Zhenghe's Blog

April 4, 2011

Notes 11: Renormalization (III)-Convergence of Renormalization

I’ve been back to Evanston from Toronto. But I guess I still have 10 more notes to post. It will take me a very long time to finish.

In this post we will discuss the convergence of renormalization. As pointed out in last post, it’s necessary that we should assume zero Lyapunov to get convergence. In fact, we will assume L^2-conjugacy to rotations. This is somehow natural because Kotani theory tells us that in the Schrodinger cocycle case, for almost every energy, zero Lyapunov exponent implies L^2-conjugacy. More concretely, we assume for A\in C^{\omega}(\mathbb R/\mathbb Z, SL(2,\mathbb R)), \alpha\in\mathbb R\setminus\mathbb Q that (\alpha,A) is L^2-conjugate to SO(2,\mathbb R)-valued cocycles. Namely, there is a measurable map B:\mathbb R/\mathbb Z\rightarrow SO(2,\mathbb R) such that \int_{\mathbb R/\mathbb Z}\|B(x)\|^2dx<\infty and B(x+\alpha)A(x)B(x)^{-1}\in SO(2,\mathbb R) for almost every x\in\mathbb R/\mathbb Z.  Let S(x)=\sup_{n\geq 1}\frac{1}{n}\sum^{n-1}_{k=0}\|B(x+k\alpha)\|^2, which is finite almost everywhere by the Maximal Ergodic Theorem. WLOG, we can assume that A can be holomorphically extended to \Omega_{\delta}=\{z\in\mathbb C/\mathbb Z: |\Im z|<\delta\} which is also Lipschitz in \Omega_{\delta}. Then we have the following lemma.

Lemma 1: There exists C>0 such that for almost every x_0, we have for every n\geq 1 and x\in\Omega_{\delta}

                       \|A_n(x_0)^{-1}(A_n(x)-A_n(x_0))\|\leq Ce^{C\|B(x_0)\|^2S(x_0)n|x-x_0|}.

Proof: As explained in Notes 2, A_k(x) is bounded in L^1. In fact, \|A_k(x)\|\leq\|B(x+k\alpha)\|\|B(x)\| for almost every x. Let x_0 be such a point. Note A_n(x_0)^{-1}A_n(x)=A(x_0)^{-1}\cdots A(x_0+(n-1)\alpha)^{-1}A(x+(n-1)\alpha)\cdots A(x). If we let A(x_0+k\alpha)^{-1}A(x+k\alpha)=H_k(x)+id, then by Lipschitz condition H_k(x)\leq C|x-x_0|. Obviously, A_n(x_0)^{-1}A_n(x)=A_{n-1}(x_0)^{-1}H_{n-1}(x)A_{n-1}(x)+A_{n-1}(x_0)^{-1}A_{n-1}(x). Hence by induction we have

                               A_n(x_0)^{-1}A_n(x)=id+\sum^{n-1}_{k=0}A_k(x_0)^{-1}H_k(x)A_k(x).

This obviously implies that

                              \|A_n(x_0)^{-1}A_n(x)-id\|\leq e^{\sum^{n-1}_{k=0}\|A_k(x_0)\|^2\|H_k(x)\|}-1.

Thus we obtain

                              \|A_n(x_0)^{-1}A_n(x)-id\|\leq e^{\sum^{n-1}_{k=0}\|B(x_0+k\alpha)\|^2\|B(x_0)\|^2\|H_k(x)\|}-1.

Hence,

                              \|A_n(x_0)^{-1}A_n(x)-id\|\leq Ce^{S(x_0)n\|B(x_0)\|^2C|x-x_0|},

which completes the proof. \square

Assume further that x_0 is a measurable continuity point of S and B. Here for example, x_0 is a measurable continuity point of S, if it is a Lebesgue density point of S^{-1}(S(x_0)-\epsilon, S(x_0)+\epsilon) for every \epsilon. It’s standard result that this is a full measure set since S is measurable and almost everywhere finite. Same definition can be applied to \|B(x)\|. By definition, it’s easy to see the portion of x that S(x) is close to S(x_0) is getting larger and larger in smaller and smaller neighborhood of x_0. Let x_0 be that S(x_0)<\infty, which is a full measure condition. Then Lemma 1 implies the following estimate

Lemma 2:  Let x_0 be as above. Then for for every d>0, there exist a n_0(d)>0 such that

                             \|A_{(-1)^nq_n}(x)\|\leq \inf\limits_{x'-x_0\in [-\frac{d}{q_n}, \frac{d}{q_n}]}C(x_0)e^{C(x_0)q_n|x-x'|}

 as long as n\geq n_0(d).

