilon_0 \leqslant \epsilon_0'\bigg( \frac{1}{l_1},\frac{1}{l_2}\bigg) \end{align} $$
and:
-
• if
$\rho (\alpha , Ae^f\!)$
is Diophantine with respect to
$\alpha $
:
$\rho (\alpha ,Ae^f\!) \in \mathrm {DC}_d ^\alpha (\gamma ,\tau )$
, then there exists two constants
$C_1=C_1(\gamma ,\kappa ,\tau ,d,k,\|A\|,\sigma )$
,
$C_2=C_2(\gamma ,\kappa ,\tau ,d,k, \|A\|,\sigma )$
and
$B_1\in C^{k_0}(2\mathbb {T}^d, SL(2,\mathbb {R}))$
with
$k_0 \in \mathbb {N}$
,
$k_0\leqslant ({k-10\tau -3})/({1+s})$
such that (27)with estimates
$$ \begin{align} B_1(\theta+\alpha)(Ae^{f(\theta)})B^{-1}_1(\theta)=R_\phi \in SL(2,\mathbb{R}), \quad \phi \notin \mathbb{Z}, \end{align} $$
(28)
$$ \begin{align} \lVert B_1 \rVert_{k_0} \leqslant C_1, \quad \lvert\deg B_1 \rvert \leqslant C_2; \end{align} $$
-
• if
$\rho (\alpha ,Ae^f\!)$
is rational with respect to
$\alpha $
:
$2\rho (\alpha ,Ae^f\!) =\langle m_0,\alpha \rangle \,\mod \mathbb {Z}$
for some
$m_0\in \mathbb {Z}^d$
, then there exists
$B_2\in C^{k_0}(2\mathbb {T}^d, SL(2,\mathbb {R}))$
with
$k_0 \in \mathbb {N}$
,
$k_0\leqslant ({k-10\tau -3})/({1+s})$
such that (29)with
$$ \begin{align} B_2(\theta+\alpha)(Ae^{f(\theta)})B^{-1}_2(\theta)=\tilde A_2 \in SL(2,\mathbb{R}), \end{align} $$
(30)
$$ \begin{align} \rho(\alpha, \tilde A_2)=0. \end{align} $$
Remark 3.9. In the inequality
$k_0\leqslant ({k-10\tau -3})/({1+s})$
, due to the precise choice of
$k, \tau , s$
, we can also pick
$k_0 = [k-10\tau -3]$
. Here, ‘
$[x]$
’ stands for the integer part of x.
Proof. (Diophantine case) By equation (26) and Proposition 3.5, take
$\epsilon _0 = \bar \epsilon $
as in Proposition 3.5 and apply it to cocycle
$(\alpha ,Ae^{f(\theta )})$
. Then there exists
$B_{l_j}\in C^{\omega }_{{1}/{l_{j+1}}}(2\mathbb {T}^d, SL(2, \mathbb {R}))$
,
$A_{l_j}\in SL(2,\mathbb {R})$
, and
$\overline f_{l_j}\in C^{k_0}(\mathbb {T}^d, sl(2,\mathbb {R}))$
with
$k_0 \in \mathbb {N}$
,
$k_0\leqslant ({k-10\tau -3})/({1+s})$
such that
$$ \begin{align} B_{l_j}(\theta+\alpha)(Ae^{f(\theta)})B^{-1}_{l_j}(\theta)=A_{l_j}e^{\overline f_{l_j}(\theta)}, \end{align} $$
with estimates
$$ \begin{align} \lvert B_{l_j}(\theta)\rvert_{{1}/{l_{j+1}}}&\leqslant \epsilon_{l_{j}}^{-{\sigma}/{2}-s}, \quad \lVert B_{l_j}(\theta)\rVert_0\leqslant \epsilon_{l_{j}}^{-{\sigma}/{2}}, \quad \lvert\deg B_{l_j}\rvert \leqslant 4l_j \ln \frac{1}{\epsilon_{l_{j}}}, \end{align} $$
$$ \begin{align} \lVert A_{l_j} \rVert &\leqslant 2\lVert A \rVert,\quad \lVert \overline f_{l_j}(\theta) \rVert_{k_0}\leqslant \epsilon_{l_{j}}^{{3}/{D}}. \quad \end{align} $$
In the last part of the analytic approximation in equation (8), taking a telescoping sum from j to
$+\infty $
, we get
$$ \begin{align} \lVert f(\theta)-f_{l_j}(\theta)\rVert_0\leqslant \frac{c}{(2\lVert A\rVert)^{\tilde{D}}l_1^{D\tau+{1}/{2}}l_{j}^{k-1}}, \end{align} $$
$$ \begin{align} \lVert f(\theta)\rVert_0+\lVert f_{l_j}(\theta)\rVert_0\leqslant \frac{c}{(2\lVert A\rVert)^{\tilde{D}}M^{D\tau+{1}/{2}}}+\frac{\tilde c}{(2\lVert A\rVert)^{\tilde{D}}M^{D\tau+{1}/{2}}}. \end{align} $$
Thus, by equations (31)–(35), we have
