# Zhenghe's Blog

## January 26, 2011

### Notes 2: Kotani theory (I)-Zero Lyapunov Exponents and L^2 rotation conjugacy

This and next posts will be about Kotani theory, which is part of my syllabus. Kotani theory is first found by Shinichi Kotani when he studies spectrum theory of $Schr\ddot odinger$ operator with ergodic potentials. It gives a complete decription of  the absolutely continuous part of spectrum of  $Schr\ddot odinger$ operators in terms of Lyapunov exponents of the  corresponding of $Schr\ddot odinger$ cocycles. Thus it builds a deep and beautiful relation between operator theory and dynamical systems.

Later Artur Avila and Raphael Krikorian generalize Kotani theory to more general coycle dynamics setting and it becomes a powerful tool to study Schrodinger operators. What I am going to introduce here are main results and some preliminary stuff. Let’s start with one dimensional (1D) discrete $Schr\ddot odinger$ operators and $Schr\ddot odinger$ cocyles. From now on I set $L(E)=L(f,A^{(E-v)})$.

For simplicity, I will continue to use the triple $(X,f,\mu)$ as in last post. Assume $v\in C(X,R)$ be  a contiuous function. Then we can define a cocycle map via

$A^{(v)}(x)=\begin{pmatrix}v(x)&-1\\1&0\end{pmatrix}.$

The correspoding cocycle dynamics $(f, A^{(v)})$ is called $Schr\ddot odinger$ cocyles. It arises from the following $Schr\ddot odinger$ operator $H_{f,v,x}$ on $l^2(\mathbb Z)$

$(H_{f,v,x}u)_n=u_{n+1}+u_{n-1}+v(f^n(x))u_n,$

where $u=(u_n)_{n\in\mathbb Z}\in l^2(\mathbb Z).$ Then $u\in\mathbb C^{\mathbb Z}$ solving the eigenfunction equation $H_{f,v,x}u=Eu, E\in\mathbb R$ if and only if

$A^{(E-v)}(f^n(x))\binom{u_n}{u_{n-1}}=\binom{u_{n+1}}{u_n}.$

Thus the growth rate of $|u_n|$ with respect to $n$ characterizes both the spectrum type of the energy $E$ and the dynamics of cocycle $(f,A^{(E-v)})$, which allows one to go back and forth between spectrum theory and dynamical systems. Let $\Sigma_x$ be the spectrum of the bounded linear selfadjoint operator $H_{f,v,x}$, then the first basic fact relate operator and cocycle is for a.e. $x,$

$\Sigma_x=\{E: (f, A^{(E-v)})\notin \mathcal U\mathcal H\}.$

If furthermore $f$ is minimal, then the above relation is in fact true for all $x.$ Thus  for a.e. $x,$ the spectrum $\Sigma_x$ is independent of $x\in X,$ let’s denote it as $\Sigma$. For each $x,$ we can further decompose $\Sigma_x$ as $\Sigma_x=\Sigma_{x,pp}\cup\Sigma_{x,ac}\cup\Sigma_{x,sc}$, which correspond to pure point, absolutely continuous and singular continuous part of the spectrum of the operator $H_{f,v,x}$. These are defined by the spectral measure of the operator, which one can find in any standard functional analysis book.  It turns out that in our case, there exists sets $\Sigma_{\bullet}, \bullet\in\{pp, ac, sc\}$ such that $\Sigma_{x,\bullet}=\Sigma_{\bullet}$ for a.e. $x$ and $\bullet\in\{pp, sc,ac\}.$ (If in addition $f$ is minimal, then in fact $\Sigma_{x,ac}=\Sigma_{ac},$ for all $x,$ which is in general not true for pp and sc.)

Now we denote $\mathcal Z=\{E: L(E)=0\}.$ And for any set $\mathcal S\subset\mathbb R,$ the essential support of the set $\mathcal S$ is given by

$\overline{\mathcal S}^{ess}=\{E\in\mathbb R: Leb(\mathcal S\cap (E-\epsilon, E+\epsilon))>0$ for every $\epsilon>0\}.$

Then the next deep relation between dynamics and spectrum is the following

$\Sigma_{ac}=\overline{\mathcal Z}^{ess}.$

The relation $\Sigma_{ac}\subset\overline{\mathcal Z}^{ess}$ is relatively easy since positive Lyapunov exponents give exponential growth of the solution $u$ to the eigenfunction equation which in some sense contradicts with the absolute continuity of spectrum. The part $\overline{\mathcal Z}^{ess}\subset\Sigma_{ac}$ is a rather deep result called Kotani theory. It in fact  is a theory about that, under the assumption that Lyapunov exponents are zero, when can one in some sense conjugate the $SL(2,\mathbb R)-$valued cocycles to $SO(2,\mathbb R)-$valued cocycles. Obviously, if the cocycle map $A$ takes value in $SO(2,\mathbb R)$, then all orbits $\{A_n(x)w\}_n$ are bounded. While zero Lyapunv exponent in general just means that $\|A_n(x)w\|_n$ grows subexponentially.  What Kotani theory tells us is in fact that the following theorem

