1.1 Toda flow

The Toda lattice was proposed by Toda in 1967 [Reference Toda67] as a model describing the positions and momenta of a chain, which is given by

(1.1) $$ \begin{align} \begin{cases} \dfrac{d}{dt}a_{n}(t)=a_{n}(t)(b_{n+1}(t)-b_{n}(t)), \\ \\ \dfrac{d}{dt}b_{n}(t)=2(a_{n}^{2}(t)-a_{n-1}^{2}(t)), \end{cases}\quad n\in\mathbb{Z}. \end{align} $$

Identifying $(a_{n}(t),b_{n}(t))$ as a doubly infinite dimensional Jacobi matrix $J(t)$ defined by

(1.2) $$ \begin{align} (J(t)u)_{n}=a_{n-1}(t)u_{n-1}+b_{n}(t)u_{n}+a_{n}(t)u_{n+1}. \end{align} $$

The study of Toda flow using equation (1.1) depends heavily on the Jacobi matrix $J(t)$ defined by (1.2). Teschl [Reference Teschl66, Theorem 12.6] shows that any initial condition $(a_{n}(0),b_{n}(0)) \in \ell ^{\infty }(\mathbb {Z})\times \ell ^{\infty }(\mathbb {Z})$ will lead to a unique solution $(a,b)\in C^{\infty }(\mathbb {R},\ell ^{\infty }(\mathbb {Z})\times \ell ^{\infty }(\mathbb {Z}))$ to equation (1.1). Under some hypotheses on the spectrum of operator $J(0),$ in [Reference Binder, Damanik, Lukic and Vandenboom14], Binder et al show the existence and uniqueness of almost periodic solutions to equation (1.1) with almost periodic initial value $(a_{n}(0),b_{n}(0)).$ Later, Leguil et al in [Reference Leguil, You, Zhao and Zhou48], by taking the initial value $(a_{n}(0),b_{n}(0))=(1,V(x+n\alpha )), n\in \mathbb {Z}$ with $V\in C^{\omega }(\mathbb {T},\mathbb {R}), \alpha \in \mathbb {T},$ extend the conclusions above to Avila’s subcritical regime. (For any spectrum E of Schrödinger operator $H_{V,\alpha ,x}$ defined by equation (1.3), the Schrödinger cocycle $(\alpha ,S_{E}^{V})$ is subcritical.) In particular, for the most important example of almost Mathieu operators (AMO), that is, $V(\cdot )=2\unicode{x3bb} \cos 2\pi (\cdot ),$ Leguil et al prove that there exist almost periodic solutions for all $0<\unicode{x3bb} <1.$ We will extend the conclusions in [Reference Leguil, You, Zhao and Zhou48] from finite dimensional frequencies to infinite dimensional frequencies.

Initial datum problems for other systems are in [Reference Damanik, Li and Xu31, Reference Damanik, Li and Xu32], where Damanik, Li, and Xu show the existence and uniqueness of spatially quasi-periodic solutions to the generalized KdV equation and Benjamin–Bona–Mahony equation with quasi-periodic initial data on the real line.

The proof of Theorem 1.1 depends on the discussions in [Reference Vinnikov and Yuditskii69], see Theorem 6.2. Given almost periodic initial datum $(a_{n}(0), b_{n}(0)) =(1,V(x+n\alpha )),$ two ingredients of the hypotheses of Theorem 6.2 are that the spectrum set of Schrödinger operator $H_{V,\alpha ,x}$ defined by equation (1.3) is homogeneous and pure absolutely continuous (ac). Thus, to apply this theorem, we, in §§1.2 and 1.3, will give our main results about the pure ac spectrum and homogeneous spectrum.

1.2 Absolute continuous spectrum

In this subsection, we will give discussions about the pure ac spectrum of the Schrödinger operator $H_{V,\alpha ,x},$ which is defined by

(1.3) $$ \begin{align} (H_{V,\alpha,x}u)_{n}=u_{n+1}+u_{n-1}+V(x+n\alpha)u_{n},\quad n\in \mathbb{Z}. \end{align} $$

A lot of research has been conducted by mathematicians and physicians about the almost periodic operator $H_{V,\alpha ,x},$ that is, the potential V is almost periodic, see [Reference Avron and Simon10, Reference Bellissard, Lima and Testard11, Reference Johnson and Moser46, Reference Moser52] for details. However, subsequently, much attention has been paid to the quasi-periodic and limit periodic cases and less progress made in the almost periodic case.

We first consider the limit-periodic potential $V,$ which lies in the closure of the space of periodic potentials. That is, there exists $\{V_{n}\}_{n\in \mathbb {N}},$ where $V_{n},n\in \mathbb {N},$ are periodic functions, such that

(1.4) $$ \begin{align} \lim_{n\rightarrow \infty}\|V-V_{n}\|_{\infty}=0. \end{align} $$

Even though periodic operators always exhibit pure ac spectrum, in [Reference Avila2], Avila shows the possibilities of new phenomena of the limit-periodic potentials, such as the genericity of purely singular continuous spectrum [Reference Avila2, Reference Damanik and Gan25] and the denseness of pure point spectrum [Reference Damanik and Gorodetski30]. Instead of the V defined by equation (1.4), if we restrict the limit-periodic potentials in the perturbative regime, that is, potentials V are with $q_{n}$ -periodic approximants $V_{n}$ such that

(1.5) $$ \begin{align} \lim_{n\rightarrow \infty}\mathrm{e}^{q_{n+1}b}\|V-V_{n}\|_{\infty}=0 \quad\text{for all } b>0, \end{align} $$

then in [Reference Pastur and Tkachenko55, Reference Pastur and Tkachenko56] Pastur and Tkachenko show that $H_{V,\alpha ,x}$ has pure ac spectrum, which serves as the fundamental work of other related studies.

Now, we consider the quasi-periodic potential $V.$ To study the spectral property of $H_{V,\alpha ,x},$ we introduce the coupling constant $\unicode{x3bb} \in \mathbb {R}^{+}$ in front of the potential V. In the case where $\unicode{x3bb} $ is sufficiently large, the Lyapuonv exponent of cocycle $(\alpha ,S_{E}^{\unicode{x3bb} V})$ is positive for any $E\in \mathbb {R},$ see [Reference Bourgain and Goldstein19, Reference Goldstein and Schlag42, Reference Herman44, Reference Sorets and Spencer65] and the references therein. Thus, Kotani’s theory implies there will be no ac spectrum in the spectrum set. If the coupling constant $\unicode{x3bb} $ is sufficiently small, the frequency $\alpha $ has to be restricted to the Diophantine frequencies, which we denote by $DC$ for the set of these frequencies. (For any $\gamma>0,\tau >d,$ the estimates $\|\langle k,\alpha \rangle \|_{\mathbb {R}/\mathbb {Z}}\geq \gamma |k|^{-\tau }\text { for all } 0\neq k\in \mathbb {Z}^{d}$ hold.) Under hypotheses that $\unicode{x3bb} $ is small enough and $\alpha \in DC,$ Dinaburg and Sina $\breve {\imath }$ [Reference Dinaburg and Sinaĭ35] prove that the ac spectrum set is not empty, and in [Reference Eliasson36], under the same hypotheses in [Reference Dinaburg and Sinaĭ35], Eliasson shows that the spectrum is pure ac. In the case where $\alpha \in DC,$ Avila and Jitomirskaya, based on a non-perturbative Anderson localization result, prove that there exists $\unicode{x3bb} _{1}$ independent on $\alpha $ such that the spectrum set is pure ac for all $\unicode{x3bb} \leq \unicode{x3bb} _{1},$ see [Reference Avila and Jitomirskaya8] for details. For AMO with subcritical coupling, that is, $0<\unicode{x3bb} <1,$ Avila and Damanik, in [Reference Avila and Damanik6], show that the spectrum is pure ac by proving that the integrated density of states is absolutely continuous. Moreover, Avila, Fayad, and Krikorian [Reference Avila, Fayad and Krikorian7] and Hou and You [Reference Hou and You45] consider the discrete and continuous Schrödinger operators, respectively. They construct non-standard Kolmogorov–Arnold–Moser (KAM) iterations and show that the set of ac spectrum is not empty for all $\alpha \in \mathbb {T}\setminus \mathbb {Q}$ and small enough $\unicode{x3bb} ,$ and Avila’s conclusions in [Reference Avila3, Reference Avila5] are able to show that the spectrum set is pure ac. The case that $\unicode{x3bb} $ is neither too large nor too small, which is the so-called global case, is extremely complicated. However, in the one-frequency case, Avila gives the fascinating global theory and shows that for typical analytic one-frequency Schrödinger operator, there is no singular continuous spectrum, see [Reference Avila4] for details. Our pure ac spectrum result is the following.

By the discussions above, we know that the works mentioned above are all with limit periodic or quasi-periodic potentials, and our Theorem 1.2 extends the results above to almost periodic potential cases. Since we consider the infinite dimensional frequency, a new technique is needed to overcome the difficulties brought by infinite dimensional frequencies. See the discussions in §4 for details.

1.3 Homogeneous spectrum

In this subsection, we will prove the homogeneity of the spectrum by using the exponential decay of the spectral gaps and Hölder continuity of the integrated density of states. Recall that in [Reference Carleson21], the concept of an homogeneous set is introduced by Carleson.

Definition 1.3. Given $\mu>0,$ a closed set $\mathcal {S}\subset \mathbb {R}$ is called $\mu $ -homogeneous if

$$ \begin{align*} |\mathcal{S}\cap(E-\varepsilon,E+\varepsilon)|\geq \mu\varepsilon \quad\text{for all } E\in \mathcal{S}, \text{ for all } 0<\varepsilon<\mathrm{diam}\,\mathcal{S}. \end{align*} $$

It is known that the homogeneity of closed subsets of $\mathbb {R}$ is important in inverse spectral theory, see the fundamental works of Sodin and Yuditskii [Reference Sodin and Yuditskii63, Reference Sodin and Yuditskii64]. In particular, under the hypotheses of finite total gap length along with a reflectionless condition, it is shown in [Reference Sodin and Yuditskii64] that the homogeneity of the spectrum implies the almost periodicity of the associated potentials and Gesztesy and Yuditskii [Reference Gesztesy and Yuditskii41], under the same hypotheses in [Reference Sodin and Yuditskii64], show that the Schrödinger operators have pure ac spectra. Here, we want to mention that the result of Remling and Poltoratski [Reference Poltoratski and Remling57, Corollary 2.3] shows that if S is weakly homogeneous and J is reflectionless on $S,$ then S does not support a singular spectrum, see [Reference Poltoratski and Remling57] for details.

In [Reference Fillman and Lukic38], Fillman and Lukic consider the Schrödinger operators with limit periodic potential and show the existence of homogeneous spectra by assuming the potential obeys the Pastur–Tkachenko condition, and Fillman [Reference Fillman37] proves the spectra of discrete operators with generic potential are homogeneous; here, generic potential is a dense subset of limit periodic potential. Later, Damanik and Fillman in [Reference Damanik and Fillman23] also show the existence of an homogeneous spectrum for the limit periodic potentials defined by (1.5).

As for the quasi-periodic potential case, we refer to [Reference Damanik, Goldstein and Lukic28], where Damanik, Goldstein, and Lukic consider continuous Schrödinger operators and show the existences of homogeneous spectra with Diophantine frequency and small potential $V.$ In [Reference Damanik, Goldstein, Schlag and Voda29], Damanik et al consider the discrete Schrödinger operators and show homogeneous spectra under the conditions that frequency is Diophantine and the Lyapunov exponent is positive. Moreover, to ensure the existence of an homogeneous spectrum for supercritical AMO $(\unicode{x3bb}>1),$ [Reference Damanik, Goldstein, Schlag and Voda29] shows that the strong Diophantine frequency is needed. Later, Leguil et al [Reference Leguil, You, Zhao and Zhou48] consider discrete Schrödinger operators with some Liouvillean frequencies and prove that the spectrum is homogeneous if V is small enough. Our result is as follow.

In Theorem 1.4, we extend the results above from limit periodic and quasi-periodic potentials to almost periodic potentials. In [Reference Avila, Last, Shamis and Zhou9], the authors construct the Schrödinger operator with one frequency whose spectrum is not homogeneous. Thus, the hypotheses in Theorem 1.4 are not restrictive.

2.1 Jacobi operator and Schrödinger operator

For $a=(a_{n})_{n\in \mathbb {Z}},b=(b_{n})_{n\in \mathbb {Z}}\in \ell ^{\infty }(\mathbb {Z},\mathbb {R}),$ we define the Jacobi operator J associate with $a,b$ by

(2.1) $$ \begin{align} (Ju)_{n}=a_{n-1}u_{n-1}+b_{n}u_{n}+a_{n}u_{n+1}. \end{align} $$

The Jacobi operator J arises naturally in the context of the spectral theorem, which says any bounded self-adjoint operator A with a cyclic vector is unitarily equivalent to a Jacobi operator on a half-line. It is self-adjoint since we restrict $a_{n},b_{n}\in \mathbb {R}\ \text { for all } n\in \mathbb {Z}.$ We restrict ourselves to the non-singular case, where $a_{n}>0\text { for all } n\in \mathbb {Z}.$

Let $\Sigma $ be the spectrum of the self-adjoint Jacobi matrices J defined by equation (2.1). Given any $z\notin \Sigma ,$ the Green’s function of J is the integral kernel of $(J-z)^{-1}$ :

(2.2) $$ \begin{align} G_{J}(m,n;z)=\langle e_{n}, (J-z)^{-1}e_{m}\rangle. \end{align} $$

Given any $z\in \mathbb {H}:=\{z\in \mathbb {C}: \mathrm {Im}\ z>0\},$ the difference equation $Ju=zu$ has two solutions $u^{\pm }$ (defined up to normalization) with $u_{0}^{\pm }\neq 0,$ which are in $\ell ^{2}(\mathbb {Z}^{\pm }),$ respectively. Let $m_{J}^{\pm }=\mp ({u_{\pm 1}^{\pm }}/{a_{0}u_{0}^{\pm }}).$ Then, $m_{J}^{+}$ and $m_{J}^{-}$ are Herglotz functions, i.e., they map $\mathbb {H}$ holomorphically into itself. For almost every $E\in \mathbb {R},$ the non-tangential limits $\lim _{\varepsilon \rightarrow 0^{+}}m_{J}^{\pm }(E+i\varepsilon )$ exist. Then, we have

(2.3) $$ \begin{align} G_{J}(0,0;z)=\frac{-1}{a_{0}^{2}(m_{J}^{+}(z)+m_{J}^{-}(z))},\quad z\in\mathbb{H}. \end{align} $$

Consider the linear Schrödinger operator $H_{V,\alpha ,x}$ defined by

(2.4) $$ \begin{align} (H_{V,\alpha,x}u)_{n}=u_{n+1}+u_{n-1}+V(x+n\alpha)u_{n},\quad n\in \mathbb{Z}, \end{align} $$

where $x\in \mathbb {T}^{d}, d\in \mathbb {N}\cup \{\infty \},$ is called the phase, $\alpha \in \mathbb {T}^{d}$ is called the frequency, and V is called the potential.

The Schrödinger operator $H_{V,\alpha ,x}$ defined by equation (2.4) will be self-adjoint if we assume V is real-valued and it is a special case of Jacobi operator $J(t)$ defined by equation (2.1) with $a_{n}\equiv 1\text { for all } n\in \mathbb {Z}.$ Moreover, since we assume $\alpha \in DC_{\infty }(\gamma ,\tau ), V(x+n\alpha )$ is almost periodic in $n\in \mathbb {Z},$ and we call $H_{V,\alpha ,x}$ the almost periodic Schrödinger operator.

2.2 Cocycle

Set $\mathbb {T}:=\mathbb {R}/\mathbb {Z}$ as the cycle (the base) and $SL(2,\mathbb {R})$ as the set of 2 by 2 real matrices with determinant 1 (the fiber). Thus, the smooth cocycles are diffeomorphisms on the product $\mathbb {T}^{d}\times \mathbb {C}^{2}$ of the form

$$ \begin{align*} \begin{aligned} (\alpha,A): \mathbb{T}^{d}\times \mathbb{C}^{2}& \rightarrow\mathbb{T}^{d}\times \mathbb{C}^{2},\\ &(x,y)\mapsto(x+\alpha,A(x)y), \end{aligned} \end{align*} $$

where $\alpha \in \mathbb {T}^{d}$ and $A\in C^{\infty }(\mathbb {T}^{d}, SL(2,\mathbb {R})), d\in \mathbb {N}\cup \{\infty \}.$

Now, we turn back to Schrödinger operator $H_{V,\alpha ,x}$ defined by equation (2.4). Note any formal solution $u=(u_{n})_{n\in \mathbb {Z}}$ of $H_{V,\alpha ,x}u=Eu$ can be rewritten as

$$ \begin{align*} \begin{aligned} \left( \begin{matrix} u_{n+1} \\ u_{n} \end{matrix} \right)=S_{E}^{V}(x+n\alpha)\left( \begin{matrix} u_{n} \\ u_{n-1} \end{matrix} \right)\!, \end{aligned} \end{align*} $$

where

$$ \begin{align*} \begin{aligned} S_{E}^{V}(x)=\left( \begin{matrix} E-V(x)\ \ \ \ -1 \\ 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \end{matrix} \right)\!. \end{aligned} \end{align*} $$

We call $(\alpha ,S_{E}^{V}(x))$ the Schrödinger cocycle. The iterations of $(\alpha ,S_{E}^{V}(\cdot ))$ are of the form $(\alpha ,S_{E}^{V}(\cdot ))^n=(n\alpha , S_{E,n}^{V}(\cdot )),$ where $S_{E,n}^{V}(\cdot )$ is called the transfer matrix and defined by

$$ \begin{align*} S_{E,n}^{V}(\cdot):= \begin{cases} S_{E,V}(\cdot+(n-1)\alpha) \cdots S_{E,V}(\cdot+\alpha) S_{E,V}(\cdot), & n\geq 0,\\[1mm] S_{E,V}^{-1}(\cdot+n\alpha) S_{E,V}^{-1}(\cdot+(n+1)\alpha) \cdots S_{E,V}^{-1}(\cdot-\alpha), & n <0, \end{cases} \end{align*} $$

then, we have

$$ \begin{align*} \begin{aligned} \left( \begin{matrix} u_{n+1} \\ u_{n} \end{matrix} \right)=S_{E,n}^{V}(x)\left( \begin{matrix} u_{1} \\ u_{0} \end{matrix} \right)\!. \end{aligned} \end{align*} $$

2.3 Integrated density of states and Lyapunov exponent

The integrated density of states (IDS) $N_{V}:\mathbb {R}\rightarrow [0,1]$ of $H_{V,\alpha ,x}$ is defined as

$$ \begin{align*} N_{V}(E)=\int_{\mathbb{T}^{d}}\mu_{V,x}(-\infty,E] \,{d}x, \end{align*} $$

where $\mu _{V,x}$ is the spectral measure of $H_{V,\alpha ,x}$ .

