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 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 , a continuous and real valued function, the cocycle map, the Lyapunov exponent and the unstable (stable) direction. We further assume that Then the first theorem is that

**Theorem 1: **For a.e. , for a.e. .

**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. implies there exists invariant sections and which are measurable and integrable in some sense.

We didn’t prove the existence of , but it follows from the same argument of the existence of Indeed, we only need to replace by Then it’s easy to see that and everything follows from the same argument of the existence of .

Since is real, we see obviously that and are also invariant sections. If for a positive measure of , then by invariance of and and ergodicity we have for a.e. Then there exists a send to for some positive This implies that fix both and under mobius transformation. This happens only when in

On the other hand, from last post, we know is monotonic in in the sense that for any fixed and is monotonic in Thus if for a positive measure of , then it contradicts with monotonicity under some microscopic computation.

The second theorem is

**Theorem 2: **Let be an open interval (then it’s also bounded by boundedness of the operator). If on then there exists a map such that it’s contiuous on both variable, holomophic in and

**Remark**: This theorem in particular implies that the spectrum of the corresponding operator are purely absolutely continuous on the interval The generalized eigenfunctions are all wave like (not just in some sense.)

**Proof:** Define funcition such that

and

Then by the proof of Theorem 1, we have

for a.e. and a.e. .

Thus by Morera’s theorem, for a.e. extends through to be a holomorphic function on On the other hand, if we conformal transform to we see that is a normal family. Thus for any compacta in is uniformly Lipschitz in Namely, there exists a constant depends only on the compacts such that

for any

Since we can for any pick a sequence converging to Then we get a holomophic function On the other hand, for we know that is continous in since they are invariant sections of systems. Thus by holomophicity, is uniquely defined for all as holomorphic function of Since is obtained by taking limits, it automatically continous in

Now let’s state a determinism theorem. First we define a map such that Thus is a probability measure on is the hull in the product topology. For example,

if is periodic then the hull is finite;

if is limit periodic and not periodic, then the hull is a compact cantor group;

if the is quasiperiodic with dimensional rational independent frequency, then hull the dimensional torus .

Thus up to a homeomorphism, we can actually have that where is the left shift. Namely Then, we have the following theorem

**Theorem 3: **Assume Then for all in hull, if for all then for all

**Proof: **Here we use the following facts:

(a) for each for a.e. where is as in Theorem 2. This follows from Theorem 2 and Fatou’s theorem via a contradiction argument.

(b) determines This is not difficult to see by a contradiction argument.

(c) are uniquely determined by their limits on a positive measure set of

Now for each obviously determines But determines by fact (a) and (c). Finally, determines by fact (b).

**Remark:** The conclusion of this theorem is exactly the definition of the deterministic process. Namely a stochastic process is said to be deterministic if for all is determined by Otherwise it is nondeterministic. Obviously, I.I.d. process are far away from deterministic. In particular, if the process is i.i.d, then for a.e.

Next Theorem is that

**Theorem 4: **If takes finitely many values and then is periodic (or equivalently, hull is finite).

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

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