Sharp \frac{1}{2}-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles

Sharp 12-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles

Abstract

We consider a similar type of scenario for the disappearance of uniform of hyperbolicity as in Bjerklöv and Saprykina (2008 Nonlinearity 21), where it was proved that the minimum distance between invariant stable and unstable bundles has a linear power law dependence on parameters. In this scenario we prove that the Lyapunov exponent is sharp -Hölder continuous.

In particular, we show that the Lyapunov exponent of Schrödinger cocycles with a potential having a unique non-degenerate minimum, is sharp -Hölder continuous below the lowest energy of the spectrum, in the large coupling regime.

1 Introduction.

We consider quasi-periodic cocycles

 {T×R2⟶T×R2(θ,v)⟶(θ+ω,A(θ)v) (1)

where is an irrational number and . A cocycle is uniformly hyperbolic if there exists two continuous maps , that are invariant, , with and

 ∣∣∣∫Tlog|λi(θ)|dθ∣∣∣>0. (2)

Since we consider only cocycles, one of these integrals will be positive, and the other integral negative (simply minus the first one). Without loss of generality, we may assume that

 ∫Tlog|λ1(θ)|dθ>0, (3)

and we call the quantity , the (maximal) Lyapunov exponent.

In this paper we tackle the problem of Hölder continuity of the Lyapunov exponent of one-parametric families of quasi-periodic cocyles with transfer matrix of the form

 AE(θ)=(V(θ)−E−110),

where the function has a non-degenerate minimum. Since these cocycles arise from the study of the spectrum of discrete Schrödinger operators of the form

 (Hθx)n=xn+1+V(θ+nω)xn+xn−1, (4)

it is known that the set of parameters such that the cocycle is not uniformly hyperbolic is a compact set on the real line. From now on, we will denote by the bottom edge of this set.

More concretely, in this paper we prove that on the whole interval the Lyapunov exponent is -Hölder continuous but not -Hölder continuous for any positive . Since the Lyapunov exponent is real analytic in the interval it implies that its asymptotic behaviour at is square root like. This problem is part of the open conjectures appearing in [7] on the asymptotics of disappearance of normally hyperbolic invariant tori in quasi-periodically forced systems (see also[3, 4, 5] for further numerical studies in different contexts). There, the authors numerically study one-parameter families of quasi-periodic skew product systems, and conjectured that the normal dynamics around the invariant tori satisfy that the minimum distance between the invariant bundles and (defined above) satisfies

 dist(E)∼a(E−E0), (5)

and that the (maximal) Lyapunov exponent of , satisfies

 L(E)∼L(E0)+b√E0−E, (6)

where are constants depending on the system and .

A step forward towards solving this problem was taken in [2], where the authors proved that the distance is asymptotically linear in the case of linear Hénon maps. These are one-parameter linear cocycles of the form (1) with transfer matrix

 AE(θ)=(V(θ)−E110),

with being any function close to , and being a parameter.

Remark 1.1.

A similar result as in [2] with the same type of potential as in this paper appears in [12].

In the literature there are plenty of results about the Hölder continuity of the Lyapunov exponent. Among them, in [9] the authors prove that, if the cocycle is analytic, then the Lyapunov exponent is continuous with respect the parameter . In [6] they established that it is Hölder continuous.

Proving asymptotics for the distance and Lyapunov exponents when the potential is close to zero is quite easy: both are square root. This can be proved by noticing that the collision of the invariant stable and unstable bundles is smooth, so producing a saddle-node bifurcation, which implies that both the distance between the invariant bundles and the Lyapunov exponent have square root asymptotics.

We prove, under the assumption that the minimum distance satisfies linear asymptotics, as in (5), together with some other general assumptions, that the Lyapunov exponent has (almost) square-root asymptotic behaviour, as in (6), but where the constant is allowed to fluctuate between two fixed, positive constants.

2 Statement of the results

Suppose that is an analytic family of quasi-periodic cocycles , given by

 At(θ)=A(θ)etw, (7)

where is a real parameter, , and . The results in this paper hold for any such family of cocycles that satisfies the following list of assumptions. The (maximal) Lyapunov exponent of is the limit

 L(t)=limn→∞1n∫Tlog∥Ant(θ)∥dθ

where . If there are two continuous maps (), that are invariant, , as in (2), then equals the biggest of the expressions

As in the introduction, we may assume that the biggest of them has . Then, it follows that

 L(t)=∫Tlog|λ1(θ)|dθ.