Define the finite Lyapunov exponent as

$$ \begin{align*} L_n(\alpha,S_{E}^{V})= \frac{1}{n}\int_{\mathbb{T}^{d}}\ln\|S_{E,n}^{V}(x)\|\,dx,\end{align*} $$

then by Kingman’s subadditive ergodic theorem, the Lyapunov exponent of $(\alpha , S_{E}^{V})$ is defined as

(2.5) $$ \begin{align} L(\alpha,S_{E}^{V})=\lim_{n\rightarrow\infty}L_n(\alpha,S_{E}^{V})= \inf_{n>0} L_n(\alpha,S_{E}^{V})\geq0. \end{align} $$

Note that in our case, $d=\infty .$ It turns out that if $\alpha \in DC_{\infty }(\gamma ,\tau )$ (thus, $\alpha $ is uniquely ergodic), then

$$ \begin{align*} L(\alpha,S_{E}^{V})=\lim_{n\rightarrow\infty} \frac{1}{n}\ln\|S_{E,n}^{V}(x)\|,\quad \mathrm{a.e.}\ x\in\mathbb{T}^{\infty}. \end{align*} $$

By the Thouless formula, the relation between the IDS and the Lyapunov exponent defined by equation (2.5) is

$$ \begin{align*} L(\alpha,S_{E}^{V})=\int\ln|E-E'|\,{d}N_{V}(E'). \end{align*} $$

2.4 Rotation number

Assume that $A\in \mathcal {H}_{\sigma }(\mathbb {T}^{d},SL(2,\mathbb {R}))$ is homotopic to identity and introduce the map:

$$ \begin{align*} F:\mathbb{T}^{d}\times S^{1} \rightarrow \mathbb{T}^{d}\times S^{1},\quad (x,v)\mapsto \bigg(x+\alpha,\frac{A(x)v}{\|A(x)v\|}\bigg), \end{align*} $$

which admits a continuous lift $\widetilde {F}:\mathbb {T}^{d}\times \mathbb {R} \rightarrow \mathbb {T}^{d}\times \mathbb {R}$ of the form $\widetilde {F}(x,y)=(x+\alpha ,y+f(x,y))$ such that $f(x,y+1)=f(x,y)$ and $\pi (y+f(x,y))={A(x)\pi (y)}/{\|A(x)\pi (y)\|}$ . We call that $\widetilde {F}$ is a lift for $(\alpha ,A)$ . Since $x\mapsto x+\alpha $ is uniquely ergodic on $\mathbb {T}^{d}$ , we can invoke a theorem by Herman [Reference Herman44] and Johnson and Moser [Reference Johnson and Moser46]: for every $(x,y)\in \mathbb {T}^{d}\times \mathbb {R}$ , the limit

$$ \begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n} \sum_{k=0}^{n-1}f(\widetilde{F}^{k}(x,y)) \end{align*} $$

exists, is independent of $(x,y)$ , and the convergence is uniform in $(x,y)$ ; the class of this number in $\mathbb {T}$ (which is independent of the chosen lift) is called the fibered rotation number of $(\alpha , A)$ , which is denoted by $\rho (\alpha , A)$ . Moreover, the rotation number $\rho _{f}(\alpha ,A)$ relates density of states $N_{V}$ as follows:

(2.6) $$ \begin{align} N_{V}(E)=1-2\rho(\alpha,A). \end{align} $$

For any $C\in SL(2,\mathbb {R})$ , it is immediate from the definition that

(2.7) $$ \begin{align} |\rho(\alpha,A)-\rho(\alpha,C)|\leq\|A(x)-C\|_{C^{0}}^{{1}/{2}}. \end{align} $$

In addition to the conclusion given by equation (2.7), we also have two conclusions below.

Lemma 2.2. [Reference Krikorian47]

The rotation number is invariant under the conjugation map which is homotopic to the identity. More precisely, if $A, B:\mathbb {T}^{d}\rightarrow SL(2, \mathbb {R})$ is continuous and homotopic to the identity, then

$$ \begin{align*} \rho(\alpha,B(\cdot+\alpha)^{-1}A(\cdot)B(\cdot))= \rho(\alpha, A). \end{align*} $$

Proposition 2.3. If $A:\mathbb {T}^{\infty }\rightarrow SL(2,\mathbb {R})$ is continuous and homotopic to the identity, and $E:2\mathbb {T}^{\infty }\rightarrow SL(2,\mathbb {R})$ is defined by

$$ \begin{align*} \begin{aligned} E(x)=\left( \begin{matrix} \cos (\pi\langle r,x\rangle )\ \ -\sin (\pi\langle r,x\rangle) \\ \sin (\pi\langle r,x\rangle)\ \ \ \ \cos (\pi\langle r,x\rangle) \end{matrix} \right)\!, \end{aligned} \end{align*} $$

then

$$ \begin{align*} \rho((0,E)\circ(\alpha,A)\circ(0,E^{-1}))=\rho(\alpha,A)+\frac{\langle r,\alpha\rangle}{2}\ \mod\! 1. \end{align*} $$

Proof. It is known that $\rho ((0,B)\circ (\alpha ,A)\circ (0,B^{-1}))=\rho (\alpha ,A)\mod \! 1$ if $B:2\mathbb {T}^{\infty }\rightarrow SL(2,\mathbb {R})$ is homotopic to the identity. In this case, the lift of $(\alpha , E(x+\alpha )A(x)E^{-1}(x))$ is given by

$$ \begin{align*} \widetilde{G}(x,y)=\bigg(x+\alpha,y+f\bigg(x,y-\frac{\langle r,x\rangle}{2}\bigg)+\frac{\langle r,\alpha\rangle}{2}\bigg). \end{align*} $$

Define

$$ \begin{align*} \begin{aligned} g(x,y)&=f\bigg(x,y-\frac{ \langle r,x\rangle}{2}\bigg)+\frac{\langle r,\alpha\rangle}{2},\\ g_{k}(x,y)&=\sum\limits_{i=0}^{k-1}g(\widetilde{G}^{i}(x,y)),\ f_{k}(x,y)=\sum\limits_{i=0}^{k-1}f(\widetilde{F}^{i}(x,y)). \end{aligned} \end{align*} $$

Using mathematical induction, then, $g_{k}(x,y)=f_{k}(x,y-{\langle r,x\rangle }/{2})+{k\langle r,\alpha \rangle }/{2}$ . Note the fact of the convergence of the Birkhoff means

$$ \begin{align*} \rho((0,E)\circ(\alpha,A)\circ(0,E^{-1}))=\rho(\alpha,A)+\frac{\langle r,\alpha\rangle}{2}\ \mod\! 1.\\[-42pt] \end{align*} $$

3.1 Decomposition along resonances

In this subsection, parameters $\rho $ , $\varepsilon $ , N, will be fixed. Define the resonant case as that where $k_{*}\in \mathbb {Z}_{*}^{\infty }$ with $0<|k_{*}|_{\eta }\leq N$ , such that

$$ \begin{align*} \|2\rho-\langle k_{*},\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}<\varepsilon^{{1}/{12}}. \end{align*} $$

The vector $k_{*}$ will be referred to as a ‘resonant site’.

After here, the constant c values are bounded uniformly but are different in different places. Moreover, for any $K>0$ , define the truncating operators $\mathcal {T}_{K}$ on $\mathcal {H}_{\sigma }(\mathbb {T}^{\infty },X)$ as

$$ \begin{align*} (\mathcal{T}_{K} f)(x)=\sum_{k\in\mathbb{Z}_{*}^{\infty},|k|_{\eta}<K} \widehat{f}(k) e^{i2\pi\langle k,x\rangle}, \end{align*} $$

and projection operator $\mathcal {R}_{K}$ as

$$ \begin{align*} (\mathcal{R}_{K} f)(x)=\sum_{k \in \mathbb{Z}_{*}^{\infty},|k|_{\eta}\geq K} \widehat{f}(k) e^{i2\pi\langle k,x\rangle}. \end{align*} $$

Decomposition of the space $\mathcal {H}_{\sigma }(\mathbb {T}^{\infty },su(1,1))$ is defined as follows: for any given ${\xi> 0}$ , $\alpha \in \mathbb {T}^{\infty }$ , $A\in SU(1,1)$ , we decompose $\mathcal {B}_{\sigma }=\mathcal {H}_{\sigma }(\mathbb {T}^{\infty },su(1,1))=\mathcal {B}_{\sigma }^{nre}(\xi )\oplus \mathcal {B}_{\sigma }^{re}(\xi )$ in such a way that for any $Y\in \mathcal {B}_{\sigma }^{nre}(\xi )$ ,

(3.1) $$ \begin{align} A^{-1}Y(x+\alpha)A\in \mathcal{B}_{\sigma}^{nre}(\xi),\ \ \ |A^{-1}Y(x+\alpha)A-Y(x)|_{\sigma}\geq \xi|Y(x)|_{\sigma}. \end{align} $$

Let $\mathcal {P}_{nre},\mathcal {P}_{re}$ be the standard projections from $\mathcal {B}_{\sigma }$ onto $\mathcal {B}_{\sigma }^{nre}(\xi )$ and $\mathcal {B}_{\sigma }^{re}(\xi )$ .

Lemma 3.1. [Reference Cai, Chavaudret, You and Zhou20]

Assume that $\varepsilon \leq (4\| A\|)^{-4}$ , $\xi \geq 13\| A\|^{2}\varepsilon ^{1/2}$ . For any $f\in \mathcal {B}_{\sigma }$ with $|f|_{\sigma }\leq \varepsilon $ , there exists $Y\in \mathcal {B}_{\sigma }$ , $f^{re}\in \mathcal {B}_{\sigma }^{re}(\xi )$ such that

$$ \begin{align*} e^{Y(x+\alpha)}(Ae^{f(x)})e^{-Y(x)}=Ae^{f^{re}(x)},\quad |Y|_{\sigma}\leq\varepsilon^{{1}/{2}},\quad |f^{re}|_{\sigma}\leq2\varepsilon. \end{align*} $$

Referring to [Reference Cai, Chavaudret, You and Zhou20, Remark 3.1], Lemma 3.1 sets up the fact that $\mathcal {B}_{\sigma }$ is a Banach space. Before giving the induction proposition, we introduce the following well-known results, which control the estimate of the small divisor.

Lemma 3.2. [Reference Montalto and Procesi51]

Let $\mu _{1}, \mu _{2}\geq 1$ . We have the following estimate for $N\gg 1$ :

(3.2) $$ \begin{align} \sup_{k\in \mathbb{Z}_{*}^{\infty},|k|_{\eta}\leq N}\prod_{j\in\mathbb{N}}(1+|k_{j}|^{\mu_{1}}\langle j\rangle^{\mu_{2}})\leq (1+N)^{c(\eta,\mu_{1}, \mu_{2})N^{{1}/({1+\eta})}} \end{align} $$

for some constant $C(\eta ,\mu _{1},\mu _{2})>0$ .

For the given $c,$ assume that $D(c)$ is the smallest one such that

(3.3) $$ \begin{align} (\ln D(c))^{{\eta}/({4+4\eta})} \geq c\ln\ln D(c). \end{align} $$

Proposition 3.3. Let $\alpha \in DC_{\infty }(\gamma ,\tau )$ , $\gamma>0$ , $0<\sigma <{1}/{10}$ , $\tau>1$ , $\eta>0$ . Consider the cocycle $(\alpha , Ae^{f(x)})$ , where $A\in SU(1,1)$ with eigenvalues $\{e^{i2\pi \rho },e^{-i2\pi \rho }\}$ and ${f\in \mathcal {H}_{\sigma }(\mathbb {T}^{\infty },su(1,1))}$ with the estimate

(3.4) $$ \begin{align} |f|_{\sigma}\leq\varepsilon\leq D(c)^{-1} \exp\{-[(\sigma-\delta)\pi] ^{({-2(2+\eta)})/{\eta}}\}, \end{align} $$

where $\delta \in (0,\sigma )$ and $D(c)$ is the one defined by equation (3.3) with $ c=c(\gamma ,\tau ,\eta ,\|A\|).$ Then, there exists $ B \in \mathcal {H}_{\delta }(2\mathbb {T}^{\infty },SU(1,1))$ , $f_{+}\in \mathcal {H}_{\delta }(\mathbb {T}^{\infty },su(1,1)),$ and $A_{+}\in SU(1,1)$ , such that

(3.5) $$ \begin{align} B(x+\alpha)(Ae^{f(x)})B(x)^{-1}=A_{+}e^{f_{+}(x)}. \end{align} $$

Moreover, let $N=\{(\sigma -\delta )\pi \}^{-1}\ln \varepsilon ^{-1}$ , then we distinguish the conclusions into two cases. Non-resonant case: if for any $0<|k|_{\eta }\leq N$ , $k\in \mathbb {Z}_{*}^{\infty }$ ,

(3.6) $$ \begin{align} \|2\rho-\langle k,\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}\geq\varepsilon^{{1}/{12}}, \end{align} $$

then

(3.7) $$ \begin{align} |B-\mathrm{Id}|_{\delta}\leq\varepsilon^{{1}/{2}},\quad \ |f_{+}|_{\delta}\leq4\varepsilon^{3},\quad \|A_{+}-A\|\leq2\varepsilon. \end{align} $$

Resonant case: if there exists $ k_{*}\in \mathbb {Z}_{*}^{\infty }$ , $0<|k_{*}|_{\eta }\leq N$ , such that

$$ \begin{align*} \|2\rho-\langle k_{*},\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}<\varepsilon^{{1}/{12}}, \end{align*} $$

then

(3.8) $$ \begin{align} \begin{aligned} |B|_{\delta}\leq e^{\pi\delta|k_{*}|_{\eta}}{\varepsilon}^{-{1}/{480}},\quad \|B\|\leq\varepsilon^{-{1}/{480}},\quad |f_{+}(x)|_{\delta}\ll \varepsilon^{10},\\\rho(\alpha,A_{+}e^{f_{+}(x)})=\rho(\alpha,Ae^{f(x)})+\frac{\langle k_{*},\alpha\rangle}{2}, \quad\ \ \qquad\end{aligned} \end{align} $$

and $A_{+}= e^{A^{\prime \prime }_{+}}, \ \|A^{\prime \prime }_{+}\|<4\varepsilon ^{1/12},$

$$ \begin{align*} \begin{aligned} A^{\prime\prime}_{+}&=\left( \begin{matrix} it_{+}\ \ \ \ \ \ \ v_{+} \\ \bar{v}_{+}\ \ \ \ -it_{+} \end{matrix} \right) \end{aligned} \end{align*} $$

with $|t_{+}|\leq 4\varepsilon ^{{1}/{12}},\ |v_{+}|\leq 2\varepsilon ^{1-{1}/{240}}e^{-2\pi | k_{*}|_{\eta }\sigma }.$

Proof. Non-resonant case. Before giving the proof, we give the estimate of small divisor in the following claim.

