# Zhenghe's Blog

## February 9, 2011

### Notes 4: Application of Kotani Theory (I)-Determinism

Filed under: Schrodinger Cocycles — Zhenghe @ 4:34 am
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In this post, I will give several theorems which are applications of Kotani Theory. But I will not give complete proof of all of them. These theorems are basically all Kotani’s work, among which Theorem 1-3 are in his original paper of Kotani Theory Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional $Schr\ddot odinger$ operators.’  And theorem 4 is in his paper Jacobi matrices with random potentials taking finitely many values.’

I will use the notations of  previous blogs. E.g. the triple $(X, f,\mu)$, $v\in C(X,\mathbb R)$ a continuous and real valued function, $A^{(E-v)}$ the $Schr\ddot odinger$ cocycle map,  $L(E)$ the Lyapunov exponent  and $u (s)$ the unstable (stable) direction.  We further assume that $supp({\mu})=X.$ Then the first theorem is that

Theorem 1: For a.e. $E$, $L(E)=0\Rightarrow u(E,x)=\bar s(E,x)$ for a.e. $x$.

Remark: Unstable direction comes from the past and stable direction comes from the future (one can take a look at the post ‘A simple proof of HAB formula’ : how to use invariant cone field to obtain invariant direction). Thus this theorem tells us that zero Lyapunov exponents implies that past determines the future, which is kind of determinism.

Idea of Proof: From the proof of Main theorem of Kotani theory we know that for a.e. $E\in\mathbb R,$ $L(E)=0$ implies there exists invariant sections $u(E,x)\in\mathbb H$ and $s(E,x)\in\mathbb H_-$ which are measurable and integrable in some sense.

We didn’t prove the existence of $s$, but it follows from the same argument of the existence of $u.$  Indeed, we only need to replace $A^{(E+i\epsilon-v)}$ by $(A^{(E+\epsilon-v)})^{-1}.$ Then it’s easy to see that $\overline{(A^{(E-v)})^{-1}(x)\cdot\mathbb H_-}\subset\mathbb H_-$ and everything follows from the same argument  of the existence of $u(E,x)$.

Since $A^{(E-v)}, E\in\mathbb R$ is real, we see obviously that $\bar u$ and $\bar s$ are also invariant sections. If $u(x)\neq\bar s(x)$ for a positive measure of  $x$, then by invariance of $u$ and $\bar s$ and ergodicity we have $u(x)\neq\bar s(x)$ for a.e. $x.$ Then there exists a $B:X\rightarrow PSL(2,\mathbb R)$ send $u, \bar s$ to $i, ti$ for some positive $t\neq 1.$  This implies that $B(f(x))A(x)B(x)^{-1}$ fix both $i$ and $ti$ under mobius transformation. This happens only when $B(f(x))A(x)B(x)^{-1}=Id$ in $PSL(2,\mathbb R).$

On the other hand, from last post, we know $A^{(E-v)}$ is monotonic in $E$ in the sense that for any fixed $x\in X$ and $u\in\mathbb R,$ $A^{(E-v)}(x)\cdot u$ is monotonic in $E.$ Thus if $B(E,f(x))A^{(E-v)}(x)B(E,x)^{-1}=Id$ for a positive measure of $E$, then it contradicts with monotonicity under some microscopic computation. $\square$

The second theorem is

Theorem 2: Let $I\in\mathbb R$ be an open interval (then it’s also bounded by boundedness of  the $Schr\ddot odinger$ operator). If $L(E)=0$ on $I,$ then there exists a map $B:I\times X\rightarrow SL(2,\mathbb R)$ such that it’s contiuous on both variable, holomophic in $I$ and $B(E,f(x))A^{(E-v)}(x)B(E,x)^{-1}\in SO(2,\mathbb R).$

Remark: This theorem in particular implies that the spectrum of the corresponding $Schr\ddot odinger$ operator are purely absolutely continuous on the interval $I.$ The generalized eigenfunctions are all wave like (not just in some $L^2$ sense.)

Proof: Define funcition $m:(\mathbb H\cup\mathbb H_-)\times X\rightarrow \mathbb H$ such that

$m(E,x)=u(E,x), for\ E\in\mathbb H$ and
$m(E,x)=\bar s(E,x), for\ E\in\mathbb H_-.$

Then by the proof of Theorem 1, we have

$\lim\limits_{\epsilon\rightarrow 0}m(E+i\epsilon,x)=\lim\limits_{\epsilon\rightarrow 0}m(E-i\epsilon,x)$ for a.e. $x.$ and a.e. $E\in I$.

Thus by Morera’s theorem, for a.e.$x,$ $m(\cdot,x)$ extends through $I$ to be a holomorphic function on $\mathbb C\setminus (\mathbb R\setminus I).$ On the other hand, if we conformal transform $\mathbb H$ to $\mathbb D,$ we see that $m(E,x)_{x}$ is a normal family. Thus for any compacta $\mathcal K$ in $\mathbb C\setminus (\mathbb R\setminus I),$ $m(\cdot,x)$ is uniformly Lipschitz in $E.$  Namely, there exists a constant $c=c(\mathcal K)$ depends only on the compacts such that

$|m(E_1,x)-m(E_2,x)|\leq c|E_1-E_2|$ for any $E_1, E_2\in\mathcal K.$

Since $supp(\mu)=X,$ we can for any $x\in X$ pick a sequence $x_n\in X$ converging to $x.$ Then we get a holomophic function $m(\cdot,x)=\lim\limits_{x_n\rightarrow x}m(\cdot,x_n).$ On the other hand, for $E\notin I,$ we know that $m(E,\cdot)$ is continous in $X$ since they are invariant sections of $\mathcal U\mathcal H$ systems. Thus by holomophicity, $m(\cdot,x)$ is uniquely defined for all $x$ as holomorphic function of $E.$ Since $m(E,\cdot)$ is obtained by taking limits, it automatically continous in $x.$ $\square$

Now let’s state a determinism theorem. First we define a map $\phi:X\rightarrow l^{\infty}(\mathbb Z)$ such that $\phi(x)=(v(f^{n}(x))).$ Thus $\tilde{\mu}=\phi_*\mu$ is a probability measure on $l^{\infty}(\mathbb Z).$ $supp(\tilde{\mu})$ is the hull in the product topology. For example,

if $(v(f^n(x)))_n$ is periodic then the hull is finite;
if $(v(f^n(x)))_n$ is limit periodic and not periodic, then the hull is a compact cantor group;
if the $(v(f^n(x)))_n$ is quasiperiodic with $d$ dimensional rational independent frequency, then hull the $d$ dimensional torus $\mathbb T^{d}$.

Thus up to a homeomorphism, we can actually have that $( X,\mu,f)=(supp(\tilde{\mu}), \tilde{\mu},\sigma),$ where $\sigma$ is the left shift. Namely $(\sigma(x))_n=x_{n+1}.$ Then, we have the following theorem

Theorem 3: Assume $Leb\{E:L(E)=0\}>0.$ Then for all $y, \hat y$ in hull, if $y_n=\hat y_n$ for all $n\leq 0,$ then $y_n=\hat y_n$ for all $n\in\mathbb Z.$

Proof: Here we use the following facts:

(a) for each $x\in X,$ $\lim\limits_{\epsilon\rightarrow 0^+}m(E+i\epsilon,x)=\lim\limits_{\epsilon\rightarrow 0^+}m(E-i\epsilon,x)$ for a.e. $E,$ where $m$ is as in Theorem 2. This follows from Theorem 2 and Fatou’s theorem via a contradiction argument.

(b) $u(E) (s(E)), E\in\mathbb H$ determines $(v(n))_{n\leq0} ((v(n))_{n\geq1}).$ This is not difficult to see by a contradiction argument.

(c) $u(E) (s(E))$ are uniquely determined by their limits on a positive measure set of $E\in\mathbb R.$

Now for each $x\in X,$  $v(f^n(x)), n\leq 0$ obviously determines $u(E,x), E\in\mathbb H.$ But $u(\cdot, x):\mathbb H\rightarrow\mathbb H$ determines $s(\cdot, x):\mathbb H\rightarrow\mathbb H_-$ by fact (a) and (c). Finally, $s(E,x):\mathbb H\rightarrow\mathbb H_-$ determines $v(f^n(x)), n>0$ by fact (b). $\square$

Remark: The conclusion of this theorem is exactly the definition of the deterministic process. Namely a stochastic process $\{v_n(x)\}_{n\in\mathbb Z}$ is said to be deterministic if for all $y\in supp(\tilde{\mu})$ $(y_n)_{n\in\mathbb Z}$ is determined by $\{y_n\}_{n\leq 0}.$  Otherwise it is nondeterministic. Obviously, I.I.d. process are far away from deterministic. In particular, if the process $v_n(x)$ is i.i.d, then for a.e. $E.$ $L(E)>0.$

Next Theorem is that

Theorem 4: If $v$ takes finitely many values and  $Leb\{E:L(E)=0\}>0,$ then $v$ is periodic (or equivalently, hull is finite).

Proof: This is an immediate consequence of Theorem 3. Indeed if $v$ is not periodic then $v$ is nondeterministic. But this cannot happen by our assumption and Theorem 3. $\square$