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.

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