Claim 3.4. For any $\alpha \in DC_{\infty }(\gamma ,\tau )$ , and parameters $N, \eta ,\tau ,\varepsilon $ given above, the following estimates hold:

(3.9) $$ \begin{align} \|\langle k,\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}\geq \gamma(1+N)^{-c(\eta,\tau)N^{{1}/({1+\eta})}}>2\varepsilon^{1/240}. \end{align} $$

Proof. The fact that $\alpha \in DC_{\infty }(\gamma ,\tau )$ and equation (3.2) yield, for any $0<|k|_{\eta }\leq N$ ,

$$ \begin{align*} \|\langle k,\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}\geq\gamma\prod_{j\in\mathbb{N}}\frac{1}{(1+|k_{j}|^{\tau}\langle j\rangle^{\tau})}\geq\gamma(1+N)^{-c(\eta,\tau)N^{{1}/({1+\eta})}}. \end{align*} $$

Now, we give the proof of the last inequality in equation (3.9). First, equation (3.4) shows

(3.10) $$ \begin{align} \ln\varepsilon^{-1} \geq\max\{[(\sigma-\delta) \pi]^{-{2(2+\eta)}/{\eta}}, [c\ln\ln\varepsilon^{-1}]^{2(1+\eta)/\eta}\}, \end{align} $$

where c depends on the parameters $\eta ,\tau ,\gamma .$ The inequality above yields

(3.11) $$ \begin{align} (\ln\varepsilon^{-1})^{{\eta}/({1+\eta})} =(\ln\varepsilon^{-1})^{{\eta}/({2(1+\eta)})} (\ln\varepsilon^{-1})^{{\eta}/({2(1+\eta)})}>c[(\sigma-\delta) \pi]^{{-1}/({1+\eta})} \ln\ln\varepsilon^{-1}. \end{align} $$

Then,

(3.12) $$ \begin{align} \begin{aligned} \ln\varepsilon^{-1}&>(\ln\varepsilon^{-1})^{{1}/({1+\eta})}c[(\sigma-\delta) \pi]^{{-1}/({1+\eta})} \ln\ln\varepsilon^{-1}\\ &>c\{[(\sigma-\delta) \pi]^{-1}\ln\varepsilon^{-1}\}^{{1}/({1+\eta})}(\ln[(\sigma-\delta) \pi]^{-1}+\ln\ln\varepsilon^{-1}), \end{aligned} \end{align} $$

where the first and second inequalities are from equations (3.11) and (3.10), respectively. The last inequality in equation (3.12) and the fact $N=\{(\sigma -\delta )\pi \}^{-1}\ln \varepsilon ^{-1}$ yield

$$ \begin{align*} \ln\varepsilon^{-1}>{cN^{{1}/({1+\eta})}}\ln N, \end{align*} $$

which yields

$$ \begin{align*} \varepsilon^{{-1}/{240}}>2\gamma^{-1}(1+N)^{c(\eta,\tau)N^{{1}/({1+\eta})}}.\\[-36pt] \end{align*} $$

Define

(3.13) $$ \begin{align} \Lambda_{N}=\bigg\{f\in\mathcal{H}_{\sigma}(\mathbb{T}^{\infty},su(1,1))| ~f(x)=\sum_{k\in \mathbb{Z}_{*}^{\infty},0<|k|_{\eta}\leq N}\widehat{f}(k)e^{i2\pi\langle k,x\rangle}\bigg\}. \end{align} $$

By equations (3.6) and (3.9), a simple computation shows that if $Y\in \Lambda _{N},~A\in SU(1,1)$ , then

$$ \begin{align*} |A^{-1}Y(x+\alpha)A-Y(x)|_{\sigma} \gtrsim\varepsilon^{1/4}|Y(x)|_{\sigma}. \end{align*} $$

Thus, $\Lambda _{N}\subset \mathcal {B}_{\sigma }^{nre}(\varepsilon ^{1/4})$ . Note $\varepsilon \leq (4\| A\|)^{-4}$ and $\varepsilon ^{1/4}>13\| A\|^{2}\varepsilon ^{1/2}$ , then by Lemma 3.1 with $\varepsilon ^{1/4}$ in place of $\xi $ , we know that there exist $Y\in \mathcal {B}_{\sigma }$ , $f^{re}\in \mathcal {B}_{\sigma }^{re}(\varepsilon ^{1/4})$ such that

(3.14) $$ \begin{align} e^{Y(x+\alpha)}(Ae^{f(x)})e^{-Y(x)}=Ae^{f^{re}(x)}, \end{align} $$

where $|Y|_{\sigma }\leq \varepsilon ^{1/2}$ , $|f^{re}|_{\sigma }\leq 2\varepsilon $ . Notice $f^{re}\in \mathcal {B}_{\sigma }^{re}(\varepsilon ^{1/4}),$ then by equation (3.13),

(3.15) $$ \begin{align} f^{re}(x)=\widehat{f}^{re}(0)+\mathcal{R}_{N}f^{re}(x),~\ \|\widehat{f}^{re}(0)\|\leq2\varepsilon. \end{align} $$

Moreover, we also have

(3.16) $$ \begin{align} \begin{aligned} |\mathcal{R}_{N}f^{re}(x)|_{\delta}&=\sum_{k\in \mathbb{Z}_{*}^{\infty},|k|_{\eta}>N}\|\hat{f}^{re}(k)\|e^{2\pi|k|_{\eta}\delta}\\ &\leq\|f^{re}\|_{\sigma}\sup_{|k|_{\eta}>N} e^{-2\pi|k|_{\eta}(\sigma-\delta)} \leq e^{-2\pi N(\sigma-\delta)}2\varepsilon\leq 2\varepsilon^{3}. \end{aligned} \end{align} $$

Set

$$ \begin{align*} e^{f_{+}(x)}:=e^{-\widehat{f}^{re}(0)}e^{f^{re}(x)},\quad A_{+}:=Ae^{\widehat{f}^{re}(0)},\quad B(x):=e^{Y(x)}. \end{align*} $$

Thus, the cocycle $(\alpha , Ae^{f^{re}(x)})$ can be written as $(\alpha , A_{+}e^{f_{+}(x)})$ and $(0,B)$ changes $(\alpha , Ae^{f(x)})$ to $(\alpha , A_{+}e^{f_{+}(x)}).$ Notice that $e^{A}e^{E}=e^{A+E+D},$ where D is a sum of terms of order at least 2 in $A, E,$ then by equations (3.15) and (3.16), we get

$$ \begin{align*} |f_{+}(x)|_{\delta}\leq4\varepsilon^{3},\quad \|A_{+}-A\|\leq\|A\|\|\mathrm{Id}-e^{\widehat{f}^{re}(0)}\|\leq 2\varepsilon,\quad |B-\mathrm{Id}|_{\delta}\leq\varepsilon^{{1}/{2}}. \end{align*} $$

Resonant case. Note that we only need to consider the case in which $A\in SU(1,1)$ is elliptic with eigenvalues $\{e^{i2\pi \rho },e^{-i2\pi \rho }\}$ for $\rho \in \mathbb {R}\setminus \{0\}$ , because if $\rho =ib$ , with $b\in \mathbb {R}$ , then equation (3.9) implies $\|i2b-\langle k_{*},\alpha \rangle \|_{\mathbb {R}/\mathbb {Z}}>2\varepsilon ^{1/240}$ .

Claim 3.5. Assume that $k_{*}$ is the resonant site with

$$ \begin{align*} 0<|k_{*}|_{\eta}\leq N, \end{align*} $$

then there is no other resonant site $k^{\prime }_{*}$ with $|k^{\prime }_{*}|\leq N^{1+({\eta }/{2})}$ .

Proof. Assume that there exists $k^{\prime }_{*}\neq k_{*}$ with $|k^{\prime }_{*}|_{\eta }\leq N^{1+({\eta }/{2})}$ , satisfying ${\|2\rho -\langle k^{\prime }_{*},\alpha \rangle \|_{\mathbb {R}/\mathbb {Z}}<\varepsilon ^{1/12}}$ , then by the Diophantine condition of $\alpha $ , we have

$$ \begin{align*} \gamma\prod_{j\in\mathbb{N}}\frac{1}{(1+|k^{\prime}_{*j}-k_{*j}|^{\tau}\langle j\rangle^{\tau})}\leq \|\langle k^{\prime}_{*},\alpha\rangle-\langle k_{*},\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}<2\varepsilon^{1/12}. \end{align*} $$

Moreover, note $|k^{\prime }_{*}-k_{*}|_{\eta }\leq 2N^{1+({\eta }/{2})}$ , then by equation (3.2) and the inequalities above,

$$ \begin{align*} \frac{\gamma}{2}\varepsilon^{-{1}/{12}}<\prod_{j\in\mathbb{N}}(1+|k^{\prime}_{*j}-k_{*j}|^{\tau}\langle j\rangle^{\tau})\leq(1+2N^{1+({\eta}/{2})})^{c(\eta,\tau)(2N^{1+({\eta}/{2})})^{{1}/({1+\eta})}}, \end{align*} $$

which, together with the fact $N=\{(\sigma -\delta )\pi \}^{-1}\ln \varepsilon ^{-1},$ implies

(3.17) $$ \begin{align} (\ln\varepsilon^{-1})^{{\eta}/({2+2\eta})}<c(\gamma,\tau,\eta)[(\sigma-\delta)\pi]^{{-(2+\eta)}/({2+2\eta})}\ln\ln\varepsilon^{-1}. \end{align} $$

However, equation (3.10) implies $[(\sigma -\delta )\pi ]^{{-(2+\eta )}/({2+2\eta })}<(\ln \varepsilon ^{-1})^{{\eta }/({4+4\eta })}$ , which, together with equation (3.17), implies

$$ \begin{align*} (\ln\varepsilon^{-1})^{{\eta}/({4+4\eta})} <c(\gamma,\tau,\eta) \ln\ln\varepsilon^{-1}. \end{align*} $$

The inequality above is contradictory to equations (3.4) and (3.3).

Note that $\mathrm {tr}\ A=e^{i2\pi \rho }+e^{-i2\pi \rho }=2\cos 2\pi \rho>-2$ , so there exists $A'\in su(1,1)$ such that $A=e^{A'}$ with $\mathrm {spec}\ (A')=\{i2\pi \rho ,-i2\pi \rho \}$ , $\rho \in (0,\tfrac 12)$ . In this resonant case, the fact that there exist $k_{*}$ with $|k_{*}|_{\eta }<N$ such that

$$ \begin{align*} \|2\rho-\langle k_{*},\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}<\varepsilon^{{1}/{12}} \end{align*} $$

together with similar calculations above yield

(3.18) $$ \begin{align} |\rho|\geq\varepsilon^{{1}/{252}}. \end{align} $$

Moreover, [Reference Hou and You45, Lemma 8.1] implies that there exists $P\in SU(1,1)$ such that

$$ \begin{align*} PAP^{-1}=\mathrm{diag}(e^{i2\pi\rho},e^{-i2\pi\rho}) := A_{*}, \end{align*} $$

where

(3.19) $$ \begin{align} \|P\|\leq2(\|A'\||2\pi\rho|^{-1})^{{1}/{2}}\leq\varepsilon^{-{1}/{480}}, \end{align} $$

here, the second inequality is by equation (3.18). Furthermore, set $h(x)=Pf(x)P^{-1},$ then the cocycle $(\alpha , Ae^{f(x)})$ is changed into $(\alpha , A_{*}e^{h(x)})$ with

(3.20) $$ \begin{align} |h|_{\sigma}\leq\varepsilon^{-{1}/{240}}\cdot\varepsilon\triangleq\varepsilon'. \end{align} $$

The estimate $\varepsilon ^{1/12}\geq 13\|A_{*}\|^2\varepsilon ^{\prime 1/2} $ enables us to apply Lemma 3.1 to the cocycle $(\alpha ,A_{*}e^{h(x)})$ to remove all the non-resonant terms of h, that is, there exists $Y\in \mathcal {B}_{\sigma }$ , $h^{re}\in \mathcal {B}_{\sigma }^{re}(\varepsilon ^{1/12})$ such that

$$ \begin{align*} e^{Y(x+\alpha)}(A_{*}e^{h(x)})e^{-Y(x)}=A_{*}e^{h^{re}(x)}, \end{align*} $$

with $|Y|_{\sigma }\leq \varepsilon ^{\prime 1/2}$ , $|h^{re}|_{\sigma }\leq 2\varepsilon '$ . Thus, we get the cocycle $(\alpha ,A_{*}e^{h^{re}(x)}).$

Now define

$$ \begin{align*} \begin{aligned} \Lambda_{1}(\varepsilon^{1/12})&=\{k\in\mathbb{Z}_{*}^{\infty}:\|\langle k,\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}\geq\varepsilon^{1/12}\},\\ \Lambda_{2}(\varepsilon^{1/12})&=\{k\in\mathbb{Z}_{*}^{\infty}:\| 2\rho-\langle k,\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}\geq\varepsilon^{1/12}\}, \end{aligned} \end{align*} $$

and define $\mathcal {B}_{\sigma }^{nre}(\varepsilon ^{1/12})$ as in equation (3.1) with A being substituted by $A_{*}.$ Then, we can compute that any $Y\in \mathcal {B}_{\sigma }^{nre}(\varepsilon ^{1/12})$ takes the precise form:

$$ \begin{align*} \begin{aligned} Y(x)&=\left( \begin{matrix} it(x)\ \ \ \ \ v(x) \\ \bar{v}(x)\ -it(x) \end{matrix} \right)\\ &=\sum_{k\in\Lambda_{1}(\varepsilon^{1/12})}\left( \begin{matrix} i\widehat{t}(k)\ \ \ \ \ 0 \\ 0\ -i\widehat{t}(k) \end{matrix} \right)e^{i2\pi\langle k,x\rangle}+\sum_{k\in\Lambda_{2}(\varepsilon^{1/12})} \left( \begin{matrix} 0\ \ \ \ \ \ \widehat{v}(k)e^{i2\pi\langle k,x\rangle} \\ \bar{\widehat{v}}(k)e^{-i2\pi\langle k,x\rangle}\ \ \ \ \ \ 0 \end{matrix} \right)\!, \end{aligned} \end{align*} $$

where $t(x)\in \mathbb {R}$ , $v(x)\in \mathbb {C}$ .

Combining with the fact that $\alpha \in DC_{\infty }(\gamma ,\tau )$ and the Claim 3.5, we have

$$ \begin{align*} \begin{aligned} \{\mathbb{Z}_{*}^{\infty}\setminus\Lambda_{1}(\varepsilon^{1/12})\}\cap\{k\in\mathbb{Z}_{*}^{\infty}:|k|_{\eta}\leq N'\}&=\{0\},\\ \{\mathbb{Z}_{*}^{\infty}\setminus\Lambda_{2}(\varepsilon^{1/12})\}\cap\{k\in\mathbb{Z}_{*}^{\infty}:|k|_{\eta}\leq N'\}&=\{k_{*}\}, \end{aligned} \end{align*} $$

where $N'\triangleq 2N^{1+({\eta }/{2})}-N$ . Thus, $h^{re}(x)\in \mathcal {B}_{\sigma }^{re}(\varepsilon ^{1/12})$ can be rewritten as

$$ \begin{align*} \begin{aligned} h^{re}(x)&=h^{re}_{0}(x)+h^{re}_{1}(x)+h^{re}_{2}(x)\\ &=\left( \begin{matrix} i\widehat{t}(0)\ \ \ \ \ \ \ 0 \\ 0\ \ \ -i\widehat{t}(0) \end{matrix} \right)+\ \left( \begin{matrix} 0\ \ \ \ \ \ \widehat{v}(k_{*})e^{i2\pi\langle k_{*},x\rangle} \\ \overline{\widehat{v}}(k_{*})e^{-i2\pi\langle k_{*},x\rangle}\ \ \ \ \ \ 0 \end{matrix} \right)+\sum_{|k|_{\eta}>N'}\widehat{h}^{re}(k)e^{i2\pi\langle k,x\rangle} \end{aligned} \end{align*} $$

with

(3.21) $$ \begin{align} \|h^{re}_{j}\|_{\sigma}\leq\|h^{re}\|_{\sigma}\leq 2\varepsilon',\quad j=0,1,2. \end{align} $$

Define $Q:2\mathbb {T}^{\infty }\rightarrow SU(1,1)$ as

$$ \begin{align*} \begin{aligned} Q(x)=\left( \begin{matrix} e^{-i\pi\langle k_{*},x\rangle}\ \ \ \ \ \ 0 \\ 0\ \ \ \ \ \ \ e^{i\pi\langle k_{*},x\rangle} \end{matrix} \right)\!, \end{aligned} \end{align*} $$

so we have

(3.22) $$ \begin{align} |Q(x)|_{\delta}\leq e^{\pi\delta|k_{*}|_{\eta}}\leq \varepsilon^{{-\delta}/({\sigma-\delta})}, \end{align} $$

where the last inequality is from $|k_{*}|_{\eta }\leq N=\{(\sigma -\delta )\pi \}^{-1}\ln \varepsilon ^{-1}.$ One can also show

$$ \begin{align*} Q(x+\alpha)(A_{*}e^{h^{re}(x)})Q(x)^{-1}=A^{\prime}_{*}e^{h^{\prime re}(x)}, \end{align*} $$

where

$$ \begin{align*} \begin{aligned} A^{\prime}_{*}=Q(x+\alpha)A_{*}Q(x)^{-1}=\left( \begin{matrix} e^{i2\pi(\rho-{\langle k_{*},\alpha\rangle}/{2})}\ \ \ \ \ \ \ \ \ \ 0 \\ 0\ \ \ \ \ \ \ \ \ \ e^{-i2\pi(\rho-{\langle k_{*},\alpha\rangle}/{2})} \end{matrix} \right)\!, \end{aligned} \end{align*} $$

and

$$ \begin{align*} \begin{aligned} h^{\prime re}(x)=Q(x)h^{re}(x)Q(x)^{-1}= \sum_{j=0}^{2}Q(x)h_{j}^{re}(x)Q(x)^{-1}. \end{aligned} \end{align*} $$

Moreover,

$$ \begin{align*} \begin{aligned} Q(x)h_{0}^{re}(x)Q(x)^{-1}=h_{0}^{re}(x),\quad \ Q(x)h_{1}^{re}(x)Q(x)^{-1}=\left( \begin{matrix} 0\ \ \ \ \ \ \widehat{v}(k_{*}) \\ \bar{\widehat{v}}(k_{*})\ \ \ \ \ \ 0 \end{matrix} \right)\!, \end{aligned} \end{align*} $$

$$ \begin{align*} \begin{aligned} M^{-1}Q(x)M=\left( \begin{matrix} \cos\pi\langle k_{*},x\rangle \ \ \ \ -\sin\pi\langle k_{*},x\rangle \\ \sin\pi\langle k_{*},x\rangle\ \ \ \ \ \ \cos\pi\langle k_{*},x\rangle \end{matrix} \right)\!. \end{aligned} \end{align*} $$

