Zhenghe's Blog

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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