$$ \begin{align} \lVert \overline f_{l_j}(\theta)\rVert_0 &\leqslant \lVert f_{l_j}^{'}(\theta)\rVert_0 + \lVert A^{-1}_{l_j}\rVert \lVert B_{l_j}(\theta+\alpha)(Ae^{f(\theta)}-Ae^{f_{l_j}(\theta)})B^{-1}_{l_j}(\theta)\rVert_0 \nonumber\\ &\leqslant \epsilon_{l_j}^{2-{8}/{D}}+2\lVert A\rVert \times \epsilon_{l_j}^{-\sigma} \times \frac{c}{(2\lVert A\rVert)^{\tilde{D}}l_1^{D \tau + {1}/{2}}l_j^{k-1}} \\ &\leqslant \epsilon_{l_j}^{1+s}.\nonumber \end{align} $$
Since
$\rho (\alpha ,Ae^f\!) \in \mathrm {DC}_d ^\alpha (\gamma ,\tau )$
, for any
$m \in \mathbb {Z}^d$
, we have
$$ \begin{align*} &\lVert 2\rho(\alpha,A_{l_j}e^{\overline f_{l_j}(\theta)})-\langle m,\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}} \\&\quad= \lVert 2\rho(\alpha,Ae^{f(\theta)}) - \langle \deg B_{l_j}, \alpha \rangle -\langle m,\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}} \\&\quad\geqslant \frac{\gamma}{(|m+\deg B_{l_j}|+1)^\tau} \\&\quad\geqslant \frac{\gamma (1+\lvert\deg B_{l_j}\rvert)^{-\tau}}{(|m|+1)^\tau}, \end{align*} $$
which implies
$\rho (\alpha ,A_{l_j}e^{\overline f_{l_j}(\theta )}) \in \mathrm {DC}_d ^\alpha (\gamma (1+\lvert \deg B_{l_j}\rvert )^{-\tau },\tau )$
.
By equation (32), we can obtain
$$ \begin{align*} \epsilon_{l_j}^{({(1+s) \sigma})/{2}} (1+\lvert\deg B_{l_j}\rvert)^\tau \leqslant \bigg[ \frac{c}{(2\lVert A\rVert)^{\tilde{D}}{l_j}^{D\tau+{1}/{2}}} \bigg]^{({(1+s) \sigma})/{2}} \bigg( 1+4l_j \ln \frac{1}{\epsilon_{l_j}}\bigg) ^\tau. \end{align*} $$
Obviously, let
$j \to \infty $
and then the right-hand side of the inequality goes to zero. Therefore, for any given
$\gamma> 0$
, there exists
$j'=j'(\gamma ) \in \mathbb {N}$
such that
$$ \begin{align*} \epsilon_{l_{j'}}^{({(1+s) \sigma})/{2}} (1+\lvert\deg B_{l_{j'}}\rvert)^\tau \leqslant \gamma \end{align*} $$
and then
$$ \begin{align} \gamma (1+\lvert\deg B_{l_{j'}}\rvert)^{-\tau} \geqslant \epsilon_{l_{j'}}^{({(1+s) \sigma})/{2}} = \epsilon_{l_{j'+1}}^{{\sigma}/{2}}. \end{align} $$
However, for
$0<|m| \leqslant N_{l_{j'+1}}=({2}/({{1}/{l_{j'+1}}-{1}/{l_{j'+2}}})) \lvert \ln \epsilon _{l_{j'+1}} \rvert $
, we have
$$ \begin{align} \frac{1}{(|m|+1)^\tau} \geqslant \frac{1}{(({2}/({{1}/{l_{j'+1}}-{1}/{l_{j'+2}}})) \ln ({1}/{\epsilon_{l_{j'+1}}})+1)^ \tau} \geqslant 2\epsilon_{l_{j'+1}}^{{\sigma}/{2}}. \end{align} $$
Then by equations (37) and (38), we can get
$$ \begin{align*} &\lVert 2\rho(\alpha,A_{l_{j'}})-\langle m,\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}} \\ &\quad\geqslant \lVert 2\rho(\alpha,A_{l_{j'}}e^{\overline f_{l_{j'}}(\theta)})-\langle m,\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}}-| 2\rho(\alpha,A_{l_{j'}}e^{\overline f_{l_{j'}}(\theta)})-2\rho(\alpha,A_{l_{j'}})| \\ &\quad\geqslant \frac{\gamma (1+\lvert\deg B_{l_{j'}}\rvert)^{-\tau}}{(|m|+1)^\tau}-2\epsilon_{l_{j'}}^{({1+s})/{2}} \\ &\quad\geqslant \epsilon_{l_{j'+1}}^{{\sigma}/{2}} \cdot 2\epsilon_{l_{j'+1}}^{{\sigma}/{2}} - 2\epsilon_{l_{j'+1}}^{{1}/{2}} \\ &\quad\geqslant \epsilon_{l_{j'+1}}^\sigma, \end{align*} $$
which means the
$(j'+1)$
th step is non-resonant with