Denote

$$ \begin{align*} \begin{aligned} L&=Q(x)h_{0}^{re}(x)Q(x)^{-1}+Q(x)h_{1}^{re}(x)Q(x)^{-1},\\ F&=Q(x)h_{2}^{re}(x)Q(x)^{-1} ,\\ B&=Q(x)e^{Y(x)}P:2\mathbb{T}^{\infty}\rightarrow SU(1,1). \end{aligned} \end{align*} $$

Then, by the discussions above, we have

$$ \begin{align*} \begin{aligned} B(x+\alpha)(Ae^{f(x)})B(x)^{-1}=A^{\prime}_{*}e^{h^{\prime re}(x)}:= A_{+}e^{f_{+}(x)}, \end{aligned} \end{align*} $$

where

(3.23) $$ \begin{align} \begin{aligned} A_{+}&=A^{\prime}_{*}e^{L}:=e^{A_{+}"}\\ &=\left( \begin{matrix} e^{i2\pi(\rho-{\langle k_{*},\alpha\rangle}/{2})}\ \ \ \ \ \ \ \ \ \ 0 \\ 0\ \ \ \ \ \ \ \ \ \ e^{-i2\pi(\rho-{\langle k_{*},\alpha\rangle}/{2})} \end{matrix} \right)\exp \left( \begin{matrix} i\widehat{t}(0)\ \ \ \ \ \ \widehat{v}(k_{*}) \\ \bar{\widehat{v}}(k_{*})\ \ \ -i\widehat{t}(0) \end{matrix} \right)\!,\\ e^{f_{+}(x)}&=e^{-L}e^{h^{\prime re}(x)}=e^{-L}e^{L+F}. \end{aligned} \end{align} $$

Now, we give the estimates of B and the cocycle $(\alpha ,A_{+}e^{f_{+}(x)}).$ First, Proposition 2.3 shows

$$ \begin{align*} \begin{aligned} \rho(\alpha,A_{+}e^{f_{+}(x)})=\rho(\alpha,Ae^{f(x)})+\frac{\langle k_{*},\alpha\rangle}{2}, \end{aligned} \end{align*} $$

and the estimates in equations (3.19)–(3.22) yield

$$ \begin{align*} \begin{aligned} |B|_{\delta}&\leq e^{\pi\delta|k_{*}|_{\eta}}{\varepsilon}^{-{1}/{480}},~\|B\|\leq\varepsilon^{-{1}/{480}},\\ |\widehat{t}(0)|&\leq 2\varepsilon',~ |\widehat{v}(k_{*})|\leq2\varepsilon'e^{-2\pi\sigma| k_{*}|_{\eta}},\\ |F|_{\delta}&\leq\varepsilon^{-{2\delta}/({\sigma-\delta})}|h_{2}^{re}|_{\delta}\leq\varepsilon^{-{2\delta}/({\sigma-\delta})}e^{-2\pi N'(\sigma-\delta)}|h_{2}^{re}|_{\sigma}\\ &\leq2\varepsilon^{1-{1}/{240}}\varepsilon^{4N^{{\eta}/{2}}}\varepsilon^{-{2\sigma}/({\sigma-\delta})}\ll \varepsilon^{10}. \end{aligned} \end{align*} $$

The estimates above imply that the $B,f_{+}$ and $A_{+}^{"}$ defined by equation (3.23) are those we need.

3.2 Reducibility of almost-periodic cocycle

Given any $\sigma \in (0,\sigma _{0})$ , define

(3.24) $$ \begin{align} \sigma_{j+1}=\sigma_{j}-\frac{\sigma_{0}-\sigma}{12(j+1)^2},\quad j\geq1, \end{align} $$

then,

$$ \begin{align*} \sigma_{\infty}=\sigma_{0}-\sum\limits_{i=0}^{+\infty}(\sigma_{i}-\sigma_{i+1})\geq\frac{5\sigma_{0}+\sigma}{6}>\sigma. \end{align*} $$

Let

(3.25) $$ \begin{align} \begin{aligned} \varepsilon_{0}<\varepsilon_{*}:=D(c)^{-1} \exp\{-\{[12(N^{*}+1)^2][(\sigma_{0}-\sigma)\pi]^{-1}\} ^{{2(2+\eta)}/{\eta}}\}, \end{aligned} \end{align} $$

where $D(c)$ is the one in equation (3.4) and

(3.26) $$ \begin{align} N^{*}=(3^{{\eta}/{4(2+\eta)}}-1)^{-1}. \end{align} $$

Moreover, we also define

(3.27) $$ \begin{align} \varepsilon_{j+1}=4\varepsilon_{j}^3,\quad N_{j}=\{(\sigma_{j}-\sigma_{j+1})\pi\}^{-1} \ln\varepsilon_{j}^{-1},\quad j\geq 0. \end{align} $$

Remark 3.6. Assume we arrive at the $(j+1)$ th step’s cocycle $(\alpha ,A_{j}e^{f_{j}})$ satisfying the hypotheses of Proposition 3.3 with $\sigma ,\delta $ and $A, f$ being replaced by $\sigma _{j},\sigma _{j+1}$ and $A_{j}, f_{j},$ then we can apply Proposition 3.3 and get a new cocycle $(\alpha ,A_{j+1}e^{f_{j+1}}).$ The conclusions of Proposition 3.3 show that $|f_{j+1}|_{\sigma _{j+1}}\leq \varepsilon _{j+1}$ for both the resonant and non-resonant cases. The choice of $\varepsilon _{*}$ defined by equation (3.25) with $N^{*}$ being given by equation (3.26) ensures that

(3.28) $$ \begin{align} \varepsilon_{j+1}\leq D(c)^{-1} \exp\{-[(\sigma_{j+1}-\sigma_{j+2})\pi] ^{({-2(2+\eta)})/{\eta}}\}. \end{align} $$

That is, $(\alpha ,A_{j+1}e^{f_{j+1}})$ satisfies the hypotheses of Proposition 3.3 with $\sigma ,\delta $ and $A, f$ being replaced by $\sigma _{j+1},\sigma _{j+2}$ and $A_{j+1}, f_{j+1}.$ That is, we can apply Proposition 3.3 to $(\alpha ,A_{j}e^{f_{j}}) \text { for all } j\geq 0.$

Proposition 3.7. Let $\alpha \in DC_{\infty }(\gamma ,\tau )$ , $\gamma>0$ , $0<\sigma _{0}<({1}/{10})$ , $\tau>1$ , $\eta>0$ ,

(3.29) $$ \begin{align} K_{j}&= \{ E\in\Sigma~| \text{ there exists } k_{j-1}\in\mathbb{Z}_{*}^{\infty},\text{ with}\ 0<|k_{j-1}|_{\eta}\leq N_{j-1} \nonumber\\&\qquad\text{such that}~\|2\rho(A_{j-1})-\langle k_{j-1},\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}<\varepsilon_{j-1}^{1/12} \}, \quad j\geq1. \end{align} $$

Suppose that $A_{0}\in SL(2,\mathbb {R})$ , $f_{0}\in \mathcal {H}_{\sigma _{0}}(\mathbb {T}^{\infty },sl(2,\mathbb {R}))$ , and $ |f|_{\sigma _{0}}\leq \varepsilon _{0}.$ Then, the following conclusions hold. (1) The system $(\alpha ,A_{0}e^{f_{0}(x)})$ is almost reducible in the strip $|\Im x|<\sigma $ . Moreover, there exists $\widetilde {B}_{j-1} \in \mathcal {H}_{\sigma _{j}}(2\mathbb {T}^{\infty },SL(2,\mathbb {R})),$ such that

(3.30) $$ \begin{align} \widetilde{B}_{j-1}(x+\alpha)A_{0}e^{f_{0}(x)} \widetilde{B}_{j-1}(x)^{-1}=A_{j}e^{f_{j}(x)}, \quad j\geq1 \end{align} $$

with estimates

(3.31) $$ \begin{align} \|\widetilde{B}_{j-1}(x)\|\leq \varepsilon_{j-1}^{-{1}/{320}},\ \ |f_{j}|_{\sigma_{j}}\leq\varepsilon_{j}, \quad j\geq1. \end{align} $$

Assume the $(j_{\ell }+1)$ th KAM step is the $\ell $ th resonant step with $\ell =i,i+1,$ then

(3.32) $$ \begin{align} N_{j_{i}}^{({4+4\eta})/({4+3\eta})}<|k_{j_{i+1}}|_{\eta}\leq N_{j_{i+1}}. \end{align} $$

(2) There exist unitary matrices $U_{j}\in SL(2, \mathbb {C})$ such that

(3.33) $$ \begin{align} \begin{aligned} U_{j}A_{j}U_{j}^{-1}=\left( \begin{matrix} e^{i2\pi\rho_{j}}\ \ \ \ \ c_{j} \\ 0\ \ \ \ \ e^{-i2\pi\rho_{j}} \end{matrix} \right)\!\quad\text{for all } j\in\mathbb{N} \end{aligned} \end{align} $$

and

(3.34) $$ \begin{align} \begin{aligned} \|\widetilde{B}_{j-1}\|^2\cdot|c_{j}|\leq 8\|A_{0}\|\quad \text{for all } j\in\mathbb{N}, \end{aligned} \end{align} $$

where $\widetilde {B}_{j-1}$ is the one in equation (3.30). Moreover, if $E\in K_{j},$ then

(3.35) $$ \begin{align} \begin{aligned} \rho(\alpha,A_{j}e^{f_{j}(x)})&= \rho(\alpha,A_{j-1}e^{f_{j-1}(x)})+2^{-1}\langle k_{j-1},\alpha\rangle,\\ |c_{j}|&< 4\varepsilon^{1/12}_{j-1}. \end{aligned} \end{align} $$

In this case, set $ A_{j}= e^{A_{j}"}$ , then

(3.36) $$ \begin{align} \begin{aligned} MA_{j}"M^{-1}=\left( \begin{matrix} it_{j}\ \ \ \ \ \ \ v_{j} \\ \bar{v}_{j}\ \ \ \ -it_{j} \end{matrix} \right) \end{aligned} \end{align} $$

with

(3.37) $$ \begin{align} \begin{aligned} |t_{j}|\leq4\varepsilon_{j-1}^{1/12},\quad |v_{j}|\leq2\varepsilon_{j-1}^{1-{1}/{240}}e^{-2\pi| k_{j-1}|_{\eta}\sigma_{j-1}},\quad \|A_{j}"\|\leq4\varepsilon_{j-1}^{1/12}. \end{aligned} \end{align} $$

(3) If there exists $ k\in \mathbb {Z}_{*}^{\infty }\setminus \{0\}$ , such that

(3.38) $$ \begin{align} \rho(\alpha,A_{0}e^{f_{0}(x)})=2^{-1}\langle k,\alpha\rangle, \end{align} $$

and $(\alpha ,A_{0}e^{f_{0}(x)})$ is not uniformly hyperbolic, then the resonant case defined in Proposition 3.3 only happens finite times. Moreover, there exists $\widetilde {B} \in \mathcal {H}_{\sigma }(2\mathbb {T}^{\infty },SL(2,\mathbb {R}))$ such that

$$ \begin{align*} \begin{aligned} \widetilde{B}(x+\alpha)A_{0}e^{f_{0}(x)}\widetilde{B}(x)^{-1} =\left( \begin{matrix} 1 \ \ \ \zeta \\ 0\ \ \ 1 \end{matrix} \right)\!, \end{aligned} \end{align*} $$

with estimate $|\zeta |\leq 2\varepsilon _{j_{n}}^{1-{1}/{120}}e^{-({25}/{16})\pi \sigma _{0} |k|_{\eta }}$ , and $|\widetilde {B}(x)|_{\sigma }\leq 4 {\varepsilon _{j_{n}}}^{-{1}/{320}} e^{(8/7)\pi \sigma |k|_{\eta }}$ , where $j_{n}$ is the index of last resonant site $k_{j_{n}}$ .

Proof. We give the conclusions in part (1) by induction.

First step. The estimate of $\varepsilon _{0}$ defined by equation (3.25) enables us to apply Proposition 3.3 to $(\alpha ,A_{0}e^{f_{0}(x)})$ and obtain the following. There exists $\widetilde {B}_{0}\in \mathcal {H}_{\sigma _{1}}(2\mathbb {T}^{\infty }, SL(2,\mathbb {R}))$ such that

$$ \begin{align*} \widetilde{B}_{0}(x+\alpha)(A_{0}e^{f_{0}(x)}) \widetilde{B}_{0}(x)^{-1}=A_{1}e^{f_{1}(x)} \end{align*} $$

with the following estimates:

$$ \begin{align*} |f_{1}|_{\sigma_{1}}\leq\varepsilon_{1},\quad \|\widetilde{B}_{0}\|\leq\varepsilon_{0}^{-{1}/{320}}. \end{align*} $$

That is, we get a new cocycle $(\alpha ,A_{1}e^{f_{1}(x)})$ with estimates (3.30) and (3.31) with $j=1$ . Moreover, if there exists $ k_{0}\in \mathbb {Z}_{*}^{\infty }$ , $0<|k_{0}|_{\eta }\leq N_{0}$ , such that $\|2\rho -\langle k_{0},\alpha \rangle \|_{\mathbb {R}/\mathbb {Z}}<\varepsilon _{0}^{1/12},$ then

$$ \begin{align*} \rho(\alpha,A_{1}e^{f_{1}(x)})=\rho(\alpha,A_{0}e^{f_{0}(x)})+\frac{\langle k_{0},\alpha\rangle}{2}. \end{align*} $$

By Remark 3.6, we know that equation (3.28) holds with $j=0.$ That is, $(\alpha ,A_{1}e^{f_{1}(x)})$ satisfies the hypotheses of Proposition 3.3 with $\sigma ,\delta $ and $A, f$ being replaced by $\sigma _{1},\sigma _{2}$ and $A_{1}, f_{1},$ respectively.

Inductive step. Assume that we have completed the lth step and are at the $(l+1)$ th KAM step with $l\geq 1$ , that is, we already construct $\widetilde {B}_{l-1} \in \mathcal {H}_{\sigma _{l}}(2\mathbb {T}^{\infty },SL(2,\mathbb {R}))$ such that the estimates in equations (3.30) and (3.31) hold with $j=1,\ldots ,l$ . Now we consider the $(l+1)$ th step. By Remark 3.6, we know that equation (3.28) holds with $j=l-1.$ Then, by applying Proposition 3.3, we know that there exists $B_{l} \in \mathcal {H}_{\sigma _{l+1}}(2\mathbb {T}^{\infty },SL(2,\mathbb {R}))$ , $A_{l+1}\in SL(2,\mathbb {R})$ , and $f_{l+1}\in \mathcal {H}_{\sigma _{l+1}}(\mathbb {T}^{\infty },sl(2,\mathbb {R}))$ such that

$$ \begin{align*} B_{l}(x+\alpha)(A_{l}e^{f_{l}(x)})B_{l}(x)^{-1}=A_{l+1}e^{f_{l+1}(x)}. \end{align*} $$

If $\|2\rho _{l}-\langle k,\alpha \rangle \|_{\mathbb {R}/\mathbb {Z}}\geq \varepsilon _{l}^{1/12}$ holds for any $k\in \mathbb {Z}_{*}^{\infty }$ with $0<|k|_{\eta }\leq N_{l},$ then the conclusions of the $\textit {non-resonant case}$ in Proposition 3.3 show

(3.39) $$ \begin{align} |B_{l}-\mathrm{Id}|_{\sigma_{l+1}}\leq\varepsilon_{j}^{{1}/{2}},\quad |f_{l+1}|_{\sigma_{l+1}}\leq4\varepsilon_{l}^3=\varepsilon_{l+1},\quad \|A_{l+1}-A_{l}\|\leq2\varepsilon_{l}. \end{align} $$

However, if there exists $k_{l}\in \mathbb {Z}_{*}^{\infty }$ with $0<|k_{l}|_{\eta }\leq N_{l}$ such that

(3.40) $$ \begin{align} \|2\rho_{l}-\langle k_{l},\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}< \varepsilon_{l}^{1/12}, \end{align} $$

then the conclusions of the $\textit {resonant case}$ in Proposition 3.3 show

(3.41) $$ \begin{align} |B_{l}|_{\sigma_{l+1}}\leq e^{\pi\sigma_{l+1}|k_{l}|_{\eta}}{\varepsilon_{l}}^{-{1}/{480}},\quad \|B_{l}\|\leq\varepsilon_{l}^{-{1}/{480}},\quad |f_{l+1}(x)|_{\sigma_{l+1}}<\varepsilon_{l+1}, \end{align} $$

$$ \begin{align*} \rho(\alpha,A_{l+1}e^{f_{l+1}(x)})=\rho(\alpha,A_{l}e^{f_{l}(x)})+\frac{\langle k_{l},\alpha\rangle}{2}, \end{align*} $$

and

(3.42) $$ \begin{align} A_{l+1}= e^{A_{l+1}"}, \quad \|A_{l+1}"\|<4\varepsilon_{l}^{1/12}, \end{align} $$

(3.43) $$ \begin{align} \begin{aligned} MA_{l+1}"M^{-1}=\left( \begin{matrix} it_{l+1}\ \ \ \ \ \ \ v_{l+1} \\ \bar{v}_{l+1}\ \ \ \ -it_{l+1} \end{matrix} \right) \end{aligned} \end{align} $$

with

(3.44) $$ \begin{align} |t_{l+1}|\leq4\varepsilon_{l}^{1/12},\quad |v_{l+1}|\leq2\varepsilon_{l}^{1-{1}/{240}}e^{-2\pi| k_{l}|_{\eta}\sigma_{l}}. \end{align} $$

Let $\widetilde {B}_{l}=B_{l}\widetilde {B}_{l-1}.$ Then, we get equation (3.30) with $j=l+1.$ Moreover, the estimates in equation (3.31) with $j=l$ and equations (3.39), (3.41), yield

$$ \begin{align*} \|\widetilde{B}_{l}(x)\|\leq\|B_{l}(x)\|\|\widetilde{B}_{l-1}(x)\|\leq \varepsilon_{l}^{-{1}/{320}}. \end{align*} $$

That is the estimates in equation (3.31) hold with $j=l+1$ .

