# Zhenghe's Blog

## January 22, 2011

### Notes of Artur Avila’s Course in Fields Institute, Toronto 1: A Simple Proof of Herman-Avila-Bochi formula

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:

$\int_{\mathbb R/\mathbb Z}L(f,R_{\theta}A)d\theta=\int_{X}\ln\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}d\mu.$

Here is the setting: $X$ is compact metric space with a probability measure $\mu$ and $f$ is a homeomorphism on it preserving $\mu;$ $A:X\rightarrow SL(2,\mathbb R)$ is a continuous cocyle map. Thus $(f,A):X\times \mathbb R^2\rightarrow X\times \mathbb R^2$ gives a $SL(2,\mathbb R)$-valued cocyle dynamics over base dynamics $( X,f)$, namely, $(x,w)\rightarrow (f(x), A(x)w)$ for $(x,w)\in X\times \mathbb R^2.$ $L(f, A)$ stands for the Lyapunov exponent of this dynamical systems. If I denote $(f,A)^n$ by $(f^n, A_n)$, then it is given by

$L(f,A)=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\int_{X} \ln\|A_n(x)\|d\mu.$

The limit exists since $\{\int_{X} \ln\|A_n(x)\|d\mu\}_{n\geq 1}$ is a subadditive sequence. $R_{\theta}$ is the rotation matrix with rotation angle $2\pi\theta$.  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

$\int_{R/Z}L(f,R_{\theta}A)d\theta\geq\int_{X}\ln\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}d\mu$

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 $z=e^{2\pi i\theta}$ and write $R_{\theta}$ as $R_z$; let $\mathcal U\mathcal H$ stands for uniformly hyperbolic systems,  then the main steps are in the following:

1. Extending z from the unit circle $S$ to unit disk $\overline{\mathbb D}.$ Then for any $z \in \mathbb D\setminus\{0\}$, $(f,R_zA(x))\in\mathcal U\mathcal H.$
2. $L(f, zR_{z}A)=\ln |z|+L(f, R_zA)$ is harmonic in $z$ for $z\in \mathbb D\setminus\{0\}$ and bounded and  well-defined for all $z\in \overline{\mathbb D}.$ It obviously agrees with $L(f,R_{z_o}A)$ for $z_0\in S$. Furthermore, $L(f, zR_{z}A)$ converges to $L(f, z_0R_{z_0}A)$ nontangentially.
3. $L(f,0R_0A)=\int_{X}\ln\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}d\mu.$
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 $SL(2,\mathbb C)-$valued cocyle case, i.e. $A:X\rightarrow SL(2,\mathbb C)$ which is continuous. Let $\mathbb C\mathbb P^1$ be the Riemann surface $\mathbb C\cup\{\infty\}$ and $A$ acts on it as Mobius transforamtion, i.e. for $A=\begin{pmatrix}a& b\\c&d\end{pmatrix}$, $A\cdot z=\frac{az+b}{cz+d}$.

Defintion 1.  $(f, A)$ is said to be uniformly hyperbolic if the are two contiuous map $u, s:X\rightarrow \mathbb C\mathbb P^1$ such that
1. they are invariant in the sense that $A(x)\cdot u(x)=u(f(x))$ and $A(x)\cdot s(x)=s(f(x))$.
2. there exist constant $C>0, 0<\lambda<1$ such that for each vector $w_s\in s(x), \|A_n(x)w_s\|\leq C\lambda^n\|w_s\|$ for $n\geq 1$,  and for each $w_u\in s(x), \|A_n(x)w_u\|\leq C\lambda^{-n}\|w_u\|$ for $n\leq 0$.