2.1 Assumptions

The first assumption we make is that the cocycle is uniformly hyperbolic, up to some (critical) parameter, and that there is an appropriate invariant cone. From the first assumption, it is not a priori clear that is a bifurcation parameter, but it will follow from the second assumption. We remark that invariant cone condition could be relaxed somewhat, but would require further technical arguments. Here, we use projective coordinates, but other coordinates could be used instead.

1. Suppose that the cocycles are uniformly hyperbolic for every . Moreover, suppose that there are two continuous functions , where is independent of , such that

 (rut(θ)1) and (rst(θ)1)

are invariant directions for , and that they lie in the subspaces and , respectively.

We write the difference between the two functions and at the point as

 d(θ)=rut(θ)−rst(θ).

Because of the fibred structure of the cocycle, the minimum of is the minimum distance between the invariant bundles (in projective coordinates). Whenever , there is a unique such that

 d(θ+jω)=d(θ+iω)⋅Di,j(θ).

That is, measures how the difference changes from step to , where the base step 0 is taken to be at . We will make some assumptions about the function , and some global estimates for the growth of . Let be the constant in 1.

1. Suppose that, for every , there is a distinguished interval , and a distinguished point , satisfying the following conditions.

1. Suppose that we are given stopping times such that, for every and , we have

 D0,j−1(θ) ≥eaj, and (8) D−(k−1),0(θ) ≤e−ak, (9)

where is independent of both and .

2. There is a unique (global) minimum distance, it is attained at , and satisfies the linear asymptotics

 d(θc)=minθ∈Td(θ)=C0|t−t0|+o(|t−t0|), (10)

for some constant independent of .

3. There is a constant , independent of both and , such that for every the distance is approximately quadratic:

 1C1(θ−θc)2≤d(θ)−d(θc)≤C1(θ−θc)2. (11)
4. The length of the interval satisfies

 |I|≥2C2⋅√d(θc), (12)

where is some constant independent of .

5. For every , the difference satisfies

 d(θ)≥√d(θc). (13)

It is important to stress that (10) implies that as , and that is indeed a bifurcation point where uniform hyperbolicity is lost. That is, we have a torus collision at the critical parameter .

Lastly, we impose a continuity condition on the Lyapunov exponent.

1. Suppose that is continuous on the interval . That is,

 limt↗t0L(t)=L(t0). (14)

Since the parameter dependence is analytic, the Lyapunov exponent is continuous at parameters satisfying that is uniformly hyperbolic. Therefore, the assumption is in fact only that is left-continuous at .

We will discuss these assumptions in Section 5.

2.2 Main results

In this paper we prove

Theorem 2.1.

Given a one-parameter family of cocycles of the form in 7, and satisfying the assumptions 1 and 1, for some and , there exist two positive constants and , such that

 K1√|t−t0|≤ddtL(t)≤K2√|t−t0|, (15)

for every .

A direct result of this is the following.

Corollary 2.2.

Under the same assumptions as in Theorem 2.1, together with the assumption 1, the Lyapunov exponent in the range is -Hölder continuous but not -Hölder continuous at for any . More specifically, we have the asymptotics

 K1√|t−t0|≤L(t)−L(t0)≤K2√|t−t0|,

for every .

Proof.

Since the system is uniformly hyperbolic in , the Lyapunov exponent is analytic there. The only point of interest is . Let . Since the Lyapunov exponent is left-continuous at , by assumption 1, we may simply integrate the derivative from to to get the result. ∎

In the setting of Schrödinger cocycles, we have the following corollary. Recall that an irrational is called Diophantine if there are constants and such that

 infp∈Z|nω−p|≥κ|n|τ,

for every .

Corollary 2.3.

Suppose that is a family of Schrödinger cocycles where is a Diophantine irrational,

 AE=(λV(θ)−E110),

and has a unique, non-degenerate minimum. Then there is a such that if , the Lyapunov exponent satisfies the asymptotics

 K1√|E−E0|≤L(E)−L(E0)≤K2√|E−E0|,

as , where are some positive constants and is the lowest energy of the spectrum.

Proof.

In this setting, the Lyapunov exponent is continuous up to the bottom edge of the spectrum, that is, 1 holds. This fact can be found in [10], where it is proved in the continuous case, but it applies also to the discrete one.