Inductively, we have proved that estimates in equations (3.30) and (3.31) hold for all $j\in \mathbb {N},$ which imply that $(\alpha ,A_{0}e^{f_{0}(x)})$ is almost reducible in the strip $|\Im x|<\sigma $ ( $\sigma _{\infty }>\sigma $ ).

Now, we will verify equation (3.32). We give the lower bound of $\rho _{j_{i+1}}$ first. Suppose there are two resonance sites $k_{j_{i}}$ , $k_{j_{i+1}}$ , which happen at the KAM steps $(j_{i}+1)$ , $(j_{i+1}+1)$ , respectively. Thus, $\|2\rho _{j_{i+1}}-\langle k_{j_{i+1}},\alpha \rangle \|_{\mathbb {R}/\mathbb {Z}}<\varepsilon ^{1/12}_{j_{i+1}},$ which yields

(3.45) $$ \begin{align} \begin{aligned} 2\rho_{j_{i+1}}&>\|\langle k_{j_{i+1}},\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}-\varepsilon^{1/12}_{j_{i+1}}\\ &>2^{-1}\|\langle k_{j_{i+1}},\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}} > \frac{\gamma}{2}(1+|k_{j_{i+1}}|_{\eta})^{-c(\eta,\tau)|k_{j_{i+1}}|_{\eta}^{{1}/({1+\eta})}}, \end{aligned} \end{align} $$

where the last two inequalities above are given by equations (3.9) and (3.2).

We consider the upper bound of $\rho _{j_{i+1}}.$ According to Proposition 3.3, after the $(j_{i}+1)$ th KAM step, we get cocycle $(\alpha ,A_{j_{i}+1}e^{f_{j_{i}+1}}),$ then Shur theorem implies that there exists a unitary matrix $U_{j_{i}+1}$ such that

$$ \begin{align*} \begin{aligned} U_{j_{i}+1}A^{\prime\prime}_{j_{i}+1}U^{-1}_{j_{i}+1}=\left( \begin{matrix} i2\pi\rho_{j_{i}+1}\ \ \ \ \ \ \ \ c^{\prime\prime}_{j_{i}+1} \\ 0\ \ \ \ \ \ -i2\pi\rho_{j_{i}+1} \end{matrix} \right)\!, \end{aligned} \end{align*} $$

which, together with equation (3.42) yields (see the details to obtain equation (3.47))

$$ \begin{align*} |2\pi\rho_{j_{i}+1}| \leq\|A^{\prime\prime}_{j_{i}+1}\|<4\varepsilon_{j_{i}}^{1/12}. \end{align*} $$

Moreover, equations (2.7) and (3.39) imply

(3.46) $$ \begin{align} \begin{aligned} |\rho_{j_{i+1}}-\rho_{j_{i}+1}|& \leq \sum^{j_{i+1}-1}_{m=j_{i}+1}\|A_{m+1}-A_{m}\|^{{1}/{2}}\leq c\varepsilon_{j_{i}}^{{3}/{2}}. \end{aligned} \end{align} $$

The inequalities (3.9), (3.45), and (3.46) yield

$$ \begin{align*} \frac{\gamma}{4}(1+|k_{j_{i+1}}|_{\eta})^{-c(\eta,\tau) |k_{j_{i+1}}|_{\eta}^{{1}/({1+\eta})}} &<\rho_{j_{i+1}}<2\varepsilon_{j_{i}}^{1/12} <\frac{\gamma}{4}\varepsilon_{j_{i}}^{{1}/{24}} (1+N_{j_{i}})^{-10c(\eta,\tau)N_{j_{i}}^{{1}/({1+\eta})}}. \end{align*} $$

Then, we have $N_{j_{i}}^{({4+4\eta })/({4+3\eta })}<|k_{j_{i+1}}|_{\eta },$ and equation (3.32) is proved.

Now, we will verify the conclusions in $\textrm {(2)}$ . Obviously, equation (3.33) holds. Now, we assume $E\in K_{j}$ , then the estimates in equations (3.42)–(3.44) with $l=j-1$ yield the conclusions in equations (3.36) and (3.37). Moreover, for the unitary matrix $U_{j}$ in equation (3.33), we get

$$ \begin{align*} \begin{aligned} U_{j}A^{\prime\prime}_{j}U^{-1}_{j}=\left( \begin{matrix} i2\pi\rho_{j}\ \ \ \ \ \ \ \ c^{\prime\prime}_{j} \\ 0\ \ \ \ \ \ -i2\pi\rho_{j} \end{matrix} \right)\!. \end{aligned} \end{align*} $$

Thus, the estimates in equation (3.42) with $l=j-1$ yield

(3.47) $$ \begin{align} |c^{\prime\prime}_{j}|,|2\pi\rho_{j}|\leq\|A_{j}"\|<4\varepsilon_{j-1}^{1/12}. \end{align} $$

Thus,

$$ \begin{align*} \begin{aligned} U_{j}A_{j}U^{-1}_{j}=e^{U_{j}A^{\prime\prime}_{j}U^{-1}_{j}}=\left( \begin{matrix} e^{i2\pi\rho_{j}}\ \ \ \ \ \ \ \ c_{j} \\ 0\ \ \ \ \ \ \ \ e^{-i2\pi\rho_{j}} \end{matrix} \right)\!,\\ \end{aligned} \end{align*} $$

where $c_{j}=\sum \nolimits _{n=0}^{+\infty }({1}/{(2n+1)!})c^{\prime \prime }_{j}(i2\pi \rho _{j})^{2n}$ . Then, $|c_{j}|\leq |c^{\prime \prime }_{j}|< 4\varepsilon ^{1/12}_{j-1}$ . That is, we get the estimate about $c_{j}$ in equation (3.35). The equality about the rotation number in equation (3.35) holds obviously, we omit the details.

Now, we come to the inequality in equation (3.34). Since the steps between $j_{i}+1$ , $j_{i+1}+1$ are all non-resonant steps, then equation (3.31) with $j_{i}=j-1,$ equation (3.42) with $l=j_{i},$ and equation (3.39) yield, for all $ j_{i}+1< j< j_{i+1}+1,$

(3.48) $$ \begin{align}\begin{aligned} \|A^{\prime\prime}_{j}\|\leq\|A^{\prime\prime}_{j_{i}+1}\|+\sum^{j-1}_{m=j_{i}+1}\|A_{m+1}-A_{m}\|^{{1}/{2}}\leq5\varepsilon_{j_{i}}^{1/12},\\|\widetilde{B}_{j-1}(x)|_{0}\leq2\varepsilon_{j_{i}}^{-{1}/{320}}. \qquad\qquad\ \ \qquad\end{aligned} \end{align} $$

The inequality above, together with equation (3.33), yields

(3.49) $$ \begin{align} |c_{j}|\leq5\varepsilon_{j_{i}}^{1/12},\quad j_{i}+1< j< j_{i+1}+1. \end{align} $$

The estimates (3.31), (3.35), (3.48), and (3.49) enable us to get the following conclusions.

1. (I) After each resonant step j, equations (3.31) and (3.35) yield

   $$ \begin{align*} \|\widetilde{B}_{j-1}(x)\|^2|c_{j}|\leq 4\varepsilon_{j}^{{37}/{480}}\leq8\|A_{0}\|. \end{align*} $$

2. (II) After each non-resonant step j, we are able to use the estimates of the last resonant step $j_{i}+1$ if it exists. Moreover, equations (3.48) and (3.49) imply

   $$ \begin{align*} \|\widetilde{B}_{j-1}(x)\|^2|c_{j}|\leq20\varepsilon_{j_{i}}^{{37}/{480}}\leq8\|A_{0}\|. \end{align*} $$

3. (III) It is possible that no resonant steps happened within the first j steps. In this case, each step is non-resonant and thus we can use the estimate $\| \widetilde {B}_{j-1} \| < 2$ . Then, $|c_{j}|\leq \|A_{j}\|\leq 2\|A_{0}\|$ , which implies $\|\widetilde {B}_{j-1}\|^2|c_{j}|\leq 8\|A_{0}\|$ .

The discussions in conclusions (I)–(III) yield equation (3.34).

Finally, we come to part (3). Assume that the resonance occurs at $j_{l}+1$ th step’s cocycle $(\alpha ,A_{j_{l}}e^{f_{j_{l}}}),l\in \mathbb {N}$ . For the k in equation (3.38), if there exists $p\in \mathbb {N}$ such that

(3.50) $$ \begin{align} N_{j_{p}-1}<|k|_{\eta}\leq N_{j_{p}}, \end{align} $$

then, in the following, we will prove that there is no $j_{p+1}$ .

Assume there exists such $j_{p+1}$ with $0<|k_{j_{p+1}}|_{\eta }<\infty $ satisfying

$$ \begin{align*} \|\langle k_{j_{p+1}},\alpha\rangle-2\rho_{j_{p+1}}\|_{\mathbb{R}/\mathbb{Z}}<\varepsilon_{j_{p+1}}^{1/12}. \end{align*} $$

In this case, the estimates in equation (3.47) with $j=j_{p+1}+1$ and equation (2.7) show

$$ \begin{align*} |\rho(\alpha,A_{j_{p+1}+1})|<\varepsilon_{j_{p+1}}^{1/12} \end{align*} $$

and

$$ \begin{align*} |2\rho(\alpha,A_{j_{p+1}+1}e^{f_{j_{p+1}+1}})-2\rho(\alpha,A_{j_{p+1}+1})| \leq 2\varepsilon_{j_{p+1}+1}^{{1}/{2}}, \end{align*} $$

respectively. The two inequalities above yield

(3.51) $$ \begin{align} |2\rho(\alpha,A_{j_{p+1}+1}e^{f_{j_{p+1}+1}})|\leq4\varepsilon_{j_{p+1}}^{1/12}. \end{align} $$

However, the two inequalities in equation (3.32) imply, for $1\leq m\leq p+1$ ,

(3.52) $$ \begin{align} \sum\limits_{l=1}^{m} |k_{j_{l}}|_{\eta}\leq\sum\limits_{l=1}^{m-1} |N_{j_{l}}|_{\eta}+|k_{j_{m}}|_{\eta}\leq \sum\limits_{i=0}^{m-1} |k_{j_{m}}|_{\eta}^{(({4+3\eta})/({4+4\eta}))^{i}} \leq \frac{31}{30}|k_{j_{m}}|_{\eta}. \end{align} $$

Set $\widetilde {k}:=k+\sum _{l=1}^{p+1} k_{j_{l}},$ then equations (3.52), (3.50), and (3.32) yield

(3.53) $$ \begin{align} \frac{16}{15}N_{j_{p+1}}\geq|\widetilde{k}|_{\eta}=\bigg|k_{j_{p+1}}+k+\sum\limits_{l=1}^{p} k_{j_{l}}\bigg|_{\eta}\geq |k_{j_{p+1}}|_{\eta}-3N_{j_{p}}>\frac{14}{15}N_{j_{p}}. \end{align} $$

Moreover, the estimates in equationsa (3.38) and (3.35) yield

$$ \begin{align*} 2\rho(\alpha,A_{j_{p+1}+1}e^{f_{j_{p+1}+1}})=\bigg\langle k+\sum_{l=1}^{p+1} k_{i_{l}},\alpha\bigg\rangle=\langle \widetilde{k},\alpha\rangle, \end{align*} $$

which, together with equations (3.53) and (3.9), yields

$$ \begin{align*} 2|\rho(\alpha,A_{j_{p+1}+1}e^{f_{j_{p+1}+1}})|=\|\langle \widetilde{k},\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}} \geq\gamma(1+2N_{j_{p+1}})^{-c(\eta,\tau)N_{j_{p+1}}^{{1}/({1+\eta})}}>\varepsilon_{j_{p+1}}^{1/120}. \end{align*} $$

The inequality above, together with equation (3.51), implies

$$ \begin{align*} \varepsilon_{j_{p+1}}^{1/120}<2\rho(\alpha,A_{j_{p+1}+1}e^{f_{j_{p+1}+1}})\leq 4\varepsilon_{j_{p+1}}^{1/12}, \end{align*} $$

which is a contradiction, so there is no $j_{p+1}.$ Thus, $2\rho (\alpha ,A_{j_{p}+1}e^{f_{j_{p}+1}})=0$ , otherwise, there exists the $(p+1)$ th resonance.

The discussions above mean that the resonant case occurs only finitely many times in the above almost reducibility procedure. By the estimates of $B_{j}$ in equation (3.39) and the sequence $(\sigma _{j})_{j\in \mathbb {N}}$ given in equation (3.24), we see that the product $\prod _{l=0}^{j}B_{l}$ converges to some $B\in \mathcal {H}_{\sigma }(2\mathbb {T}^{\infty },SL(2,\mathbb {R}))$ such that $B(x+\alpha ) A_{0} e^{f_{0}(x)} B(x)^{-1} = A_{\infty },$ with

(3.54) $$ \begin{align} \rho(\alpha,A_{\infty})=\rho(\alpha,A_{j_{p}+1}e^{f_{j_{p}+1}})=0. \end{align} $$

Assuming that there are actually n times resonant steps, associated with integers vectors

$$ \begin{align*} k_{j_{l}}\in\mathbb{Z}_{*}^{\infty}\quad \text{with } 0<|k_{j_{l}}|_{\eta}\leq N_{j_{l}}, l=1,\ldots,n, \end{align*} $$

then, $k=k_{j_{1}}+\cdots +k_{j_{n}}$ . In view of inequality (3.52), equation (3.32) with similar calculation of equation (3.53), we get

(3.55) $$ \begin{align} \frac{14}{15}|k_{j_{n}}|_{\eta}\leq|k|_{\eta}\leq \frac{16}{15} |k_{j_{n}}|_{\eta}. \end{align} $$

Now we estimate the constant matrix $A_{\infty }$ . The fact that we have assumed that the initial cocycle $(\alpha ,A_{0}e^{f_{0}(x)})$ is not hyperbolic and equation (3.54) imply $A_{\infty }$ is a parabolic matrix. As $A_{\infty }\in SL(2,\mathbb {R})$ , we have $A_{\infty }=e^{A_{\infty }"}$ with $A_{\infty }"\in sl(2,\mathbb {R})$ and $\mathrm {det}A_{\infty }"=0$ . Assume that $A_{\infty }"=( \begin {smallmatrix} a_{11} \ \ \ \ \ a_{12} \\ a_{21}\ \ -a_{11} \end {smallmatrix} )$ , then there exists $\phi \in \mathbb {T} $ such that $R_{\phi }A_{\infty }"R_{-\phi }=( \begin {smallmatrix} 0\ \ \ a_{12}-a_{21} \\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ 0 \end {smallmatrix} )$ . Let $\widetilde {B}=R_{\phi }B$ , and $\zeta =a_{12}-a_{21}$ , we can see that the cocycle $(\alpha ,A_{0}e^{f_{0}(x)})$ is conjugated to $A_{\infty }=( \begin {smallmatrix} 1 \ \ \ \zeta \\ 0\ \ \ 1 \end {smallmatrix} )$ by $\widetilde {B}(x)$ .

To estimate $|\zeta |$ , let us focus on $(\alpha ,A_{j_{n}+1}e^{f_{j_{n}+1}(x)})$ , which is obtained by the last resonant step. In the following, we will estimate the constant matrix $A_{\infty }$ . Obviously,

$$ \begin{align*} \begin{aligned} A_{\infty}=e^{A^{\prime\prime}_{\infty}}=M^{-1}\exp\left( \begin{matrix} i \beta_{11}\ \ \ \ \ \ \ \beta_{12} \\ \bar{\beta}_{12} \ \ \ \ \ -i \beta_{11} \end{matrix} \right)M, \end{aligned} \end{align*} $$

where $\beta _{11}\in \mathbb {R}$ , $\beta _{12}\in \mathbb {C}$ . Since by equations (3.36) and (3.37) with $j_{n}+1$ in place of j, and Proposition 3.3, it follows that

$$ \begin{align*} \begin{aligned} |\beta_{12}|&=|(M(A_{\infty}"-A_{j_{n}+1}") M^{-1})_{12}+(M A_{j_{n}+1}" M^{-1})_{12}|\\ & \leq 16\varepsilon_{j_{n}}^{3}+2\varepsilon_{j_{n}}^{1-{1}/{240}}e^{-2\pi| k_{j_{n}}|_{\eta}\sigma_{j_{n}}} \leq \varepsilon_{j_{n}}^{1-{1}/{120}}e^{-2\pi| k_{j_{n}}|_{\eta}\sigma_{j_{n}}}. \end{aligned} \end{align*} $$

Then, we have $|\beta _{11}|\leq \varepsilon _{j_{n}}^{1-{1}/{120}}e^{-2\pi | k_{j_{n}}|_{\eta }\sigma _{j_{n}}}$ since $\mathrm {det}A_{\infty }"=0$ . Thus, by equation (3.55),

$$ \begin{align*} \begin{aligned} |\zeta|=|a_{12}-a_{21}| &\leq|\beta_{11}|+|\beta_{12}| \leq2\varepsilon_{j_{n}}^{1-{1}/{120}}e^{-2\pi| k_{j_{n}}|_{\eta}\sigma_{j_{n}}}\\ &\leq 2\varepsilon_{j_{n}}^{1-{1}/{120}}e^{-({25}/{16})\pi\sigma_{0} |k|_{\eta}}<2^{-1}, \end{aligned} \end{align*} $$

where the third inequality is by $|k|_{\eta } \leq {16}/{15} |k_{j_{n}}|_{\eta }$ in equation (3.55) and $\sigma _{\infty }>5\sigma _{0}/6.$ In view of Proposition 3.7, there exists $\widetilde {B}_{j_{n}} \in \mathcal {H}_{\sigma _{j_{n}+1}}(2\mathbb {T}^{\infty },SL(2,\mathbb {R}))$ , such that

$$ \begin{align*} \widetilde{B}_{j_{n}}(x+\alpha)A_{0} e^{f_{0}(x)}\widetilde{B}_{j_{n}}(x)^{-1} =A_{j_{n}+1}e^{f_{j_{n}+1}(x)}. \end{align*} $$

By equation (3.8) and Proposition 3.7 with $\sigma _{\infty }>\sigma ,$ we get

$$ \begin{align*} |\widetilde{B}_{j_{n}}(x)|_{\sigma}\leq 2 \prod_{i=1}^{n} |B_{j_{i}}(x)|_{\sigma} \leq2 {\varepsilon_{j_{n}}}^{-{1}/{320}} e^{({8}/{7})\pi\sigma|k|_{\eta}}. \end{align*} $$

Thus, $|\widetilde {B}(x)|_{\sigma }\leq 4 {\varepsilon _{j_{n}}}^{-{1}/{320}} e^{(8/7)\pi \sigma |k|_{\eta }}$ .