One well-known equivalent condition for $\mathcal U\mathcal H$ systems is the existence of invariant cone field. More concretely, $(f, A)\in \mathcal U\mathcal H$ if and only if for each $x\in X,$ there is open disk $U(x)\subset \mathbb C\mathbb P^1$ such that there exists positive integer $N$

$\overline{A_n(x)\cdot U(x)}\subset U(f^n(x)), \forall x\in X, \forall n\geq N.$

The existence of $U(x)$  implies the existence of a corresponding $S(x).$ Using this it easy to obtain the following lemma

Lemma 2. Consider a holomorphic 1-parameter family of $\mathcal U\mathcal H$ systems $(f, A_{\lambda})$, $\lambda$ is in some open region in complex plane. Holomorphicity in the sense that for each fixed $x,$ $A_{\lambda}(x)$ is  holomorphic in $\lambda.$ Then $u(x,\lambda)$ and $s(x,\lambda)$ are  both holomorphic in $\lambda$.
Proof:  It’s easy to see by the equivalent condition above,
$u(x,\lambda)=\lim\limits_{n\rightarrow\infty}(A_{\lambda})_n(f^{-n}(x))\cdot u_{-n}$,
for any $u_{-n}\in U(f^{-n}(x))$ and the convergence is uniform in $x$ and $\lambda$. Thus the result follows . $\square$

By this lemma, we can show the following theorem,

Theorem 3. $L(f,A)$ is pluriharmonic in $\mathcal U\mathcal H$ (Pluriharmonicity means that for $(f, A_{\lambda})$ in the above Lemma , $L(f,A_{\lambda})$ is harmonic in $\lambda$).
Proof: Define a map $B_{\lambda}:X\rightarrow SL(2,\mathbb C)$ with column vectors in $u_{\lambda}(x)$ and $s_{\lambda}(x)$. Then $B_{\lambda}(f(x))A_{\lambda}(x)B_{\lambda}(x)^{-1}=diag(\gamma_{\lambda}(x), \gamma_{\lambda}(x)^{-1}),$
where all maps are holomorphical in $\lambda$. Then by Birkhoff Ergodic Theorem, we have
$L(f, A_{\lambda})=\int_X\ln|\gamma_{\lambda}(x)|d\mu.$
Hence, the result follows easily. $\square$

Now what left in the main steps are 1 and 3. Step1 is based on a key observation such that $\overline{R_{i\theta}\cdot\mathbb H_-}\subset\mathbb H_-,$ where $\theta>0$ and $\mathbb H_-$ is the low-half plane in $\mathbb C\mathbb P^1.$ An easy way to see the above fact is to Mobius transform $\mathbb H$ to $\mathbb D$ via $Q=\frac{-1}{1+i}\begin{pmatrix}1&-i\\1&i\end{pmatrix}\in \mathbb U(2).$ Then the action of $R_{\theta}$ on $\mathbb H$ is conjugated to the action of $\hat{R_{\theta}}=QR_{\theta}Q^*=diag(e^{-2\pi i\theta}, e^{2\pi i\theta})$ on $\mathbb D.$ Then it’s easy to see that $\hat{R_{i\theta }}$ contracts $\overline{\mathbb D}^c$ into $\overline{\mathbb D}^c.$ Hence, the above fact follows. Now an easy appliction of the equivalent condition for $\mathcal U\mathcal H$ systems implies the results in step1.

The proof of step3 is a straightforward computation. Indeed, it’s easy to see $P=0R_0=\frac{1}{2}\begin{pmatrix}1&-i\\i&1\end{pmatrix}$ and $P\cdot(\mathbb H^c)=-i$, thus $PA\cdot (-i)=-i$ for all $A\in SL(2, \mathbb R),$ which means that $-i$ is a constant invariant section of the projective dynamics$(f, PA)$ on $X\times CP^1.$ If $PA(x)\binom{-i}{1}=\gamma(x)\binom{-i}{1},$ then it’s easy to see $L(f,PA)=\int_X\ln|\gamma(x)|d\mu$ and $|\gamma(x)|=\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}$.
Hence, step3 follows.

Finally, mean value formula for harmonic functions implies

$L(f,0R_0A)=\int_{R/Z}L(f, zR_zA)d\theta=\int_{R/Z}L(f, R_{\theta}A)d\theta,$

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.