Notes 12: Distributions of Eigenvalues of Ergodic 1D discrete Schrodinger operators in a.c. spectrum region

This post will be something about the distribution of eigenvalues of one dimensional discrete Schrodinger operators in absolutely continuous spectrum region. Namely, given a triple and a potential function Then these together generates a family of operators which is given by

for

For eigenfunction equations there is an associated Schrodinger cocycle dynamics The cocycle map is given by

Let be it’s Lyapunov exponent (see notes 1 for definition). One way to consider the distribution of eigenvalues is to consider the finite approximation. Namely, as in Notes 7, we will restrict the operator to subinterval and consider Dirichlet boundary condition, i.e.   Denote it by .  Let be the eigenvalues of  Then we consider the sequence of measure It’s standard result that as in measure  for Here as a function of energy, is the so-called integrated density of states (IDS). Obviously is nondecreasing. In fact, the relation between IDS and fibered rotation number is that

Lemma 1: 

Thus by notes 3 we know, is flat on resolvent set. Namely, the support of the measure are precisely the spectrum of for .

Proof: To see why we let  which a polynomial in of degree We set and Then by induction it’s easy to see that

 and

Thus is an eigenvector of if and only Hence, if and only if where

Then we note the following facts:

(1)  For near , it’s easy to see there is constant invariant cone field. Thus there is no rotation for   for any and for any .

(2) for near . Because lies in resolvent set.

(3) is monotonic in .

Thus as goes from to , measures the averaged times of the vector passing through So as is some doubled fibered rotation number. Since is nondecreasing and for near it’s necessary that by Notes 3.

Now I would like to Sketch a rough proof the following main theorem of this post, which is contained in the paper ‘Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with A.C. Spectrum‘ by Artur Avila, Yoram Last and Barry Simon.

Theorem 2: Assume . Then for Leb a.e. with , we have that tends to be an arithematic progression. Namely, it’s going to be something like:

And is nothing other than . The convergence is independent of .

Remark: This theorem tells that if we look at a window around with size , where we expect finitely many eigenvalues. Then if we rescale it by , we will find arithematic progression as goes to infinity. This is nothing other than the local distribution of eigenvalues in absolutely continuous spectrum region. Now let’s sketch a proof.

Proof: Note that by the proof of Lemma 1, we know that is an eigenvalue of   if and only if . By kotani theory (see Notes 2 and 3), for a.e. E with , we have a map such that

.

Thus

where . In fact, for sufficiently large, what we need to find are these such that

.

Assume that is a measurable continuity point for both maps and (see Notes 11 for definition and properties of measurable continuity points). Then it suffices to show that in some sense as .

In fact, replacing by ,  the same arguments of Lemma1-Lemma4 give the following results:

(1) .
(2) Complexifying at with magnitude , we get .
(3) Then converges to some entire function with .
(4) implies that sufficiently close to for large .
(5) By passing to a suitable subsequence, we get that

for all as .
(6) Thus we can write it as . By (3) we have , which implies that is affine.

Finally, we want to show that .  We need to compute the following

The above formula lies in

Since converges to , we only need to take care the part. Then we have

Let . Then it’s easy to see that the above formula becomes

.

Since for , we’ve scaled by , we get

where is the Hilbert-Schmit norm (see notes 2). This completes the proof.  

Here we used the fact that . See Theorem 4 of Artur Avila and David Damanik‘s paper ‘Absolute continuity of the IDS for the almost Mathieu operator with non-critical coupling‘ for detailed information.

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Notes 11: Renormalization (III)-Convergence of Renormalization

I’ve been back to Evanston from Toronto. But I guess I still have 10 more notes to post. It will take me a very long time to finish.

In this post we will discuss the convergence of renormalization. As pointed out in last post, it’s necessary that we should assume zero Lyapunov to get convergence. In fact, we will assume -conjugacy to rotations. This is somehow natural because Kotani theory tells us that in the Schrodinger cocycle case, for almost every energy, zero Lyapunov exponent implies -conjugacy. More concretely, we assume for  that  is -conjugate to -valued cocycles. Namely, there is a measurable map such that and for almost every   Let which is finite almost everywhere by the Maximal Ergodic Theorem. WLOG, we can assume that can be holomorphically extended to which is also Lipschitz in Then we have the following lemma.

Lemma 1: There exists such that for almost every  we have for every  and

                      

Proof: As explained in Notes 2, is bounded in In fact, for almost every  Let  be such a point. Note If we let then by Lipschitz condition Obviously, Hence by induction we have

                               

This obviously implies that

                              

Thus we obtain

                              

Hence,

                             

which completes the proof.