Proof: If n is sufficiently large, measurable continuity hyperthesis implies that for every x'\in [x_0-\frac{d}{q_n}, x_0+\frac{d}{q_n}], we can find some x_0' with |x_0'-x'|\leq\frac{1}{q_n} and such that B(x_0'), B(x_0'+\beta_n) are close to B(x_0) and S(x_0'), S(x_0'+\beta_n) are close to S(x_0). WLOG, we can assume n is even. Then Lemma 1 together with our choice of x_0 implies that

                             \|A_{q_n}(x_0')^{-1}(A_{q_n}(x)-A_{q_n}(x_0'))\|\leq Ce^{C\|B(x_0)\|^2S(x_0)n|x-x_0'|}. 

And we can of course assume x_0' such that \|A_{q_n}(x_0')\|\leq\|B(x_0')\|\|B(x_0'+\beta_{n})\|.  Since \|A^{-1}B\|\geq\|B\|\|A\|^{-1} and |x_0'-x'|<\frac{1}{q_n}. Combined these together we get the estimate we want. For n odd, we apply the same discussion to \|A_{-q_n}(x_0')^{-1}(A_{-q_n}(x)^{-1}-A_{-q_n}(x_0')^{-1})\|. \square

If we renormalize around x_0, we know from last post that

\mathcal R^n(\Phi)(1,0)=(1,A_{(-1)^{n-1}q_{n-1}}(x_0+\beta_{n-1}(\cdot)))=(1,A^{(n,0)}) and
\mathcal R^n(\Phi)(0,1)=(\alpha_n, A_{(-1)^{n}q_{n}}(x_0+\beta_{n-1}(\cdot)))=(\alpha_n, A^{(n,1)}).

Note q_{n-1}<q_{n}<\beta_{n-1}^{-1}. Then we have the following obvious corollary from Lemma 2

Corollary 3: Let \Omega_{\delta/\beta_{n-1}}(\mathbb R)=\{z\in\mathbb C: |\Im z|<\delta/\beta_{n-1}\}. Then

              \|A^{(n,i)}(x)\|\leq\inf\limits_{x'\in [-d,d]}Ce^{C|x-x'|}, where i=0,1 and x\in\Omega_{\delta/\beta_{n-1}}(\mathbb R).

Thus by homorphicity, A^{(n,i)} are precompact in C^{\omega}. So we can take limit along some subsequence. Denote the limit by \tilde{A}. Then by estimate in Corollary 3 we  get \|\tilde A(z)\|\leq Ce^{|\Im z|}.

We in fact also have that B(x_0)\tilde A(x) B(x_0)^{-1}\in SO(2,\mathbb R) for x\in \mathbb R. This is given by the following lemma

Lemma 4: Let A, x_0 be as above, then for every d>0 and every \epsilon>0, there exists a n_0(d,\epsilon) such that if n>n_0(d, \epsilon) and \|\alpha n\|_{\mathbb R/\mathbb Z}\leq\frac{d}{n}, then

          B(x_0)A_n(x)B(x_0)^{-1} is \epsilon close to SO(2,\mathbb R) for every x\in[x_0-\frac{d}{n}, x_0+\frac{d}{n}].

Proof: For n sufficiently large, for every x\in[x_0-\frac{d}{n}, x_0+\frac{d}{n}], as in proof of Lemma 2, we can find some x' \frac{\epsilon}{n} which is close to x and S(x'), B(x'), B(x'+n\alpha)) are \epsilon close to S(x_0), B(x_0). Then the same argument of proof of Lemma 2 implies that A_n(x) and A_n(x') are \epsilon close.

