Traveling fronts for Fisher-KPP lattice equations in almost periodic media

This paper investigates the existence of almost periodic traveling fronts for Fisher-KPP lattice equations in one-dimensional almost periodic media. By the Lyapunov exponent of the linearized operator near the unstable steady state, we give sufficient condition of the existence of minimal speed of traveling fronts. Furthermore, it is showed that almost periodic traveling fronts share the same recurrence property as the structure of the media. As applications, we give some typical examples which have minimal speed, and the proof of this depends on dynamical system approach to almost periodic Schrodinger operator.


Background and main results.
Since the pioneer works of [7,23,34], the traveling fronts of reaction-diffusion equations in unbounded domain have become an important branch of the theory of equations with diffusion. Specially, traveling fronts of the Fisher-KPP type equation in continuous media (1.1) u t − (a(x)u x ) x = c(x)u(1 − u), t ∈ R, x ∈ R and more general reaction-diffusion equations in heterogeneous media received intense attention in the last few decades. As a simplest heterogenous case, traveling fronts in spatially periodic media were considered widely. First, the definition of spatially periodic traveling waves was provided by [53,58] independently, and then [26] proved the existence of spatially periodic traveling waves of Fisher-KPP equations in the distributional sense. Then, in the series of works [10,11], the authors investigated traveling fronts of Fisher-KPP type equations in high-dimensional periodic media deeply. The traveling fronts of spatially periodic Fisher-KPP type equations in discrete lattice (1.2) u t (t, n) − u(t, n + 1) − u(t, n − 1) + 2u(t, n) = c(n)u(t, n)(1 − u(t, n)), were also studied in [24,39]. Besides above works, a more general framework was given by [39,57] to study traveling fronts for Fisher-KPP type equations and more general diffusion systems.
However, few works on traveling waves of Fisher-KPP equations exist in more complicated media. Matano [41] first gave a definition of spatially almost periodic traveling waves and provided some sufficient conditions on the existence of spatially almost periodic traveling fronts of reaction-diffusion equations with the bistable nonlinearity. In [38], Liang showed the existence and uniqueness of the spatially almost periodic traveling front of Fisher-KPP equations in one-dimensional almost periodic media with free boundary. We also notice that the propagation problems of (temporally) nonautonomous reaction-diffusion equations were studied by Shen in a series of works [18,51,52].
In this paper, we are concerned with the almost periodic traveling fronts of the Fisher-KPP equation (1.2). To introduce the definition of the almost periodic traveling fronts, let we first recall the definition of classical and periodic traveling fronts. In the case where the media is homogeneous, that is, c is a constant sequence, the classical traveling fronts are defined by a solution u(t, n) = U (n−wt) with an invariant profile U and a speed w; in the case where the media is periodic, that is, c is a periodic sequence with period N , the periodic traveling fronts are defined by a solution u(t, n) = U (n − wt, n) with a periodic profile U , U (ξ, n) = U (ξ, n + N ) and an average speed w. Then it is natural to consider the recurrence of the profile depending on the structure of the media if one tries to generalize the definition of traveling fronts in heterogeneous media. To be exact, in the almost periodic media, the generalized traveling fronts need to inherit the almost periodicity of the media.
Before introducing the definition of generalized traveling fronts with almost periodic recurrence, we give some notions and background. A sequence f : Z → R is almost periodic if {f (· + k)|k ∈ Z} has a compact closure in l ∞ (Z). Denote by H(f ) the hull of the almost periodic sequence f , i.e., H(f ) = {f (n + k)|k ∈ Z}, the closure in l ∞ (Z). We also denote g · k = g(· + k), g ∈ H(f ), k ∈ Z, and let l ∞ loc (Z) be the set of all sequences in Z with pointwise topology and C 0 (R × Z) be the set of all continuous functions with locally uniform topology.
Remark 1.1. The speed w(g) is nonzero. Indeed, t(1; g · k) = t(1 + k; g) − t(k; g) is bounded with respect to k, since {t(k; g)|g ∈ H(c)} is a one-cover of H(c) in l ∞ loc (Z). Combining this with the definition of w(g), we deduce that w(g) is nonzero. It needs to point out that a more general extension of traveling fronts, socalled generalized transition front, was presented by Berestycki and Hamel [8]. For (1.2), the generalized transition front is defined as below.
Definition 1.2. A generalized transition front of (1.2) is an entire solution u = u(t, n) for which there exists a function N : R → Z such that (1.4) lim n→−∞ u(t, n + N (t)) = 1, lim n→+∞ u(t, n + N (t)) = 0, uniformly in t ∈ R. We say u has an average speed w ∈ R, provided lim t−s→+∞ Remark 1.3. In the periodic case, the almost periodic traveling front in Definition 1.1 is exactly the classical periodic traveling front [24]. However, there may exist a generalized transition front (Definition 1.2) which is not a classical traveling front [8].
A generalized transition front u = u(t, x) of (1.1) can be defined in the same way by assuming N = N (t) : R → R. Nadin and Rossi [45] investigated the existence of the generalized transition front of (1.1) when a, c are almost periodic. Motivated by the works [11,24,39,43], especially the work of Nadin and Rossi [45], we want to construct the almost periodic traveling fronts via the eigenvalue problem of the linearized operator of (1.2) near the equilibrium state u ≡ 0: (1.5) (L g u)(n) := u(n+1)+u(n−1)−2u(n)+g(n)u(n) = Eu(n), g ∈ H(c).
Here the subcript g is to emphasize the dependence of g. We will always shorten the notation L c = L.
One novelty of the paper is that we will use method from dynamical systems to study the operator L g , thus to study (1.3). Note that (1.5) can be rewritten as u(n + 1) u(n) = A(n) u(n) u(n − 1) , where A(n) = E + 2 − g(n) −1 1 0 . Let A n (g) = A(n − 1) · · · A(0) be the transfer matrix. Then the Lyapunov exponent of L g at energy E is denoted by L(E) and given by where µ is the Haar measure on H(c). It is known that L(E) is identical for g ∈ H(c) (see Proposition 3.4). The Lyapunov exponent characterizes the decay rate of any solutions of (1.5), and is also a fundamental topic in smooth dynamical systems. We always use Σ(L) to denote the spectrum of L, and denote λ 1 = max Σ(L). Once we have this, we can state our main results as follows: .
Then for (1.3), the following statements hold: (1) If w * < w, then for any w ∈ (w * , w), there exists a time-increasing almost periodic traveling front with average wave speed w; (2) If w * < w, then there exists a time-increasing generalized transition front with average speed w * ; (3) There is no generalized transition fronts with average speed w < w * .
Note that the sufficient condition for the existence result in Theorem 1.1 (1) is fulfilled up to a constant perturbation of g (Lemma 4.1), which was first observed in [45]. Since adding a constant to the potential doesn't affect the spectral property of L, in what follows we always assume w * < w.
We also should point out that the established almost periodic traveling fronts share the same recurrence property as the potential c(n). To make it precisely, if the frequency of the almost periodic sequence c(n) is α, then the frequency of v(·, ·; c) is also α, as we will show in Corollary 1.1 and Corollary 1.3.

