**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 Part of the contents of this course are even from Artur’s unpublished work.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.

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