Thus we can reduce the proof to the case B(x_0)A_n(x')B(x_0)^{-1} is \epsilon close to SO(2,\mathbb R). But this is clear since we can furthermore choose x' such that B(x'+\alpha n)A_n(x')B(x')^{-1}\in SO(2,\mathbb R), and we’ve already had B(x'), B(x'+n\alpha) are \epsilon close to B(x_0). \square

Thus we can write \tilde A(z)=B(x_0)^{-1}R_{\phi (z)}B(x_0), where \phi:\mathbb C\rightarrow\mathbb C satisfying |\Im{\phi(z)}|\leq C+C|\Im z| is an entire function.  Thus \phi must be linear. We’ve basically proved the following theorem

Theorem 5: If the real analytic cocycle dynamics (\alpha, A) is L^2 conjugate to rotations, then for almost every x_0\in\mathbb R/\mathbb Z there exists B(x_0)\in SL(2,\mathbb R), and a sequence of affine functions with bounded coefficients \phi^{(n,0)}, \phi^{(n,1)}:\mathbb R\rightarrow\mathbb R such that 

                       R_{-\phi^{(n,0)}(x)}BA^{(n,0)}(x)B^{-1}\rightarrow id and R_{-\phi^{(n,1)}(x)}BA^{(n,1)}(x)B^{-1}\rightarrow id.

as n\rightarrow\infty.

To conclude, Theorem 5 implies the following final version of renormalization convergence theorem

Theorem 6: Let (\alpha, A) be as in Theorem 5; let deg be the topological degree of map A. Then there exists a sequence of renomalization representatives (\alpha_n, A^{(n)}) and \theta_n\in\mathbb R, such that

                                    R_{-\theta_n-(-1)^ndeg x}A^{(n)}(x)\rightarrow id in C^{\omega} as n\rightarrow\infty

Proof:  Let B, \phi^{(n,0)}(x)=a_{n,0}x+b_{n,0} and \phi^{(n,1)}(x)=a_{n,1}x+b_{n,1} be as in Theorem 5. Let n be large and let \tilde B(x)=R_{a_{n,0}\frac{x^2-x}{2}+b_{n,0}x}B. Then \tilde A(x)=\tilde B(x+1)A^{(n,0)}(x)\tilde B(x)^{-1} is C^{\omega} close to identity and \tilde B(x+\alpha_n)A^{(n,1)}(x)\tilde B(x)^{-1} is C^{\omega} close to R_{\psi^n(x)}, where \psi^n(x)=(a_{n,0}\alpha_n+a_{n,1})x+\frac{\alpha_{n}^2-\alpha_n}{2}+b_{n,0}+b_{n,1}. By Lemma 1 of Notes 10, we know there exists C\in C^{\omega}(\mathbb R, SL(2,\mathbb R)) which is C^{\omega} close to identity such that C(x+1)\tilde A(x)C(x)^{-1}=id.

Thus B^{(n)}=C\tilde B is a normalizing map for A^{(n,0)} and A^{(n)}(x)=B^{(n)}(x+\alpha_n)A^{(n,1)}(x)B^{(n)}(x)^{-1} is C^{\omega} close to some R_{\psi_n(x)}, where \psi_n is linear. Since the way we get renormalization representatives preserving homotopic relation and the degree of n-th renormalization representative of (\alpha, R_{deg x}) is (-1)^n deg, we get that degree of (\alpha_n, A_n) is (-1)^ndeg. Thus the linear coefficient of \psi_n must be close to (-1)^ndeg and A^{(n)} must be close to R_{\theta_n-(-1)^ndeg x} for some \theta_n\in\mathbb R. \square

I’ve finished the serial posts about renormalization. It’s a powerful technique in the way that we can use it to reduce global problem to local problem and apply perturbation theory like KAM thoerem. More precisely, we can start with L^2 conjugacy to rotations and end up as C^{\omega} close to rotations. If degree is zero and \alpha satisfying some arithematic properties, we can then apply standard KAM theorem to get reducibility.

For these posts, I am following Artur‘s course and he and Krikorian‘s papers. Here I only do the C^{\omega} case while they’ve considered smooth cases in there papers.

Next post will be something about distribution of eigenvalues of the our old friend: One dimensional discrete Schrodinger operator:)

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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 post`Proof 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.