1.2.
Applications. Of course, the interesting thing is to give concrete examples in which we can establish almost periodic traveling fronts for any average wave speed w > w * (i.e., w = ∞). Based on the work of [33], Nadin and Rossi [45] showed that if a, c are finitely differentiable quasi-periodic function with Diophantine frequency α, and c is small enough (the smallness must depend on α), then the operator Lu = (a(x)u x ) x + c(x)u has a positive almost periodic function. Consequently (1.1) has a time-increasing generalized transition front with average speed w ∈ (w * , ∞). Here we recall Therefore, the natural question is that whether one can remove the arithmetic condition of α, or whether one can remove the smallness of c. Now we answer this question as follows: where α ∈ T d is rationally independent, V : T d → R is a positive real analytic function. Then the following statements hold: (1) (1.2) has a time increasing almost periodic traveling front with average wave speed w ∈ (w * , λ 1 L(λ 1 ) ). (2) The traveling front u(t, n; c) can be rewritten as u(t, n; c) = U (t + T (n), nα), where U ∈ C 0 (R × T d ), T ∈ ∞ loc (Z). Corollary 1.1 (1) reduces w = ∞ to L(λ 1 ) = 0. For example, based on former results [2,59], Corollary 5.5 shows that for any irrational α ∈ R\Q, if V is analytic and close to constant, even the closeness is independent of α. Hence combining Proposition 3.5 and the continuity of the Lyapunov exponent [14,15], L(λ 1 ) = 0. On the other hand, Corollary 1.1 (2) states that if c(n) is quasi-periodic with frequency α, then the resulting traveling front is also quasi-periodic with frequency α. Corollary 1.1 (1) follows from Theorem 1.1 and the continuity of the Lyapunov exponent. Thus the assumption that V is analytic is necessary, since the Lyapunov exponent might be discontinuous in smooth topology [56]. Furthermore, Corollary 5.4 also states the results for V is just a finitely differentiable quasi-periodic function as in [45]. To be exact, if α ∈ DC d (γ, τ ), V ∈ C s (T d , R) with s > 6τ + 2, and V is small enough, then (1.2) has a time increasing almost periodic traveling front with average wave speed w ∈ (w * , ∞). However, it is widely believed that if the regularity is worser, then for generic V , the spectrum of (1.5) has no absolutely continuous component [4]. Therefore, most probably L(λ 1 ) > 0 by the well-known Kotani's theory [35].
The AMO was first introduced by Peierls [48], as a model for an electron on a 2D lattice, acted on by a homogeneous magnetic field [27,50]. Now, if V (θ) = 2κ cos θ + C, then we have the following: Suppose that c(n) = 2κ cos(2πnα) + C where α ∈ R\Q and C is a specified constant such that c has a positive infimum. Then the following statements hold: (1) If |κ| ≤ 1, then (1.3) has a time-increasing quasi-periodic traveling front with average wave speed w ∈ (w * , ∞). (2) If |κ| > 1, then (1.3) has a time-increasing quasi-periodic traveling front with average speed w ∈ (w * , λ 1 | ln κ| ). Note that AMO plays the central role in the Thouless et al theory of the integer quantum Hall effect [54]. This model has been extensively studied not only because of its importance in physics [46], but also as a fascinating mathematical object [5,6,31,32]. For us, the example is interesting since Nadin and Rossi [45] asked is it possible to construct a rigorous example where lim E→λ 1 µ(E) > 0? Then Corollary 1.2 (2) gives an affirmative answer to their question in the discrete setting. However, it is still open whether in this case, (1.2) has a time-increasing almost periodic traveling front with average wave speed w ∈ (w * , ∞).
From this aspect, we should mention that compared to [45], we are mainly concerned with Fisher-KPP lattice equations, since the main results we mentioned above [2,3,14,15,31] only work in the discrete case. Whether the corresponding results are valid for the continuous case are widely open.
We also remark that we are mainly concerned with almost periodic traveling fronts of (1.2) with average wave speed w ∈ (w * , ∞), while Nadin and Rossi's example [45] and Corollary 1.1 only give examples where c are quasi-periodic. Therefore, it is interesting and important to give concrete examples of real almost periodic sequence c(n), such that (1.2) has almost periodic traveling front with average wave speed w ∈ (w * , ∞). To state the result clearly, let's show how to construct the desired almost periodic sequence.
We assume that the frequency α = (α j ) j∈N belongs to the infinite dimensional cube R 0 := [1,2] N , and we endow R 0 with the probability measure P induced by the product measure of the infinite dimensional cube R 0 . We now define the set of Diophantine frequencies that was first developed by Bourgain [13]: 13]). Given γ ∈ (0, 1), τ > 1, we denote by DC ∞ (γ, τ ) the set of Diophantine frequencies α ∈ R 0 such that where j := max{1, |j|}.
As proved in [13,16], for any τ > 1, Diophantine frequencies DC ∞ (γ, τ ) are typical in the set R 0 in the sense that there exists a positive constant C(τ ) such that P(R 0 \DC ∞ (γ, τ )) ≤ C(τ )γ. Now for any α ∈ DC ∞ (γ, τ ), we say that c(n) : Z → R is an almost periodic sequence with frequency α and analytic in the strip r > 0 if we may write it in totally convergent Fourier series where Z ∞ * := {k ∈ Z ∞ : |k| 1 := j∈N j |k j | < ∞} denotes the set of infinite integer vectors with finite support. Since α is rationally independent, H(c) = T ∞ = i∈N T 1 with infinite product topology. Once we have these, we can introduce our precise results as follows: . Suppose that c(n) is an almost periodic sequence with frequency α and analytic in the strip r > 0. Furthermore, suppose that there exists = (γ, τ, r) such that Then following statements hold: (1) (1.2) has a time increasing almost periodic traveling front with average wave speed w ∈ (w * , ∞) (2) The front u(t, n; c) can be rewritten as u(t, n; c) = U (t + T (n), nα), where U ∈ C 0 (R × T ∞ ), T ∈ ∞ loc (Z). Finally let's outline the novelty of the proof of these applications. The proof depends crucially on dynamical approach to almost-periodic Schödinger operator, i.e., in order to study the spectral property of Schrödinger operator (1.5), one only needs to study the corresponding Schrödinger cocycle (c.f. section 2.6). For analytic quasi-periodic potentials (Corollary 1.1 and Corollary 1.2), the result follows from the continuity of the Lyapunov exponent for analytic cocycles. For the almost-periodic case, we need to prove the existence of positive almost-periodic functions. The key observation is Lemma 5.1, which says that if the Schrödinger cocycle is reduced to a constant parabolic cocycle, and the conjugacy is close to the identity, then the corresponding Schrödinger equation has a positive almost-periodic solution.
Here reducibility means the cocycle can be conjugated to a constant cocycle (c.f. section 5). From this aspect, the powerful method is KAM. For almost-periodic Hamiltonian systems, KAM was first developed by Pöschel [49]. One can consult [13,16,42] for more study on similar objects. However, it is well-known traditional KAM method only works for positive measure parameters, and here we need to fix the energy to be the supremum of the spectrum, thus the corresponding cocycle is fixed. To solve the difficulty, the method is to make good use of fibred rotation number which was first developed by Herman [28] for quasiperiodic cocycles (not necessarily quasiperiodic Schrödinger cocycle), and it can be extended to the almost periodic setting.
1.3. Structure of the paper. In the section 2, we introduce some preliminary knowledge which will be needed in our proof. In the section 3, we introduce and investigate some properties of the generalized principal eigenvalue and the Lyapunov exponent which turn out to be powerful techniques in constructing the almost periodic traveling front with any average wave speed w > w * . Moreover, we also show the reason why the existence of positive almost periodic solution of (1.5) implies w = ∞, c.f. Proposition 3.5.
In the section 4, we prove Theorem 1.1 by the following steps: First we establish the almost traveling front with any average wave speed w > w * by constructing super-sub solution, and get the monotonicity of the fronts in t, thus proves (1) in Theorem 1.1. Next we make use of the properties of spreading speed to deduce that even the generalized transition fronts with the average speed w < w * can not exist, and then (2) is proved. Lastly, we construct a generalized transition front with the critical wave speed w * by pulling back the solution of the Cauchy problem associated with the initial datum Heaviside function, and then we finish the proof.
In the section 5, we use KAM method to get the positive quasi-periodic (almost-periodic) solution of (1.5) with the positive infimum when c = V (·α + θ) is very close to a constant, where V ∈ C s (T d , R)(V ∈ C ω (T ∞ , R)). This will help us to prove Corollary 1.3. At last, we finish all the proofs of applications.

