Rotation number – github-backend

旧博客原文

原题：Rotation number

Consider compact 1 dimension dynamic system.

We focus on $S_1$ , it does not mean $S_1$ is the only compact 1 dimensional system , but it is a typical example.

$T: S_1\to S_1$ .

If $T$ is a homomorphism then $T$ stay the order of $S_1$ (by continuous and the zero point theorem). That is just mean:

(may be do a reflexion $e^{2\pi i\theta}\to e^{-2\pi i\theta}$ ).

In the homomorphism case. We try to define the rotation number to describe the expending rate of the dynamic system.

$T: \mathbb S_1\to \mathbb S_1$ i.e. $T:\mathbb R/\mathbb Z\to \mathbb R/\mathbb Z$ . Lifting to,

$\hat T:\mathbb R\to \mathbb R$ .

How to realize the lifting?

Step1: Periodic extend $T:\mathbb S_1\to \mathbb S_1$ to $T': \mathbb R\to \mathbb S_1$ . (Regard $\mathbb S_1$ as $[0,2\pi]$ ).

Step2: Consider the “flow” of $T':\mathbb R\to \mathbb S_1$ . we get $\hat T:\mathbb R\to \mathbb R$ .

The rotation number is defined as:

$\rho (T)=\limsup_{n\to \infty}\frac{\hat T^n(x)}{n}$ .

Following we will shall it is independent of the choice of $x$ and $\rho (T)=\lim_{n\to \infty}\frac{\hat T^n(x)}{n}$ in fact.

It is not difficult to proved the following property:

Property:

1.If $T'$ is conjugate (in fact semi-conjugate is enough ). Then 1.If $T'$ is conjugate (in fact semi-conjugate is enough ). Thenrotation number of $T'$ equal to rotation number of $T$ .

2.If $T'$ is conjugate to $T$ . Then rotation number of $T'$ equal to
rotation number of $T$ .

$T'(y)=\Psi\circ T\circ \Psi^{-1}(y)=\Psi(\Psi^{-1}(y)+t(\Psi^{-1}(y)))$ )

Example: $T: x\to x+\alpha$ , $\alpha\in \mathbb R$ . It is not difficult to prove the rotation number of $T$ is $\alpha$ .

Propersion:

1)For $n\geq 1$ we have that $\rho(T^n)=\rho(T) (mod 1)$ .

2).If $T$ has a periodic point, i.e. $x\in S_1,\exists n\in \mathbb N^*, T^{n}(x)=x$ . Then $latex\rho (T)$ is rational.

3) $T:\mathbb R/\mathbb Z \to \mathbb R/\mathbb Z$ has no periodic point then $\rho(T)$ is irrational.

4) The limit actually exists and we have: $\rho(T)=\lim_{n\to \infty}\frac{\hat T^n(x)}{n} (mod 1)$ .

pf of 1):

$\rho(T^n)=\lim_{k\to \infty}\frac{(\hat T^n)^{k}(x)}{k}$

$=\lim_{k\to \infty}n\frac{(\hat T)^{nk}(x)}{nk}$

$=n\rho (T)$ .

Used the property $|x-y|<k \leftrightarrow |T^{\omega}(x)-T^{\omega}(y)|<k+1, \forall \omega\in N^*, \forall k\in \mathbb Z^{+}$ .

pf of 2):

It is not difficult to prove $\rho(T)$ is independent with the choice of $x$ . So choose $x$ to be the periodic point.

Remark: but the inverse of 2) is not true. For example:

$x\to x+\frac{1}{2}+\frac{1}{100}sin(4\pi x)$ .

This dynamic system has both periodic points( $\{0,\frac{1}{2}\},\{\frac{1}{4},\frac{3}{4}\}$ ) and non-periodic pint (maybe orbits generated by $\{\frac{1}{\sqrt{2}}\})$ .

pf of 3):

If not. Assume $\rho(T)$ is rational number $\frac{q}{p}$ . Take any point $x\in \mathbb S_1$ , then:

$\lim_{n\to \infty}\frac{\hat T^n(x)}{n}=\frac{q}{p}$ .

$\Longrightarrow \lim_{n\to \infty}\frac{(\hat T^p)(x)}{n}=q$ .

$\Longrightarrow \lim_{n\to \infty}\frac{(\hat T^p-q)^n(x)}{n}=0$ .

Now assume $\hat T^p-q=\widetilde T$ .

Then $\widetilde x>x$ . $\forall x\in \mathbb S_1$ (if $\widetilde x<x, \forall x\in \mathbb S_1$ , take reflection $x\to -x$ ).

