# Zhenghe's Blog

## March 3, 2011

### Notes 9: Averaging and Renormalization(I)-Defining the Renormalization Operator

This is post will be a breif introduction about some averaging and renormalization procedures in Schrodinger cocycles. Renormalization is everywhere in dynamical systems and it has been used by many people in Schrodinger cocycles for a while. Lot’s of interesting results have been obtained. As for averaging, I guess it has not been very widely used in Schrodinger cocycles yet. Although it has been a powerful technique in Hamiltonian dynamics for a very long time in studying longtime evolution of action variables. Artur has already used them to construct counter-examples to Kotani-Last Conjecture and Schrodinger Conjecture. We’ve introduced Kotani-Last Conjecture in previous post, namely, the counter-example is that the base dynamics is not almost periodic but the corresponding Schrodinger operator admits absolutely continuous spectrum. For Schrodinger conjecture, the counter example is a Schrodinger operator whose absolutely continous spectrum admits unbounded generalized eigenfunctions.

I am not familar with both averaging and renormalization.  So I may need a little bit longer time to finish this post. But I will try my best to give a brief and clear introduction.

(I) Averaging

Let the frequency $\alpha$ be very Louville number. For instance, let $\alpha=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots}}}}$ be the continued fraction expansion. Thus $\frac{p_n}{q_n}=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots+\frac{1}{a_n}}}}}$ be the n’th step approximant. We let $a_n,$ hence $q_n,$ grows sufficiently fast (see my third post for introduction of Liouville number). Let the potential $v$ be a very small real analytic function. Combining averaging procedure with some renormalization and KAM procedure,  Avila, Fayad and Krikorian proved in their paper ‘A KAM scheme for $SL(2,\mathbb R)$ cocycles with Liouvillean frequencies’ the following theorem

Theorem:  Let $v: \mathbb R/\mathbb Z\rightarrow\mathbb R$ be analytic and close to a constant. For every $\alpha\in\mathbb R,$ there exists a positive measure set of $E\in\mathbb R$ such that $( \alpha, A^{(v,E)})$ is conjugate to a cocycle of rotations.

The main difficulty is Liouvillean frequency case, where they used the idea of averaging. We will not prove this theorem. We are going to give an idea how does the averaging procedure look like in Schrodinger cocycle.

Let’s start with the base dynamics $x\mapsto x.$ Namely, we consider one-parameter family of cocycles over fixed point. We have two different ways to consider the spectrum, which is the following

$I=\cup_x\Sigma_x:\{E\in\mathbb R: |tr(A^{(E-v)}(x))|=|E-v(x)|\leq 2$ for at least one $x \},$
$I^0=\cap_x\Sigma_x:\{E\in\mathbb R: |tr(A^{(E-v)}(x))|=|E-v(x)|\leq 2$ for all $x \}.$

Let’s start with $E\in I^0.$ For simplicity, set $A(x)=A^{(E-v)}(x)$ and we omit the dependence on $E.$ Let $u(x)$ be the invariant direction of $A(x).$ Let $B^0:\mathbb R/\mathbb Z\rightarrow SL(2,\mathbb R)$ be that $B^0(x)\cdot u(x)=i.$ Thus $B^0(x)A(x)B^0(x)^{-1}=R_{\psi^0(x)}$  for some $\psi^0:\mathbb R/\mathbb Z\rightarrow\mathbb R$ analytic. Thus there exists a constant $C_0$ such that

$B^0(x+\frac{p}{q})A(x)B^0(x)^{-1}=R_{\psi^0(x)}+\frac{C_0}{q}.$

Then for any closed interval $I^1$ inside the interior of $I^0,$ we can choose large $q$ such that $R_{\psi^0(x)}+\frac{C_0}{q}$ is again elliptic. Hence similarly, there is some $B^1:\mathbb R/\mathbb Z\rightarrow SL(2,\mathbb R)$ which is $\frac{1}{q}$ close to identity and $\psi^1:\mathbb R/\mathbb Z\rightarrow\mathbb R$ such that

$B^1(x)(R_{\psi^0(x)}+\frac{C_0}{q})B^1(x)^{-1}=R_{\psi^1(x)}.$ Hence

$B^1(x+\frac{1}{q})(R_{\psi^0(x)}+\frac{C_0}{q})B^1(x)^{-1}=R_{\psi^1(x)}+\frac{C_1}{q^2}$ for some constant $C_1.$