Preliminaries
2.1. Maximum principle, existence and uniqueness for the Cauchy Problem. The maximum principle on the whole space can be stated as follows: Proposition 2.1 (Maximum principle [18]). Let v ∈ ∞ (Z). Assume that for any bounded interval I = [0, The following Harnack inequality is a very useful technique when we study the properties of the solution of (1.2). We will present it here for the reader's convenience.
Remark 2.1. The proof of the Harnack inequality could be found in [40] with the initial value u(0, n) having a finite support. However, we should notice that the argument can be applied to (2.1) similarly with minor modification.
Combining the Harnack inequality with the maximum principle, we can deduce the strong maximum principle as follows: Corollary 2.1 (Strong maximum principle [40]). Under the assumption of Proposition 2.1, either u ≡ 0 or u > 0 in I × Z.
The comparison principle is a consequence of the strong maximum principle, and it is useful for us to construct the almost periodic traveling front. To state it, we first give the definition of super-sub solutions: Letū, u ∈ C(R × Z) be two bounded functions. We say thatū is a supersolution of (1.2) if for any given n ∈ Z,ū is absolutely continuous in t and satisfies u t −ū(n + 1) −ū(n − 1) + 2ū(n) − cū(1 −ū) ≥ 0 for a.e. t ∈ (0, ∞), and u is a subsolution if for any given n ∈ Z, u is absolutely continuous in t and satisfies The strong comparison principle is given by Proposition 2.3 (Strong comparison principle). Let u and u be a supersolution and a subsolution of (1.2) repectively. If u(0, n) ≤ u(0, n) in Z, then u < u or u ≡ u in (0, ∞) × Z.
Usually, well-behaved Cauchy Problem possesses the property that it admits a unique global solution. Moreover, existence and uniqueness is vital for us when we construct the generalized transition front with minimal speed w * . Theorem 2.1 ( [47]). For any initial value ϕ(n) ∈ ∞ (Z), there exists a unique u ∈ C 0 (R × Z) with u(t, ·) ∈ ∞ (Z) for any t ∈ (0, ∞) such that We denote by l(ϕ) := l(ϕ, ϕ) the induced quadratic form on l c (Z). Furthermore, we write l ≥ 0 if l(ϕ) ≥ 0 for all ϕ ∈ l c (Z).

Definition 2.1 ([37]
). Let l be a quadratic form l associated with the Schrödinger operator −L such that l ≥ 0. We say the form is critical if there does not exist a positive ∈ l ∞ loc (Z)(i.e. ≥ 0 and The operator L is said to be critical, if the quadratic form l associated with −L is critical. The following well known formula reveals the connection between the operator L and the associated quadratic form l. That is, l(ϕ, ψ) = −Lϕ, ψ .
The critical operator has the following important property which is useful for us to reveal the connection between the Lyapunov exponent and positive almost periodic solution (c.f. Proposition 3.5): Proposition 2.4. [37] Let L be a critical operator. Then there exists a unique positive function in l ∞ loc (Z) such that Lu ≤ 0 (up to scalar multiplication).

2.3.
Properties of the almost periodic traveling front. Almost periodic traveling front also implies the almost periodicity of the time recurrence in the sense of the following lemma.
Lemma 2.2. Assume u(t, n; g) is an almost periodic traveling front of (1.3). Then for any > 0, there exist relative dense sets {t k } ⊂ R and {n k } ⊂ Z such that sup n∈Z |u(t k , n + n k ; g) − u(0, n; g)| ≤ .
Since {n k } is a relative dense set, i.e., there exists L ≥ 0 such that for any point n in Z, one has [n−L, n+L]∩{n k } = ∅. Denote M := max −L≤n≤L,g∈H(c) t(n; g).
Since {t(n, g)} is a one-cover of H(c) in l ∞ loc (Z) and H(c) is compact, M can be obtained. Hence for any t ∈ R, one has [−M + t, M + t] ∩ {t(n k ; g)} = ∅, as desired.
The average wave speed of almost periodic traveling front relates with the recurrence of c in the following way. Proof. Recall that w(g) = lim |n−k|→∞ n−k t(n;g)−t(k;g) exists, and it is not zero.
|n−k|→∞ t(n;g)−t(k;g) n−k exists. By Definition 1.1, {t(1; g · k)|k ∈ Z} has a compact closure in l ∞ loc (Z). Hence it is an almost periodic sequence. Then for any k ∈ Z, The limit on the right-hand side exists, and it is independent of g ∈ H(c). Then the proof is complete.
An observation is that an almost traveling front is also a generalized transition front.
Proposition 2.6. An almost periodic traveling front of (1.3) with average wave speed w(g) is a generalized transition front of (1.3) with average speed w(g).
Proof. From Definition 1.1(3), w(g) = lim |n−k|→∞ n−k t(n;g)−t(k;g) exists. Without loss of generality, we will always assume that w(g) > 0. For any k ∈ Z, there exists an absolute constant L such that for any |n| ≥ L, Thus for n > L, for n < −L.
The Lyapunov exponent is defined as We say an SL(2, R) cocycle (T, A) is uniformly hyperbolic if, for every x ∈ X, there exists a continuous splitting for some constants C, c > 0, and it holds that If L(T, A) > 0 and the splitting is not continuous, then we call (T, A) is non-uniformly hyperbolic.
2.5. Fibered rotation number. Assume X is compact, and (X, T ) is uniquely ergodic with respect to its unique invariant probability measure. For this kind of dynamically defined cocycles (T, A), one can define the rotation number of the cocycle. Let S 1 be the set of unit vectors of R 2 , consider a projective cocycle F T,A on X × S 1 : If A : X → SL(2, R) is continuous and homotopic to the identity, then there exists a liftF (T, n converges uniformly to ρ in (x, y) ∈ X × R, and it is independent on the lift of F T,A , up to an addition of an integer [28]. Then ρ is called fibered rotation number of (T, A), and we denote it as rot(T, A).