4.1 Auxiliary lemmas

Before giving the proof of Theorem 1.2, we give some auxiliary lemmas to get some necessary estimates such as the growth of cocycle, the estimates of integrated density of states, Lyapunov exponent, and rotation number. Moreover, we will apply Lemmas 4.6–4.8 without proof in our topology since we equip $\mathbb {T}^{\infty }$ with the product topology of $\mathbb {T},$ which is similar to the topology in [Reference Avila1]. Here and subsequently, we denote $\Sigma _{V,\alpha }$ by the spectrum of $H_{V,\alpha ,x}$ and we acquiesce in the equality $S_{E}^{V}(x)=A_{0}e^{f_{0}(x)}.$

Lemma 4.1. Let $K_{j}, j\in \mathbb {N}$ be the sets defined by equation (3.29) and set $E\in K_{j}.$ Then, there exists $m\in \mathbb {Z}_{*}^{\infty }$ with the estimate $0<|m|_{\eta }< 2N_{j-1}$ such that

(4.1) $$ \begin{align} \|2\rho(\alpha,A_{0}e^{f_{0}})-\langle m,\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}\leq\varepsilon_{j-1}^{1/12}. \end{align} $$

Moreover,

(4.2) $$ \begin{align} \sup\limits_{0\leq n\leq c\varepsilon^{-1/12}_{j-1}}\|S_{E,n}^{V}\|\leq c\varepsilon_{j-1}^{-{1}/{160}}. \end{align} $$

Proof. For $l\in \mathbb {N},$ we assume that the lth resonance occurs at the $(j_{l}+1)$ th step of KAM iteration. For a given $j,$ there exists $p\in \mathbb {N}$ such that $j_{p}+1=j$ . We just consider the case $p\geq 2$ since in the case $p=1,$ we just take $m=k_{j_{1}}.$

Now, we assume $p\geq 2.$ Set $m=k_{j_{p}} -\sum _{l=1}^{p-1} k_{j_{l}} $ and note

$$ \begin{align*} 2\rho(\alpha,A_{j_{p}}e^{f_{j_{p}}(x)}) =2\rho(\alpha,A_{0}e^{f_{0}(x)})+\sum_{l=1}^{p-1}{\langle k_{j_{l}},\alpha\rangle}. \end{align*} $$

Moreover, equations (2.7), (3.7), and (3.29) yield

$$ \begin{align*} \|2\rho(\alpha,A_{j_{p}}e^{f_{j_{p}}(x)})-\langle k_{j_{p}},\alpha\rangle\kern-1.5pt\|_{\mathbb{R}/\mathbb{Z}}<2\varepsilon_{j_{p}}^{1/12}. \end{align*} $$

The inequalities above imply

$$ \begin{align*} \|2\rho(\alpha,A_{0}e^{f_{0}(x)})-\langle m,\alpha\rangle\kern-1.5pt\|_{\mathbb{R}/\mathbb{Z}}<2\varepsilon_{j_{p}}^{1/12}. \end{align*} $$

Moreover, for the m defined above, equation (3.32) yields

$$ \begin{align*} |m|_{\eta}\leq\sum\limits_{l=1}^{p} |k_{j_{l}}|_{\eta}< \sum\limits_{i=0}^{p-1} N_{j_{p}}^{(({4+3\eta})/({4+4\eta}))^{i}} \leq 2N_{j_{p}}. \end{align*} $$

Now we come to equation (4.2). First, equation (3.30) shows that

$$ \begin{align*} A_{0}e^{f_{0}(x)}=\widetilde{B}_{j-1}(x+\alpha)^{-1} A_{j}e^{f_{j}(x)}\widetilde{B}_{j-1}(x). \end{align*} $$

Moreover, equation (3.37) implies $\|A_{j}e^{f_{j}(x)}\|\leq 1+\varepsilon _{j-1}^{1/12}.$ The discussions above, together with equation (3.31), yield

$$ \begin{align*} \sup_{0\leq n\leq c\varepsilon^{-1/12}_{j-1}}\|S_{E,n}^{V}\| \leq(1+\varepsilon_{j-1}^{1/12})^{c\varepsilon_{j-1}^{-1/12}} \varepsilon_{j-1}^{-{1}/{160}}\leq c\varepsilon_{j-1}^{-{1}/{160}}.\\[-50pt] \end{align*} $$

Lemma 4.2. The integrated density of states is $\tfrac 12\text {-}H\ddot {o} lder$ for every $0<\varepsilon \leq \varepsilon _{0}^{1/4}$ . Moreover, if $E\in \Sigma _{V,\alpha },$ then

(4.3) $$ \begin{align} L(\alpha,S_{E}^{V})=0. \end{align} $$

Proof. We prove equation (4.3) first, which can be derived from the inequality

(4.4) $$ \begin{align} \|S_{E,n}^{V}\|\leq n^{c},\quad n\geq 1,\quad E\in\Sigma_{V,\alpha}, \end{align} $$

where c is a positive constant. We distinguish the proof of equation (4.4) into two cases.

Case 1. If $(\alpha ,A_{0}e^{f_{0}(x)})$ is reducible, then there exists $\widetilde {B}\in \mathcal {H}_{\sigma }(2\mathbb {T}^{\infty },SL(2,\mathbb {R}))$ such that

$$ \begin{align*} \widetilde{B}(x+\alpha) A_{0} e^{f_{0}(x)} \widetilde{B}(x)^{-1} = A_{\infty}. \end{align*} $$

Since $E\in \Sigma _{V,\alpha }$ , the cocycle $(\alpha ,A_{0}e^{f_{0}(x)})$ is not uniformly hyperbolic [Reference Johnson and Moser46], and hence

$$ \begin{align*} \begin{aligned} A_{\infty}=\left( \begin{matrix} e^{i2\pi\rho}\ \ \ \ \ \ \ \ \ c \\ 0 \ \ \ \ \ \ \ e^{-i2\pi\rho} \end{matrix} \right) \end{aligned} \end{align*} $$

with $\rho \in \mathbb {R}$ . Then, $\|A_{\infty }^{n}\|\leq nc+1$ , which implies that

$$ \begin{align*} \|S_{E,n}^{V}(x)\|\leq\|A_{\infty}^{n}\| \|\widetilde{B}(x)\|^2\leq(nc+1)C\leq Cn\quad \text{for all } n\geq 1. \end{align*} $$

Thus, equation (4.3) holds.

Case 2. If $(\alpha ,A_{0}e^{f_{0}(x)})$ is not reducible but almost reducible, we need the following lemma to describe the growth of the cocycle.

Lemma 4.3. Suppose $E\in \Sigma _{V,\alpha }$ , then $\text { for all } j\geq 1$ , there exists $\widetilde {B}^{\prime }_{j-1}$ , such that

(4.5) $$ \begin{align} \widetilde{B}_{j-1}'(x + \alpha) A_{0} e^{f_{0}(x)} \widetilde{B}_{j-1}'(x)^{-1} =\left( \begin{matrix} e^{i2\pi\rho_{j}}\ \ \ \ \ \ \ c_{j} \\ 0\ \ \ \ \ \ \ e^{-i2\pi\rho_{j}} \end{matrix} \right)+\widetilde{F}_{j}(x), \end{align} $$

where $\rho _{j}\in \mathbb {R} $ , with

(4.6) $$ \begin{align} \|\widetilde{F}_{j}(x)\|\leq\varepsilon_{j}^{{1}/{4}},\quad \|\widetilde{B}^{\prime}_{j-1}(x)\|\leq \varepsilon_{j-1}^{-{1}/{320}},\quad \|\widetilde{B}^{\prime}_{j-1}(x)\|^2|c_{j}|\leq 8\|A_{0}\|. \end{align} $$

Proof. The conclusions in part (1) of Proposition 3.7 show that there exists $\widetilde {B}_{j-1}\in \mathcal {H}_{\sigma _{j}}(2\mathbb {T}^{\infty },SL(2,\mathbb {R}))$ , such that equation (3.30) holds with equation (3.31). Then, there exists unitary matrices $U_{j}$ such that

$$ \begin{align*} \begin{aligned} U_{j}A_{j}e^{f_{j}(x)}U_{j}^{-1}=\left( \begin{matrix} e^{i2\pi\rho_{j}}\ \ \ \ \ \ \ c_{j} \\ 0\ \ \ \ \ \ \ e^{-i2\pi\rho_{j}} \end{matrix} \right)+ F_{j}(x), \end{aligned} \end{align*} $$

where $\rho _{j}\in \mathbb {R} \cup i\mathbb {R}$ with $ |F_{j}(x)|_{\sigma _{j}} < \varepsilon _{j}$ and the estimates in equations (3.31) and (3.34) hold.

Case 1: $\rho _{j}\in \mathbb {R}.$ Let $\widetilde {B}^{\prime }_{j-1}(x)=U_{j}\widetilde {B}_{j-1}(x)$ , $\widetilde {F}_{j}=F_{j}$ , then

$$ \begin{align*} \widetilde{B}_{j-1}'(x + \alpha) A_{0} e^{f_{0}(x)} \widetilde{B}_{j-1}'(x)^{-1} =\left( \begin{matrix} e^{i2\pi\rho_{j}}\ \ \ \ \ \ \ c_{j} \\ 0\ \ \ \ \ \ \ e^{-i2\pi\rho_{j}} \end{matrix} \right)+\widetilde{F}_{j}(x). \end{align*} $$

Moreover, the estimates in equation (4.6) are also satisfied.

Case 2: $\rho _{j}\in i\mathbb {R}.$ We first assume $|\rho _{j}|>\varepsilon _{j}^{1/4}.$ Let

$$ \begin{align*} Q_{j}=\left( \begin{matrix} q_{j}\ \ \ \ \ 0 \\ 0 \ \ \ \ q_{j}^{-1} \end{matrix} \right)\!, \end{align*} $$

where $q_{j}=\|\widetilde {B}_{j-1}\|\varepsilon _{j}^{1/4}$ . Then, we have

$$ \begin{align*} \begin{aligned} Q_{j}\left[ \left( \begin{matrix} e^{i2\pi\rho_{j}}\ \ \ \ \ \ \ c_{j} \\ 0\ \ \ \ \ \ \ e^{-i2\pi\rho_{j}} \end{matrix} \right)+ F_{j}(x)\right]Q_{j}^{-1}&=\left( \begin{matrix} e^{i2\pi\rho_{j}}\ \ \ \ \ \ \ 0 \\ 0\ \ \ \ \ \ \ e^{-i2\pi\rho_{j}} \end{matrix} \right)+F^{\prime}_{j}(x), \end{aligned} \end{align*} $$

where

$$ \begin{align*} \begin{aligned} F^{\prime}_{j}(x)=\left( \begin{matrix} 0\ \ \ c_{j}q_{j}^2 \\ 0\ \ \ \ \ \ \ 0 \end{matrix} \right)+Q_{j}F_{j}(x)Q_{j}^{-1}. \end{aligned} \end{align*} $$

Moreover, equation (3.34) yields

$$ \begin{align*} |c_{j}|q_{j}^{2}=|c_{j}| \|\widetilde{B}_{j-1}\|^{2} \varepsilon_{j}^{{1}/{2}}\leq 8\|A_{0}\|\varepsilon_{j}^{{1}/{2}},\quad \|Q_{j} F_{j} Q_{j}^{-1}\| \leq \varepsilon_{j}^{{1}/{2}}, \end{align*} $$

then we have $\|{F}_{j}'\|\leq c \varepsilon _{j}^{1/2}.$ We will show that this implies that the system is uniformly hyperbolic.

More precisely, given a non-zero vector $( \begin {smallmatrix} a \\ b \end {smallmatrix} ) \in \mathbb {R}^2 $ with $ |a| \geq |b|$ , let

$$ \begin{align*} \left( \begin{matrix} a' \\ b' \end{matrix} \right)=\left[ \left( \begin{matrix} e^{i2\pi\rho_{j}}\ \ \ \ \ \ 0 \\ 0 \ \ \ \ \ \ \ e^{-i2\pi\rho_{j}} \end{matrix} \right)+{F}_{j}'(x)\right] \left( \begin{matrix} a \\ b \end{matrix} \right)=\left( \begin{matrix} e^{i2\pi\rho_{j}}a \\ e^{-i2\pi\rho_{j}}b \end{matrix} \right)+{F}_{j}'(x) \left( \begin{matrix} a \\ b \end{matrix} \right)\!. \end{align*} $$

Without loss of generality, assume $ i2\pi \rho _{j}> 0 $ , then

$$ \begin{align*} |a'| \geq (e^{i2\pi\rho_{j}}-2\|{F}_{j}'\|) |a|, \ \ \ |b'| \leq e^{-i2\pi\rho_{j}}|b|+2\|{F}_{j}'\||a|. \end{align*} $$

Therefore, $|a'|-|b'|\geq (i4\pi \rho _{j} - 4c\varepsilon _{j}^{1/2})|a|>0,$ which, together with the cone field criterion (compare, e.g., [Reference Yoccoz71]), implies $(\alpha ,A_{0}e^{f_{0}})$ is uniformly hyperbolic. This conflicts with our assumption that $E\in \Sigma _{V,\alpha }$ . So we have $|\rho _{j}| \leq \varepsilon _{j}^{1/4}.$ We put it into the perturbation to obtain the following:

$$ \begin{align*} \widetilde{B}_{j-1}'(x + \alpha) A_{0} e^{f_{0}(x)} \widetilde{B}_{j-1}'(x)^{-1} =\left( \begin{matrix} 1\ \ \ c_{j} \\ 0\ \ \ \ 1 \end{matrix} \right)+\widetilde{F}_{j}(x) \end{align*} $$

with $\|\widetilde {F}_{j}\| \leq \varepsilon _{j}^{1/4}$ . Thus, we get the new cocycle with $\rho _{j}=0$ and the perturbation being given above. So we have $\rho \in \mathbb {R}$ .

To control the growth of the cocycle, we need the following lemma proved by Avila, Fayad, and Krikorian [Reference Avila, Fayad and Krikorian7].

Lemma 4.4. We have that

$$ \begin{align*} M_{l}(\mathrm{id}+\xi_{l}) \cdots M_{0}(\mathrm{id}+\xi_{0})=M^{(l)}(\mathrm{id}+\xi^{(l)}), \end{align*} $$

where $M^{(l)}=M_{l} \cdots M_{0}$ and

$$ \begin{align*} \|\xi^{(l)}\| \leqslant e^{\sum_{k=0}^{l}\|M^{(k)}\|^{2}\|\xi_{k}\|}-1. \end{align*} $$

Now, we come back to the proof of Case 2 of Lemma 4.2. Let

$$ \begin{align*}M_{k}=U_{j}A_{j}U_{j}^{-1}=\left(\begin{array}{@{}cc@{}}e^{i2 \pi \rho_{j}} & c_{j} \\ 0 & e^{-i2 \pi \rho_{j}}\end{array}\right)\!,\quad \xi_{k}=U_{j}A_{j}^{-1}U_{j}^{-1} \widetilde{F}_{j}(x+k\alpha).\end{align*} $$

Apply Lemma 4.4 to the $M_{k}$ given above and equation (4.5), so that we obtain

$$ \begin{align*} \begin{aligned} S_{E,n}^{V}(x)=\widetilde{B}^{\prime -1}_{j-1}(x+(n+1)\alpha) U_{j}A_{j}^{n}U_{j}^{-1}(\mathrm{id}+\xi^{(n)}) \widetilde{B}^{\prime}_{j-1}(x+\alpha), \end{aligned} \end{align*} $$

where $\|\xi ^{(n)}\| \leq e^{\sum _{k=1}^{n}\|U_{j}A_{j}^{k}U_{j}^{-1}\|^{2}\|\xi _{k}\|}-1$ . Since $\rho _{j}\in \mathbb {R}$ , we have $\|U_{j}A_{j}^{k}U_{j}^{-1}\|\leq 1+k|c_{j}|,$ which, together with $|\widetilde {F}_{j}| \leq \varepsilon _{j}^{1/4},$ yields

$$ \begin{align*} \begin{aligned} \|S_{E,n}^{V}\| &\leq\|\widetilde{B}^{\prime}_{j-1}\|^{2}(1+n|c_{j}|)\cdot e^{\sum_{k=1}^{n}(1+k|c_{j}|)^{2}(1+|c_{j}|) \varepsilon_{j}^{{1}/{4}}}\\ &\leq\|\widetilde{B}^{\prime}_{j-1}\|^{2}(1+n|c_{j}|) e^{n^{3} \varepsilon_{j}^{{1}/{4}}}. \end{aligned} \end{align*} $$

The inequality above, together with equation (4.6), implies

(4.7) $$ \begin{align} \sup_{0\leq n\leq c\varepsilon^{-1/12}_{j-1}}\|S_{E,n}^{V}\|\leq 2\|\widetilde{B}_{j-1}\|^{2}(1+n|c_{j}|) \leq 2 \varepsilon_{j-1}^{-{1}/{160}}+16 n. \end{align} $$

Since we are at the almost reducible situation, for any fixed large $n\in \mathbb {N}$ , there exists i such that $n\in [\varepsilon ^{-1/36}_{i-1},\varepsilon ^{-1/12}_{i-1}]$ , then equation (4.7) shows that

$$ \begin{align*} \sup\limits_{0\leq n\leq c\varepsilon^{-1/12}_{i-1}}\|S_{E,n}^{V}\|\leq 2 n^{{9}/{40}}+16 n\leq 17n. \end{align*} $$

The above equation yields

$$ \begin{align*} \lim\limits_{n\rightarrow\infty}\frac{1}{n}\ln \|S_{E,n}^{V}(x)\|\leq \lim\limits_{n\rightarrow\infty}\frac{1}{n}\ln (17n)=0. \end{align*} $$

So we get equation (4.3) for $E\in \Sigma _{V,\alpha }$ .