Theorem 1: For almost every $E$, if $L(E)=0,$ then there exists a map $B:X\rightarrow SL(2,\mathbb R)$ such that$B(f(x))A^{(E-v)}(x)B(x)^{-1}\in SO(2,\mathbb R)$ and $\int_X\|B(x)\|^2d\mu<\infty.$

Thus the generalized eigenfunctions for absolutely continous spectrum of the operators in question oscillate in some $L^2$ sense. Which are kind of wave like solutions and far from eigenfunctions of real eigenvalues, which decay in $l^2$ sense. As I said this is obviously stronger then zero Lyapunov exponents. Indeed, in this case we have

$\frac{1}{n}\int_X\ln\|A_n(x)\|d\mu\leq\frac{1}{n}\int_X\|A_n(x)\|d\mu\leq\frac{1}{n}\int_X\|B(f^n(x))^{-1}\|\|B(x)\|d\mu$
$\leq\frac{1}{n}\int_X\|B(x)\|^2d\mu\rightarrow 0,$ as $n\rightarrow\infty$

Now let me explain how can one construct the above $B:X\rightarrow SL(2,\mathbb R).$  We first prove the following key lemma

Lemma 2: Assume $E\in\mathbb R$ satisfying $L(E+i\epsilon)=O(\epsilon),$ for $\epsilon>0$ small. Then $(f,A^{(E-v)})$ is $L^2-$conjugate to $SO(2,\mathbb R)-$valued cocycles.
Proof: First we note by the same reason that $(f,R_{i\theta}A)\in\mathcal U\mathcal H$ as in last blog, we have $(f,A^{(E+i\epsilon-v)})\in\mathcal U\mathcal H$. Thus there is invariant section $u^{\epsilon}:X\rightarrow \mathcal H$ which is the unstable direction. There are different ways to calculate Lyapunov exponents of cocylce dynamics via invaiant section of corresponding projective dynamics. We use the following: $-\frac{1}{2}$ of the contraction rate measured in $Poincar\acute e$ metric of mobius transformation at $u^{\epsilon}$. In our case we need to consider the following composition of map. Let $\mathbb H_{\epsilon}=\{E:\Im E>\epsilon\}$ with standard $Poincar\acute e$ metric, then the composition is

$\mathbb H\overset{A^{(E+i\epsilon-v)}}{\longrightarrow}\mathbb H_{\epsilon}\overset{i_{\epsilon}}{\longrightarrow}\mathbb H,$

where the first map is a isometry and the second one (which is the inclusion map) is a contraction. Thus we consider the contraction of the second map at invariant section. Then the Lyapunov exponents is given by

$L(E+i\epsilon)=-\frac{1}{2}\int_X\ln(1-\frac{\epsilon}{\Im u^{\epsilon}})d\mu.$

For simplicity, let me first assume that:

$\lim\limits_{\epsilon\rightarrow 0}u^{\epsilon}(x)$ exists for $a.e. x$ and we denote it by $u(x)$.
(then obviously, $u(x)$ is invariant for a.e. $x$, i.e. $A^{(E-v)}(x)\cdot u(x)=u(f(x))$ for a.e.$x$.)

Assuming this, we have the following straightforward estimate via Fatou’s lemma and our assumption in Lemma :

$\frac{1}{2}\int_X\frac{1}{\Im u}d\mu\leq\liminf\limits_{\epsilon\rightarrow 0}-\frac{1}{2\epsilon}\int_X\ln(1-\frac{\epsilon}{\Im u^{\epsilon}})d\mu=O(1)<\infty.$

Since $u(f(x))=E-v(x)-\frac{1}{u(x)}$, we have

$\frac{1}{2}\int_X\frac{1}{\Im u}d\mu=\frac{1}{2}\int_X\frac{1}{\Im u(f(x)}d\mu=\frac{1}{2}\int_X\frac{|u|^2}{\Im u}d\mu<\infty$.

