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}<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.

Advertisements

2 Comments »

  1. So many connections between these things.
    The set of Liouville numbers has zero Lebesgue measure. Could its Hausdorff dimension be positive?

    Comment by Pengfei — March 2, 2010 @ 1:53 pm | Reply

  2. The Hausdorff dimension is zero.

    Comment by zhenghe — March 2, 2010 @ 2:38 pm | Reply


RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Blog at WordPress.com.

%d bloggers like this: