Rotation number

Consider compact 1 dimension dynamic system.

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

$latex T: S_1\to S_1$.

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

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

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

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

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

How to realize the lifting?

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

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

The rotation number is defined as:

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

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

It is not difficult to proved the following property:

Property:

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

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

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

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

Propersion:

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

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

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

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

pf of 1):

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

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

$latex =n\rho (T)$.

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

pf of 2):

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

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

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

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

pf of 3):

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

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

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

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

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

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

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

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

pf of 4):

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

Corollary:

Assume $latex \rho(T)$ is irrational.

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

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

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

Denjoy’s theorem

Proposition:

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

leave as a ex.

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

$latex Var(log(|T’|))=$

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

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

Denjoy’s theorem:

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

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

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

Sublemma1:

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

Sublemma2:

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

Graph:

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

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

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

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

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

Problem:

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