January 22, 2011

Notes of Artur Avila’s Course in Fields Institute, Toronto 1: A Simple Proof of Herman-Avila-Bochi formula

I am visiting my coadvisor Artur Avila in Fields Institute in Toronto from Jan.18 to Apr.01. He is giving a course entitled `Ergodic and Spectral Theory of Quasiperiodic Cocycles ‘ and I am working on some related problems with him. This is the first time he gives such a  graduate level course.

Since his way of math always fasinates me, I plan to take careful notes of his lecture and  latex them. I will post some of them here. I will also post some topics I promised in the last blog. In this post, I am starting with a simple proof of the following nice so-called Herman-Avila-Bochi formula:

\int_{\mathbb R/\mathbb Z}L(f,R_{\theta}A)d\theta=\int_{X}\ln\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}d\mu.

Here is the setting: X is compact metric space with a probability measure \mu and f is a homeomorphism on it preserving \mu; A:X\rightarrow SL(2,\mathbb R) is a continuous cocyle map. Thus (f,A):X\times \mathbb R^2\rightarrow X\times \mathbb R^2 gives a SL(2,\mathbb R)-valued cocyle dynamics over base dynamics ( X,f) , namely, (x,w)\rightarrow (f(x), A(x)w) for (x,w)\in X\times \mathbb R^2. L(f, A) stands for the Lyapunov exponent of this dynamical systems. If I denote (f,A)^n by (f^n, A_n), then it is given by

L(f,A)=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\int_{X} \ln\|A_n(x)\|d\mu.

The limit exists since \{\int_{X} \ln\|A_n(x)\|d\mu\}_{n\geq 1} is a subadditive sequence. R_{\theta} is the rotation matrix with rotation angle 2\pi\theta.  Thus the above formula gives the averaged Lyapunov exponent of a one-parameter family of cocycle dynamics.

I forgot to say a word about the history of this formula: It’s first proved by Michael Herman as an inequality

\int_{R/Z}L(f,R_{\theta}A)d\theta\geq\int_{X}\ln\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}d\mu

in his famous 1983 paper in CMH  ‘Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2’. Artur Avila and Jairo Bochi made it an equality in their 2002 paper `A formula with some applications to the theory of Lyapunov exponents‘. The proof I am going to post here is a simpler version which Artur did in his class.

What lies behind this formula is in fact the mean value formula for harmonic functions.   Let z=e^{2\pi i\theta} and write R_{\theta} as R_z; let \mathcal U\mathcal H stands for uniformly hyperbolic systems,  then the main steps are in the following:

1. Extending z from the unit circle S to unit disk \overline{\mathbb D}. Then for any z \in \mathbb D\setminus\{0\}, (f,R_zA(x))\in\mathcal U\mathcal H.
2. L(f, zR_{z}A)=\ln |z|+L(f, R_zA) is harmonic in z for z\in \mathbb D\setminus\{0\} and bounded and  well-defined for all z\in \overline{\mathbb D}. It obviously agrees with L(f,R_{z_o}A) for z_0\in S. Furthermore, L(f, zR_{z}A) converges to L(f, z_0R_{z_0}A) nontangentially.
3. L(f,0R_0A)=\int_{X}\ln\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}d\mu.
4. Mean value formula implies the formula in question.

There are quite a few notions, observations and standard results need to be explained. Let’s do them one by one. First is the defintion of uniformly hyperbolic cocycle dynamcal systems, let’s consider the more general SL(2,\mathbb C)-valued cocyle case, i.e. A:X\rightarrow SL(2,\mathbb C) which is continuous. Let \mathbb C\mathbb P^1 be the Riemann surface \mathbb C\cup\{\infty\} and A acts on it as Mobius transforamtion, i.e. for A=\begin{pmatrix}a& b\\c&d\end{pmatrix}, A\cdot z=\frac{az+b}{cz+d}.

Defintion 1.  (f, A) is said to be uniformly hyperbolic if the are two contiuous map u, s:X\rightarrow \mathbb C\mathbb P^1 such that
1. they are invariant in the sense that A(x)\cdot u(x)=u(f(x)) and A(x)\cdot s(x)=s(f(x)).
2. there exist constant C>0, 0<\lambda<1 such that for each vector w_s\in s(x), \|A_n(x)w_s\|\leq C\lambda^n\|w_s\| for n\geq 1,  and for each w_u\in s(x), \|A_n(x)w_u\|\leq C\lambda^{-n}\|w_u\| for n\leq 0.