Assume further that is a measurable continuity point of and Here for example, is a measurable continuity point of  if it is a Lebesgue density point of for every It’s standard result that this is a full measure set since is measurable and almost everywhere finite. Same definition can be applied to By definition, it’s easy to see the portion of that  is close to  is getting larger and larger in smaller and smaller neighborhood of  Let be that which is a full measure condition. Then Lemma 1 implies the following estimate

Lemma 2:  Let be as above. Then for for every there exist a such that

                             

 as long as

Proof: If is sufficiently large, measurable continuity hyperthesis implies that for every we can find some with and such that are close to and are close to WLOG, we can assume is even. Then Lemma 1 together with our choice of implies that

                              

And we can of course assume such that   Since and  Combined these together we get the estimate we want. For odd, we apply the same discussion to

If we renormalize around  we know from last post that

and

Note Then we have the following obvious corollary from Lemma 2

Corollary 3: Let Then

               where and

Thus by homorphicity, are precompact in So we can take limit along some subsequence. Denote the limit by Then by estimate in Corollary 3 we  get

We in fact also have that for This is given by the following lemma

Lemma 4: Let be as above, then for every and every there exists a such that if and then

           is close to for every

Proof: For sufficiently large, for every as in proof of Lemma 2, we can find some   which is close to and are close to . Then the same argument of proof of Lemma 2 implies that and are close.

Thus we can reduce the proof to the case is close to But this is clear since we can furthermore choose such that and we’ve already had are close to

Thus we can write where satisfying  is an entire function.  Thus must be linear. We’ve basically proved the following theorem

Theorem 5: If the real analytic cocycle dynamics is conjugate to rotations, then for almost every there exists and a sequence of affine functions with bounded coefficients such that 

                        and

as

To conclude, Theorem 5 implies the following final version of renormalization convergence theorem

Theorem 6: Let be as in Theorem 5; let deg be the topological degree of map Then there exists a sequence of renomalization representatives and such that

                                     in as

Proof:  Let and be as in Theorem 5. Let be large and let Then is close to identity and is close to where By Lemma 1 of Notes 10, we know there exists which is close to identity such that

Thus is a normalizing map for and is close to some where is linear. Since the way we get renormalization representatives preserving homotopic relation and the degree of n-th renormalization representative of is we get that degree of is Thus the linear coefficient of must be close to and must be close to for some

I’ve finished the serial posts about renormalization. It’s a powerful technique in the way that we can use it to reduce global problem to local problem and apply perturbation theory like KAM thoerem. More precisely, we can start with conjugacy to rotations and end up as  close to rotations. If degree is zero and  satisfying some arithematic properties, we can then apply standard KAM theorem to get reducibility.

For these posts, I am following Artur‘s course and he and Krikorian‘s papers. Here I only do the case while they’ve considered smooth cases in there papers.

Next post will be something about distribution of eigenvalues of the our old friend: One dimensional discrete Schrodinger operator:)

Notes 10: Renormalization (II)-Renormalization Representatives

Recall in the last post we define Then given a SL(2,R) cocycle dynamics is equivalent to a action such that

We’ve also defined the renormalization operator around such that where is rescaling operator and is the basing change operator. By definition we have

and

Obviously, we are looking at smaller and smaller space scale but larger and larger time scale. Thus if we want to study the limit of renormalization process, it is necessary to assume zero Lyapunov exponent to start with. See last post for all the details.

In this post we will first analytically normalize to be for Because then the correponding is well-defined on by commuting relation. Secondly, we will explain the relation between the dynamics of original system and these of renormalized systems.

(I) Normalizing to be

We state the normalizing result in the following Lemma

Lemma 1: For any action with we can conjugate it to be that Here Here is the projection to the first coordinate.

Proof: Denote Then it suffices to find some such that Then automatically,  is well-defined on

First let’s consider the case Obviously, is equivalent to Thus it’s sufficient to define on an arbitrary interval with length biger than 1.  One way to do it is to let around a small interval around 0, hence around 1, then extends it to an open connected interval containing It’s easy to see that if is  close to identity on then we can choose such that so is on for some

The case needs a little bit more computation. We will first deal with case is close to identity near a neighborhood of in It’s enough to find some  where is from above and is something satisfying

and (this obviously imlies that D’ is holomorphic).

Thus it’s necessary

Let’s first note that the analyticity of  implies the periodicity of Indeed, we have By analyticity, we get  Combined with we get

Thus we can define And we can of course extend to for some  It’s easy to see that  is sufficiently close to zero if is suffciently close to identity. Let’s introduce the following operator

where is the Cauchy transform.

It’s a standard result that  inverts A direct computation show that   and  Thus is invertible near zero function. Thus if is sufficiently small, we can find some such that Then can be our choice and is near identity. This addresses the local case. Note we in this argument we only need the smallness of

For the global case, we can first find some B such that   and then approximate in topology by some Then is close to identity and we can apply the local argument. 

(I omited some details, one can find the detailed proof in Avila and Krikorian‘s paper ‘Reducibility or nonuniform hyperbolicity for quasiperiodic Schrodinger cocylces‘)

Back to the case  Make the notation  and Let be the normalizing map such that Let  Then

 is called a representative of the n-th renomalization of

By the proof of Lemma 1, it’s not difficult to see that representative is not unique but all of them are conjugate. In fact, any element of conjugacy classs of

(II) One of the relation between the dynamics of  and the .