$$ \begin{align*} B_{l_{j'+1}}=\widetilde{B}_{l_{j'}} \circ B_{l_{j'}}, \quad \lvert \widetilde B_{l_{j'}}(\theta)- \mathrm{Id}\rvert_{{1}/{l_{j'+2}}} \leqslant \epsilon_{l_{j'+1}}^{1-{8}/{D}}, \quad \deg B_{l_{j'+1}}=\deg B_{l_{j'}}. \end{align*} $$
Assume that for
$l_n$
,
$j'+1 \leqslant n \leqslant j_0$
, we have
$$ \begin{align*} B_{l_n}(\theta+\alpha)(Ae^{f_{l_{n}}(\theta)})B^{-1}_{l_n}(\theta)=A_{l_n}e^{f_{l_n}'(\theta)}, \end{align*} $$
which is equivalent to
$$ \begin{align*} B_{l_n}(\theta+\alpha)(Ae^{f(\theta)})B^{-1}_{l_n}(\theta)=A_{l_n}e^{f_{l_n}'(\theta)}+B_{l_n}(\theta+\alpha)(Ae^{f(\theta)}-Ae^{f_{l_{n}(\theta)}})B^{-1}_{l_n}(\theta), \end{align*} $$
rewrite that
$$ \begin{align} B_{l_n}(\theta+\alpha)(Ae^{f(\theta)})B^{-1}_{l_n}(\theta)=A_{l_n}e^{\overline f_{l_n}(\theta)}, \end{align} $$
with estimates
$$ \begin{align} B_{l_{n}}=\widetilde{B}_{l_{n-1}} \circ B_{l_{n-1}}, \quad \lvert \widetilde B_{l_{n-1}}(\theta)- \mathrm{Id}\rvert_{{1}/{l_{n+1}}} \leqslant \epsilon_{l_n}^{1-{8}/{D}}, \quad \deg B_{l_{n-1}}=\deg B_{l_{j'}}. \end{align} $$
Therefore, by equation (40) and the Diophantine condition on
$\rho (\alpha ,Ae^f\!)$
, for
$0<|m| \leqslant ({2}/{({1}/{l_{n+1}}-{1}/{l_{n+2}})}) \lvert \ln \epsilon _{l_{n+1}} \!\rvert $
, we obtain
$$ \begin{align*} &\lVert 2\rho(\alpha,A_{l_n})-\langle m,\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}} \\ &\quad\geqslant \lVert 2\rho(\alpha,A_{l_n}e^{\overline f_{l_n}(\theta)})-\langle m,\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}}-| 2\rho(\alpha,A_{l_n}e^{\overline f_{l_n}(\theta)})-2\rho(\alpha,A_{l_n})| \\ &\quad\geqslant \frac{\gamma (1+\lvert\deg B_{l_{j'}}\rvert)^{-\tau}}{(|m|+1)^\tau}-2\epsilon_{l_n}^{({1+s})/{2}} \\ &\quad\geqslant \epsilon_{l_{j'+1}}^{{\sigma}/{2}} \cdot 2\epsilon_{l_{n+1}}^{{\sigma}/{2}} - 2\epsilon_{l_{n+1}}^{{1}/{2}} \\ &\quad\geqslant \epsilon_{l_{n+1}}^\sigma \quad \text{ for all } j'+1 \leqslant n \leqslant j_0. \end{align*} $$
This means the
$(j_0 + 1)$
th step is still non-resonant with estimates
$$ \begin{align*} B_{l_{j_0 + 1}}=\widetilde{B}_{l_{j_0}} \circ B_{l_{j_0}}, \quad \lvert \widetilde B_{l_{j_0}}(\theta)- \mathrm{Id}\rvert_{{1}/{l_{j_0 + 2}}} \leqslant \epsilon_{l_{j_0 + 1}}^{1-{8}/{D}}, \quad \deg B_{l_{j_0}}=\deg B_{l_{j'}}. \end{align*} $$
In conclusion, we know that there are at most finitely many resonant steps in the iteration process. Assume that
$n_q$
is the last resonant step, then
$\text {for all } j> n_q$
, we have
$$ \begin{align*} B_{l_j}(\theta+\alpha)(Ae^{f_{l_{j}}(\theta)})B^{-1}_{l_j}(\theta)=A_{l_j}e^{f_{l_j}^{'}(\theta)}, \end{align*} $$
with estimates
$$ \begin{align} B_{l_{j}}=\widetilde{B}_{l_{j-1}} \circ B_{l_{j-1}}, \quad \lvert \widetilde B_{l_{j-1}}(\theta)- \mathrm{Id}\rvert_{{1}/{l_{j+1}}} \leqslant \epsilon_{l_j}^{1-{8}/{D}}, \quad \deg B_{l_{j-1}}=\deg B_{l_{n_q}}. \end{align} $$
Denote
$B_1=\lim \nolimits _{j\to \infty }B_{l_j}$
,
$\tilde A_1=\lim \nolimits _{j\to \infty }A_{l_j} \in SL(2,\mathbb {R})$
. Notice that
$\rho (\alpha , \tilde A_1) \neq 0$
, otherwise it will contradict
$\rho (\alpha ,Ae^f\!) \in \mathrm {DC}_d ^\alpha (\gamma ,\tau )$
. Thus,
$\tilde A_1$
can only be standard rotation in
$SL(2,\mathbb {R})$
, which is the case of equation (27).