Next, we will prove the conclusion of Lemma 4.2 about the integrated density of states. To this end, let $I_{j}\triangleq (\varepsilon _{j}^{1/4}, \varepsilon _{j-1}^{{1}/{15}}) $ , $j\geq 1$ . Then, for any small $0<\varepsilon <\varepsilon _{0}^{{1}/{15}}$ , there exists j such that $\varepsilon \in I_{j}$ . Moreover, we also need the lemma below.

Lemma 4.5. There exists $W\in \mathcal {H}_{\sigma }(2\mathbb {T}^{\infty },GL(2,\mathbb {C}))$ such that

(4.8) $$ \begin{align} Q(x) = W(x+\alpha)A_{0}e^{f_{0}(x)}W(x)^{-1}, \end{align} $$

with

(4.9) $$ \begin{align} \|W\|\leq\varepsilon ^{-{1}/{4}},\quad \|Q\|\leq1+c \varepsilon^{{1}/{2}}. \end{align} $$

Proof. Let $\widetilde {B}^{\prime }_{j-1}$ and $\widetilde {F}_{j}$ be those in Lemma 4.3, so

$$ \begin{align*} \widetilde{B}_{j-1}'(x + \alpha) A_{0} e^{f_{0}(x)} \widetilde{B}_{j-1}'(x)^{-1} =\left( \begin{matrix} e^{i2\pi\rho_{j}}\ \ \ \ \ \ \ c_{j} \\ 0\ \ \ \ \ \ \ e^{-i2\pi\rho_{j}} \end{matrix} \right)+\widetilde{F}_{j}(x). \end{align*} $$

Let $D=\mathrm {diag}\{d,\ d^{-1}\},\ W(x) = D\widetilde {B}_{j-1}'$ with $d = \varepsilon ^{1/4}\|\widetilde {B}_{j-1}'\|.$ Then, we have

$$ \begin{align*} {W}(x+\alpha)A_{0}e^{f_{0}(x)}{W}(x)^{-1} = \left( \begin{matrix} e^{i2\pi\rho_{j}}\ \ \ \ \ \ \ \ \ 0 \\ 0 \ \ \ \ \ \ \ e^{-i2\pi\rho_{j}} \end{matrix} \right)+ \widetilde{F}_{j}'(x), \end{align*} $$

where

$$ \begin{align*} \widetilde{F}_{j}^{'}(x) = \left( \begin{matrix} 0\ \ \ \|\widetilde{B}_{j-1}'\|^{2}\varepsilon ^{{1}/{2}}c_{j} \\ 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \end{matrix} \right)+ D\widetilde{F}_{j}D^{-1}. \end{align*} $$

Since $\varepsilon \in I_{j}$ , we have $d<1$ , which implies $\|W\|\leq \varepsilon ^{-1/4}$ . Moreover, equation (4.6) and the fact that $\varepsilon \in I_{j}$ imply $\|\widetilde {F}_{j}^{'}(x)\|<c\varepsilon ^{1/2}$ and $\rho _{j}\in \mathbb {R}$ . Let

$$ \begin{align*} Q(x) = W(x+\alpha)A_{0}e^{f_{0}(x)}W(x)^{-1}, \end{align*} $$

then $\|Q\|\leq 1+c\varepsilon ^{1/2}$ .

Now, we prove conclusions about the integrated density of states. Notice

$$ \begin{align*} L(\alpha,S_{E}^{V})= \lim\limits_{n\rightarrow\infty}\frac{1}{n}\ln\|S_{E,n}^{V}(x)\|. \end{align*} $$

Abbreviate $L(\alpha ,S_{E}^{V})$ to $L(E,V)$ and by the Thouless formula, we obtain

(4.10) $$ \begin{align} \begin{aligned} L(E+i \varepsilon,V) - L(E,V) &= \int_{\mathbb{R}}\frac{1}{2}\ln\bigg(1+\frac{\varepsilon^{2}}{(E'-E)^{2}}\bigg) \mathrm{d} N(E')\\ &\geq \int_{E-\varepsilon}^{E+\varepsilon}\bigg(\frac{1}{2}\ln2\bigg)~\mathrm{d} N(E') ,\\ &\geq \frac{1}{2}\ln2(N(E+\varepsilon)-N(E-\varepsilon)). \end{aligned} \end{align} $$

So it is enough to show that for $ 0<\varepsilon \leq \varepsilon _{0}^{1/4}$ , $ L(E+i \varepsilon )<C\varepsilon ^{1/2}$ .

Notice that

$$ \begin{align*} \begin{aligned} S_{E+i \varepsilon}^{V}(x)=\left( \begin{matrix} E+i\varepsilon-V(x) \ \ \ \ -1 \\ \ \ \ \ 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \end{matrix} \right)=S_{E}^{V}(x)+ \left( \begin{matrix} i \varepsilon \ \ \ \ \ \ \ 0 \\ 0 \ \ \ \ \ \ \ \ 0 \end{matrix} \right) \end{aligned} \end{align*} $$

and there exists $I_{j}$ such that $\varepsilon \in I_{j}$ , then by equations (4.8) and (4.9), we get

$$ \begin{align*} \begin{aligned} W(x+\alpha)S_{E+i \varepsilon}^{V}(x)W(x)^{-1}&=Q(x)+W(x+\alpha) \left( \begin{matrix} i \varepsilon \ \ \ \ \ \ \ 0 \\ 0 \ \ \ \ \ \ \ \ 0 \end{matrix} \right)W(x)^{-1},\\ &\triangleq \widetilde{S}(x). \end{aligned} \end{align*} $$

The Lyapunov exponent is clearly invariant under conjugacies, so we have ${L(E+i\varepsilon ,V)= L(\alpha ,S_{E+i \varepsilon }^{V})=L(\alpha ,\widetilde {S})}$ . Thus,

$$ \begin{align*} \|\widetilde{S}(x)\| \leq 1+ c\varepsilon ^{{1}/{2}}+ \varepsilon \cdot \varepsilon ^{-{1}/{2}} = 1+ c\varepsilon ^{{1}/{2}}, \end{align*} $$

which implies

$$ \begin{align*} L(\alpha,\widetilde{S}) = \lim_{n\rightarrow\infty} \dfrac{1}{n}\ln\|\widetilde{S}_{n}(x)\|\leq\lim_{n\rightarrow\infty} \frac{1}{n} \sum^{n}_{i=1}\ln\|\widetilde{S}(x+(i-1)\alpha)\|\leq c\varepsilon ^{{1}/{2}}, \end{align*} $$

which, together with equation (4.10), implies that integrated density of states is $\tfrac 12$ - Hölder continuous.

Let $\mathcal {B}'$ be the set of $E\in \Sigma _{V,\alpha }$ such that $(\alpha ,S_{E}^{V}(x))$ is bounded, and $\mathcal {R}'$ be the set of $E\in \Sigma _{V,\alpha }$ such that $(\alpha ,S_{E}^{V}(x))$ is reducible. Then, we have two lemmas below, which are the basis of the proof of Theorem 1.2.

Proof. If $E\in \mathcal {R}'$ , then there exist at most finitely many resonance sites $k_{j_{0}},\ldots ,k_{j_{l}}$ . Since there exists $B\in \mathcal {H}_{\sigma }(2\mathbb {T}^{\infty },SL(2,\mathbb {R}))$ such that

$$ \begin{align*} B(x+\alpha) A_{0} e^{f_{0}(x)} B(x)^{-1} = A_{\infty}, \end{align*} $$

with $\|B\|\leq 2\varepsilon _{j_{l}}^{-{1}/{320}}$ , then by equation (3.35) and $E\notin \mathcal {B}'$ , we have

$$ \begin{align*} \rho(\alpha,A_{\infty})=\rho(\alpha,A_{0}e^{f_{0}(x)})+\sum_{i=0}^{l}\frac{\langle k_{j_{i}},\alpha\rangle}{2}=0, \end{align*} $$

which yields $2\rho (\alpha ,S_{E}^{V}(x))=\langle k,\alpha \rangle \mod \! 1$ for some $k\in \mathbb {Z}_{*}^{\infty }$ .

4.2 Proof of Theorem 1.2

Once we get the conclusions in §4.1, we are ready to prove Theorem 1.2. First, Lemma 4.8 shows that to prove Theorem 1.2, it is enough to prove $\mu (\Sigma _{V,\alpha }\setminus \mathcal {B}')=0.$ Moreover, Lemma 4.9 shows that $\mathcal {R}'\setminus \mathcal {B}'$ is countable. The definition of $\mathcal {R}'$ implies that if $E\in \mathcal {R}'$ , any non-zero solution $H_{V,\alpha ,x}u=Eu$ satisfies $\inf \nolimits _{n\in \mathbb {Z}} \{|u_{n}|^2+|u_{n+1}|^2\}>0$ . Thus, if, furthermore, $E\in \mathcal {R}'\setminus \mathcal {B}',$ then $u\notin \ell ^{2}(\mathbb {Z}).$ That is, there are no eigenvalues in $\mathcal {R}'\setminus \mathcal {B}'$ and $\mu (\mathcal {R}'\setminus \mathcal {B}')=0$ . So it is enough to prove that ${\mu (\Sigma _{V,\alpha }\setminus \mathcal {R}')=0.}$

Actually, $E\in \Sigma _{V,\alpha }\setminus \mathcal {R}'$ and $\mu (\Sigma _{V,\alpha }\setminus \mathcal {R}')=0$ are equivalent to $ E\in \lim \sup \nolimits _{j\rightarrow \infty }K_{j}$ and $ \mu (\lim \sup \nolimits _{j\rightarrow \infty }K_{j}) = 0,$ respectively. Moreover, by the Borel–Cantelli lemma, $\mu (\lim \sup \nolimits _{j\rightarrow \infty }K_{j}) = 0$ is equivalent to $\sum \nolimits _{j} \mu (\overline {K_{j}}) < \infty .$ So, the last thing is to show

(4.11) $$ \begin{align} \sum\limits_{j} \mu (\overline{K_{j}}) < \infty. \end{align} $$

We will devote ourselves to proving equation (4.11) in the rest of this section.

Let $J_{j}(E)=(E-c \varepsilon _{j-1}^{{1}/{18}},E+c \varepsilon _{j-1}^{{1}/{18}})$ , $j\geq 1$ . Lemma 4.6 and $E\in K_{j}$ yield

(4.12) $$ \begin{align} \mu(J_{j}(E))\leq c \varepsilon_{j-1}^{{1}/{18}}\sup\limits_{0\leq n\leq c \varepsilon_{j-1}^{-{1}/{18}}}\|S_{E,n}^{V}\|^2\leq c \varepsilon_{j-1}^{{31}/{720}}. \end{align} $$

Thus, $\overline {K_{j}}\subset \bigcup _{l=0}^{r}J_{j}(E_{l})$ since $\overline {K_{j}}$ is compact, where $E_{l}\in K_{j}, l=0,\ldots ,r.$ Refine this subcover if necessary, then we can assume that any $x\in \mathbb {R}$ is contained in at most two different $J_{j}(E_{l})$ . Thus, $N(\overline {K_{j}})\subset \bigcup _{i=0}^{r}N(J_{j}(E_{i}))$ .

We see that $N(E) = 1-2\rho (\alpha ,S_{E}^{V})$ and equation (4.1) yield

(4.13) $$ \begin{align} \|N(E)-\langle m,\alpha\rangle\|_{\mathbb{R}/\mathbb{Z}}\leq\varepsilon_{j-1}^{1/12} \quad \text{ for all } E\in K_{j}. \end{align} $$

For $m(l)\in \mathbb {Z}_{*}^{\infty }$ satisfying equation (4.13), set

(4.14) $$ \begin{align} T_{l}=(\langle m(l),\alpha\rangle-\varepsilon_{j-1}^{{1}/{12}}, \langle m(l),\alpha\rangle+\varepsilon_{j-1}^{{1}/{12}}). \end{align} $$

For the fixed $T_{l}$ defined above, we have

(4.15) $$ \begin{align} |T_{l}|=2\varepsilon_{j-1}^{{1}/{12}}=2c^{-1}|J_{j}(E)| ^{{3}/{2}} \leq2c^{-1}|N(J_{j}(E))|\ \text{ for all } E\in K_{j}, \end{align} $$

where the last inequality is by Lemma 4.7. For the constant c in equation (4.15), set $N_{c}=[4(c^{-1}+1)]+1,$ where $[\cdot ]$ denotes the integer part. Then, for the fixed $T_{l},$ select $\{E_{l,s}\}_{s=1}^{N_{c}}\subset K_{j}$ such that $N(J_{j}(E_{l,s})), s=1,\ldots ,N_{c},$ intersects $T_{l}$ with $J_{j}(E_{l,s})\in \{J_{j}(E_{i})\}_{i=0}^{r}.$ Thus,

(4.16) $$ \begin{align} N(\overline{K_{j}})\subset\bigcup_{i=0}^{r}N(J_{j}(E_{i})) \subset\bigcup_{l=0}^{\widetilde{N}} \bigcup_{s=1}^{N_{c}}N(J_{j}(E_{l,s})), \end{align} $$

where $\widetilde {N}$ is the number of such $m(l)$ satisfying equation (4.13).

Note for fixed $E_{i}\in K_{j},$ there exists $l\in \mathbb {N}$ such that $N(E_{i})\in T_{l},$ and thus $N(J_{j}(E_{i}))$ intersects $T_{l},$ which, together with the selection of $\{E_{l,s}\}_{s=1}^{N_{c}}$ above, yields the second relation in equation (4.16).

We, in the lemma below, give the upper bound of $\widetilde {N},$ which is the number of such $m(l)$ satisfying equation (4.13).