Remark 2.2.
In the following, we will always take X = T d where d ∈ N + or ∞ (we endow it with the product topology if d = ∞), and consider the quasi-periodic (or almost-periodic) cocycle (α, A).
The following two lemmas are are useful for us: 36]). The rotation number is invariant under the conjugation map which is homotopic to the identity. More precisely, if A, B : T d → SL(2, R) is continuous and homotopic to the identity, then rot(α, B(· + α) −1 A(·)B(·)) = rot(α, A).

2.6.
Almost periodic Schrödinger operator. For the almost periodic sequence, we also have the following: where Ω is a compact abelian group, ω ∈ Ω, f : Ω → R is continuous, and T : Ω → Ω is a minimal translation, say T = · + α.
As result, we define the almost periodic Schrödinger operator as a selfadjoint operator on l 2 (Z) : It's well known that the spectrum Σ(L f,T,ω ) is a compact set of R. Moreover Σ(L f,T,ω ) is independent of ω, and we shorten the notation as Σ(L f,T ). In We say θ is the phase, V is the potential, and α is the frequency.
For almost-periodic Schrödinger operator, one can define the integrated density of states (shorten as IDS), denoted by k(E), as follows: where L L is the restriction of L V,α,ω to the set I = {1, · · · , L − 1} with boundary conditions u(0) u(1) = cotθ, u(L) = 0. The integrated density of states will be crucial for our study of the positive almost periodic solution. Furthermore, we have the following elementary fact: Suppose E is at the rightmost of the spectrum. Then by the well known characterization of the biggest eigenvalue λ L and E λ L = sup the rightmost of the spectrum E is always large than λ L . Thus the number of eigenvalues of L L less than or equal to E is always L − 1. By the definition of the IDS, k(E) = 1.
Note that a sequence (u n ) n∈Z is a formal solution of the eigenvalue equation L f,T,ω u = Eu, if and only if We call (T, S f E ) an almost-periodic Schrödinger cocycle. In this paper, we will mainly consider the following two kinds of Schrödinger cocycles.
The study of the spectral properties of the almost-periodic Schrödinger operator is closely related to the dynamics of almost-periodic Schrödinger cocycle. For example, we will need the following two important facts: 30]). IDS of the Schrödinger operator relates with the fibered rotation number of Schrödinger operator as follows: 29]). Let L f,T,ω be an almost periodic Schrödinger operator. Then we have the following:

Properties of the Linearized Problem
3.1. Generalized principal eigenvalue for more general operator. In this section, we will define and study the properties of generalized principal eigenvalue of a more general operator since it will be needed in our proof.
Remarkably, generalized principal eigenvalue theory for elliptic operator is of its own interest, and it turns out to be very useful in studying maximum principle [9]. Now we consider the operator with a, b, c being almost periodic sequences and inf a > 0, inf b > 0. In particular, M 1,1,g−2 = L g defined in (1.5).
For any (maybe unbounded) interval I ⊂ Z, we define the generalized principal eigenvalue for M a,b,c as where M a,b,c;I is the restriction of M a,b,c to the set I ⊂ Z. If I is bounded, it is exactly the classical principal eigenvalue (the largest eigenvalue). In the case I = Z, a = b ≡ 1, c is replaced by c − 2, we will show below that it coincides with λ 1 = max Σ(L). From (3.1), λ 1 (M a,b,c , I) is nondecreasing with respect to the inclusion of intervals I.
Proof. For simplicity, we shorten the notation M a,b,c; We only prove the case I = Z. Since b(n) = a(n − 1), it is clear that M a,b,c;N is a bounded self-adjoint operator. Then for any N ∈ N + , here · denotes the norm in the Banach space of linear bounded operators from l 2 (Z) to l 2 (Z).
Since a, b, c is almost periodic, hence bounded. By direct calculation, From (3.4) and Proposition 3.1, we have In particular, λ 1 (M a,b,c , Z) lies in the rightmost of the spectrum.
Other generalizations of principal eigenvalue will be listed below, and they are all indispensable to our proof. Define the following quantities: We denote the hull of triple (a, b, c) by H(a, b, c): Proposition 3.3. The following hold: is a constant function with respect to g on H(c).
(2). Analogous to the above argument, for any sequences Hence for any δ > 0, there exists δ such that | δ u δ · n − λ| ≤ δ holds for any n. Thereby Note that u ε is almost periodic for any > 0. Passing along a subsequence i k → ∞, for any δ > 0, it follows from (1) that Finally, letting δ → 0, If a(n) = b(n + 1), apply the above argument to λ 1 (M a,b,c ), λ 1 (M a,b,c ), then we are done.
3.2. The Lyapunov exponent of the linearized operator. The Lyapunov exponent is crucial to determine the average wave speed of the almost periodic traveling front. We will discuss its properties here and illustrate the connection with the existence of positive almost periodic solution. Recall and L = L c . Note that Proposition 3.3 tells us λ 1 (L g , Z) = λ 1 (L, Z), for any g ∈ H(c). As a consequence of Lemma 3.2, λ 1 = max Σ(L) = λ 1 (L, Z). Hence we do not distinguish them with a slight abuse of notation in the forthcoming paragraphs. First we state the following technical lemma which will be frequently used, and it is an immediate consequence of [40, Lemma 6.2].

