This post is about the main theorem of **Artur Avila **and **David Damanik**‘s paper **`Generic singular spectrum for ergodic operators’. **It is another application of Kotani Theory, which also has its own interests. I will break the proof of main theorem down to several lemmas and point out the key ideas. I will also try to carry out all the details. **I am always grateful to Artur who is always willing to explain to me ideas and details whenever I want.**

In this post, I will use to denote the Lyapunov exponents of the system The main theorem is the following

**Theorem: **Assume is not periodic ( for any ). Then for generic (generic means residual), for a.e.

**Remark: **By Kotani theory this Theorem implies that for generic continuous for a.e. Thus for generic continuous potentials, the spectrum is of singular type, i.e. singular continuous and pure points.

Let’s consider the unstable direction as a function where are nonpositive integers and stands for bounded real-valued sequence on . Let

where and is the ball around zero function in with radius Now let’s state and prove our lemmas.

**Lemma 1:** For any bounded subset is bounded. Here boundedness in is with respect to hyperbolic metric.

**Proof:** WLOG, I can assume for any and for some Let

and

Then we show that of which the latter is a bounded set in Here I use to denote the map

Indeed, since We can of course take limit along even integers. Thus the inclusion is obvious. For boundedness it’s easy to see that for any

and

**Lemma 2: **Assume for some . Let as pointwisely. Then in compact open topology as functions on (i.e. uniform convergence in any compacta in )

**Proof:** By the proof of Theorem 1, we actually see that converge to in compact open topology and the convergence is independent of . This is due to the same reason of the proof of Theorem 2 of last post. Namely, are holomorphic functions takes value in , thus they are normal family. The independence with respect to is due to the uniform shrink rate of invariant cone field under projectivized action.

Now we have

Thus for any we can choose large such the first and third terms in the summation above are both less than on any compacta in . For this fixed for any large enough and in any compacta.

**Lemma 3: **For any fixed the function is continuous.

**Proof:** By passing to subsequence we can assume such that and pointwisely as

Thus for a.e. we have the converges to pointwisely as By Lemma 1 this implies that converges to for a.e. . And they lie in a compact set in by our choice of the ball Hence by Bounded Convergence Theorem, we have

**Remark: **Note what we need for in Lemma 3 are just uniform boundedness and pointwise convergence.

**Lemma 4: **Fix any interval Then the function is continuous.

**Remark:** As a function, behave badly when . The only thing that is always true is that it’s upper semicontinuous and plurisubharmonic. This Lemma shows that after taking an appropriate integral, it behaves nicely in sense.

**Proof: **Consider the semicircle such that where is upperhalf part of the circle centered at origin with radius Pick a point inside Then there is a harmonic measure on such that

By our assumption, Lemma 3 and bounded convergence theorem, it’s easy to see that and are continuous. Thus is also continuous. Via the conformal transformation which transform the unit disk to the region inside it’s not difficult to see that is also continuous.

Recall that We define a function such that . Then

**Lemma 5:** is upper semicontinuous.

**Proof:** We only need to restrict our self on Because outside this interval is always in resolvent set hence the systems are always in and Lyapunov exponents are positive.

It’s obvious that it suffices to show for any such that and any there exists a such that whenever and we have

Let’s show how to choose such a By definition of we can choose a such that where Then by Lemma 4 we can choose such that with Indeed we then have

Which implies that

Which implies that on thus This obviously implies what we want to show.

**Lemma 6:** There exists a dense set such that if then

(1) takes finitely many values;

(2) is not periodic for a.e.

**Proof: **(**This proof is due to Artur. It’s simpler than the one in their paper**) For a simple function let’s define to be the period of the sequence Then by ergodicity for a.e. Again by ergodicity it’s not difficult to see that if then Thus by our assumption on the triple if we can always change the value on a small measure set to produce a new simple function sufficiently close to in such that

**Lemma 7: **For any is dense in in topology.

**Proof: **Fix abitrary By Lemma 6, we can choose such that is sufficiently close to in By Theorem 4 of last post, we have By standard real analysis theorem and by Lemma 5, we can choose such that is sufficiently close to both in and such that

Now we are ready to prove our main Theorem

**Proof of Theorem: **For any we have by Lemma 5 and Lemma 7 is both open and dense. Thus is residual, which completes the proof.

**Corollary: **There is generic set such that for each

Where is the rotation matrix with rotation angle

**Sketch of Proof: **Use HAB formula we can get the corresponding Lemma 4 in terms of the function hence Lemma 5 in terms of the map On the other hand, Kotani theory can be carried over to this one parameter family thus we can have similar Theorem 4 of last post, hence similar result of Lemma 7 for Then the conclusion follows easily.

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