This post is about the density of positive Lyapunov exponents for cocylces in any regular class. It’s again from **Artur Avila‘**s** **course here in **Fields institute, Toronto** and from his paper `**Density of Positive Lyapunov Exponents for cocylces**.’ Since there are very detailed descriptions and proofs in the paper, I will only state one of the main theorems and give the idea of proof and point out how are they related to previous posts.

Recall that the Corollary of last post is a stronger result, but only in class. Obviously, it’s more difficult to obtain density results in higher regularity class.

Again I will use the base space assume and is not periodic. I will use to denote the Lyapunov exponent of the corresponding cocyle map Let’s first introduce a concept to state the main theorem.

**Definition: **A topological space is * ample *if there exists some dense vector space , endowed with some finer (than uniform) topological vector space structure, such that for every for every and the map from to is continous.

**Remark**: Note that if is a manifold, then is ample. Namely we can take

The main theorem is the following

**Theorem 1**: Let be ample. Then the Lyapunov exponent is positive for a dense subset of

**Remark: **This is basically an optimal result for density of positive Lyapunov exponents for cocycles.

The key theorem lead to Theorem 1 is the next theorem. Let denote the sup norm in and and for let and be the correponding -balls. Then

**Theorem 2**: There exists such that if is -close to then for and every the map

is an analytic function, which depends contiously (as an analytic function) on

**Remark: **It will be clear later why this leads to Theorem 1. All the main ingredients for proving Theorem 2 have in fact already been included in previous posts.

**Idea of Proof of Theorem 2**: The key point is to find such that we can check:

1. For and for is provided

2. For is provided

On the other hand, we can write down the explicit conformal transformation such that where Notice that Let’s denote Once we have these facts, by pluriharmonic theorem in the post`**Proof of HAB formula**‘ and **mean value formula **for harmonic functions, we have

a. for by fact 2; thus

b. but

c. is pluriharmonic for by fact 1,

from which plus some additional direct computation will establish the result of Theorem 2. This argument is similar to Lemma 4 of last post and the proof of HAB formula.

The idea to obtain facts 1 and 2 is to check that for we define function and check that at points inside for in facts 1 or 2. Thus will be an invariant conefield for for any and small, which implies This is similar to the cases in Kotani theory or HAB formula, where when we complexify or we get For detailed proof see Artur’s paper.

**Proof of Theorem 1: **We must show that for every there exists a sufficiently close to in and

For any Let Then by subharmonicity of , more concretely, by upper semicontinuity and sub-mean value property, we can choose suitable closed path to see that if then

Since is ample, we can choose suffciently small and some as in Theorem 2 such that and is sufficient close to in for every Then by above observation and Corollary of last post we can find some such that

Again by the assumption that is ample and the analyticity of the map we can assume such that

By Theorem 2, the function is analyic in Since , we have for every sufficiently small s>0, Thus we can choose sufficiently small and some such that .

For me it’s very interesting to see how Kotani Theory, Uniform Hyperbolicity and Mean value formula lie at the bottom of this density result.

Let me mention an interesting application of Theorem 1. Consider the case where and is irrational.

Let’s consider the cocylce space endowed with some inductive limit topology via subspace Here and is the space of real analytic cocycle maps which can be extended to

Then there is a theorem started with the Schrodinger cocycles in the regime of positive Lyapunov exponents and Diophantine frequencies in **Goldstein** and **Schlag**‘s paper **‘Holder Continuity of the IDS for quasi-periodic Schrodinger equations and averages of shifts of subharmonic functions’ **, continued as all irrational frequencies and all Lyapunov exponents Schrodinger case in **Bourgain** and **Jitomirskaya**‘s paper **‘Continuity of the Lyapunov Exponent for Quasiperiodic Operators with Analytic Potential**‘ and ended up as the general real analytic cocyle case in **Jitomirkaya, Koslover **and** Schulteis**‘ paper **‘Continuity of the Lyapunov Exponent for analytic quasiperiodic cocycles’ **such that

**Theorem: **The Lyapunov exponent is jointly continuous.

(Artur will talk about the proof of this theorem in future classes, so maybe I will post the idea of proof in the future ). Combining with Theorem 1 we obviously have the following Corollary

**Corollary: **For any fixed irrational frequency Lyapunov exponent is positive for an open and dense subset of

To finish topics closely related to Kotani Theory, let’s mention the follow **Kotani-Last Conjecture:**

**In Schrodinger cocycle case: Almost Periodicity of the base dynamics.**

Recall that by Notes 4, we know determinism. On the other hand almost periodicity is stronger then determinsim, which has the following equivalent description:

**For any and such that **

** **

It means that sufficiently precise finite information determines the whole potential to specified precision. It is obviously stronger than determinism. Thus it seems natural to pose above conjecture. Unfortunately, it turns out this is not true. Artur already has a counter example.

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