One well-known equivalent condition for \mathcal U\mathcal H systems is the existence of invariant cone field. More concretely, (f, A)\in \mathcal U\mathcal H if and only if for each x\in X, there is open disk U(x)\subset \mathbb C\mathbb P^1 such that there exists positive integer N

\overline{A_n(x)\cdot U(x)}\subset U(f^n(x)), \forall x\in X, \forall n\geq N.

The existence of U(x)  implies the existence of a corresponding S(x). Using this it easy to obtain the following lemma

Lemma 2. Consider a holomorphic 1-parameter family of \mathcal U\mathcal H systems (f, A_{\lambda}), \lambda is in some open region in complex plane. Holomorphicity in the sense that for each fixed x, A_{\lambda}(x) is  holomorphic in \lambda. Then u(x,\lambda) and s(x,\lambda) are  both holomorphic in \lambda.
Proof:  It’s easy to see by the equivalent condition above,
u(x,\lambda)=\lim\limits_{n\rightarrow\infty}(A_{\lambda})_n(f^{-n}(x))\cdot u_{-n},
for any u_{-n}\in U(f^{-n}(x)) and the convergence is uniform in x and \lambda. Thus the result follows . \square

By this lemma, we can show the following theorem,

Theorem 3. L(f,A) is pluriharmonic in \mathcal U\mathcal H (Pluriharmonicity means that for (f, A_{\lambda}) in the above Lemma , L(f,A_{\lambda}) is harmonic in \lambda).
Proof: Define a map B_{\lambda}:X\rightarrow SL(2,\mathbb C) with column vectors in u_{\lambda}(x) and s_{\lambda}(x). Then B_{\lambda}(f(x))A_{\lambda}(x)B_{\lambda}(x)^{-1}=diag(\gamma_{\lambda}(x), \gamma_{\lambda}(x)^{-1}),
where all maps are holomorphical in \lambda. Then by Birkhoff Ergodic Theorem, we have
L(f, A_{\lambda})=\int_X\ln|\gamma_{\lambda}(x)|d\mu.
Hence, the result follows easily. \square

Now what left in the main steps are 1 and 3. Step1 is based on a key observation such that \overline{R_{i\theta}\cdot\mathbb H_-}\subset\mathbb H_-, where \theta>0 and \mathbb H_- is the low-half plane in \mathbb C\mathbb P^1. An easy way to see the above fact is to Mobius transform \mathbb H to \mathbb D via Q=\frac{-1}{1+i}\begin{pmatrix}1&-i\\1&i\end{pmatrix}\in \mathbb U(2). Then the action of R_{\theta} on \mathbb H is conjugated to the action of \hat{R_{\theta}}=QR_{\theta}Q^*=diag(e^{-2\pi i\theta}, e^{2\pi i\theta}) on \mathbb D. Then it’s easy to see that \hat{R_{i\theta }} contracts \overline{\mathbb D}^c into \overline{\mathbb D}^c. Hence, the above fact follows. Now an easy appliction of the equivalent condition for \mathcal U\mathcal H systems implies the results in step1.

The proof of step3 is a straightforward computation. Indeed, it’s easy to see P=0R_0=\frac{1}{2}\begin{pmatrix}1&-i\\i&1\end{pmatrix} and P\cdot(\mathbb H^c)=-i, thus PA\cdot (-i)=-i for all A\in SL(2, \mathbb R), which means that -i is a constant invariant section of the projective dynamics(f, PA) on X\times CP^1. If PA(x)\binom{-i}{1}=\gamma(x)\binom{-i}{1}, then it’s easy to see L(f,PA)=\int_X\ln|\gamma(x)|d\mu and |\gamma(x)|=\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}.
Hence, step3 follows.

Finally, mean value formula for harmonic functions implies

L(f,0R_0A)=\int_{R/Z}L(f, zR_zA)d\theta=\int_{R/Z}L(f, R_{\theta}A)d\theta,

which establishes the formula we want to show.

I’ve already given a self-contained and complete proof of Herman-Avila-Bochi formula in this post. I like this proof very much. Because it’s very conceptul and few computation is involved. One can also see how does mean value formula  for harmonic functions gets into dynamical systems.