Hence

$\int_X\frac{1+|u|^2}{\Im u}d\mu<\infty.$

Let $\phi(u)=\frac{1+|u|^2}{\Im u}$. Now we define $B:X\rightarrow SL(2,\mathbb R)$ such that $B(x)\cdot u=i,$ thus $B(f(x))A(x)B(x)^{-1}\cdot i=i$ which implies $B(f(x))A(x)B(x)^{-1}\in SO(2,\mathbb R)$. On the other hand, it’s easy to see for quite general reason $\|B(x)\|_{HS}^2=\phi(u).$ Thus $\int_X\|B(x)\|_{HS}^2d\mu<\infty.$
Here for $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL(2,\mathbb R),$ $\|A\|_{HS}^2$ is the Hilbert-Schmit norm of $A$ which is just $a^2+b^2+c^2+d^2.$ It’s easy to see that if $A\cdot z=i,$ then $\|A\|_{HS}^2=\phi(z)$. Since all norms of $SL(2,R)$ are equivalent, we complete the proof of the Lemma up to the assumption.    $\square$

To get around the assumption, we need conformal barycenter, which is a Borelian function $\mathcal B:\mathcal M\rightarrow\mathbb H,$ where $\mathcal M$ is the space of probability measures on $\mathbb H.$ This function is equivariant with respect to $SL(2,\mathbb R)$ change of coordinates. i. e. for

$A\in SL(2,\mathbb R), \nu\in\mathcal M,$ we have $\mathcal B(A_*\nu)=A\cdot\mathcal B(\nu).$

Now let’s consider the probability measures $\nu_{\epsilon}=\mu\otimes\delta_{u^{\epsilon}}$ on $X\times\overline{\mathbb H}$. Then there is a subsequence converging to some $\nu=\mu\otimes\nu_x$ such that $\int_{X\times\overline{\mathbb H}}\phi d\nu<\infty$. Then apply conformal barycenter to $\nu_x$, we get a point $u(x)=\mathcal B(\nu_x)\in\mathbb H$ such that $u$ is an invariant section and

$\int_X\phi(u(x))d\mu=\int_{X\times\overline{\mathbb H}}\phi d\delta_{u(x)}d\mu\leq\int_{X\times\overline{\mathbb H}}\phi d\nu<\infty.$

Now we can construct the map $B$ as  in the proof of lemma.

Although we introduce conformal barycenter to complete the proof of Lemma, to prove the Theorem we actually only need to apply Fubini and Fatou theorem, to pass the existence of limits from For every $x$, converges for  a.e. $E$  to  For a.e. $E$, converges for a.e. $x$. Then we get the conclusion in assumption for a.e. E.

In the next post, I will prove how can we get the condition in Lemma under the condtion in Theorem, which will complete the proof of theorem.

## 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.

## April 21, 2010

### Syllabus of my qualifying exam: Topics in 1D discrete quasiperiodic Schrodinger cocycles

Filed under: Schrodinger Cocycles — Zhenghe @ 9:35 pm
Tags: , ,

Here is the my qualifying exam syllabus. I like these topics, because they  lie in the intersection of dynamical systems and operator theory,  both of which are my favorite areas.

Since operators are quasiperiodic, the corresponding cocycle dynamics are over irrational rotation on unit circle, consequently, the spectrum of Schrodinger operators depend in a very subtle way on the arithematic property of irrational rotation numbers, which I introduced in the last blog.

The syllabus is based on some papers I’ve read, some of which I know in detail and some I just went through roughly. I am going to introduce some of the them in future blogs.

1. Spectral measure of self-adjoint operators.
1.1. Continuous functional calculus of normal operators.
1.2. Spectral measure of self-adjoint operators.

2. Irrational rotation on unit circle.
2.1. Strict ergodicity.
2.2  Continued fraction expansion
2.2. Diophantine, Brjuno and Liouville numbers.

3.Dynamics of quasiperiodic SL(2,R)-cocyles.
3.1. Lyapunov exponents and Oseledec’s theorem.
3.2. Hyperbolicity and reducibility of SL(2,R)-cocycles.
3.3. Some global results.
3.3.1. Density of uniformly hyperbolic systems in $C^0$-topology.
3.3.2. Genericity of zero Lyapunov exponents in $C^0$-topology.
3.3.3. Continuity of Lyapunov exponents in analytic topology.
3.3.4. Ubiquity of nonuniformly hyperbolic systems.

4. 1D discrete quasiperiodic Schrodinger operators and Schrodinger cocycles.
4.1. Correspondence between uniform hyperbolicity and resolvent set.
4.2. Cantor spectrum and density of uniform hyperbolicity
4.3. Kotani theory: reducibility and absolutely continuous spectrum.
4.4. Anderson localization and nonuniform hyperbolicity.
4.5. Almost Mathieu Operator.

## 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}, 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, 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 $f'_{a_{1}}(p_{1}(a_{1}))=-1$then:

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 of$C^r$maps$(r\geq1)$(Kozlosky, Shen and Von Stier, 2005).

## February 17, 2010

### This should be my math blog:)

Filed under: Uncategorized — Zhenghe @ 1:28 am

Hahaha

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