It’s given by the next lemma

Lemma 2: If a -renormalization representative is conjugate to rotations, then is conjugate to rotations. Here again

Proof:  Recall

and

And is the normalizing map. By assumption and dicussion following the proof of Lemma 1, we can assume that for every  If we set  then it’s not difficult to see that the above choice of implies that

for every  which in turn implies that for every

Consider the commuting pair and A simpler version of proof of Lemma 1 implies that we can choose some normalizaing map for Thus if we set we have and it is 1-periodic. This completes the proof.

Next post, I will do some computation concerning the limit of renormalization process in Schrodinger case in zero Lyapunov exponents regime.

Notes 9: Averaging and Renormalization(I)-Defining the Renormalization Operator

This is post will be a breif introduction about some averaging and renormalization procedures in Schrodinger cocycles. Renormalization is everywhere in dynamical systems and it has been used by many people in Schrodinger cocycles for a while. Lot’s of interesting results have been obtained. As for averaging, I guess it has not been very widely used in Schrodinger cocycles yet. Although it has been a powerful technique in Hamiltonian dynamics for a very long time in studying longtime evolution of action variables. Artur has already used them to construct counter-examples to Kotani-Last Conjecture and Schrodinger Conjecture. We’ve introduced Kotani-Last Conjecture in previous post, namely, the counter-example is that the base dynamics is not almost periodic but the corresponding Schrodinger operator admits absolutely continuous spectrum. For Schrodinger conjecture, the counter example is a Schrodinger operator whose absolutely continous spectrum admits unbounded generalized eigenfunctions.

I am not familar with both averaging and renormalization.  So I may need a little bit longer time to finish this post. But I will try my best to give a brief and clear introduction.

(I) Averaging

Let the frequency be very Louville number. For instance, let be the continued fraction expansion. Thus be the n’th step approximant. We let hence grows sufficiently fast (see my third post for introduction of Liouville number). Let the potential be a very small real analytic function. Combining averaging procedure with some renormalization and KAM procedure,  Avila, Fayad and Krikorian proved in their paper ‘A KAM scheme for  cocycles with Liouvillean frequencies’ the following theorem

Theorem:  Let  be analytic and close to a constant. For every  there exists a positive measure set of  such that is conjugate to a cocycle of rotations.

The main difficulty is Liouvillean frequency case, where they used the idea of averaging. We will not prove this theorem. We are going to give an idea how does the averaging procedure look like in Schrodinger cocycle.

Let’s start with the base dynamics  Namely, we consider one-parameter family of cocycles over fixed point. We have two different ways to consider the spectrum, which is the following

for at least one 
for all

Let’s start with  For simplicity, set and we omit the dependence on  Let be the invariant direction of Let be that  Thus   for some analytic. Thus there exists a constant such that

 

Then for any closed interval inside the interior of we can choose large  such that is again elliptic. Hence similarly, there is some which is close to identity and such that

Hence

for some constant

  Thus is conjugate to Iterating times we get where is the n-th Birkhoff sume It’s not difficult to see that  is close to its averaging  Thus except those bad such that for some stay ellipitc. By monotonicity of the Schrodinger cocycle with respect to energy and suitable choice of these bad are of arbitrary small measure.

(II) Renormalization

Here we only introduce renormalization in quasiperiodic -cocycle where the frequency is one dimensional (i.e. the base dynamics is one dimensional). Let’s denote the renormalization operator by This will be a operator defined on the space of all cocycle dynamics. More concretely, it’s defined on the space of action on cocycle dynamics. We will continue to use the continued fraction expansion and approximants as in Averaging.

A natural way to see that the cocycle dynamics is to see it as which commutes with Because it’s easy to check that

 This is similar to the following case. Denote the orientation preserving diffeomorphism on by  Then is orientation preserving diffeomorphism if and only if it can be lifted to and commutes with where More generally, consider a commuting pair  where G has no fixed point. Then define a dynamical systems on  Because preserving the equivalent relation. Here is diffeomorphic to but not in a canonical way.

Thus it’s more natural to view the pair as a group action, which is the following group homomorphism

with and

This automatically encodes the commuting relation.

The reason we introduce above procedure is that after renormalization, we will have no canonical way to glue into

Now let’s define the renormalization operator step by step. We will renormalize around

 (1) Define the -action.
Let be subgroup of Then for satisfying  we can define action such that Let

(2) Define two operations on the above -action.
The first is the rescaling operator which is given by

The second is base changing operator For it’s given by

Obviously these two operations commutes with each other.

(3) More facts about continued fraction expansion.
Let be as in averaging. Let be the Gauss map  Denote Thus where comes from the continued fraction expansion. Define a map

for

Let Then it’s easy to see that we also have

(4) Define the renormalization operator.
is given by

where is from

It easy to see that

and

Thus geometrically, if we  look at  it looks like we glue and via  Then by commuting relation define a dynamical systems on Then we rescale the first coordinate to be again. Finally we get the new dynamics

Let  Then it’s easy to see that

Thus we are looking at smaller and smaller space scale but larger and larger time scale.

Next post we will show how are the dynamics of original cocycle and these of renormalized cocycles related. And we will normalize so that and do some computation concerning the limit of the renormalzation.