February 25, 2010

Why golden mean is optimal: an introduction to Diophantine, Brjuno and Liouville number

Irrational rotations on unit circle S^1=R/Z is one of most important examples in dynamical systems, which I am going to denote it by: \alpha: S^1\rightarrow S^1, x\mapsto x+\alpha, where \alpha\in [0, 1]\setminus Q.

Then we have the following facts:  

Theorem 1: Dynamical systems defined above are minimal and uniquely ergodic.
(Th0se who are not familiar with these concepts can google them or take a look at Peter Walters’ “An introduction to ergodic theory”.)

The subtlety of these dynamical systems lie in the way the orbit Orb(x) of a point x recurrent to itself, in particular, we can fix x=0. It turns out the recurrence of 0 is closely related to the arithmetic property of the rotation angle \alpha, which lead to interesting classification of irrational numbers. WLOG, I am going to restrict myself to I=[0,1].

Definition 2: \alpha satisfies a Diophantine condition DC(\gamma, \tau), \gamma>0, \tau>0, if |q\alpha-p|>\gamma |q|^{-\tau}, (p, q)\in Z^2, q\neq 0.

Let DC=\cup_{\gamma>0, \tau>0}DC(\gamma, \tau) be the so-called Diophantine number, then it is well known that DC_{\tau}=\cup_{\gamma>0}DC(\gamma, \tau) has full Lebesgue measure if \tau>1. In some sense, Diophantine condition means that \alpha cannot be approximated by rational numbers too fast, which in dynamics language means 0 cannot be recurrent to itself too fast under rotation \alpha

There is actually another equivalent way to define Diophantine number in terms of continued fraction expansion of \alpha.
Let \alpha=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots}}}} be the continued fraction expansion of \alpha, and \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, then we have the following properties:

(1)\frac{1}{q_n}>|q_{n-1}\alpha-p_{n-1}|>\frac{1}{q_n+q_{n-1}},
(2)\parallel k\alpha\parallel_{R/Z}>|q_{n-1}\alpha-p_{n-1}|, \forall q_{n-1}+1\leq k\leq q_n-1.
where \parallel\cdot\parallel_{R/Z} denote the distance to the nearest integer.

Thus, to define Diophantine condition, we need only care about the continued fraction approximants, where then Diophantine conditions are actually given by the growth rate of sequence (q_n)_{n\in Z^+}. Use above inequalities and definition 2, it’s not difficult to see that:

\alpha\in DC_{\tau} iff q_{n+1}<cq_{n}^{\tau}, where c is some constant.

But what about the growth rate of (q_n)_{n\in Z^+} in general? In fact, we have the following nice induction formula for (q_n)_{n\in Z}:

q_n=a_nq_{n-1}+q_{n-2}.

Thus, the slowest growth is when a_n=1 for all n, where the corresponding \alpha is the golden mean and (q_n)_{n\in Z} is the Fibonacci sequence if we set q_0=1, q_{-1}=0! So in some sense, golden mean is the most irrational irrational number, because it is approximated by rational numbers in the slowest rate.

For Fibonacci sequence b_n, we can find the exact expression by generating function, which is 

b_n=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{n+1}-(\frac{1-\sqrt{5}}{2})^{n+1}).

Hence, sequence (q_n)_{n\in Z^+} grows at least exponentially fast.

Definition 3: \alpha is called a Brjuno number if the associated sequence (q_n)_{n\in Z^+} satisfies \sum_n \frac{log q_{n+1}}{q_n}<\infty.

By the equivalent condition of Diophantine number and the slowest growth of (q_n)_{n\in Z^+}, Brjuno number contains all Diophantine number so it’s a full Lebesgue measure set. We can define another number which decribe the growth rate of (q_n)_{n\in Z^+} outside a set contains Brjuno numbers.

\beta(\alpha)=\limsup\limits_{n\rightarrow\infty}\frac{log q_{n+1}}{q_n}.

Then obviously we have:

DC\subset Brjuno Numbers \subset\{\alpha: \beta(\alpha)=0\}.