And there do not exists $n\in \mathbb N^*$ such that $\widetilde T^nx>x+1$ . If not, we could prove rotation number is large than $\frac{1}{n}$ lead a contradiction.

So $\{\widetilde T^nx\}_{n=1}^{\infty}$ is a bounded monotonically increasing sequences in $\mathbb S_1$ , it limits point $z\in \mathbb S_1$ must satisfied $\widetilde T^n (z)=z$ .

pf of 4):

Using the point wise approximation inequality induced from the monotonically and stay ordering property of $\mathbb S_1$ by $T$ .

Corollary:

Assume $\rho(T)$ is irrational.

1. Let $n_1,n_2,m_1,m_2\in \mathbb Z$ , and $x,y\in \mathbb R$ . If $\hat T^{n_1}(x)+m_1<\hat T^{n_2}(x)+m_2$ , then $hat T^{n_1}(y)+m_1<\hat T^{n_2}(y)+m_2$ .

2. The bijection $n\rho (T)+m\to \hat T^n(0)+m$ between the set $\Omega=\{n\rho(T)+m| n,m\in \mathbb Z\}$ and $\Gamma=\{\hat T^{n}(0)+m,n,m\in \mathbb Z\}$ precise the natural ordering on $\mathbb R$ .

This corollary is not difficult to prove use the established property.

Denjoy’s theorem

Proposition:

If $T: \mathbb R/\mathbb Z\to \mathbb R/\mathbb Z$ is a minimal orientation presenving homomorphism with irrational rotation number $\rho$ then $T$ is topologically conjugate to the standard rotation $R_{\rho}: \mathbb R/\mathbb Z\to \mathbb R/\mathbb Z$ .

leave as a ex.

For $T: \mathbb R/\mathbb Z\to \mathbb R/\mathbb Z$ , $T': \mathbb R/\mathbb Z\to \mathbb R$ . We define the variation of $log|T'|: \mathbb R/\mathbb Z\to \mathbb R$ by:

$Var(log(|T'|))=$

$sup\{\sum_{i=0}^{n-1}|log|T'|(x_{i+1})-log|T'|(x_i)|: 0=x_0<x_1<...<x_n=1\}$

We say that the logarithm of $|T'|$ has bounded variation if this value $Var(log|T'|)$ is finite.

Denjoy’s theorem:

If $T: \mathbb R/\mathbb Z\to \mathbb R/\mathbb Z$ is a $C^1$ orientation preserving homomorphism of the circle with derivative of standard variation and irrational rotation number $\rho=\rho(T)$ then $T:\mathbb R/\mathbb Z\to \mathbb R/\mathbb Z$ is topologically conjugate to the standard rotation :

$R_{\rho}:\mathbb R/\mathbb Z\to \mathbb R/\mathbb Z$ .

Due to the upper proposition we only need show $T:\mathbb R/\mathbb Z\to \mathbb R/\mathbb Z$ is minimal. Proof pf minimal is splitting to following two sub lemmas.

Sublemma1:

If $T$ has irrational rotation number and there are a constant $C>0$ and a sequences of integers $q_n\to \infty$ such that the map: $T: \mathbb R/\mathbb Z\to \mathbb R/\mathbb Z$ Satisfy : $|(T^{q_n})'(x)||(T^{-q_n})'(x)|\geq C$ Then $T: \mathbb R/\mathbb Z\to \mathbb R/\mathbb Z$ is minimal.

Sublemma2:

Fix $x\in \mathbb R/\mathbb Z$ and write $x_n=T^n(x)$ , for $x\in \mathbb Z$ There exists an increasing sequences $q_n\to \infty$ of natural number such that the intervals $(x_0,x_{q_n}),(x_1,x_{q_n+1}),...,(x_i,x_{q_n+i}),...,(x_{q_n},x_{2q_n})$ are all disjoint.

Paradox and problem

Graph:

$\rho(T)>0$ because of existence of fix point.

$T_{\alpha}=T+\alpha$ for $\alpha< sup_x|T_x-x|$ .

Is $\rho(T_{\alpha})=0$ always true for $\alpha \in R$ ?

If it is right, then there is a contradiction with argument $(*)$ , but for what type of dynamic system $T$ ?

$T_{\alpha}=T+\alpha$ satisfied $\rho(T_{\alpha})=\rho(T)+\alpha$ . for all $\alpha\in \mathbb R$ ?

Problem:

If $T$ is not homomorphism but $T:x\to x+g(x)$ induced $g(x)=x-f(x)$ , $f(x)$ is striating increasing, Is the limit of $\lim_{x\to \infty}\frac{\hat T(x)}{n}$ always exists? it could not be increase with $x$ .