Moreover, lim
and the limit the convergence is uniform in g ∈ H(c).
First we need the following lemma: is a compact Abelian group and T is minimal.
Since E > λ 1 , by Theorem 2.3, (T, S E ) is uniformly hyperbolic. Then as a consequence of Lemma 3.2, the limit L E := lim n→∞ 1 n ln S E n (g) exists, and it is independent of g ∈ H(c). Moreover, the convergence is uniform in g ∈ H(c). Thus we can deduce that for any g ∈ H(c), where the last equality follows from the direct examination of the definition. Note that in the proof of Lemma 3.2, we can deduce that the limit lim n→∞ 1 n ln |S E n (g) · v| exists for any v ∈ R 2 \{0}. Now it follows from (2.6) that uniform hyperbolicity implies that for any g ∈ H(c), and every n ≥ 0, Denote < 0.
Hence v g E ∈ E s (g), and lim Then we can deduce that where M depends on g l ∞ . It follows that Inserting this inequality into (3.11) and (3.12), L(E) = lim n→±∞ − ln φ E (n;g) n follows directly. Then the proof is complete.
The concavity and monotonicity of the Lyapunov exponent L(E) will be needed in our construction of the almost periodic traveling front, and it is given by: +∞) is concave, nondecreasing and there exists C > 0 such that, for E > λ 1 , where δ was given in Lemma 3.1 and L(E) > L := lim Proof. L(E) is nondecreasing and concave follows from [45,Lemma 2.5].
The existence of positive almost periodic solution always implies the Lyapunov exponent L(E) will decay to 0 as E λ 1 , and it is crucial for us to determine in which case we can establish the almost periodic traveling front with average wave speed w ∈ (w * , ∞) (c.f. Corollaries 1.2 and 1.3). First we need a preliminary lemma about critical operator. Lemma 3.4. Suppose that L admits a positive bounded solution ϕ of Lφ = Eφ. Then the associated eigenvalue is the generalized principal eigenvalue λ 1 and there hold: Proof. Taking ϕ as the test function in (3.6), it follows from Proposition 3.3 the associated eigenvalue is exactly the generalized principal eigenvalue λ 1 .
(1). Assume by contradiction that L − λ 1 is not critical. Denote the quadratic form associated with λ 1 −L by h. Then by Lemma 2.1, there exists a positive function ∈ l ∞ loc (Z) such that h(u) = (λ 1 − L)u, u ≥ u, u for any u ∈ l c . That is, L − λ 1 ± are bounded above by 0(the supreme of the spectrum of L − λ 1 , see Proposition 3.2, i.e. the ground state energe of L − λ 1 ). It follows from [20,Theorem 4] that ≡ 0, this is impossible since positive. Hence L − λ 1 is critical.
Although the positive solution of Lφ = Eφ may not be almost periodic, almost periodicity can still be revealed in the following way: Once we have this, the following result follows: Proof. Denote φ E (·) := φ E (·; c) which can be obtained in Proposition 3.4. Consider the analogueφ E of φ E but with the initial condition as follows: The functionφ E (−n) shares the same properties with φ E . Particularly, the limitL(E) := lim n→±∞ 1 n lnφ E (n) exists and it is positive. Let ϕ E := φ E φ E . Then by the direct computation that Thus, We claim that q E converges uniformly to 0 in R as E → λ 1 . Otherwise, there exist > 0 and two sequences {E i } i and {n i } i such that E i → λ 1 and |q E i (n i )| ≥ for all i ∈ N. According to Lemma 3.5, φ E (·+1) φ E (·) is almost periodic, hence {q E i } is uniformly bounded. Passing along a subsequence n i k in (3.14), converges pointwise to a positive solution ϕ * of L g * admits a supersolution which is not a solution. And from the assumption, Lemma 3.4 (2),(3) and Proposition 2.4, L g * admits a unique positive supersolution. However, as will see, ϕ and φ are both positive supersolutions. This is impossible! Therefore we have q E → 0 unifromly as E → λ 1 . Finally we havẽ tends to 0 as E → λ 1 . Then they both tend to 0 as E → λ 1 due to the positivity of L(E) andL(E).

Construction of the fronts
4.1. Construction of almost periodic fronts. In this section, we start to prove Theorem 1.1. First we will consider (1) in Theorem 1.1 in this subsection. The basic idea is to apply the super-sub solution method. From now on, we set σ E (n; g) := − ln φ E (n + 1; g) φ E (n; g) for any n ∈ Z, g ∈ H(c), where φ E (n; g) is given by Proposition 3.4. Notice that σ E (n; g) is almost periodic by Lemma 3.5. Before constructing the the almost periodic traveling front, we should first notice the following facts: Proof. The proofs can be referred to [45, Lemmas 1.6 and 3.2]. Now afterwards, we take w ∈ (w * , w), and let E > λ 1 be as in Lemma 4.1. For a given almost periodic sequence σ, we define the operator = e −σ(n) φ(n + 1) + e σ(n−1) φ(n − 1) + g(n)φ(n).
. = φ 1+ E (n; g), the positive function ζ satisfies Then it follows that It is clear that the choice of θ is not unique. Let us now define {θ(·; g)} g∈H(c) in the following certain way. Note that the function θ = e u with | u −λ| < δ − δ in Proposition 4.1 is the unique almost periodic solution (see [40, Indeed, almost periodicity follows from the inequality Define for all (t, n) ∈ R × Z, g ∈ H(c): u(t, n; g) := min{1, φ E (n; g)e Et }, u(t, n; g) := max{0, φ E (n; g)e Et − Aθ(n; g)φ 1+ E (n; g)e (1+ )Et }, where and θ(·; g) are given by Propositions 4.1 and A is a positive constant that is to be specified. Notice that is independent of g. Denote S g = {ũ is an entire solution of (1.3)|u(t, n; g) ≤ũ(t, n) ≤ u(t, n; g) in R×Z.} Proposition 4.2. There exists a solution u of (1.3) satisfying u ∈ S g . Moreover, u = u(t, n; g) is increasing in t.
Proof. By the calculation, φ E (n; g)e Et is a supersolution on the R × Z of (1.2). Then u is also a supersolution of the equation (1.2). Take (t, n) ∈ R×Z so that u(t, n; g) > 0 and set ζ := φ E (n; g)e Et . Then we have: u t (t, n; g) − u(t, n + 1; g) − u(t, n − 1; g) + 2u(t, n; g) − g(n)u(t, n; g) Therefore, as 0 obviously solves (1.2), for u to be a subsolution it is sufficient to choose A so large that, for all (t, n) such that u(t, n; g) > 0, one has Aδθ(·; g)ζ 1+ ≥ gζ 2 .
Define the sequence {u i } as follows: u i is the solution of (1.3) for t > −i with initial condition u i (−i, n; g) = u(−i, n; g). By the comparison principle Proposition 2.3, u i satisfies ∀t > −i, n ∈ Z, u(t, n; g) ≤ u i (t, n; g) ≤ u(t, n; g).
Thus, for i, j ∈ N with j < i and for any 0 < h < 1, using the monotonicity of u, we will get u j (−j, n; g) = u(−j, n; g) ≥ u(−j − h, n; g) ≥ u i (−j − h, n; g).
Note that u i (· − h, ·; g) is also a solution of (1.3). The comparison principle Proposition 2.3 gives us ∀j < i, 0 < h < 1, t > −j, n ∈ Z, g ∈ H(c) u j (t, n; g) ≥ u i (t − h, n; g). Now by the arguments before, we can prove {u i } i converges locally uniformly to a global function u ≤ u ≤ u of (1.3). Then passing to the limit as i, j → ∞, u(t, n; g) ≥ u(t − h, n; g) for all (t, n) ∈ R × Z and 0 < h < 1, g ∈ H(c). This means that u is nondecreasing in t. If the monotonicity were not strict, then the parabolic maximum principle Corollary 2.1 would imply that u is constant in time, which contradicts u ≤ u ≤ u. Then we finish the proof.
Before proving this proposition, we give an observation about one-cover.
Remark 4.1. We claim that {Φ g |g ∈ H(c)} is said to be a one-cover of H(c) in some metric space X if and only if In fact, (4.4) holds straightforward if {Φ g |g ∈ H(c)} is a one-cover. Assume that g k → g * and is the metric of X. Then for any k ∈ Z, there exists {n k } such that (Φ g·n k , Φ g k ) + g · n k − g k ∞ < 1/k. Therefore, g · n k → g * since g k → g * , and thus Φ g·n k → Φ g * by (4.4). Hence Φ is continuous since Proof of Proposition 4.4. (1) follows from the uniqueness of φ E (n; g) which follows from Proposition 3.4.
(2). By Remark 4.1, it is sufficient to prove Φ g·n k → Φ g * provided that g· n k → g * . For g·n k → g * , φ E (n; g·n k ) converges locally uniformly, up to some subsequence, to some functionφ E (n). Moreover, we haveφ E (n) ≤ Ce −δn for n > 0 since φ E (n; g · n k ) ≤ Ce −δn from Lemma 3.1. Thereforeφ satisfies which yields thatφ = φ E (n; g * ) by uniqueness. That is to say, all the convergence subsequences converge to the same limit. Hence φ E (n; g · n k ) → φ E (n; g * ) locally uniformly in n ∈ Z.
With this at hand, we can prove that u satisfies (4) of Definition 1.1.
To prove u satisfying (2) of Definition 1.1, we will consider u(t, 0; g · n).  Proof. For any (t, n) ∈ R × Z, we have where the last inequality follows from θ(n + k; g) = θ(n; g · k). Therefore, for some positive constant M . From the second inequality, we deduce that lim t→−∞ u(t, 0; g · n) = 0 uniformly in n ∈ Z.
Now we need the uniform convergence in R × Z to explain why u(t, n; g) is an almost periodic traveling front. Before that, the following lemma is needed.

