This and next posts will be about Kotani theory, which is part of my syllabus. Kotani theory is first found by **Shinichi Kotani** when he studies spectrum theory of operator with ergodic potentials. It gives a complete decription of the absolutely continuous part of spectrum of operators in terms of Lyapunov exponents of the corresponding of cocycles. Thus it builds a deep and beautiful relation between operator theory and dynamical systems.

Later **Artur Avila** and **Raphael Krikorian** generalize Kotani theory to more general coycle dynamics setting and it becomes a powerful tool to study Schrodinger operators. What I am going to introduce here are main results and some preliminary stuff. Let’s start with one dimensional (1D) discrete operators and cocyles. From now on I set .

For simplicity, I will continue to use the triple as in last post. Assume be a contiuous function. Then we can define a cocycle map via

The correspoding cocycle dynamics is called cocyles. It arises from the following operator on

where Then solving the eigenfunction equation if and only if

Thus the growth rate of with respect to characterizes both the spectrum type of the energy and the dynamics of cocycle , which allows one to go back and forth between spectrum theory and dynamical systems. Let be the spectrum of the bounded linear selfadjoint operator , then the first basic fact relate operator and cocycle is for a.e.

If furthermore is minimal, then the above relation is in fact true for all Thus for a.e. the spectrum is independent of let’s denote it as . For each we can further decompose as , which correspond to pure point, absolutely continuous and singular continuous part of the spectrum of the operator . These are defined by the spectral measure of the operator, which one can find in any standard functional analysis book. It turns out that in our case, there exists sets such that for a.e. and (If in addition is minimal, then in fact for all which is in general not true for pp and sc.)

Now we denote And for any set the essential support of the set is given by

for every

Then the next deep relation between dynamics and spectrum is the following

The relation is relatively easy since positive Lyapunov exponents give exponential growth of the solution to the eigenfunction equation which in some sense contradicts with the absolute continuity of spectrum. The part is a rather deep result called **Kotani theory**. It in fact is a theory about that, under the assumption that Lyapunov exponents are zero, when can one in some sense conjugate the valued cocycles to valued cocycles. Obviously, if the cocycle map takes value in , then all orbits are bounded. While zero Lyapunv exponent in general just means that grows subexponentially. What Kotani theory tells us is in fact that the following theorem

**Theorem 1:** For almost every , if then there exists a map such that and

Thus the generalized eigenfunctions for absolutely continous spectrum of the operators in question oscillate in some sense. Which are kind of wave like solutions and far from eigenfunctions of real eigenvalues, which decay in sense. As I said this is obviously stronger then zero Lyapunov exponents. Indeed, in this case we have

as

Now let me explain how can one construct the above We first prove the following key lemma

**Lemma 2**: Assume satisfying for small. Then is conjugate to valued cocycles.

**Proof:** First we note by the same reason that as in last blog, we have . Thus there is invariant section which is the unstable direction. There are different ways to calculate Lyapunov exponents of cocylce dynamics via invaiant section of corresponding projective dynamics. We use the following: of the contraction rate measured in metric of mobius transformation at . In our case we need to consider the following composition of map. Let with standard metric, then the composition is

where the first map is a isometry and the second one (which is the inclusion map) is a contraction. Thus we consider the contraction of the second map at invariant section. Then the Lyapunov exponents is given by

For simplicity, let me first assume that:

* exists for and we denote it by .
(then obviously, is invariant for a.e. , i.e. for a.e..)*

Assuming this, we have the following straightforward estimate via Fatou’s lemma and our assumption in Lemma :

Since , we have

.

Hence

Let . Now we define such that thus which implies . On the other hand, it’s easy to see for quite general reason Thus

Here for is the Hilbert-Schmit norm of which is just It’s easy to see that if then . Since all norms of are equivalent, we complete the proof of the Lemma up to the assumption.

To get around the assumption, we need conformal barycenter, which is a Borelian function where is the space of probability measures on This function is equivariant with respect to change of coordinates. i. e. for

we have

Now let’s consider the probability measures on . Then there is a subsequence converging to some such that . Then apply conformal barycenter to , we get a point such that is an invariant section and

Now we can construct the map as in the proof of lemma.

Although we introduce conformal barycenter to complete the proof of Lemma, to prove the Theorem we actually only need to apply Fubini and Fatou theorem, to pass the existence of limits from **For every , converges for a.e. ** to **For a.e. , converges for a.e. **. Then we get the conclusion in assumption for a.e. E.

In the next post, I will prove how can we get the condition in Lemma under the condtion in Theorem, which will complete the proof of theorem.

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