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:)