Notes 8:Examples(II)-Limit Periodic Potentials and Almost Mathieu Operator

This notes will be some further examples which are some natural generalization of periodic potentials. Example 3 will be limit periodic potential, where we will construct a potential with positive measure set of absolutely continuous spectrum. Example 4 will be quasi-periodic potential, in fact, the Almost Mathieu operator. I will only state some main results for the quasi-periodic example.

Example 3: Limit periodic potentials

There are two different equivalent ways to define limit periodic potentials. The first is that consider the where is a compact Cantor group, is a minimal translation and is Haar measure. Let be continuous. Then this potential is limit periodic. The equivalent say it’s limit periodic is that, start with any triple and consider the sequence It’s limit periodic if it can be approximated in by periodic sequence. In fact if so , the hull of   in will be compact cantor group. Thus it’s not very difficult to see the equivalence between these two descriptions.

For simplicity, let’s consider the the Cantor group of 2-adic integers Where the topology is induced from product topology. Taking minimal translation Note in this topology is dense and as  Then potential on this space can be approximated by sequence of potentials of period Let’s start with some Define by induction as follows

and

where is sufficiently small. Thus

So if then remains for suitable  Here  can be any compact set. On the other hand, in the interior of each band we have no control for two types of points: boundary points, where the matrices is parabolic; these such that Because then and small perturbation may lead to the appearance of new gaps. But we can always ignore a small interval around these Thus a large part of each band persists (note each band can be  broken into two bands).

Each time we choose a suitable smaller perturbation so that as Eventually, we can get spectrum such that which may also be a Cantor set.

On the other hand, for each let be the invariant direction of  Then it’s easy to see   are invariant section of cocylce Let be the projection. By above induction procedure, it’s not difficult to see that for each and  there are some such that

 Obviously,

These imply that is an invariant section of which takes values in Thus Lyapunov exponents stay zero through Thus by Kotani Theory we get absolutely continous spetrum.

Example 4: Quasiperiodic Potentials-Amost Mathieu Operator

This type of potentials are of most interest. For simplicity, let’s focus on one dimensional frequency case, where the triple is As in the Corollary of Notes 6, is the frequency. Let which is the quasiperiodic potential. Let’s recall the operator is for

The cocycle associated with the family of spectral equation is

where is defined as

One of the mostly studied model is the so-called Almost Mathieu Operator, where and is the coupling constant. Here are some of the Theorems concerning this model

Theorem 1 (Bourgain-Jitomirskaya 2002): for all and

Theorem 2 (Avila-Jitmirskaya 2009): is a Cantor set for all and 

Theorem 3 (Avila-Krikorian 2006): for al and

Other results about this model will appear in future posts.

From now on we will mostly focus on quasiperiodic potential case. Next posts I will do some averaging and renormalization procedures.

Notes 7: Examples (I)-Periodic Potentials

From now on, I will mainly focus on the Schrodinger cocyles case.

Example 1: Constant Potentials

Let’s start with the simplest example: potential over fixed point. Equivalently, we consider constant – matrix .

Since eigenvalues are invariant under conjugacy, we can up to a conjugacy assume  is one of  the following

Under mobius transformation, they have invariant directions in  as Recall in Notes 3 we define functon

such that

In particular if  is the unstable invariant direction of then  As the above cases,  can be But (eigenvalue) is invariant under conjugacy. Thus we can always move the computation to unit disk to avoid  For simplicity let’s always denote it as Then for above matrices, we get

(note for each fixed is well-defined but not uniquely determined).

Consider and the function  Let’s use to move to unit disk Then becomes the unit circle  Let’s consider the function (see notes 3) instead of  It’s easy to see that for nonelliptic matrices, is integer valued. By monotonicity of Notes 3, is not well-defined. But it’s well-defined as   By the proof of Lemma 2 of Notes 3, we know when changes from to so is some   Thus the change of correponding angle of the point in is exactly This implies that WLOG, we can set Thus

The above argument can be applied to any Combining the equivalent description of spectrum in Notes 2 give the following description of the corresponding operator:

– the spectrum is
–  
– is analytic and strictly decreasing in the interior of the spectrum

Example 2: Periodic Potentials

Next let’s consider periodic potential case. Namely, where and is the averaged counting measure. Let  hence

For any we have for any  This implies that

-the eigenvalue of  is independent of Denote it by 
–

Here for any bounded linear operator on any Banach space, is the spectral radius of   From the above facts we get

We also have the following obvious equivalent relations

and   Elliptic,

where stands for trace. For the second case we also have

Back to Schrodinger case, there is a nice graph of the function (which is also for elliptic case). It’s obviously a polynomial in of degree It has exactly critical points with critical values satisfying We have in fact the following description of the spectrum of the correponding operator

Theorem: consists of bands and there are spectral gaps ( bands may touch at the boundary points, or equivalently, gap may collapse).

Proof: By the same argument for Schrodinger cocycle over fixed point above and the fact , we have the following easy facts for as a function on real line:

-it’s continuous on  and is analytic in the interior of spectrum;
-it’s -valued outside the interior of spectrum thus constant in each connected components of the resolvent set;
– and and nonincreasing.

These together implies that:

– consists of  compact intervals. Some of them may touch at the boundary points. These are spectrum bands. on
-between each two bands is the so-called spectrum gap and there are of them (some of them may collasped). On each of them These are labelings of spectral gaps.