By equation (41) and Cauchy estimates,
$\text {for all } j> n_q$
, we can calculate
$$ \begin{align*} \lVert \widetilde B_{l_{j}} - \mathrm{Id} \rVert_{k_0} &\leqslant \sup_{\substack{\lvert l\rvert\leqslant k_0, \theta \in \mathbb{T}^{d}}}\lVert (\partial^{l_1}_{\theta_1} \cdots \partial^{l_d}_{\theta_d}) (\widetilde B_{l_{j}} - \mathrm{Id}) \rVert \\ &\leqslant (k_0)!(l_{j+2})^{k_0}\lvert \widetilde B_{l_{j}} - \mathrm{Id} \rvert_{{1}/{l_{j+2}}} \\ &\leqslant (k_0)!(l_{j+1})^{(1+s)k_0} \epsilon_{l_{j+1}}^{1-{8}/{D}} \\ &\leqslant \frac{C}{l_{j+1}^{(1-{8}/{D})(D\tau+{1}/{2})-(1+s)k_0}}, \end{align*} $$
where C does not depend on j.
Note that while we write the assumption as
$k>(D+2)\tau +2$
, in the actual operational process, we simply choose
$k=[(D+2)\tau +2]+1$
. Therefore, the value of k is entirely determined by the parameter D.
Thus, if we pick
$k_0 = [k-10\tau -3]$
, we have
$$ \begin{align*} \lVert \widetilde B_{l_{j}} \rVert_{k_0} \leqslant 1+ \frac{C}{l_{j+1}^{{1}/{6}}}. \end{align*} $$
Since we pick
$l_1=M$
sufficiently large, then
$$ \begin{align*} \lVert B_1 \rVert_{k_0} &\leqslant \lVert \prod\limits_{j=n_q+1}^{\infty} \widetilde B_{l_{j}} \rVert_{k_0} \lVert B_{l_{n_q}} \rVert_{k_0} \\ &\leqslant \bigg( \prod\limits_{j=n_q+1}^{\infty} \bigg( 1+ \frac{C}{l_{j+1}^{{1}/{6}}}\bigg) \bigg) \sup_{\substack{\lvert l\rvert\leqslant k_0, \theta \in \mathbb{T}^{d}}}\lVert (\partial^{l_1}_{\theta_1} \cdots \partial^{l_d}_{\theta_d}) (B_{l_{n_q}}(\theta)) \rVert \\ &\leqslant 2(k_0)!(l_{n_q+1})^{k_0}\lvert B_{l_{n_q}}\rvert_{{1}/{l_{n_q+1}}} \\ &\leqslant {l_{n_q}}^{({\sigma}/{2}+s)(D\tau+{1}/{2})+(1+s)k_0}. \end{align*} $$
The estimate of
$\deg B_1$
is clearly valid, which gives equation (28).
(Rational case) By equation (26) and Proposition 3.5, take
$\epsilon _0 = \bar \epsilon $
as in Proposition 3.5 and apply it to cocycle
$(\alpha ,Ae^{f(\theta )})$
. Then there exists
$B_{l_j}\in C^{\omega }_{{1}/{l_{j+1}}}(2\mathbb {T}^d, SL(2,\mathbb {R}))$
,
$A_{l_j}\in SL(2,\mathbb {R})$
, and
$\overline f_{l_j}\in C^{k_0}(\mathbb {T}^d, sl(2,\mathbb {R}))$
with
$k_0 \in \mathbb {N}$
,
$k_0\leqslant ({k-10\tau -3})/({1+s})$
such that
$$ \begin{align} B_{l_j}(\theta+\alpha)(Ae^{f(\theta)})B^{-1}_{l_j}(\theta)=A_{l_j}e^{\overline f_{l_j}(\theta)}, \end{align} $$
with estimates
$$ \begin{align} \lvert B_{l_j}(\theta)\rvert_{{1}/{l_{j+1}}}&\leqslant \epsilon_{l_{j}}^{-{\sigma}/{2}-s}, \quad \lVert B_{l_j}(\theta)\rVert_0\leqslant \epsilon_{l_{j}}^{-{\sigma}/{2}}, \quad \lvert\deg B_{l_j}\rvert \leqslant 4l_j \ln \frac{1}{\epsilon_{l_{j}}}, \end{align} $$
$$ \begin{align} \lVert A_{l_j} \rVert &\leqslant 2\lVert A \rVert,\quad \lVert \overline f_{l_j}(\theta) \rVert_{k_0}\leqslant \epsilon_{l_{j}}^{{3}/{D}}. \end{align} $$
Since
$\rho (\alpha ,Ae^f\!) = {\langle m_0,\alpha \rangle }/{2}\, \mod {\mathbb {Z}}/{2}$
,
$m_0\in \mathbb {Z}^d$
, we have
$$ \begin{align*} \rho(\alpha,A_{l_j}e^{\overline f_{l_j}(\theta)}) & = \rho(\alpha,Ae^{f(\theta)}) - \frac{\langle \deg B_{l_j},\alpha \rangle}{2} \quad\mod \frac{\mathbb{Z}}{2} \\ & = \frac{\langle m_0 - \deg B_{l_j},\alpha \rangle}{2} \quad\mod \frac{\mathbb{Z}}{2}. \end{align*} $$
From now on, we omit ‘
$\mod ({\mathbb {Z}}/{2})$
’ for simplicity. By equation (6) and the proof of Proposition 3.5, we know the rotation number is invariant in the non-resonant case.