Thus $(\frac{1}{q},A)$ is conjugate to $(\frac{1}{q}, R_{\psi^1(x)}+\frac{C_1}{q^2}).$ Iterating $q$ times we get $R_{\psi_{q}^1(x)}+\frac{C_1}{q},$ where $\psi_{q}^1(x)$ is the n-th Birkhoff sume $\sum^{q-1}_{j=0}\psi(x+\frac{j}{q}).$ It’s not difficult to see that $\psi^1$ is close to its averaging $q\hat{\psi}(0).$ Thus except those bad $E$ such that $\psi_{q}^{1}(x)=\pm id$ for some $x,$ $(\frac{1}{q}, A^{(E-v)})$ stay ellipitc. By monotonicity of the Schrodinger cocycle with respect to energy and suitable choice of $q,$ these bad $E$ are of arbitrary small measure.

(II) Renormalization

Here we only introduce renormalization in quasiperiodic $SL(2,\mathbb R)$-cocycle where the frequency is one dimensional (i.e. the base dynamics is one dimensional). Let’s denote the renormalization operator by $\mathcal R.$ This will be a operator defined on the space of all cocycle dynamics. More concretely, it’s defined on the space of $\mathbb Z^2$ action on cocycle dynamics. We will continue to use the continued fraction expansion and approximants as in Averaging.

A natural way to see that the cocycle dynamics $(\alpha, A):(\mathbb R/\mathbb Z)\times\mathbb R^2\rightarrow (\mathbb R/\mathbb Z)\times\mathbb R^2$ is to see it as $(\alpha, A):\mathbb R\times\mathbb R^2\rightarrow \mathbb R\times\mathbb R^2,$ which commutes with $(1, id):\mathbb R\times\mathbb R^2\rightarrow \mathbb R\times\mathbb R^2.$ Because it’s easy to check that

$A(x)=A(x+1)\Leftrightarrow (\alpha,A)\cdot(1, id)=(1, id)\cdot(\alpha,A).$

This is similar to the following case. Denote the orientation preserving diffeomorphism on $\mathbb R$ by $Diffeo_+(\mathbb R).$ Then $F:\mathbb R/\mathbb Z\rightarrow\mathbb R/\mathbb Z$ is orientation preserving diffeomorphism if and only if it can be lifted to $Diffeo_+(\mathbb R)$ and commutes with $G:\mathbb R\rightarrow\mathbb R,$ where $G(x)=x+1.$ More generally, consider a commuting pair $(F,G)\in Diffeo_+(\mathbb R)$ where G has no fixed point. Then $F$ define a dynamical systems on $\mathbb R/G.$ Because $F(G(x))=G(F(x))\sim F(x)$ preserving the equivalent relation. Here $\mathbb R/G$ is diffeomorphic to $\mathbb R/\mathbb Z,$ but not in a canonical way.

Thus it’s more natural to view the pair $(F,G)$ as a $\mathbb Z^2$ group action, which is the following group homomorphism

$\Phi:\mathbb Z^2\rightarrow Diffeo_+(\mathbb R)$ with $\Phi(1,0)=G$ and $\Phi(0,1)=F.$

This automatically encodes the commuting relation.

The reason we introduce above procedure is that after renormalization, we will have no canonical way to glue $\mathbb R$ into $\mathbb R/\mathbb Z.$

Now let’s define the renormalization operator $\mathcal R$ step by step. We will renormalize around $0\in\mathbb R/\mathbb Z.$

(1) Define the $\mathbb Z^2$-action.
Let $\Lambda^r=\mathbb R\times C^r(\mathbb R,SL(2,\mathbb R))$ be subgroup of $Diff^r(\mathbb R\times\mathbb R^2).$ Then for $(\alpha,A)\in\Lambda^r$ satisfying $A(x)=A(x+1),$ we can define $\mathbb Z^2$ action $\Phi:\mathbb Z^2\rightarrow \Lambda^r$ such that $\Phi(1,0)=(1,id), \Phi(0,1)=(\alpha,A).$ Let $\Phi(n,m)=(\alpha_{n,m}, A_{n,m}).$

(2) Define two operations on the above $\mathbb Z^2$-action.
The first is the rescaling operator $M_{\lambda},$ which is given by

$M_{\lambda}(\Phi)(n,m)=(\lambda^{-1}\alpha_{n,m}, x\mapsto A_{n,m}(\lambda x)).$

The second is base changing operator $N_{U}.$ For $U\in GL(2,\mathbb Z),$ it’s given by

$N_{U}(\Phi)(n.m)=\Phi(n',m'), \binom{n'}{m'}=U^{-1}\binom{n}{m}.$

Obviously these two operations commutes with each other.