The proof obviously implies the properties of the polynomial

A natural question is that when are all gaps open (not collapsed). Obviously by the proof of Theorem, all gaps are open if and only if all the roots of polynomials are simple. Let’s give another description of these polynomials. Let’s restrict the operator to  with three different type of boundary conditions. Let

1. First let’s restrict to any subinterval and consider Dirichlet boundary condition, i.e. Denote it by . In this case it can be represented as a symmetric matrix

Let Then by induction it’s easy to see that

Hence  In case 2 and 3 we will only restrict to

2. Periodic boundary condition, i.e.  Denote it by  then it is

3. Antipeiodic boundary condition, i.e. Denote it by  then it is

Then by case 1 it’s not difficult to see that for some integer

and 

Thus all the spectral gaps are open if and only if all eigenvalues of the operators and are simple. An easy case is that assume are sufficiently large and distinct. Then after scaling, all nondigonal coefficients of and are sufficiently small. Namely, and are small perturbation of digonal matrices with distinct eigenvalues. Then so are and themself. Which implies that all gaps are open.

For simplicity, consider even. Denote eigenvalues of by and by Then spectrum bands are

odd; even.

And spectral gaps are

 even;  odd.

Finally let’s consider Then it’s easy to see that it can be analytically extends through interior of each bands and through gaps. But they cannot be globally defined, since there are nontrivial winding of the invariant direction  around each This winding comes from the parabolicity. Indeed, if we instead consider the eigenvalue of then

and

of which the derivative has singularity at parabolicity. This also explain why and as functions on the whole real line can at most be continous.

Note for periodic potential and in spectrum bands, it’s always since they are just parabolic and elliptic matrices. Thus by Theorem 2 of Notes 4, all the spectrum are purely absolutely continous.

Assume potentials are uniformly bounded. Then it’s interesting to note that as there are more and more bands which are also thiner and thiner. Thus in quasiperiodic potential case, it’s natural to expect the spectrum is Cantor set under some assumption. We will give an example in next post.

Notes 6: Density of Positive Lyapunov Exponents for SL(2,R)-cocycles in any Regularity Class

This post is about the density of positive Lyapunov exponents for cocylces in any regular class. It’s again from Artur Avila‘s course here in Fields institute, Toronto and from his paper `Density of Positive Lyapunov Exponents for cocylces.’ Since there are very detailed descriptions and proofs in the paper, I will only state one of the main theorems and give the idea of proof and point out how are they related to previous posts.

Recall that the Corollary of last post is a stronger result, but only in class. Obviously, it’s more difficult to obtain density results in higher regularity class.

Again I will use the base space assume and is not periodic. I will use to denote the Lyapunov exponent of the corresponding cocyle map Let’s first introduce a concept to state the main theorem.

Definition: A topological space is ample if there exists some dense vector space , endowed with some finer (than uniform) topological vector space structure, such that for every for every and the map from to is continous.

Remark: Note that if is a  manifold, then is ample. Namely we can take

The main theorem is the following

Theorem 1: Let  be ample. Then the Lyapunov exponent is positive for a dense subset of

Remark: This is basically an optimal result for density of positive Lyapunov exponents for cocycles.

The key theorem lead to Theorem 1 is the next theorem. Let denote the sup norm in and and for let and be the correponding -balls. Then

Theorem 2: There exists such that if is -close to then for and every the map 

              

is an analytic function, which depends contiously (as an analytic function) on

Remark: It will be clear later why this leads to Theorem 1. All the main ingredients for proving Theorem 2 have in fact already been included in previous posts.

Idea of Proof of Theorem 2: The key point is to find such that we can check:
1. For and for   is provided
2. For is provided

On the other hand, we can write down the explicit conformal transformation such that where  Notice that  Let’s denote Once we have these facts, by pluriharmonic theorem in the post`Proof of HAB formula‘ and mean value formula for harmonic functions, we have

a. for  by fact 2; thus
b. but
c. is pluriharmonic for by fact 1,

from which plus some additional direct computation will establish the result of Theorem 2. This argument is similar to Lemma 4 of last post and the proof of HAB formula.

The idea to obtain facts 1 and 2 is to check that for we define function and check that at points inside for in facts 1 or 2. Thus will be an invariant conefield for for any and small, which implies This is similar to the cases in Kotani theory or HAB formula, where when we complexify or we get For detailed proof see Artur’s paper.

Proof of Theorem 1: We must show that for every   there exists a sufficiently close to in  and

For any Let  Then by subharmonicity of , more concretely, by upper semicontinuity and sub-mean value property, we can choose suitable closed path to see that if then

Since is ample, we can  choose suffciently small and some as in Theorem 2 such that  and  is sufficient close to in for every  Then by above observation and Corollary of last post we can find some such that

Again by the assumption that is ample and the analyticity of the map we can assume  such that 

By Theorem 2, the function is analyic in Since , we have for every sufficiently small s>0, Thus we can choose sufficiently small  and some such that .

For me it’s very interesting to see how Kotani Theory, Uniform Hyperbolicity and Mean value formula lie at the bottom of this density result.