Let
$m_{l_{n_i}} \in \mathbb {Z}^d, i=1,2,3, \ldots $
represent resonant sites of the
$(n_i)$
th step with
$0<|m_{l_{n_i}}| \leqslant N_{l_{n_i}}={2}/{({1}/{l_{n_i}})-({1}/{l_{n_i+1}})} \lvert \ln \epsilon _{l_{n_i}}\!\rvert $
. By in [Reference Cai10, Claim 1], we have
$N_{l_{n_{i+1}}} \gg 4N_{l_{n_i}}, i=1,2,3, \ldots $
. So there must exist
$j \in \mathbb {Z}$
sufficiently large, provided that there are
$q-1$
resonant steps before the jth step, such that
$$ \begin{align*} m_0 - (m_{l_{n_1}}+m_{l_{n_2}}+ \cdots +m_{l_{n_{q-1}}}) = m', \end{align*} $$
where
$m' \in \mathbb {Z}^d$
with
$0<|m'| \leqslant N_{l_j}=({2}/{({1}/{l_{j}}-{1}/{l_{j+1}})}) \lvert \ln \epsilon _{l_{j}}\!\rvert $
and
$N_{l_j} \gg 2N_{l_{n_{q-1}}} \gg m_{l_{n_1}}+m_{l_{n_2}}+ \cdots +m_{l_{n_{q-1}}}$
. Then,
$$ \begin{align} \rho(\alpha,A_{l_j}e^{\overline f_{l_j}(\theta)}) &= \frac{\langle m_0 - \deg B_{l_j},\alpha \rangle}{2} \nonumber\\ &= \frac{\langle m_0 - (m_{l_{n_1}}+m_{l_{n_2}}+ \cdots +m_{l_{n_{q-1}}}+m'),\alpha \rangle}{2} \\ &= 0.\nonumber \end{align} $$
Since all the steps between the
$(n_{q-1})$
th step and jth step are non-resonant and by equations (6), (36), and (45), we have
$$ \begin{align*} & \lVert 2\rho(\alpha,A_{l_{j-1}})-\langle m',\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}} \\ &\quad\leqslant \lVert 2\rho(\alpha,A_{l_{j-1}}e^{\overline f_{l_{j-1}}(\theta)})-\langle m',\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}} + |2\rho(\alpha,A_{l_{j-1}}e^{\overline f_{l_{j-1}}(\theta)})-2\rho(\alpha,A_{l_{j-1}})| \\ &\quad\leqslant |2\rho(\alpha,A_{l_{j}}e^{\overline f_{l_j}(\theta)})| + 2\epsilon_{l_{j-1}}^{({1+s})/{2}} \\ &\quad= 0 + 2\epsilon_{l_{j}}^{{1}/{2}} \\ &\quad< \epsilon_{l_{j}}^\sigma. \end{align*} $$
Thus, the jth step is the
$(n_q)$
th resonant step and
$m'$
is the unique resonant site
$m_{l_{n_q}}$
with
$0<|m_{l_{n_q}}| \leqslant N_{l_{n_q}}=({2}/{({1}/{l_{n_q}}-{1}/{l_{n_q+1}})}) \lvert \ln \epsilon _{l_{n_q}}\!\rvert $
.
Now we apply Proposition 3.5 to cocycle
$(\alpha ,A_{l_{n_q}}e^{\widetilde f_{l_{n_q}}(\theta )})$
, then we can get
$$ \begin{align*} \widetilde{B}_{l_{n_q}}(\theta+\alpha)(A_{l_{n_q}}e^{\widetilde f_{l_{n_q}}(\theta)})\widetilde{B}_{l_{n_q}}^{-1}(\theta)=A_{l_{{n_q}+1}}e^{f_{l_{{n_q}+1}}^{'}(\theta)}, \end{align*} $$
which gives
$$ \begin{align*} {B}_{l_{{n_q}+1}}(\theta+\alpha)(Ae^{{f_{l_{{n_q}+1}}}(\theta)}){B}_{l_{{n_q}+1}}^{-1}(\theta)=A_{l_{{n_q}+1}}e^{f_{l_{{n_q}+1}}^{'}(\theta)}, \end{align*} $$
where
$B_{l_{n_q+1}}=\widetilde {B}_{l_{n_q}} \circ B_{l_{n_q}} \in C^{\omega }_{{1}/{l_{n_q+2}}}(2\mathbb {T}^{d},SL(2,\mathbb {R}))$
.