The assumptions 1 and 1 are proved in [12], for , and . Specifically, there is an interval for each energy , where is the lowest energy of the spectrum. The functions and are the functions and in that paper, respectively. In that paper, (1) corresponds to 1, and (2) corresponds to the assumptions 1b to 1d. Moreover, the first estimate in (3) (b) is the same one as in (8) for . Similarly, the first estimate in (3) (c) is the one in (9). That shows 1a. The last assumption 1e corresponds to [12, Lemma 7.6]. ∎

The proof of Theorem 2.1 appears in Section 4, but first we prove a small lemma.

3 The derivative of the Lyapunov exponent and Avila’s lemma

One of the key tools for proving Theorem 2.1 is expressing the derivative of the Lyapunov exponent as an integral with respect to the stable and unstable directions, and the difference between them. Since, for parameters below , the cocycle is uniformly hyperbolic, it means that there exists a , with the same regularity as the cocycle,

 Bt(θ)=(αt(θ)βt(θ)γt(θ)δt(θ))

such that is the diagonal transfer matrix

 Dt(θ)=(d1,t(θ)00d2,t(θ)),

with . In fact, since the dynamics on the base is irrational, the Lyapunov exponent is given by

 L(t)=∫Tlog(d1,t(θ))dθ.
Lemma 3.1.

Given a one-parameter family of uniformly hyperbolic quasi-periodic cocycles , of the form (1), with , and , where is given by

 w(θ)=(w1(θ)w2(θ)w3(θ)−w1(θ)),

then

 ddtL|t=0=3∑i=1∫Tqi(θ)wi(θ)dθ,

where

 q1(θ)=α0(θ)δ0(θ)+β0(θ)γ0(θ),q2(θ)=γ0(θ)δ0(θ), and q3(θ)=−β0(θ)α0(θ).

This lemma appears in [1] in the case that the transfer matrix is analytic. For completeness sake, we include a slightly different proof of this lemma.

Remark 3.2.

The proof of Lemma 3.1 can be generalized mutatis mutandis for one-parameter families of uniformly hyperbolic cocycles acting on , where is a compact manifold and , with base dynamics on satisfying that its jacobian at any point has determinant 1.

Proof.

Under the assumptions of the lemma,

 B(θ+ω)−1A(θ)etw(θ)B(θ)=D(θ), (16)

where the matrix-valued maps and also depend on the parameter .

First notice that

 ddtL|t=0=∫Td′1(θ)d1(θ)dθ.

So, by differentiating (16) with respect and setting we obtain

 (B(θ+ω)−1)′A(θ)B(θ)+B(θ+ω)−1A(θ)w(θ)B(θ)+B(θ+ω)−1A(θ)B′(θ)=D′(θ). (17)

Also, by differentiating with respect we get , and using identity (16), Equation (17) is transformed into

 −B(θ+ω)−1B′(θ+ω)D(θ)+D(θ)B(θ)−1w(θ)B(θ)+D(θ)B(θ)−1B′(θ)=D′(θ). (18)

Finally, by dividing both sides by in the last equation, considering the average of the entry, and using the fact that the first and third monomials of the left-hand side cancel out, we get the desired result. ∎

4 Proof of Main Theorem

Proof of Theorem 2.1.

Suppose that all the assumptions are satisfied, and fix a . Let us drop from the notation and simply write and instead of and , respectively. In our setting,

 Bt(θ)=⎛⎜ ⎜⎝ru(θ)√ru(θ)−rs(θ)rs(θ)√ru(θ)−rs(θ)1√ru(θ)−rs(θ)1√ru(θ)−rs(θ)⎞⎟ ⎟⎠, and w(θ)=(0010).

Lemma 3.1 can be used to express the derivative of at any , simply by shifting the parameter, since the formula doesn’t depend on . That is, we can simply apply the lemma to , and get the derivative at the parameter value . The derivative of the Lyapunov exponent is therefore equal to

 −∫Tru(θ)rs(θ)ru(θ)−rs(θ)dθ.

Recall the notation

 d(θ)=ru(θ)−rs(θ)

and , where , that satisfy the relations

 d(θ+jω)=d(θ+iω)⋅Di,j(θ).

First of all, for every , we have the uniform inequalities

 12C∫T1d(θ)dθ≤∫Tru(θ)rs(θ)ru(θ)−rs(θ)dθ≤2C∫T1d(θ)dθ,

since there is a constant , independent of and , such that , for every . In particular, this means that the result follows, if we can show that the same inequalities hold for the integral

 ∫T1d(θ)dθ.

