Shrimadhi S. - Denjoy’s Theorem

Abstract

Prove Denjoy’s Theorem that $C^2$-diffeomorphisms that have irrational rotation number are topologically conjugate to rotations. Describe Denjoy counterexample of a non-minimal $C^0$-homeomorphism, which is in particular not conjugate to a rotation. Give idea of how to extend to a $C^1$-counterexample.

Talk

1. The theorems

For this talk the main statement is that under strong conditions, every map is conjugate to a rotation and under weak conditions it is semi-conjugate to a rotation.

Definition

We define the notion of a Semi-conjugate system.

Theorem

If $f$ is an orientation-preserving Homeo of $S^1$, is $C^2$ with rotation number irrational, then

1. $f$ is t.c. to a rotation by a irrational number
2. Every orbit of $f$ is dense
3. There are no wandering intervals of $f$

Those three statements are all equivalent.

Definition

An interval $I$ is called wandering if $f^n(I)$ never intersects with $f^m(I)$ Or in the case of homeos if $f^n$ never intersects with $I$.

Theorem

If $f$ is a homeo and the rotation number is irrational, then $f$ is semi-conjugate $\varphi$ to a rotation $R^{\alpha }$. This semi-conjugation $\varphi$ is

• of degree $1$ and monotone
• a topological conjugacy iff all orbits of $f$ are dense.

Definition

Let $f:\mathbb{R}\to \mathbb{R}$ be a $C^1$ function. Non-linearity is defined as $nl(F,I)=\log {\max }_{x\in I}{f}^{\prime }(x)-\log {\min }_{y\in I}{f}^{\prime }(y)$ This measures how non-linear a map is.

We use the logarithm to have (sub-)additive non-linearity.

Lemma

Let $f,g$ be $C^2$ functions of $R$. The non-linearity is subadditive for the product of two functions $f\circ g$.

Proof: The proof of this depends on the fundamental theorem of calculus which is why we need $C^2$ here. Applying the fundamental theorem to the circle we get an upper bound for non-linearity at each segment given by $K={\int }_{S^1}|\frac{f^{\prime \prime }(x)}{f^{\prime }(x)}|dx$. Some calculations gives us $nl(f^n,I_0)\le nl(f,I_0)+...+nl(f,I_n)\le K$ where $I_0$ is some interval and $I_n=f^n(I_0)$ are pairwise disjoint.

Proof that there is no wandering interval: If there were a wandering interval, we show that there is a subsequence of intervals where the non-linearity is bounded below by a factor going to infinity. This means that the sum of them can’t be bounded by some $K$.

I am way to tired to go into the details…

2. Counterexamples

We construct a counterexample to the first theorem which is not $C^2$.

Example

Our Counterexample $f$ has irrational rotation number, is $C^1$ that has a wandering interval. The idea is to take a irrational rotation and then replace each point of an orbit by a interval. (But in a way, that the circle does not become infinitely big) Constructing a $C^0$ example is quite easy but a $C^1$ can be tricky, since you need to shrink the intervals and have derivative $1$ at the edges of the intervals. (which might induce infinite second derivative??)