Since
$\rho (\alpha ,A_{l_{n_q}}e^{\overline f_{l_{n_q}}(\theta )}) = \rho (\alpha ,Ae^{\overline f_{l_{n_q}}(\theta )}) = 0$
, for
$m \in \mathbb {Z}^d$
with
$0<|m| \leqslant N_{l_{n_q+1}}=({2}/{({1}/{l_{n_q+1}}-{1}/{l_{n_q+2}})}) \lvert \ln \epsilon _{l_{n_q+1}}\rvert $
, we have
$$ \begin{align*} & \lVert 2\rho(\alpha,A_{l_{n_q}})-\langle m,\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}} \\ &\quad\geqslant \lVert 2\rho(\alpha,A_{l_{n_q}}e^{\overline f_{l_{n_q}}(\theta)})-\langle m,\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}} - |2\rho(\alpha,A_{l_{n_q}}e^{\overline f_{l_{n_q}}(\theta)})-2\rho(\alpha,A_{l_{n_q}})| \\ &\quad\geqslant \frac{\kappa}{|m|^ \tau} - 2\epsilon_{l_{n_q}}^{({1+s})/{2}} \\ &\quad\geqslant \frac{\kappa}{(( {2}/{({1}/{l_{n_q+1}}-{1}/{l_{n_q+2}})}) \ln ({1}/{\epsilon_{l_{n_q+1}}})) ^\tau} - 2\epsilon_{l_{n_q+1}}^{{1}/{2}} \\ &\quad\geqslant 2\epsilon_{l_{n_q+1}}^{{\sigma}/{2}} - 2\epsilon_{l_{n_q+1}}^{{1}/{2}} \\ &\quad\geqslant \epsilon_{l_{n_q+1}}^\sigma, \end{align*} $$
which means the
$(n_q+1)$
th step is non-resonant with
$$ \begin{align*} \deg B_{l_{n_q+1}} = \deg \widetilde B_{l_{n_q}} + \deg B_{l_{n_q}} = m_0. \end{align*} $$
Assume that for
$l_j$
,
$n_q+1 \leqslant j \leqslant j_0$
, we already have
$$ \begin{align*} B_{l_j}(\theta+\alpha)(Ae^{f_{l_{j}}(\theta)})B^{-1}_{l_j}(\theta)=A_{l_j}e^{f_{l_j}^{'}(\theta)}, \end{align*} $$
which is equivalent to
$$ \begin{align*} B_{l_j}(\theta+\alpha)(Ae^{f(\theta)})B^{-1}_{l_j}(\theta)=A_{l_j}e^{f_{l_j}^{'}(\theta)}+B_{l_j}(\theta+\alpha)(Ae^{f(\theta)}-Ae^{f_{l_{j}}})B^{-1}_{l_j}(\theta), \end{align*} $$
rewrite that
$$ \begin{align} B_{l_j}(\theta+\alpha)(Ae^{f(\theta)})B^{-1}_{l_j}(\theta)=A_{l_j}e^{\overline f_{l_j}(\theta)}, \end{align} $$
with estimates
$$ \begin{align} B_{l_{j}}=\widetilde{B}_{l_{j-1}} \circ B_{l_{j-1}},\quad \lvert \widetilde B_{l_{j-1}}(\theta)- \mathrm{Id}\rvert_{{1}/{l_{j+1}}} \leqslant \epsilon_{l_j}^{1-{8}/{D}},\quad \deg B_{l_{j-1}}= m_0. \end{align} $$
Note that equation (46) gives
$$ \begin{align*} &\bigg\lvert \rho{(\alpha,Ae^{f(\theta)})} - \frac{\langle \deg B_{l_j}, \alpha \rangle}{2} - \rho{(\alpha,A_{l_j})} \bigg\rvert \\ &\quad= \lvert \rho{(\alpha,A_{l_j}e^{\overline f_{l_j}(\theta)})} - \rho{(\alpha,A_{l_j})} \rvert \\ &\quad\leqslant \epsilon_{l_{j+1}}^{{1}/{2}} \end{align*} $$
and equation (47) implies
$$ \begin{align*} \deg B_{l_j} = \deg B_{l_{n_q}} = m_0. \end{align*} $$
Therefore, we have
$$ \begin{align} \lvert \rho{(\alpha,A_{l_{n_q}}e^{\overline f_{l_{n_q}}(\theta)})} - \rho{(\alpha,A_{l_j})} \rvert \leqslant \epsilon_{l_{j+1}}^{{1}/{2}}. \end{align} $$
By equation (48) and
$\rho (\alpha ,A_{l_{n_q}}e^{\overline f_{l_{n_q}}(\theta )}) = 0$
, for
$m \in \mathbb {Z}^d$
with
$0<|m| \leqslant N_{l_{j+1}}=({2}/({1}/{l_{j+1}}-{1}/{l_{j+2}})) \lvert \ln \epsilon _{l_{j+1}}\rvert $
, we have
$$ \begin{align*} &\lVert 2\rho(\alpha,A_{l_j})-\langle m,\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}} \\ &\quad\geqslant \lVert 2\rho(\alpha,A_{l_{n_q}}e^{\overline f_{l_{n_q}}(\theta)})-\langle m,\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}}-| 2\rho(\alpha,A_{l_{n_q}}e^{\overline f_{l_{n_q}}(\theta)})-2\rho(\alpha,A_{l_j})| \\ &\quad\geqslant \lVert \langle m,\alpha \rangle \rVert_{\mathbb{R} / \mathbb{Z}}-2\epsilon_{l_j}^{({1+s})/{2}} \\ &\quad\geqslant \frac{\kappa}{|m|^ \tau} - 2\epsilon_{l_{j+1}}^{{1}/{2}} \\ &\quad\geqslant \frac{\kappa}{(( {2}/{({1}/{l_{j+1}}-{1}/{l_{j+2}})}) \ln ({1}/{\epsilon_{l_{j+1}}})) ^\tau} - 2\epsilon_{l_{j+1}}^{{1}/{2}} \\ &\quad\geqslant 2\epsilon_{l_{j+1}}^{{\sigma}/{2}} - 2\epsilon_{l_{j+1}}^{{1}/{2}} \\ &\quad> \epsilon_{l_{j+1}}^\sigma \quad \text{for all } n_q+1 \leqslant j \leqslant j_0. \end{align*} $$
This means the
