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.

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