Again \{\alpha: \beta(\alpha)=0\} is a full Lebesgue measure set. Thus,  it seems that \{\alpha: \beta(\alpha)>0\} is a rather small set which may even be empty. But that’s not true:

Definition 4: \alpha\in{\alpha: \beta(\alpha)=\infty} are said to be Liouville number.
Remark: Some also define Liouville numbers to be the complement of Diophantine numbers in irrational numbers.

Obviously, the set of all Liouville numbers has zero Lebesgue measure and they are approximated by rational numbers really very fast. It is an interesting fact that the set of Liouville numbers is generic though it has zero Lebesgue measure. These arithmetic properties in many ways govern dynamical behaviors in many dynamical systems.

February 21, 2010

Dynamics of quadratic family: from attracting fixed point to absolutely continuous invariant probability (ACIP)

This is one of my previous notes, I post it  for testing:)

In study of Smooth Ergodic Theory, we always follow the way that geometrical properties of derivative implies statistical properties of dynamical systems.

Typical statistical behavior of dynamical systems is hyperbolicity, which includes uniform hyperbolicity, partial hyperbolicity and nonuniform hyperbolicity. Famous examples of uniformly hyperbolic systems are linear automorphisms on 2-torus and geodesic flows, while partially hyperbolic systems are given by time-1 map of Anosov flows and Frame flows. Both uniform hyperbolicity and partial hyperbolicity are open conditions.

Then how about nonuniform hyperbolicity? First of all, it is not an open condition. And there is few good example. Furthermore, we can ask ‘are typical systems nonuniformly hyperbolic?’. Here I am going to introduce an famous example related to nonuniformly hyperbolic theory, which gave rise to nice generalization.

First, let us introduce the so-called Henon map, which is given by f_{a,b}(x,y)=(x^2-a+y, bx). It was given by Henon in 1976 and he found some strange attractor when a=1.4, b=0.3. Then followed by Benedicks, Carleson 1991 and Benedicks, Young 1993’s work, we have the following theorem:

Theorem: Let f_{a,b}(x,y)=(x^{2}-a+y,bx), then \forall b\neq 0 sufficiently small, there exist\Omega^{*}(b)satisfies

(a)\forall a\in\Omega^*(b), f_{a,b} is nonuniformly hyperbolic.
(b)m(\Omega^*(b))>0, and there is no interval in \Omega^{*}(b)

Concretely, here nonuniform hyperbolicity means for every pair of a, b above, there exists an attractor \Lambda_{a,b} with ergodic SRB measure with nonzero Lypapunov exponents and \Lambda_{a,b} contains tagencies.

Henon map is the two dimensional case, its study was originated from the one dimensional case, namely, b=0 and f_a(x)=x^2-a the quadratic family, where we have the following theorem, which is known as Jacobson theorem

Theorem(Jacobson, 1981):
There exists \Omega^*, s.t. i) \forall a\in\Omega^*, f_a is nonuniformly hyperbolic
(which means it has ergodic SRB measure which is absolutely continuous w.r.t. Lebesgue measure, and positive Lyapunov exponent); ii) m(\Omega^*)>0 and contains no intervals.

Now we are going to focus our study on the quadratic family f_a(x)=x^2-a. First, we will study the change of dynamical behavior of f_a when parameter a varies.

Let f_{a}=x^{2}-a, \varepsilon>0 is a sufficiently small number, p_{1}(a)\leq p_{0}(a) are two roots of equation x^{2}-a=x and I_{a}=[-p_{0}(a), p_{0}(a)], then then we have the following description:

Let a_{0} be  the number that x^{2}-a_{0}=x has a unique solution, then

If a<a_{0}, then \forall x\in R, f_{a}^{n}(x)\rightarrow\infty when n\rightarrow\infty

If a=a_{0} then:
f_{a}^{n}(x)\rightarrow p_{1}(a), \forall x\in I_{a}
f_{a}^{n}(x)\rightarrow\infty, \forall x\notin I_{a}

If a=a_{0}+\varepsilon then:
f_{a}^{n}(x)\rightarrow p_{1}(a), \forall x\in I_{a}
f_{a}^{n}(x)\rightarrow\infty, \forall x\notin I_{a}

Let a_{1} be the number that f'_{a_{1}}(p_{1}(a_{1}))=-1then:

If a=a_{1}-\varepsilon then:
f_{a}^{n}(x)\rightarrow p_{1}(a), \forall x\in I_{a}
f_{a}^{n}(x)\rightarrow\infty, \forall x\notin I_{a}

If a=a_{1}+\varepsilon, then f'_{a_{1}}(p_{1}(a))<-1 so there is no fix point. But the graph of y=f^{2}_{a}(x) and y=x will intersect  in four points and one of them is an attracting fix point. So f_{a} will have an attracting 2-periodic orbit such that \forall x\in I, f^{n}_{a}(x) will approach this orbit.