(3) More facts about continued fraction expansion.
Let $\alpha$ be as in averaging. Let $G$ be the Gauss map $G(\alpha)=\{\alpha^{-1}\}.$ Denote $G^n(\alpha)=\alpha_n.$ Thus $a_n=[\alpha_{n-1}^{-1}],$ where $a_n$ comes from the continued fraction expansion. Define a map

$U(x)=\begin{pmatrix}[x^{-1}]&1\\1&0\end{pmatrix}$ for $x\in (0,1).$

Let $\beta_n=\alpha_{n}\cdots\alpha.$ Then it’s easy to see that we also have $\beta_n=(-1)^n(q_n\alpha-p_n)=\frac{1}{q_{n+1}+\alpha_{n+1}q_n}.$

(4) Define the renormalization operator.
$\mathcal R:\Lambda^r\rightarrow\Lambda^r$ is given by

$\mathcal R(\Phi)=M_{\alpha}(N_{U(\alpha)}(\Phi)),$ where $\alpha$ is from $\Phi(0,1)=(\alpha,A).$ $\square$

It easy to see that

$\mathcal R(\Phi)(1,0)=M_{\alpha}(\Phi)(0,1)=(1,A(\alpha(\cdot)))$ and
$\mathcal R(\Phi)(0,1)=M_{\alpha}(\Phi)(1,-a_1)=M_{\alpha}(1-a_1\alpha, A_{-a_1})=(\alpha_1,A_{-a_1}(\alpha(\cdot))).$

Thus geometrically, if we  look at $\mathbb R\times\mathbb R\mathbb P^1,$ it looks like we glue $\{x\}\times\mathbb R\mathbb P^1$ and $\{x+\alpha\}\times\mathbb R\mathbb P^1$ via $(\alpha,A).$ Then by commuting relation $(1-a_1\alpha, A_{-a_1})$ define a dynamical systems on $(\mathbb R\times \mathbb R\mathbb P^1)/(\alpha,A).$ Then we rescale the first coordinate to be $\mathbb R/\mathbb Z$ again. Finally we get the new dynamics $\mathcal R(\Phi)(0,1).$

Let $Q_n=U(\alpha_{n-1})\cdots U(\alpha).$ Then it’s easy to see that

$\mathcal R^n(\Phi)=M_{\alpha_{n-1}}\circ N_{U(\alpha_{n-1})}\circ\cdots\circ M_{\alpha}\circ N_{U_{\alpha}}(\Phi)=M_{\beta_{n-1}}(N_{Q_n}(\Phi)).$

Thus we are looking at smaller and smaller space scale but larger and larger time scale.

Next post we will show how are the dynamics of original cocycle and these of renormalized cocycles related. And we will normalize $\mathcal R^n(\Phi)$ so that $\mathcal R^n(\Phi)(1,0)=(1,id)$ and do some computation concerning the limit of the renormalzation.

## February 16, 2011

### Notes 6: Density of Positive Lyapunov Exponents for SL(2,R)-cocycles in any Regularity Class

This post is about the density of positive Lyapunov exponents for $SL(2,\mathbb R)$ cocylces in any regular class. It’s again from Artur Avilas course here in Fields institute, Toronto and from his paper Density of Positive Lyapunov Exponents for $SL(2,\mathbb R)$ 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 $C^0$ class. Obviously, it’s more difficult to obtain density results in higher regularity class.

Again I will use the base space $(X,f,\mu);$ assume $X=supp(\mu)$ and $f$ is not periodic. I will use $L(A)$ to denote the Lyapunov exponent of the corresponding cocyle map $A:X\rightarrow SL(2,\mathbb R).$ Let’s first introduce a concept to state the main theorem.