Let me mention an interesting application of Theorem 1. Consider the case where and is irrational.

Let’s consider the cocylce space endowed with some inductive limit topology via subspace   Here and is the space of  real analytic cocycle maps which can be extended to

 Then there is a theorem started with the Schrodinger cocycles in the regime of positive Lyapunov exponents and Diophantine frequencies in Goldstein and Schlag‘s paper ‘Holder Continuity of the IDS for quasi-periodic Schrodinger equations and averages of shifts of subharmonic functions’ , continued as all irrational frequencies and all Lyapunov exponents Schrodinger case in Bourgain and Jitomirskaya‘s paper ‘Continuity of the Lyapunov Exponent for Quasiperiodic Operators with Analytic Potential‘ and ended up as the general real analytic cocyle case in Jitomirkaya, Koslover and Schulteis‘ paper ‘Continuity of the Lyapunov Exponent for analytic quasiperiodic cocycles’ such that

Theorem: The Lyapunov exponent is jointly continuous.

(Artur will talk about the proof of this theorem in future classes, so maybe I will post the idea of proof in the future ). Combining with Theorem 1 we obviously have the following Corollary

Corollary: For any fixed irrational frequency Lyapunov exponent is positive for an open and dense subset of

To finish topics closely related to Kotani Theory, let’s mention the follow Kotani-Last Conjecture:

In Schrodinger cocycle case: Almost Periodicity of the base dynamics.

Recall that by Notes 4, we know determinism. On the other hand almost periodicity is stronger then determinsim, which has the following equivalent description:

For any and such that

It means that sufficiently precise finite information determines the whole potential to specified precision. It is obviously stronger than determinism. Thus it seems natural to pose above conjecture. Unfortunately, it turns out this is not true. Artur already has a counter example.

Notes 5: Application of Kotani Theory (II)-Genericity of Singular Spectrum

This post is about the main theorem of Artur Avila and David Damanik‘s paper `Generic singular spectrum for ergodic operators’. It is another application of Kotani Theory, which also has its own interests. I will break the proof of main theorem down to several lemmas and point out the key ideas. I will also try to carry out all the details. I am always grateful to Artur who is always willing to explain to me ideas and details whenever I want.

In this post, I will use to denote the Lyapunov exponents of the system The main theorem is the following

Theorem: Assume is not periodic ( for any ). Then for generic (generic means residual), for a.e.

Remark: By Kotani theory this Theorem implies that for generic continuous for a.e. Thus for generic continuous potentials, the spectrum is of singular type, i.e. singular continuous and pure points.

Let’s consider the unstable direction as a function where  are nonpositive integers and  stands for bounded real-valued sequence on .  Let

where  and is the ball around zero function in with radius  Now let’s state and prove our lemmas.

Lemma 1: For any bounded subset is bounded. Here boundedness in  is with respect to hyperbolic metric.

Proof: WLOG, I can assume for any and for some Let

and

Then we show that of which the latter is a bounded set in Here I use to denote the map

Indeed, since We can of course take limit along even integers. Thus the inclusion  is obvious. For boundedness it’s easy to see that for any

and

Lemma 2: Assume for some . Let as pointwisely. Then in compact open topology as functions on (i.e. uniform convergence in any compacta in )

Proof:  By the proof of Theorem 1, we actually see that converge to in compact open topology  and the convergence is independent of . This is due to the same reason of the proof of Theorem 2 of last post. Namely, are holomorphic functions takes value in , thus they are normal family. The independence with respect to  is due to the uniform shrink rate of invariant cone field under projectivized action.

Now we have 

Thus for any we can choose large such the first and third terms in the summation above are both less than on any compacta in . For this fixed for any large enough and in any compacta.

Lemma 3: For any fixed the function is continuous.

Proof: By passing to subsequence we can assume such that and pointwisely as

Thus for a.e. we have the converges to pointwisely as By Lemma 1 this implies that converges to for a.e. . And they lie in a compact set in by our choice of the ball Hence by Bounded Convergence Theorem, we have

Remark: Note what we need for in Lemma 3 are just uniform boundedness and pointwise convergence.

Lemma 4: Fix any interval Then the function is continuous.  

Remark: As a function, behave badly when . The only thing that is always true is that it’s upper semicontinuous and plurisubharmonic. This Lemma shows that after taking an appropriate integral, it behaves nicely in sense.

Proof: Consider the semicircle such that where is upperhalf part of  the circle centered at origin with radius Pick a point inside Then there is a harmonic measure on such that

By our assumption, Lemma 3 and bounded convergence theorem, it’s easy to see that and are continuous. Thus is also continuous. Via the conformal transformation which transform the unit disk  to the region inside   it’s not difficult to see that is also continuous.

Recall that We define a function such that . Then

Lemma 5:  is upper semicontinuous.

Proof: We only need to restrict our self on  Because outside this interval is always in resolvent set hence the systems are always in and Lyapunov exponents are positive.

It’s obvious that it suffices to show for any such that and any there exists a such that whenever and we have

Let’s show how to choose such a By definition of we can choose a such that where Then by Lemma 4 we can choose such that with Indeed we then have

Which implies that

Which implies that on thus This obviously implies what we want to show.