Then following the  idea of renormalization, we can take part of the graph of f^{2}_{a} and analyze it just as those of f_{a} for a>a_{1}. Then we will find that the attracting 2-periodic orbit will preserve until a reaches an a_{2}. For a>a_{2}, we consider f^{4}_{a}, then we will find a fix point which is an attracting 4-periodic orbit of f_{a}.

Keep going this way, we will find a sequence \{a_{i}\}_{i\geq0} such that for every a_{i}, there will be a doubling of period of attracting orbit. In the end, \{a_{i}\}_{i\geq0} will converge to a number a_{\infty}<2.

a=a_{\infty} is the so-called infinitely renormalizable case,   it was dicovered by Feigenbanm in 1970s. In this case, f_{a} has an invariant interval I, for almost all x\in I, f^{n}_{a}(x)\rightarrow C, where C is an invariant Cantor set and the dynamical behavior on C just likes the irrational rotation of unit circle.

For a=2,Ulam-Von Neumann found the behavior of f_{a}(x)=x^{2}-a on [-2, 2] is conjugate with the tent map T on [0,1] via map h(z)=2\cos(\pi z), where T is given by

T(x)=-2x+1 for 0\leq x\leq \frac{1}{2}
T(x)=2x-1 for \frac{1}{2}< x\leq1.

That is, f_{2}=h\circ T\circ h^{-1}. But we know that the Lebesgue measure m of [0, 1] is the ergodic and invariant measure of tent map T. So the smooth map h will push m forward to be a ergodic acip (absolutely continuous invariant probability) \mu of f_{2} on [-2, 2]. But for one dimensional map with single critical point, existence of acip implies positive Lyapunov exponent. So f_{2} is a nonuniformly hyperbolic system on [-2, 2] and \mu is its SRB measure (in one dimensional case SRB coinsides with acip).

When a>2, there will be no invariant interval of f_{a}, but there does exist an invariant Cantor set on I_{a} such that the dynamical behavior on it is topologically conjugate with the left shift map on \Sigma_{2}=\{0, 1\}^{N}.

Now we are going to study the parameter interval (a_{\infty}, 2), which has rich phenomena and deep results. For example, there are lots of parameters with attracting periodic orbits which give rise to many cascades of doubling bifurcation which just like the process from a_0 to a_{\infty}. Actually, Simo and Tatjer numerically found about 30 million “periodic windows” with period \leq 24, and the total length of them is about 10\% of (a_{\infty}, 2).

If we let \Omega= (a_{\infty}, 2), \Omega^-= \{a: f_a admits attracting periodic orbits\} and \Omega^+=\{a: f_a admits acip\}, then we have the following serial deep results:

Theorem(Jacobson, 1981):
m(\Omega^+)>0.

Theorem(Luzzatto, Takahashi, 2006):
m(\Omega^+)>10^{-5000}.

Theorem(Craczjk, Smotek, Lyubich, 1997):
\Omega^- is open and dense in \Omega.

Theorem(Lyubich, 2002):
\Omega^+\bigcup\Omega^- has full measure of \Omega.

Remark:
1) (a_{\infty}, 2)\setminus (\Omega^{+}\bigcup\Omega^{-}) is not empty, for example, there are infinitely renormalizable systems and systems whose physical measure support on repelling fixed point.

2) Generalization: first,  m(\Omega^+)>0 for the following one dimensional maps: Multimodel map(Tsujii, 1993), Contracting Lorenz map(Rovella, 1993), Lorenz like map with critical point(Luzzatto and Tacker 2000) and infinite critical points map(Pacifer, Parclla and Viena, 2000); second, \Omega^- is open and dense in space ofC^rmaps(r\geq1)(Kozlosky, Shen and Von Stier, 2005).

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