$(j_0 + 1)$
th step is also non-resonant with estimates
$$ \begin{align*} B_{l_{j_0 + 1}}=\widetilde{B}_{l_{j_0}} \circ B_{l_{j_0}}, \quad \lvert \widetilde B_{l_{j_0}}(\theta)- \mathrm{Id}\rvert_{{1}/{l_{j_0 + 2}}} \leqslant \epsilon_{l_{j_0 + 1}}^{1-{8}/{D}}, \quad \deg B_{l_{j_0}}= m_0. \end{align*} $$
To conclude,
$\text {for all } j> n_q$
, we have
$$ \begin{align*} B_{l_j}(\theta+\alpha)(Ae^{f_{l_{j}}(\theta)})B^{-1}_{l_j}(\theta)=A_{l_j}e^{f_{l_j}^{'}(\theta)}, \end{align*} $$
with estimates
$$ \begin{align} B_{l_{j}}=\widetilde{B}_{l_{j-1}} \circ B_{l_{j-1}}, \quad \lvert \widetilde B_{l_{j-1}}(\theta)- \mathrm{Id}\rvert_{{1}/{l_{j+1}}} \leqslant \epsilon_{l_j}^{1-{8}/{D}}, \quad \deg B_{l_{j-1}} = m_0. \end{align} $$
Denote
$B_2=\lim \nolimits _{j\to \infty }B_{l_j}$
,
$\tilde A_2=\lim \nolimits _{j\to \infty }A_{l_j} \in SL(2,\mathbb {R})$
. The rest of the process is similar to the above Diophantine case and then equations (29) and (30) hold.
This finishes our proof.
Remark 3.10. Compared with Cai and Ge [Reference Cai and Ge12], the reducibility results in this paper are certainly optimal. We have conducted a process parallel to the almost reducibility procedure. In contrast to the previous work, the conclusions in this paper are sharper in terms of regularity loss and the loss of regularity is independent of the parameter k.
4. Spectral application
With the above reducibility theorem in hand, we can prove our spectral applications of Theorems 1.3 and 1.6 as stated in the introduction from this. Let us first introduce several definitions and cite some results shown by [Reference Cai and Wang13, Reference Ge and Kachkovskiy25].
For spatial transport properties of a quantum particle on the lattice
$\mathbb {Z}$
, we are interested in studying the observable quantity associated with the position of the particle, which is represented by the unbounded self-adjoint operator
$$ \begin{align*} (Xx)_n:=(nx)_n, \quad n \in \mathbb{Z}, \end{align*} $$
with its natural domain of definition
$$ \begin{align*} \mathrm{Dom} X = \bigg\{ x_n \in \ell^2(\mathbb{Z}):\sum\limits_{n \in \mathbb{Z}} |n|^2|x_n|^2 < +\infty \bigg\}. \end{align*} $$
Our focus is on the phenomenon of ballistic motion, which informally states that the particle’s position grows linearly with time (
$X(T) \approx T$
). More precisely, we aim to investigate the following limit:
$$ \begin{align} \lim\limits_{T \to +\infty} \frac{1}{T} X(T) x, \end{align} $$
where the initial state at time
$x \in \mathrm {Dom} X$
. The limit in equation (50) can be regarded as the ‘asymptotic velocity’ of the state x as time approaches infinity, provided that the limit exists. Then we can define the asymptotic velocity operator as
$$ \begin{align} Q=\lim\limits_{T \to +\infty} \frac{1}{T} X(T) = \lim\limits_{T \to +\infty} \frac{1}{T} { \displaystyle \int_{0}^{T} e^{itH}Se^{-itH} \,dt}, \end{align} $$
where S is bounded and satisfies
$$ \begin{align} Sx_n = i(x_{n+1} - x_{n-1}). \end{align} $$
The Schrödinger operator H is said to demonstrate strong ballistic transport if the strong limit on the right-hand side of equation (51) exists, is defined on the entire
$\ell ^2(\mathbb {Z})$
, and
$\ker Q = \{0\}$
.
Definition 4.1. [Reference Ge and Kachkovskiy25]
Let
$\mathcal {K} \subset \mathbb {R}$
be a Borel subset. We say H has strong ballistic transport on
$\mathcal {K}$
if there exists a self-adjoint operator Q such that
$$ \begin{align} \lim\limits_{T \to +\infty} \frac{1}{T} { \displaystyle \int_{0}^{T} e^{itH}1_{\mathcal{K}}(H)S1_{\mathcal{K}}(H)e^{-itH} \,dt} = 1_{\mathcal{K}} Q 1_{\mathcal{K}} \end{align} $$
and
$\ker Q = \mathrm {Ran} (1_{\mathcal {K}})^{\perp }$
, where
$1_{\mathcal {K}}(\cdot )$
denotes the indicator function of
$\mathcal {K}$
.