Lemma 4.10. $\widetilde {N}$ satisfies the following inequality:

(4.17) $$ \begin{align} \widetilde{N}<\varepsilon_{j-1}^{-{1}/{40}}. \end{align} $$

Proof. For $E\in K_{j},$ Lemma 4.1 implies that there exists $m\in \mathbb {Z}_{*}^{\infty }$ such that equation (4.1) holds (thus, this m satisfies equation (4.13) since $N(E) = 1-2\rho (\alpha ,S_{E}^{V})$ ) with

(4.18) $$ \begin{align} |m|_{\eta}=\sum_{s\in\mathbb{N}}\langle s\rangle^{\eta}|m_{s}|<2N_{j-1}. \end{align} $$

Moreover, we assume j is large enough such that $1\leq \eta <\sqrt {1+\log _{2}{N_{j-1}}}-1.$

For the m satisfying equation (4.18), denote $\widetilde {M}_{m}$ by the number of its non-zero components. Then, we have the following estimates:

$$ \begin{align*} 2N_{j-1} \geq|m|_{\eta}=\sum_{j=1}^{\widetilde{M}_{m}}\langle i_{j}\rangle^{\eta}|m_{i_{j}}| \geq \sum_{j=1}^{\widetilde{M}_{m}}\langle i_{j}\rangle^{\eta} \geq \sum_{j=1}^{\widetilde{M}_{m}} j^{\eta}\approxeq\frac{1}{1+\eta} \widetilde{M}_{m}^{1+\eta}, \end{align*} $$

and thus,

(4.19) $$ \begin{align} \widetilde{M}_{m}\leq (2\eta+2)^{{1}/({1+\eta})}(2N_{j-1})^{{1}/({1+\eta})}< 2 (2N_{j-1})^{{1}/({1+\eta})} < (2N_{j-1})^{{1}/{\eta}}, \end{align} $$

where the last inequality is by $\eta <\sqrt {1+\log _{2}{N_{j-1}}}-1.$ The second inequality above shows that $2 (2N_{j-1})^{{1}/({1+\eta })}$ is the uniform bound of the $\widetilde {M}_{m}$ for all m given by equation (4.13). Set $C_{n}^{j}, n\geq j,$ to be the notation of combination, which denotes the prescribed size of taking j numbers from the given set $\{1,2,\ldots ,n\}.$

Now, we are ready for proving the inequality in equation (4.17). First, equation (4.18) shows that ${m_{s}=0}\ \text {for} \ s\geq (2N_{j-1})^{{1}/{\eta }}.$ Thus, we are restricted to the $(2N_{j-1})^{{1}/{\eta }}$ -dimensional tori, which, for the fixed $j\in \mathbb {N},$ is finite dimensional. Moreover, equation (4.19) shows that the uniform bound of the number of non-zero components of m satisfying equation (4.13) is $2(2N_{j-1})^{{1}/({1+\eta })}.$ That is, there are at least $(2N_{j-1})^{{1}/{\eta }}-2(2N_{j-1})^{{1}/({1+\eta })}$ many components being zero. So for these m satisfying equation (4.13), we just need to consider its $2(2N_{j-1})^{{1}/({1+\eta })}$ many components, which are allowed to be non-zero. Second, equation (4.18) also yields $|m_{s}|<2N_{j-1}|s|^{-\eta }, s\geq 1$ . Thus, for fixed $s\leq (2N_{j-1})^{{1}/{\eta }}, m_{s}$ can take any of the values below:

$$ \begin{align*} \{\pm[2N_{j-1}|s|^{-\eta}],\pm([2N_{j-1}|s|^{-\eta}]-1),\ldots,\pm 2,\pm1,0\}. \end{align*} $$

Set

(4.20) $$ \begin{align} \{n_{1},n_{2},\ldots,n_{2(2N_{j-1})^{{1}/({1+\eta})}}\}. \end{align} $$

We consider these m values, which satisfy equation (4.13) and whose indexes set of $2(2N_{j-1})^{{1}/({1+\eta })}$ many components, which may not be zero, is set above. By the discussions above, we know that the number of these m is less than

$$ \begin{align*} \prod_{s=1}^{2(2N_{j-1})^{{1}/({1+\eta})}}(4N_{j-1}\langle n_{s}\rangle^{-\eta}+1). \end{align*} $$

Note the elements of the index set defined in equation (4.20) are taken from the set $\{0,1,\ldots ,(2N_{j-1})^{{1}/{\eta }}\},$ and thus we have the estimate

$$ \begin{align*} \begin{aligned} \widetilde{N}&< C_{(2N_{j-1})^{{1}/{\eta}}}^{2(2N_{j-1})^{{1}/({1+\eta})}} \prod_{s=1}^{2(2N_{j-1})^{{1}/({1+\eta})}}(4N_{j-1}\langle n_{s}\rangle^{-\eta}+1)\\ &\leq C_{(2N_{j-1})^{{1}/{\eta}}}^{2(2N_{j-1})^{{1}/({1+\eta})}} \prod_{s=0}^{2(2N_{j-1})^{{1}/({1+\eta})}}(4N_{j-1}\langle s\rangle^{-\eta}+1)<\varepsilon_{j-1}^{-{1}/{40}}, \end{aligned} \end{align*} $$

where the second inequality is by the fact that $\langle n_{s}\rangle ^{-\eta }\leq \langle s\rangle ^{-\eta }$ and the last inequality is by the second inequality in equation (3.9).

Now, we continue the estimation of $\mu (\overline {K_{j}}).$ First, equation (4.16) shows that there are at most $N_{c}\widetilde {N}$ intervals $J_{j}(E_{i})$ . Then equations (4.16) and (4.17) yield

$$ \begin{align*} \mu(\overline{K_{j}}) \leq \sum^{r}_{i=0} \mu (J_{j}(E_{i}))<N_{c}\varepsilon_{j-1}^{-{1}/{40}}\cdot c \varepsilon_{j-1}^{{31}/{720}}\leq c\varepsilon_{j-1}^{{13}/{720}}, \end{align*} $$

where the second inequality is by equation (4.12).

5.1 Gap estimate

The proof of Theorem 1.4 is based on the gap estimates via the Moser–Pöschel argument, see Theorem 5.1. Thus, we will give a brief introduction about the gap estimate and the conclusion in our setting.

In [Reference Moser and Pöschel53], Moser and Pöschel consider a continuous quasi-periodic Schrödinger operator with small analytic potential V and show that $|G_{k}(V)|$ is exponentially small with $|k|$ for large enough $|k|.$ Later, in [Reference Eliasson36, Reference Puig59], Eliasson and Puig follow the work in [Reference Moser and Pöschel53] and show $G_{m}(V)$ is at least sub-exponentially small with respect to m under some hypotheses. In [Reference Hadj-Amor43, Reference Shi and Yuan60], Hadj-Amor, and Shi and Yuan consider the discrete Schrödinger operator and the extended Harper model, showing the sub-exponential smallness with respect to m of $G_{m}(V).$ Damanik and Goldstein, in [Reference Damanik and Goldstein26], give a sharper upper bound $2\varepsilon e^{-r_{0}|m|/2}$ of $|G_{k}(V)|$ . Leguil et al, in [Reference Leguil, You, Zhao and Zhou48], give the upper bound of $|G_{k}(V)|$ with the Liouvillean frequency, and, in the Diophantine frequency case, the authors extend the result in [Reference Damanik and Goldstein26] by giving a sharper upper bound $\varepsilon ^{2/3} e^{-2\pi r|m|} \text { for all } r\in (0,r_{0}).$ In [Reference Damanik and Fillman24, Reference Liu and Yuan50], Damanik, and Liu and Yuan prove the gap labeling theorem for ergodic Schrödinger operators and give the estimate of spectral gaps of AMO in the exponential regime. Our result is the following.

For any $r_{0}>0,$ under the Diophantine frequency case, the estimate of gap in [Reference Leguil, You, Zhao and Zhou48] is $\varepsilon ^{2/3} e^{-2\pi r|m|}\ \text { for all } r\in (0,r_{0}).$ However, the weight in our Theorem 5.1 is $\tfrac 54\pi \sigma _{0}.$ The main reason that we lose much more analytic radius is that, with the infinite dimensional frequency, the estimate of the resonant site given by equation (3.32) is much worse than that in [Reference Leguil, You, Zhao and Zhou48]. Consequently, we can only get much worse estimates about $\zeta $ and $\widetilde {B}$ in part $\mathrm {(3)}$ of Proposition 3.7.

5.2 Proof of Theorem 1.4

We will apply the estimate in Theorem 5.1 to prove Theorem 1.4. Set

$$ \begin{align*} \underline{E}=\min \Sigma_{V,\alpha}, \quad \overline{E}=\max \Sigma_{V,\alpha},\quad G_{0}(V)=(-\infty,\underline{E})\cup(\overline{E},+\infty). \end{align*} $$

Moreover, given any $E\in \Sigma _{V,\alpha }$ and any $\varepsilon>0$ , set

$$ \begin{align*} {\mathcal M} = {\mathcal M}(E,\varepsilon)\triangleq\{m\in\mathbb{Z}_{*}^{\infty} \setminus \{0\} ~|~ G_{m}(V)\cap(E-\varepsilon,E+\varepsilon)\neq \varnothing \}, \end{align*} $$

and let $m_{0}\in {\mathcal M}$ be such that $|m_{0}|_{\eta }=\min _{m\in {\mathcal M}}|m|_{\eta }$ . Since $E\in \Sigma _{V,\alpha }$ , it is obvious that

$$ \begin{align*} \begin{aligned} |G_{m_{0}}(V)\cap(E-\varepsilon,E+\varepsilon)|\leq\varepsilon,\\ |(-\infty,\underline{E})\cap(E-\varepsilon,E+\varepsilon)|\leq\varepsilon,\\ |(\overline{E},+\infty) \cap(E-\varepsilon,E+\varepsilon)|\leq\varepsilon. \end{aligned} \end{align*} $$

Moreover, by the definition of ${\mathcal M},$ we know that

(5.1) $$ \begin{align} \mathrm{dist}(G_{m}(V),G_{m'}(V)) \leq2\varepsilon\quad \text{for all } m,m'\in{\mathcal M}. \end{align} $$

Now, we assume $0<\varepsilon \leq \varepsilon _{0}$ and separate the discussions into the following three cases.

Case 1: $G_{0}(V)\cap (E-\varepsilon ,E+\varepsilon )= \varnothing $ . Consider two different gaps $G_{m}(V)$ and $G_{m'}(V)$ . Without loss of generality, we assume that $E_{m}^{+}\leq E_{m'}^{-}$ . Hence,

$$ \begin{align*} \mathrm{dist}(G_{m}(V),G_{m'}(V))=E_{m'}^{-}-E_{m}^{+}. \end{align*} $$

Thus, by Lemma 4.2, the equality above with equation (5.1), we have

$$ \begin{align*} |N(E_{m'}^{-})-N(E_{m}^{+})|\leq c (E_{m'}^{-}-E_{m}^{+})^{{1}/{2}}, \end{align*} $$

where c is independent of $E.$ Since $\alpha \in DC_{\infty }(\gamma ,\tau )$ , according to equation (2.6) and Lemma 4.9, gap labeling [Reference Johnson and Moser46], we have

$$ \begin{align*} |N(E_{m'}^{-})-N(E_{m}^{+})|=\|\langle m'-m,\alpha\rangle\|_{\mathbb{R}/ \mathbb{Z}}\geq\gamma(1+|m'-m|_{\eta})^{-c(\eta,\tau) (|m'-m|_{\eta})^{{1}/({1+\eta})}}. \end{align*} $$

The three inequalities above show, $\text { for all } m'\neq m\in \mathbb {Z}_{*}^{\infty }\setminus \{0\},$

(5.2) $$ \begin{align} 2\varepsilon\geq\mathrm{dist}(G_{m}(V),G_{m'}(V))\geq\frac{\gamma^2}{c^2}(1+|m'-m|_{\eta})^{-2c(\eta,\tau)(|m'-m|_{\eta})^{{1}/({1+\eta})}}. \end{align} $$

By equation (5.2) we get, for all $m\in {\mathcal M}\setminus \{m_{0}\},$

(5.3) $$ \begin{align} 2\varepsilon\geq\mathrm{dist}(G_{m}(V),G_{m_{0}}(V))\geq \frac{\gamma^2}{c^2}(1+|2m|_{\eta})^{-2c(\eta,\tau) (|2m|_{\eta})^{{1}/({1+\eta})}}, \end{align} $$

which, together with $0<\varepsilon \leq \varepsilon _{0},$ yields $|m|_{\eta }>c_{1}\ln ({1}/{\varepsilon })$ with $c_{1}=c_{1}(\eta ,\tau ,\gamma )$ . Moreover, Theorem 5.1 shows $|G_{m}(V)|\leq \varepsilon _{0}^{3/4}e^{-(5/4)\pi \sigma _{0} |m|_{\eta }}.$ Thus, with direct calculations, we get

$$ \begin{align*} \begin{aligned} \sum_{m\in{\mathcal M}\setminus\{m_{0}\}} |G_{m}(V)\cap(E-\varepsilon,E+\varepsilon)|&\leq \sum_{m\in{\mathcal M}\setminus\{m_{0}\}} |G_{m}(V)| \\ &\leq \sum_{|m|_{\eta}>c_{1}\ln({1}/{\varepsilon})} \varepsilon_{0}^{{3}/{4}}e^{-({5}/{4})\pi\sigma_{0} |m|_{\eta}}\leq\frac{\varepsilon}{4}, \end{aligned} \end{align*} $$

where the last inequality is by $\varepsilon \leq \varepsilon _{0}$ and with the similar calculations in the proof of Lemma 4.10, we omit the details. Then we have, for any $0<\varepsilon \leq \varepsilon _{0}$ ,

$$ \begin{align*} \begin{aligned} |\Sigma_{V,\alpha}\cap(E-\varepsilon,E+\varepsilon)|&\geq2\varepsilon -|G_{m_{0}}(V)\cap(E-\varepsilon,E+\varepsilon)|\\ &\quad-\sum_{m\in{\mathcal M}\setminus\{m_{0}\}} |G_{m}(V)\cap(E-\varepsilon,E+\varepsilon)|\geq\frac{3}{4}\varepsilon. \end{aligned} \end{align*} $$

Case 2: $(-\infty ,\underline {E})\cap (E-\varepsilon ,E+\varepsilon )\neq \varnothing $ . In this case, for any $m\in {\mathcal M}$ ,

(5.4) $$ \begin{align} |E_{m}^{-}-\underline{E}|\leq2\varepsilon. \end{align} $$

Using the same way to get equation (5.3), we also get

(5.5) $$ \begin{align} |E_{m}^{-}-\underline{E}|\geq\frac{\gamma^2}{c^2} (1+|m|_{\eta})^{-2c(\eta,\tau)(|m|_{\eta})^{{1}/({1+\eta})}}. \end{align} $$

Then, equations (5.4) and (5.5) imply $|m|_{\eta }>2c_{1}\ln ({1}/{\varepsilon })$ . Similarly, we have

$$ \begin{align*} \sum_{m\in{\mathcal M}} |G_{m}(V)\cap(E-\varepsilon,E+\varepsilon)|\leq \sum_{|m|_{\eta}>2c_{1}\ln({1}/{\varepsilon})} \varepsilon_{0}^{{3}/{4}}e^{-({5}/{4})\pi\sigma_{0} |m|_{\eta}}\leq\frac{\varepsilon}{4}, \end{align*} $$

provided $\varepsilon \leq \varepsilon _{0}$ . So we have, for any $0<\varepsilon \leq \varepsilon _{0}$ ,

$$ \begin{align*} \begin{aligned} |\Sigma_{V,\alpha}\cap(E-\varepsilon,E+\varepsilon)|&\geq2\varepsilon -|(-\infty,\underline{E})\cap(E-\varepsilon,E+\varepsilon)|\\ &\quad-\sum_{m\in{\mathcal M}} |G_{m}(V)\cap(E-\varepsilon,E+\varepsilon)| \geq\frac{3}{4}\varepsilon. \end{aligned} \end{align*} $$

Case 3: $(\overline {E},+\infty )\cap (E-\varepsilon ,E+\varepsilon )\neq \varnothing $ . The proof is similar to that of Case 2, so we omit the details.

Finally, we get

$$ \begin{align*} |\Sigma_{V,\alpha}\cap(E-\varepsilon,E+\varepsilon)| \geq\tfrac{3}{4}\varepsilon \quad \text{for all } 0<\varepsilon\leq\varepsilon_{0}, \text{ for all } E\in\Sigma_{V,\alpha}. \end{align*} $$

As for the case $\varepsilon \in (\varepsilon _{0},\mathrm {diam}\Sigma _{V,\alpha })$ , we have

$$ \begin{align*} |\Sigma_{V,\alpha}\cap(E-\varepsilon,E+\varepsilon)|\geq\frac{3}{4}\varepsilon_{0}\geq\frac{3\varepsilon_{0}}{4\mathrm{diam}\Sigma_{V,\alpha}}\varepsilon. \end{align*} $$

Choose $\mu =\min \{3/4,{3\varepsilon _{0}}/{4\mathrm {diam}\Sigma _{V,\alpha }}\}$ and this concludes the proof.

5.3 Proof of Theorem 5.1

In this subsection, we give the proof of Theorem 5.1. To this end, we give the lemma below.

Lemma 5.2. Let $\alpha \in DC_{\infty }(\gamma ,\tau ),\kappa \in (0,\tfrac 14)$ , and $V\in \mathcal {H}_{\sigma _{0}}(\mathbb {T}^{\infty },\mathbb {R}), \eta>0,$ be a non-constant function. Assume that $E_{k}^{+}$ is a right edge point of the spectral gap $G_{k}(V)$ , and there are $\zeta \in (0,\tfrac 12)$ and $B\in \mathcal {H}_{\sigma }(2\mathbb {T}^{\infty },SL(2,\mathbb {R}))$ such that

(5.6) $$ \begin{align} \begin{aligned} B(x+\alpha)^{-1}S_{E_{k}^{+}}^{V}(x)B(x)=X:=\left( \begin{matrix} 1 \ \ \ \ \zeta \\ 0 \ \ \ \ 1 \end{matrix} \right)\!. \end{aligned} \end{align} $$

If

(5.7) $$ \begin{align} |B|_{\sigma}^{6}\zeta^{\kappa}\leq \frac{1}{16 \widetilde{C}^2}, \end{align} $$

then $|G_{k}(V)|\leq \zeta ^{1-\kappa }$ .

We postpone the proof of Lemma 5.2 to Appendix and now we apply Lemma 5.2 to prove Theorem 5.1. First, according to the conclusions in part (3) of Proposition 3.7, we have

$$ \begin{align*} \begin{aligned} |B|_{{\sigma_{0}}/{24}}^{6}\zeta^{{1}/{5}}&\leq 4^6 ({\varepsilon_{j_{l}}})^{-{3}/{160}} e^{({48}/{7})\pi({\sigma_{0}}/{24})|k|_{\eta}}\cdot 4^{{1}/{5}}\varepsilon_{j_{l}}^{{1}/{5}-{1}/{600}}e^{-({5}/{16})\pi\sigma_{0} |k|_{\eta}},\\ &\leq 4^{6+({1}/{5})}\varepsilon_{j_{l}}^{{6}/{25}-{49}/{2400}}e^{-({3}/{112})\pi\sigma_{0} |k|_{\eta}}\leq (4\widetilde{C})^{-2}. \end{aligned} \end{align*} $$

Thus, we can apply the conclusion of Lemma 5.2 and get

$$ \begin{align*} |G_{k}(V)|\leq\zeta^{{4}/{5}}\leq 4^{{4}/{5}}\varepsilon_{j_{l}}^{{4}/{5}-{4s}/{50}}e^{-({5}/{4})\pi\sigma_{0} |k|_{\eta}}\leq \varepsilon_{0}^{{3}/{4}}e^{-({5}/{4})\pi\sigma_{0} |k|_{\eta}}. \end{align*} $$