Consider
u i k (t, n) := u i (t + t(n k ; g), n + n k ), i = 1, 2. Then the nonnegative function w k := u 2 k − u 1 k satisfies lim k→∞ w k (t 0 , 0) = 0 and where Dw k (t, n) = w k (t, n + 1) − w k (t, n − 1) + 2w k (t, n). Take M large enough such that M + sup , and this yields that Thus the proof is complete.
Using Lemma 4.4, we have the following result about uniform convergence.
Theorem 4.1. Assume that g * = lim k→∞ g · n k for some sequence {n k } k∈Z + .
We are now in the position to prove the existence of almost periodic traveling front.
Theorem 4.2. u(t, n; g) is an almost periodic traveling front with the average wave speed w ∈ (w * , w).

Now combining this with Proposition 4.4, (3) of Definition 1.1 is proved.
Finally, Lemma 4.2 gives rise to (4). Hence we finish the proof. Now we ends the proof of (1) in Theorem 1.1.

4.2.
Non-existence of Fronts with Speed Less than w * . Next we turn to prove (3) in Theorem 1.1, i.e., there is even no generalized transition front with average speed w < w * . Compared to the previous section, we only consider the generalized transition front of (1.2). However, the similar arguments can be applied to (1.3) with minor modification. From now on, for the sake of simplicity, we denote Dφ(t, n) = φ(t, n + 1) + φ(t, n − 1) − 2φ(t, n), t, n ∈ R × Z and u(t, n; s, u 0 ), t ≥ s, n ∈ Z a solution of (1.2) with initial value u 0 starting at time s. Now we begin with a lemma which will provide a lower bound of the average speed of generalized transition front.
To prove this lemma, we need Proposition 4.5. Let g ∈ H(c) and u g (t, n; 0, u 0 ) be the solution of Then for any 0 ≤ w < w * , we have lim t→∞ inf 0≤n≤wt u g (t, n; 0, u 0 ) = 1 exists uniformly in g. Proof.
That is to say, u g s (t, n; 0, u 0 ) is a subsolution of (4.10) with g replaced by g l for any l ≥ N .
Then for any 0 ≤ w < w * , we have lim t→∞ inf 0≤n≤wt u k (t, n) = 1 exists uniformly in k by Proposition 4.5. Therefore, the solution u(t, n; 0, u where u Particularly, there exists no generalized transition front with average speed w < w * . Proof. First by (1.4) and Proposition 2.3, we can check that u(t, n + N (t)) < 1.
In all, we have proved (3) in Theorem 1.1.