Definition 4.2. [Reference Ge and Kachkovskiy25]
Cocycle
$(\alpha , S^V_E)$
is said to be
$C^k$
-reducible in expectation on
$\mathcal {K}$
if it is
$C^k$
-reducible for each
$E \in \mathcal {K}$
and there exists a choice of
$L^2(\mathbb {T}^d)$
-normalized conjugations
$B(E; \cdot )$
such that
$$ \begin{align*} \displaystyle \int_{\mathcal{K}} \|B(E; \cdot)\|^4_{C^k(\mathbb{T}^d)}\,d\rho(E) < +\infty. \end{align*} $$
Lemma 4.3. [Reference Ge and Kachkovskiy25]
Let a (Borel) subset
$\mathcal {K} \subset \mathbb {R}$
,
$\{H_{V,\alpha ,\theta } \}_{\theta \in \mathbb {T}^d}$
be a quasiperiodic operator family whose cocycles are
$C^k$
-reducible in expectation on
$\mathcal {K}$
for some
$k> {5d}/{2}$
. Then the family
$\{H_{V,\alpha ,\theta } \}_{\theta \in \mathbb {T}^d}$
has strong ballistic transport on
$\mathcal {K}$
.
As a direct application of our conclusion, now we can prove Theorem 1.3 in the following.
Proof. We will not consider E in the spectral gap, because cocycle
$(\alpha , S^V_E)$
is uniformly hyperbolic in the quasi-periodic Schrödinger case. Therefore, it is always reducible under our conditions and the conclusion is naturally valid.
For
$E \in \Sigma _{V,\alpha }$
, Theorem 3.8 shows that cocyle
$(\alpha , S^V_E)$
is reducible. As
$k> 14\tau + 2$
and picking
$k_0 = [k-10\tau -3]$
, we can get
$k_0> [4\tau -1]$
and
$\tau>d$
, then condition
$k_0> {5d}/{2}$
is obviously true. Here E, which satisfies neither the Diophantine nor rational condition in the spectrum set, is a set of measure zero. Note that, in Ge and Kachkovskiy [Reference Ge and Kachkovskiy25], under the setting of Zhao [Reference Zhao41], we have strong ballistic transport on the whole spectrum. In the case of
$C^k$
in this paper, the conclusion still holds. Therefore, by Lemma 4.3, the Schrödinger operator
$H_{V,\alpha ,\theta }$
has strong ballistic transport for a.e.
$E \in \Sigma _{V,\alpha }$
.
Readers can also refer to the detailed proof of Ge and Kachkovskiy [Reference Ge and Kachkovskiy25].
This finishes the proof of Theorem 1.3.
Now, for the application of the reducibility theorem to spectral structures in Theorem 1.6, we give the proof as follows.
Proof. Referring to Cai and Wang [Reference Cai and Wang13], we know the homogeneity of the spectrum is connected with polynomial decay of gap length and Hölder continuity of IDS. Then we need the following lemmas.
Lemma 4.4. [Reference Cai and Wang13]
Let
$G_m(V)=(E^-_m, E^+_m)$
denote the gap with label m and
$\alpha \in \mathrm {DC}_d(\kappa , \tau ), V \in C^k(\mathbb {T}^d, \mathbb {R})$
with
$k \geqslant D_0 \tau $
, where
$D_0$
is a numerical constant. There exists
$\tilde \epsilon = \tilde \epsilon (\kappa , \tau , k, d)> 0$
such that if
$\lVert V \rVert _k \leqslant \tilde \epsilon $
, then
$\lvert G_m(V) \rvert _k \leqslant {\tilde \epsilon }^{1/4}\lvert m \rvert ^{- {k}/{9}}$
.
Lemma 4.5. [Reference Cai10]
Let
$\alpha \in \mathrm {DC}_d(\kappa , \tau ), V \in C^k(\mathbb {T}^d, \mathbb {R})$
with
$k>17\tau +2$
. Then there exists
$\unicode{x3bb} _0$
depending on
$V, d, \kappa , \tau , k$
such that if
$\unicode{x3bb} < \unicode{x3bb} _0$
, then
$N_{\unicode{x3bb} V, \alpha }$
is
$\tfrac 12$
-Hölder continuous:
$$ \begin{align*} N(E+\hat \epsilon)-N(E-\hat \epsilon) \leqslant C_0 \hat \epsilon^{{1}/{2}}\quad \text{ for all } \hat \epsilon>0, \text{ for all } E \in \mathbb{R}, \end{align*} $$
where
$C_0$
depends only on
$d, \kappa , \tau , k$
.
Through the quantitative almost reducibility theory and reducibility theory, we can show both of the above lemmas. Under the preconditions of Theorem 3.8 together with Lemmas 4.4 and 4.5, for any
$E \in \Sigma _{V,\alpha }$
with
$2\rho (\alpha , S^V_E) =\langle m_0,\alpha \rangle \,\mod \mathbb {Z}$
for some
$m_0\in \mathbb {Z}^d$
, we will divide E into three cases when considering
$\nu $
-homogeneous. In each case, by calculating directly, Definition 1.5 is satisfied for every
$E \in \Sigma _{V,\alpha }$
. Please refer to the detailed proof of Cai and Wang [Reference Cai and Wang13].
This finishes the proof of Theorem 1.6.