Recall the interval , and consider the transformation

 ∫I+kω1d(θ)dθ=∫I1d(θ+kω)dθ.

Thus the integral over , where , becomes

 ∫I1d(θ)D0,k(θ)dθ,

and over , where again , it becomes

 ∫ID−k,0(θ)d(θ)dθ.

Recall the stopping times from the assumptions, where . Set

 C+ ={θ+kω:θ∈I,0

Using the relations above, we conclude that

The assumption 1a yields the inequalities

 ∫I1d(θ)≤∫C1d(θ)dθ≤(1+21−e−a)∫I1d(θ)dθ.

Indeed, if we set

 S+(θ)=σ+(θ)∑k=11D0,k−1(θ), and S−(θ)=σ−(θ)∑k=1D−(k−1),0(θ),

then the bounds in (8) and (9) give us

 0≤S±(θ)≤∞∑k=1e−ak=11−e−a,

which immediately imply the above bounds. The assumption 1c implies that

 ∫I1d(θc)+1C1(θ−θc)2dθ≤∫I1d(θ)dθ≤∫I1d(θc)+C1(θ−θc)2dθ. (19)

For any , we compute the indefinite integral

 ∫1d(θc)+b⋅(θ−θc)2dθ=1√b⋅d(θc)arctan((θ−θc)√bd(θc)).

By assumption 1d, it follows that either or . Since the problem is symmetric, it doesn’t matter which inclusion holds. Computing the integral over the interval , we obtain the inequality

 1√b⋅d(θc)arctan(√b)≤∫I1d(θc)+b⋅(θ−θc)2dθ≤π√d(θc). (20)

Plugging in the constant instead of , and using the inequality in (19), results in the bounds

 √C1√d(θc)arctan(√1C1)≤∫C1d(θ)dθ≤(1+21−e−a)π√d(θc).

Finally, using the assumption 1e, the remainder of the integral can be computed as

 0≤∫T∖C1d(θ)≤1√d(θc).

In conclusion, there is a constant (uniform in ) such that

 1K√d(θc)≤∫T1d(θ)dθ≤K√d(θc).

Since , by assumption (1b), it follows that

 K1√|t−t0|≤ddtL(t)≤K2√|t−t0|,

where are independent of , provided is close enough to . ∎

5 Discussion and future directions

The first assumption 1 holds for Schrödinger cocycles that are homotopic to the identity, for energies below and above the lowest and highest energies of the spectrum, respectively. For a proof, see [8]. The important implication of this is that there is no fibre rotation, and the assumption should apply to many classes of cocycles.

It should be possible to extend the methods to include more general cases, where fibre rotation is allowed.

It is clear that the bounds

 1C√d(θc)≤∫I1d(θ)dθ≤C√d(θc),

critically depend on the assumptions 1c and 1d. (see the computations leading up to (20)). In some computed examples (see [11, 12]), the length of the interval actually satisfies much better bounds ( can be chosen arbitrarily large as ). Different local behaviour of their difference, that is replacing the square in 1c by another exponent, should produce different types of asymptotics.

The linear asymptotics for the distance in assumption 1b enters in the last step, and also affect the final asymptotics. As we said in (5), the minimum angle between the invariant directions is conjectured to be linear when a system loses uniform hyperbolicity (see for instance [7]).

Remark 5.1.

The assumptions 1 and 1b together imply that the angles between the invariant directions satisfies linear asymptotics. Indeed, this angle is simply the difference between and , and the mean-value theorem gives the linear asymptotics since .

The assumption 1a is simply a statement of exponential growth when the invariant directions are very close to each other. It can be relaxed somewhat, as long as their sums

 S+(θ)=σ+(θ)∑k=11D0,k−1(θ), and S−(θ)=σ−(θ)∑k=1D−(k−1),0(θ).

are uniformly bounded.

The assumption 1e simply ensures that directions are not too close to each other outside some critical region. In known examples satisfying these assumptions, (13) can in fact be replaced with .

As for the last assumption 1, we refer to [10] for the case of Schrödinger cocycles. There it is proved to be continuous in the open spectral gaps, for the continuous case, but it applies also to the discrete case. The Lyapunov exponent for Schrödinger cocycles is continuous in for analytic potentials (see [9]), but may fail to be so for non-analytic potentials (see for instance [13]).

Acknowledgments.

The authors acknowledge fruitful discussions with Michael Benedicks, Kristian Bjerklöv, Alex Haro, Rafael de la Llave, and Joaquim Puig.

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