4.3.
Construction of the critical fronts. At last, to verify (2) of Theorem 1.1, we only need to consider (1.3) with g = c, i.e., (1.2). First we want to construct the critical front with average speed w * . By critical front we mean that Definition 4.1. We say that an entire solution u of (1.2) with 0 < u < 1, is a critical traveling front (to the right) if for all (t 0 , n 0 ) ∈ R × Z, v is an entire solution of (1.2) such that v(t 0 , n 0 ) = u(t 0 , n 0 ) and 0 < v < 1, then Before going any further, we introduce some useful lemmas.
Proof. Suppose that the conclusion fails. Then for any k ∈ Z + , there exist t k and n k such that |u(t k , n k ) − u(t k , n k + 1)| > 1 − 1/k. After passing a subsequence, we have u(s k , m k ) − u(s k , m k + 1) > 1 − ε k or u(s k , m k + 1) − u(s k , m k ) > 1 − ε k for some s k , m k , and ε k with ε k → 0. We only prove the former case. Note that 0 < u(t, n) < 1. Then Note also that | d dt u| ≤ 4+ g l ∞ (Z) . Then there exists S which is independent of k such that u(s k −S, m k ) ≥ 3/4−ε k > 1/2 for k large. On the other hand, from Proposition 2.2, there exists a constant C(S) which only depends on S such that 1/2 < u(s k − S, m k ) ≤ C(S)u(s k , m k + 1) ≤ C(S)ε k → 0 as k → 0.
This yields a contradiction.
With Lemma 4.6 at hand, we have the following equivalent definition of generalized transition front. Lemma 4.7. Let u(t, n) be a solution of (1.2) with 0 < u < 1 and u(t, n) → 0 as n → ∞, u(t, n) → 1 as n → −∞ for any t ∈ R.
Let us now construct critical front. Fix any θ ∈ (0, 1). For any k ∈ Z + , we define Then by Lemma 4.5 and the continuity of u(t, n; 0, H k ) with respect to t, we can define s k := min{s|u(s, 0; 0, H k ) = θ} > 0. In particular, u(s k , 0; 0, H k ) = θ. Note that by Theorem 2.1, u(t, n; s, H k ) = u(t − s, n; 0, H k ) for any t ≥ s. Then u(0, 0; −s k , H k ) = θ for any k ∈ Z + . The idea is to take the limit of some subsequence from {u(t, n; −s k , H k )}, and prove the resulting function is exactly the critical traveling front. Moreover, it is exactly a generalized transition front with average speed w * . Before that, an observation about some important properties of s k is given by Lemma 4.8. s k is strictly increasing and converges to +∞.
Proof. We first show that s k is strictly increasing. Assume by contradiction, that s k ≥ s k+1 for some k ∈ Z + . Note that Then by Proposition 2.3, we have In particular, θ = u(0, 0; −s k+1 , H k+1 ) < u(s k+1 − s k , 0; −s k , H k ). Notice that u(−s k , 0; −s k , H k ) = 0. Then by intermediate value theorem there exists −s k < τ < s k+1 − s k ≤ 0 such that Therefore the definition of s k gives s k ≤ τ + s k . That is impossible since τ < 0. Hence s k < s k+1 .
Next, we prove that lim k→∞ s k = +∞. Suppose by contradiction that after passing to a subsequence, lim k→∞ s k = s ∞ < +∞. Let φ E,k be a solution of Note that for any t ≤ 0, for some constant C only depending on E, λ 1 and g l ∞ , and the last inequality follows from Lemma 3.1. Then we can take K large such that u K (t, K) ≤ θ/2 for any t < 0. Moreover, combining the Proposition 3.4, there exists K 1 such that for any t ≥ −s ∞ and n ∈ Z, Thus for any t ≥ −s K 1 and n ∈ Z, it follows from Proposition 2.3 u(t, n; −s K 1 , H K 1 ) ≤ u K (t, n + K).
As s k → ∞, there exists a subsequence of {u(t, n; −s k , H k )} such that it converges to some entire solution u(t, n). Moreover, u(t, n) is "steeper" than any other entire solution in the following sense (see [21] for continuous case). Lemma 4.9. Let u(t, n) be a limit of some subsequences of u(t, n; −s k , H k ). Assume that v is an entire solution of (1.2) with v(t, n) ∈ (0, 1) on R × Z. Then for any t ∈ R, there exists n t ∈ Z ∪ {±∞} such that First we need the following proposition whose proof can be found in [25,Lemma 4]: 25]). Consider the solution u 1 (t, n) and u 2 (t, n) of (1.2). Denote w(t, n) = u 1 (t, n)−u 2 (t, n). If w(t 0 , n 0 ) > 0 for some (t 0 , n 0 ) ∈ R×Z, w(t 0 , n) > 0 for n < n 0 , w(t 0 , n) < 0 for n > n 0 , then the following hold: (1) For any t ≥ t 0 , if w(t, n) > 0 for some n ∈ Z, then w(t, m) > 0 for any m < n.
(2) For any t ≥ t 0 , if w(t, n) < 0 for some n ∈ Z, then w(t, m) < 0 for any m > n.
Proof of Lemma 4.9. First we prove that if u(t 0 , n 0 ) < v(t 0 , n 0 ) for some (t 0 , n 0 ) ∈ R × Z, then u(t 0 , m) ≤ v(t 0 , m) for any m > n 0 . Suppose by contradiction that u(t 0 , m) > v(t 0 , m) for some m > n 0 . Then we have (4.17) u(t 0 , n 0 ; −s k , H k ) < v(t 0 , n 0 ), and u(t 0 , m; −s k , H k ) > v(t 0 , m) for some k large enough. It is clear to check that Applying Proposition 4.7 with w(t, n) := u(t, n; −s k , H k ) − v(t, n), we can conclude that for t ≥ −s k and n ∈ Z where w(t, n) < 0, one has w(t, m) < 0 for any m > n. This contradicts (4.17). Similarly, if u(t 0 , n 0 ) > v(t 0 , n 0 ) for some (t 0 , n 0 ) ∈ R×Z, then u(t 0 , m) ≥ v(t 0 , m) for any m < n 0 . Therefore the existence of n t follows directly. Now we begin to construct an entire solution by taking the limit of {u(t, n; −s k , H k )}: Lemma 4.10. The limit u(t, n) := lim k→∞ u(t, n; −s k , H k ) exists locally uniformly in (t, n) ∈ R × Z. Moreover, u is an entire solution of (1.2) with u(0, 0) = θ.
Assume, by contradiction, that there is n 1 < 0 such that w(0, n 1 ) < 0 (The proof is similar in the case where there is n 1 > 0 such that w(0, n 1 ) > 0). Note that We can deduce from Proposition 4.7 that if w(t, n) < 0 for some t > −s k and n ∈ Z, then w(t, m) < 0 for any m > n (similarly, if w(t, n) > 0 for some t > −s k and n ∈ Z, then w(t, m) > 0 for m < n). Therefore, w(0, m) < 0 for m > n 1 contradicts with w(0, 0) = 0.
That is to say, u(t, n) is a critical traveling front.
Let u w (t, n) be the almost periodic traveling front u(t, n; c) obtained in Theorem 4.2 with average wave speed w > w * . Thus by Proposition 2.6, u w (t, n) is also a generalized transition front with average speed w. Then a similar argument to that of [44,Theorem 3.1] yields that sup t∈R diam{n ∈ Z|ε ≤ u(t, n) ≤ 1 − ε} < ∞ for any ε ∈ (0, 1/2). Therefore, it follows from Lemma 4.7 that u(t, n) is a generalized transition front.
Combining this with the intermediate value theorem, there exists T 0 > τ such that u(t + T 0 , n ) = u(t , n ).
As u(t, n) is a critical traveling front, so is v(t, n) := u(t + T 0 , n). Combining u(t , n ) = v(t , n ) and the definition of critical traveling front, one has u(t , n) ≥ v(t , n) if n < n and u(t , n) ≤ v(t , n) if n > n , v(t , n) ≥ u(t , n) if n < n and v(t , n) ≥ u(t , n) if n > n .
Hence u(t , n) = v(t , n) for n ∈ Z. Now we get the conclusion that u(t, n) = v(t, n) = u(t + T 0 , n) for any (t, n) ∈ R × Z by the similar arguments in is an incomplete answer since we cannot determine whether (1.2) has an almost periodic traveling front with average wave speed w ≥ w, even the generalized transition front. However, in the case w = ∞, we can get a complete answer by Theorem 1.1. In this section, we will provide some conditions on c in (1.2) to guarantee w = ∞. The idea is to apply Proposition 3.5 and to use KAM method for constructing the positive almost periodic solution of (L V,α,θ u)(n) = u(n+1)+u(n−1)−2u(n)+V (nα+θ)u(n) = λ 1 u(n) with V, α, θ to be specified and λ 1 = max Σ(L V,α,θ ). We will first provide a simple criterion, which says that if the Schrödinger cocycle can be reduced to constant parabolic cocycle, and the conjugacy is close to the identity, then the corresponding Schrödinger equation has a positive almost-periodic solution. Recall that an almost-periodic cocycle (α, A) is said to be C s reducible where 0 ≤ s ≤ ∞, ω, if there exist B(·) ∈ C s (T d , SL(2, R)) and a constant matrixÃ such that Denote the norm in C s (T d , SL(2, R)) as F s := sup |l|≤s,θ∈T d ∂ l F (θ) . Then we have the following: is a parabolic matrix (i.e. the trace |tr(Ã)| = 2). Then has an almost-periodic positive solution.
Proof. Without loss of generality, we assume tr(Ã) = 2. Then one can find . It directly follows that
In all, the proof is complete.

