I am visiting my coadvisor **Artur Avila** in Fields Institute in Toronto from Jan.18 to Apr.01. He is giving a course entitled `**Ergodic and Spectral Theory of Quasiperiodic Cocycles ‘** and I am working on some related problems with him. This is the first time he gives such a graduate level course.

Since his way of math always fasinates me, I plan to take careful notes of his lecture and latex them. I will post some of them here. I will also post some topics I promised in the last blog. In this post, I am starting with a simple proof of the following nice so-called **Herman-Avila-Bochi **formula:

Here is the setting: is compact metric space with a probability measure and is a homeomorphism on it preserving is a continuous cocyle map. Thus gives a -valued cocyle dynamics over base dynamics , namely, for stands for the Lyapunov exponent of this dynamical systems. If I denote by , then it is given by

The limit exists since is a subadditive sequence. is the rotation matrix with rotation angle . Thus the above formula gives the averaged Lyapunov exponent of a one-parameter family of cocycle dynamics.

I forgot to say a word about the history of this formula: It’s first proved by **Michael Herman** as an inequality

in his famous 1983 paper in CMH **‘Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2’**. **Artur Avila **and **Jairo Bochi **made it an equality in their 2002 paper `**A formula with some applications to the theory of Lyapunov exponents**‘. The proof I am going to post here is a simpler version which Artur did in his class.

What lies behind this formula is in fact the **mean value formula **for harmonic functions. Let and write as ; let stands for uniformly hyperbolic systems, then the main steps are in the following:

1. Extending z from the unit circle to unit disk Then for any ,

2. is harmonic in for and bounded and well-defined for all It obviously agrees with for . Furthermore, converges to nontangentially.

3.

4. Mean value formula implies the formula in question.

There are quite a few notions, observations and standard results need to be explained. Let’s do them one by one. First is the defintion of uniformly hyperbolic cocycle dynamcal systems, let’s consider the more general valued cocyle case, i.e. which is continuous. Let be the Riemann surface and acts on it as Mobius transforamtion, i.e. for , .

**Defintion 1. ** is said to be uniformly hyperbolic if the are two contiuous map such that

1. they are invariant in the sense that and .

2. there exist constant such that for each vector for , and for each for .

One well-known equivalent condition for systems is the existence of invariant cone field. More concretely, if and only if for each there is open disk such that there exists positive integer

The existence of implies the existence of a corresponding Using this it easy to obtain the following lemma

**Lemma 2. **Consider a holomorphic 1-parameter family of systems , is in some open region in complex plane. Holomorphicity in the sense that for each fixed is holomorphic in Then and are both holomorphic in .

**Proof**: It’s easy to see by the equivalent condition above,

,

for any and the convergence is uniform in and . Thus the result follows .

By this lemma, we can show the following theorem,

**Theorem 3**. is pluriharmonic in (Pluriharmonicity means that for in the above Lemma , is harmonic in ).

**Proof**: Define a map with column vectors in and . Then

where all maps are holomorphical in . Then by Birkhoff Ergodic Theorem, we have

Hence, the result follows easily.

Now what left in the main steps are 1 and 3. Step1 is based on a key observation such that where and is the low-half plane in An easy way to see the above fact is to Mobius transform to via Then the action of on is conjugated to the action of on Then it’s easy to see that contracts into Hence, the above fact follows. Now an easy appliction of the equivalent condition for systems implies the results in step1.

The proof of step3 is a straightforward computation. Indeed, it’s easy to see and , thus for all which means that is a constant invariant section of the projective dynamics on If then it’s easy to see and .

Hence, step3 follows.

Finally, mean value formula for harmonic functions implies

which establishes the formula we want to show.

I’ve already given a self-contained and complete proof of **Herman-Avila-Bochi** formula in this post. I like this proof very much. Because it’s very conceptul and few computation is involved. One can also see how does **mean value formula** for harmonic functions gets into dynamical systems.

This post is very interesting!

Comment by Pengfei — January 24, 2011 @ 2:54 am |

Thanks:)

Comment by Zhenghe — January 24, 2011 @ 12:31 pm |