Lemma 6: There exists a dense set such that if then
(1) takes finitely many values;
(2)  is not periodic for a.e.

Proof: (This proof is due to Artur. It’s simpler than the one in their paper) For a simple function let’s define to be the period of the sequence Then by ergodicity for a.e.  Again by ergodicity it’s not difficult to see that if then Thus by our assumption on the triple if we can always change the value on a small measure set to produce a new simple function sufficiently close to in such that

Lemma 7: For any is dense in in topology.

Proof: Fix abitrary By Lemma 6, we can choose such that is sufficiently close to in By Theorem 4 of last post, we have By standard real analysis theorem and by Lemma 5, we can choose such that is sufficiently close to both in and such that

Now we are ready to prove our main Theorem

Proof of Theorem: For any we have by Lemma 5 and Lemma 7 is both open and dense. Thus is residual, which completes the proof.

Corollary: There is generic set such that for each
                                            
Where is the rotation matrix with rotation angle

Sketch of Proof: Use HAB formula we can get the corresponding Lemma 4 in terms of the function hence Lemma 5 in terms of the map On the other hand, Kotani theory can be carried over to this one parameter family thus we can have similar Theorem 4 of last post, hence similar result of Lemma 7 for Then the conclusion follows easily.

Notes 4: Application of Kotani Theory (I)-Determinism

In this post, I will give several theorems which are applications of Kotani Theory. But I will not give complete proof of all of them. These theorems are basically all Kotani’s work, among which Theorem 1-3 are in his original paper of Kotani Theory `Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional operators.’  And theorem 4 is in his paper `Jacobi matrices with random potentials taking finitely many values.’

I will use the notations of  previous blogs. E.g. the triple , a continuous and real valued function, the cocycle map,  the Lyapunov exponent  and the unstable (stable) direction.  We further assume that Then the first theorem is that

Theorem 1: For a.e. , for a.e. .

Remark: Unstable direction comes from the past and stable direction comes from the future (one can take a look at the post ‘A simple proof of HAB formula’ : how to use invariant cone field to obtain invariant direction). Thus this theorem tells us that zero Lyapunov exponents implies that past determines the future, which is kind of determinism.

Idea of Proof: From the proof of Main theorem of Kotani theory we know that for a.e. implies there exists invariant sections and which are measurable and integrable in some sense.

We didn’t prove the existence of , but it follows from the same argument of the existence of   Indeed, we only need to replace by Then it’s easy to see that and everything follows from the same argument  of the existence of .

Since is real, we see obviously that and are also invariant sections. If for a positive measure of  , then by invariance of and and ergodicity we have for a.e. Then there exists a send to for some positive   This implies that fix both and under mobius transformation. This happens only when in

On the other hand, from last post, we know is monotonic in in the sense that for any fixed and is monotonic in Thus if for a positive measure of , then it contradicts with monotonicity under some microscopic computation.

The second theorem is

Theorem 2: Let be an open interval (then it’s also bounded by boundedness of  the operator). If on then there exists a map such that it’s contiuous on both variable, holomophic in and

Remark: This theorem in particular implies that the spectrum of the corresponding operator are purely absolutely continuous on the interval The generalized eigenfunctions are all wave like (not just in some sense.)

Proof: Define funcition such that

and

Then by the proof of Theorem 1, we have

for a.e. and a.e. .

Thus by Morera’s theorem, for a.e. extends through to be a holomorphic function on On the other hand, if we conformal transform to we see that is a normal family. Thus for any compacta in is uniformly Lipschitz in   Namely, there exists a constant depends only on the compacts such that

for any

Since we can for any  pick a sequence converging to  Then we get a holomophic function  On the other hand, for we know that is continous in since they are invariant sections of systems. Thus by holomophicity, is uniquely defined for all as holomorphic function of  Since  is obtained by taking limits, it automatically continous in

Now let’s state a determinism theorem. First we define a map such that Thus is a probability measure on is the hull in the product topology. For example,

if is periodic then the hull is finite;
if is limit periodic and not periodic, then the hull is a compact cantor group;
if the is quasiperiodic with dimensional rational independent frequency, then hull the dimensional torus .

Thus up to a homeomorphism, we can actually have that where is the left shift. Namely Then, we have the following theorem

Theorem 3: Assume Then for all in hull, if  for all  then for all

Proof: Here we use the following facts:

(a) for each for a.e. where is as in Theorem 2. This follows from Theorem 2 and Fatou’s theorem via a contradiction argument.

(b)  determines This is not difficult to see by a contradiction argument.

(c) are uniquely determined by their limits on a positive measure set of

Now for each   obviously determines But determines by fact (a) and (c). Finally, determines by fact (b). 

Remark: The conclusion of this theorem is exactly the definition of the deterministic process. Namely a stochastic process is said to be deterministic if for all  is determined by   Otherwise it is nondeterministic. Obviously, I.I.d. process are far away from deterministic. In particular, if the process is i.i.d, then for a.e.

Next Theorem is that

Theorem 4: If takes finitely many values and   then is periodic (or equivalently, hull is finite).

