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.