Definition: A topological space $\mathfrak B\subset C(X,SL(2,\mathbb R))$ is ample if there exists some dense vector space $\mathfrak b\subset C(X,sl(2,\mathbb R))$, endowed with some finer (than uniform) topological vector space structure, such that for every $A\in\mathfrak B, e^{b}A\in\mathfrak B$ for every $b\in\mathfrak b,$ and the map $b\mapsto e^{b}A$ from $\mathfrak b$ to $\mathfrak B$ is continous.

Remark: Note that if $X$ is a $C^r$ manifold, $r\in\mathbb N\cup\{\infty,\omega\},$ then $C^r(X,SL(2,\mathbb R))$ is ample. Namely we can take $\mathfrak b=C^r(X,sl(2,\mathbb R)).$

The main theorem is the following

Theorem 1: Let $\mathfrak B\subset C(X, SL(2,\mathbb R))$ be ample. Then the Lyapunov exponent is positive for a dense subset of $\mathfrak B.$

Remark: This is basically an optimal result for density of positive Lyapunov exponents for $SL(2,\mathbb R)$ cocycles.

The key theorem lead to Theorem 1 is the next theorem. Let $\|\cdot\|_*$ denote the sup norm in $C(X,sl(2,\mathbb R))$ and $C(X,sl(2,\mathbb C)),$ and for $r>0$ let $\mathcal B_*(r)$ and $\mathcal B_*^{\mathbb C}(r)$ be the correponding $r$-balls. Then

Theorem 2: There exists $\eta>0$ such that if $b\in C(X,sl(2,\mathbb R))$ is $\eta$-close to $\begin{pmatrix}0&1\\-1&0\end{pmatrix},$ then for $\epsilon>0$ and every $A\in C(X,SL(2,\mathbb R)),$ the map

$\Phi(\cdot; A):\mathcal B_*(\eta)\rightarrow\mathbb R, a\mapsto\int_{-1}^{1}\frac{1-t^2}{|t^2+2it+1|}L(e^{\epsilon(tb+(1-t^2)a)}A)dt$

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

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 $\eta>0$ such that we can check:
1. For $z\in\partial{\mathbb D}\cap\mathbb H$ and for $z=(\sqrt2-1)i, e^{\epsilon(zb+(1-z^2)a)}A$  is $\mathcal U\mathcal H,$ provided $a\in\mathfrak B_*^{\mathbb C}(\eta),$
2. For $z\in\mathbb D\cap\mathbb H, e^{\epsilon(zb+(1-z^2)a)}A$ is $\mathcal U\mathcal H,$ provided $a\in\mathfrak B_*(\eta).$

On the other hand, we can write down the explicit conformal transformation $\psi:\mathbb D\rightarrow\mathbb D\cap\mathbb H$ such that $\psi(z)=\phi^{-1}(\sqrt{\phi(z)}),$ where $\phi:\mathbb D\rightarrow\mathbb H, z\mapsto i\frac{1-z}{1+z}.$ Notice that $\psi(0)=(\sqrt2-1)i.$ Let’s denote $\rho(z,a)=L(e^{\epsilon(zb+(1-z^2)a)}A).$ Once we have these facts, by pluriharmonic theorem in the postProof of HAB formula‘ and mean value formula for harmonic functions, we have

a.$\rho(\psi(0),a)=\int_0^1\rho(\psi(e^{2\pi i\theta}),a)d\theta$ for $a\in\mathfrak B_*(\eta)$ by fact 2; thus
b.$\int_0^{1/2}\rho(\psi(e^{2\pi i\theta}),a)d\theta=\rho(\psi(0),a)-\int_{1/2}^{1}\rho(\psi(e^{2\pi i\theta}),a)d\theta;$ but
c.$\rho(\psi(0),a)-\int_{1/2}^{1}\rho(\psi(e^{2\pi i\theta}),a)d\theta$ is pluriharmonic for $a\in\mathfrak B_*(\eta)$ 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 $m\in\mathbb R,$ we define function $m(\epsilon)=e^{\epsilon(zb+(1-z^2)a)}\cdot m\in\mathbb P\mathbb C^1$ and check that $\frac{d}{d\epsilon}m(\epsilon)$ at $\epsilon=0$ points inside $\mathbb H$ for $z, a$ in facts 1 or 2. Thus $\mathbb H$ will be an invariant conefield for $e^{\epsilon(zb+(1-z^2)a)}A$ for any $A\in C(X,SL(2,\mathbb R))$ and $\epsilon>0$ small, which implies $\mathcal U\mathcal H.$ This is similar to the cases in Kotani theory or HAB formula, where when we complexify $E$ or $\theta$ we get $\mathcal U\mathcal H.$ For detailed proof see Artur’s paper. $\square$