Proof: This is an immediate consequence of Theorem 3. Indeed if is not periodic then is nondeterministic. But this cannot happen by our assumption and Theorem 3.

Notes 3: Kotani Theory (II)-Fibred Rotation Number (IDS)

Up to now, all my posts started from Jan.22.2011 are based on Artur‘s course here in Fields institute, Toronto. Part of the contents of this course are even from Artur’s unpublished work.

This time let me prove the following theorem

Theorem 1: Let as last time, then for a.e. implies that for small .

As I mentioned last, this Theorem together with the Lemma of last post imply the main theorem of kotani theory.

The proof this theorem make use of the harmonicity of in upperhalf plane. In particular there is a harmonic function such that is holomophic. We are going to show this is in fact the fibred rotation number of the correspoding cocyles and which is also basically the IDS (integrated density of states) of the operators. This will be the key object of this post. We will prove the following facts about :

1.  well-defined for all and is continuous up to .
2.  is monotonic in  in some sense which implies the monotonicity of in .
3. Thus is differentiable for a.e. and Cauchy-Riemann equation will imply the conclusion of our theorem.

Let me carry out all the details.

I have to say, for me the fibred rotation number is always a subtle concept. This time I am going to expore as detailed information about it as I can.

From last post we know there exist invariant section for projective dynamics . Thus

from which it’s easy to see that another way to calculate Lyapunov Exponents via invariant section is

thus

is holomorphic.

From which we see that (Here is well-defined. Because by last post, more concretely proof of Lemma 2, it’s easy to see that . Thus there is no nontrivial loop around origin). For obvious reason, it’s convenient to instead consider  (so  is holomorphic functon). By Birkhoff Ergodic Theorem, we have for a.e. x

which implies that  is some sort of averaged rotation, i.e. a rotation number.

Before proving the next Lemma, I need to do some preparation. To consider rotation number in more general setting, we need go from the upperhalf plane to the disk  via the following matrix

It’s easy to see that  And , where is the subgroup of preserving the unit disk in under Mobius transformation. For

let

Then it’s easy to see

for i.e.  then 
 and
for then
 Let’s denote this class by
for  then 

And all these sets of , or equivalently of are multiplicative.

In the following Lemma, I always consider the equivalent dynamics  The Lemma is

Lemma 2: is well-defined for all and is continuous on .

Proof: First let’s show that, as long as the cocycle map or equivalently, , we can define via any continuous section (not necessary invariant).

Let’s define be that

Then obviously  is the unstable invariant section of , thus can be defined as

for a.e.

Then we show that can be replaced by any continuous section and the convergence is independent of the choice of such . Indeed, we always have that for any

(hence  for all ).

In fact, by our choice of cocyle map, if we denote   then

which is a disk stay away from with distance at least Thus the above estimate follows easily (Note this is not true for Lyapunov exponents, i.e. cannot necessary be bounded). Now we may fix constant section to do the remaing computation.

It’s easy to see that , so

For simplicity let Then the above formula implies that

and obviously

We then have

as

Hence, the convergence is uniform.  In the similar way, we can show that is uniform contious in . Indeed, it’s easy to see for any fixed is unform continuous on  So we can choose such that are equi-uniform continuous. Now for any  we can choose sufficiently small such that for

for all

Now for arbitrary we have Thus


for large.  Since is arbitrary, we see for .

Thus we can extend to  which is continous up to Again we denote it by   The above computation actually shows that for

Indeed, if not we may without loss of generality assume

Then we can choose sufficient close to and sufficiently large such that all the following terms are less than :

which is obvious a contradiction. 

Our next lemma is

Lemma 3: is nonincreasing.

Proof: This in fact follows from the monotonicity of the following function.  Fix arbitrary consider the function in

To make everything clear, let’s introduce another way to study the fibred rotation number. Fix , consider

A direct computation shows that Indeed,

. Thus

. So

and

Consider Then the relation between and are and Thus it’s not difficult to see that

Now since we start with the same we obviously have and

where the first inequality follows from the fact that preserves order and the second one follows from monotonicity. So by induction we have

for all

Thus   and   which completes the proof.

Now we are ready to prove the theorem of this post.

Proof of Theorem 1: By standard harmonic theorem it’s easy to see in our case, is Poisson integral of . Obviously, is again harmonic. Since is monotonic, is in fact the Poisson integral of Then Fatou’s Theorem tells us for Leb a.e. 

Now by Cauchy-Riemann equation we have that

Now since Lyapunov exponents is a nonnegative upper semicontinuous  function, it’s continuous at , where   Thus the above discussion shows that for a.e. E with we have

which completes the proof the Theorem.

Now I’ve already finished the proof, but probabily I will show that  in the future. As I said in the last post, is relatively easy. It lies in the fact that the generalized eigenfunctions of absolutely continuous spectrum grow at most polynomially fast, which obviously contradicts with positive Lyapunov exponents. And the other part due to Kotani theory has already been contained in these two posts. Let me go back to this in the future.

Next post I will give some application of Kotani theory on deterministic potential and problems concerning density of positive Lyapunov exponents.