# Zhenghe's Blog

## 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.

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