Proof of Theorem 1: We must show that for every $A_1\in\mathfrak B,$  there exists a $A_2\in\mathfrak B$ sufficiently close to $A_1$ in $\mathfrak B$ and $L(A_2)>0.$

For any $\epsilon>0.$ Let $\gamma(t)=L(e^{\epsilon(tb+(1-t^2)a)}A_1).$ Then by subharmonicity of $\gamma$, more concretely, by upper semicontinuity and sub-mean value property, we can choose suitable closed path to see that if $\gamma(0)>0,$ then $\Phi(a;A_1)>0.$

Since $\mathfrak B$ is ample, we can  choose suffciently small $\epsilon>0$ and some $b$ as in Theorem 2 such that $e^{b}A_1\in\mathfrak B$ and $e^{\epsilon tb}A_1$ is sufficient close to $A_1$ in $\mathfrak B$ for every $t\in [-1,1].$ Then by above observation and Corollary of last post we can find some $a\in\mathcal B_*(\eta)$ such that $\Phi(a;A_1)>0.$

Again by the assumption that $\mathcal B$ is ample and the analyticity of the map $\Phi(\cdot;A_1),$ we can assume $a$ such that  $e^{\epsilon(tb+(1-t^2)a)}A_1\in\mathfrak B.$

By Theorem 2, the function $\phi(s)=\Phi(sa;A_1)$ is analyic in $s\in [-1,1].$ Since $\phi(1)>0$, we have for every sufficiently small s>0, $\phi(s)>0.$ Thus we can choose sufficiently small $s_0$ and some $t_0\in[-1,1]$ such that $A_2=e^{\epsilon(t_0b+(1-t_0^2)s_0a)}A_1$. $\square$

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 $(X,f,\mu)=(\mathbb R/\mathbb Z,R_{\alpha}, Leb),$ where $R_{\alpha}:\mathbb R/\mathbb Z\rightarrow\mathbb R/\mathbb Z, x\mapsto x+\alpha$ and $\alpha$ is irrational.

Let’s consider the cocylce space $C^{\omega}(\mathbb R/\mathbb Z, SL(2,\mathbb R))$ endowed with some inductive limit topology via subspace $C^{\omega}_{\delta}(\mathbb R/\mathbb Z, SL(2,\mathbb R)).$  Here $\delta>0$ and $C^{\omega}_{\delta}(\mathbb R/\mathbb Z, SL(2,\mathbb R))$ is the space of  real analytic cocycle maps which can be extended to $\{z\in\mathbb C/\mathbb Z: |\Im z|<\delta\}.$

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 $SL(2,\mathbb R)$ cocyle case in Jitomirkaya, Koslover and Schulteispaper ‘Continuity of the Lyapunov Exponent for analytic quasiperiodic cocycles’ such that

Theorem: The Lyapunov exponent $L:(\mathbb R\setminus\mathbb Q)\times C^{\omega}(\mathbb R/\mathbb Z, SL(2,\mathbb R)), (\alpha, A)\mapsto L(\alpha,A)$ 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 $\alpha,$ Lyapunov exponent is positive for an open and dense subset of $C^{\omega}(\mathbb R/\mathbb Z, SL(2,\mathbb R)).$

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

In Schrodinger cocycle case: $Leb\{E:L(E)=0\}>0\Rightarrow$ Almost Periodicity of the base dynamics.

Recall that by Notes 4, we know $Leb\{E:L(E)=0\}>0\Rightarrow$ determinism. On the other hand almost periodicity is stronger then determinsim, which has the following equivalent description:

For any $\forall\epsilon>0, \exists\delta>0$ and $N\in\mathbb Z^+$ such that
$|v(f^n(x))-v(f^n(y))|<\delta,\forall -N\leq n\leq 0\Rightarrow |v(f^n(x))-v(f^n(y))|<\epsilon, \forall n\in\mathbb Z.$

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.

## 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.

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