5.2.
Analytic quasi-periodic potential. Motivated by Lemma 5.1, to prove the existence of positive almost periodic solution of (5.2), we only need to prove that the corresponding Schrödinger cocycle is reducible to a parabolic constant cocycle: Moreover, the conjugation is close to constant.
First we state the reducibility result for analytic quasi-periodic potential. To prove this, we first need a non-resonance cancelation lemma. The result will be the basis of our proof, and we will also use this when we deal with analytic almost periodic potentials.
Let B be a sl(2, R) valued Banach algebra. Assume that for any given η > 0, α ∈ T d where d ∈ N + ∪ {∞} and A ∈ SL(2, R), we have a decomposition of the Banach space B into non-resonant spaces and resonant spaces, i.e. B = B nre (η) ⊕ B re (η). Here B nre (η) is defined in the following way: for any Y ∈ B nre (η), we have where | · | is the norm of the Banach space B.
Once we have this, we have the following: Lemma 5.2. Assume that A ∈ SL(2, R), ≤ (4 A ) −4 and η ≥ 13 A 2 1 2 . For any F ∈ B with |F | ≤ , there exist Y ∈ B and F re ∈ B re (η) such that Moreover, we have the estimates |Y | ≤ 1 2 and |F re | ≤ 2 .
Remark 5.1. The proof of the lemma for B := C ω r (T d , su(1, 1)) with d ∈ N + could be found in [17, Lemma 3.1], and we can easily see that the proof works for any other Banach algebra.
In our application, we will set B := C ω r (T d , sl(2, R)) where d ∈ N + ∪ {∞}, r > 0 (the definition of C ω r (T ∞ , sl(2, R)) will be introduced later). Define the norm in B as |F | r := sup | θ|≤r |F (θ)|. The non-resonant space B nre will take the truncating operator T K on B: For any K > 0, we define (k)e i k,θ and define R K as Obviously, T K F + R K F = F . Now as a direct application of Lemma 5.2, we have the following: sl(2, R)). For any r ∈ (0, r), there exists c = c(γ, τ, d) such that if |rot(α, A)| ≤ 2 A 1 2 , and Moreover, we have the following estimates: Proof. We only need to apply the non-resonance cancelation lemma (Lemma 5.2). In this case, we will define where K = 2 r−r | ln |, and prove that for any Y ∈ Λ K , the operator has a bounded inverse. Thus we only need to consider the equation Without loss of generality, we assume that A = e iρ p 0 e −iρ , where e ±iρ are the two eigenvalues of A, p ∈ R, and write Taking the Fourier transformation for the above equation and comparing the Fourier coefficients, we can get Note for any k ∈ Z d with 0 < |k| ≤ K, if satisfies (5.5), we have where T K F re =F re (0), R K F re = |k|>KF re (k)e i k,θ . Moreover, we have the estimates |Y | r ≤ 1 2 , |F re | r ≤ 2 . Consequently, for any r ∈ (0, r), we have Furthermore, we can compute that eF re (0)+R K F re (θ) = eF re (0) (id +e −F re (0) O(R K F re (θ))) = eF re (0) e F (θ) , Finally, if we denote A = AeF re (0) , then we get

5.3.
Finitely differentiable quasi-periodic potential. Now we want to get the reducibility result for finitely differentiable case. Note that for any f ∈ C s (T d , sl(2, R)), by [60, Lemma 2.1], there exist an analytic sequence sl(2, R)) and a universal constant C > 1 such that The basic idea is we approximate a finitely differentiable cocycle by an analytic cocycle. If the analytic cocycle is reducible, then the finitely differentiable cocycle is also reducible. In our case, we will set (5.7) If we assume , then by (5.6) we have Consequently, We also define Then for any s > 6τ (1 + δ) 3 + 1, where 0 < δ < 1, we can compute that for any m ≥ 2 1 δ , With these parameters, we have the following: Proof. To prove this, we only need to show inductively that there exist Y l k , F l k ∈ C ω 1 l k+1 (T d , sl(2, R)) and A l k ∈ SL(2, R) such that with the estimates Once this holds, Y l k 0 ≤ |Y l k | 1 l k+1 ≤ 2 1 2 , and as a consequence of (5.12), By (5.10), we have Taking limits of (5.13), we then have the desired results. Now let's finish the iteration. First step: First by our assumption (5.11), and then by (5.9) we have and by Lemma 2.3, we have with the estimates |Y l 1 | 1 Induction step: Now at the (k + 1)-th step, first notice that if we write then we have Then by implicit function theorem, there existsF l k ∈ C ω 1 l k+1 such that G l k (θ) = eF l k (θ) .
Then by Lemma 2.3 and (5.14), we have Consequently, we can apply Proposition 5.1 to the cocycle (α, A l k eF l k ), and there existỸ l k+1 , F l k+1 ∈ C ω 1 l k+2 (T d , sl(2, R)) such that where C is a sum of terms at least 2 orders in B, D. Thus there exists 1 2 i . Moreover, by (5.12) and (5.14), we have (θ) , and thus we finish the iteration.
Proof. Write Then S E V (θ) = Ae F (θ) which is close to constant. Now we consider the energy E which lies in the extreme right endpoint of the spectrum. Since the spectrum is compact, and it is included in [−4 + inf V, sup V ], then A ≤ 6. By the assumption that s > 6τ + 2, there exists 0 < δ < 1, such that s > 6τ (1 + δ) 3 + 1. For such selected δ, we can take ≤ c .
Taking limits of (5.20), we get the desired results.
Proof. The proof is same as Theorem 5.1 if we replace Corollary 5.1 by Corollary 5.2.

5.5.
Proof of the applications. In the final subsection, we will give the applications of Theorem 1.1 in various settings, including the quasi-periodic case and almost-periodic case. First, we give the proof of Corollary 1.1.
Then the result follows from Corollary 1.1.
Next we give some typical examples such that L = lim E→λ 1 L(E) = 0 which implies that w = ∞. First we consider the quasi-periodic case, if the potential V is finitely differentiable, then we have the following: Corollary 5.4. Let α ∈ DC d (γ, τ ), γ > 0, τ > d, V ∈ C s (T d , R) with s > 6τ + 2. There exists = (γ, τ, d, s) such that if V s ≤ , then (1.2) with c(n) = V (nα) has a time increasing almost periodic traveling front with average wave speed w ∈ (w * , ∞).
Proof. By the assumption and Corollary 5.1, we know (L V,α,0 u)(n) = u(n + 1) + u(n − 1) − 2u(n) + V (nα)u(n) = λ 1 u(n) has a positive almost periodic solution. It follows that L = lim However, in some cases, we can relax the condition, and the right concept is "Almost reducible".
Definition 5.2. An analytic cocycle (α, A) is C ω -almost reducible if the closure of its analytic conjugacy class contains a constant.
Proof. By [59, Corollary 1.3] and [2, Corollary 1.2], there exists = (r) such that if |V | r ≤ , then one frequency analytic quasi-periodic Schrödinger cocycle (α, S E V ) is almost reducible. Clearly by its definition, any almostreducible cocycle is not non-uniformly hyperbolic. Thus either L(E) = 0 or (α, S E V ) is uniformly hyperbolic. Then L(λ 1 ) = 0 follows from Theorem 2.3 since we only consider λ 1 which is the right endpoint of the spectrum. By Corollary 1.1, the result follows directly.