Johannes Heißler - Rotation numbers

Abstract

Prove Lemma 5.2 using the Fekete Lemma. Sketch alternative proof of perfectness of $Home{o}_{+}(S^1)$ given in [Ghys] Proposition 5.11 comparing to that of Talk 1. Discuss example of Fuchsian groups with respect to rotation numbers.

Talk

1. Lifts to the reals

For any given (non homeo) map $f$ on $S^1$ there is a lift to $\mathbb{R}$ up to a whole number translation. Generally, we can define the degree by how often a map wraps the circle around itself but this time we are only interested in homeos whose degree is one. The lift then becomes strictly monotonic. And a orientation preserving map is increasing.

The group of lift will be called $\widetilde{Home{o}_{+}(S^1)}$. The elements are characterised by the fact that moving right by $n$ the value increases by $n$.

We denote by $\tilde{f}$ an element of the above group and by $\tilde{H}$ the group itself

Theorem

For any $\tilde{f}\in \tilde{H}$ there is a translation number $\tau$ s.t. ${\tilde{f}}^n(x)-x=\tau \cdot n+O(1)$, meaning the repetition can be approximated by a translation. A second characterisation is $\tau (\tilde{f})={\lim }_{n\to \infty }\frac{{\tilde{f}}^n(0)}{n}$ Two lifts differ by an integer. The translation numbers then project down to a Rotation number on the circle.

2. Quasi-morphisms

Definition

We define a Quasimorphism be the standard definition.

Theorem

Für jeden Quasimorphismus gibt es eine translationszahl $\tau (g)$ in Abhängigkeit von $g$, sodass $\varphi (g^n)=\tau (g)n+O(\Delta h)$ mit $O(\Delta h)\in [-\Delta h,\Delta h]$ für den Defekt des Quasimorphismus.

Proof: To prove this theorem, we will use the Lemma of Fekete, which gives us conditions for convergence of a sequence when given subadditivity. We do a nice classical analysis proof where we first prove subadditivity, then use the lemma to bound the quasimorphism from below and above. $\varphi (g^n)-c\le \tau (g)n\le \varphi (g^n)+c$ Conceptually the proof of Fekete is very richt and important but I didn’t have time to write it down.

Definition

For each QM $\varphi$ there is a unique QM that is bounded distance from $\varphi$ and homogenous, i.e. we have $\tau (g^n)=n\tau (g)$. We get them by averaging.

Theorem

Funnily enough they are invariant under conjugation, i.e. $\tau (fg{f}^{-1})=\tau (g)$ and form an abelian subgroup: $\tau (fg)=\tau (g)+\tau (h)$.

Quasi-morphisms are interesting. It is known that Simple groups dont have morphism but we might ask if they have quasimorphisms.

Quasimorphisms can be used to distinguish between Perfect group (every element can be written as a product of commutators) and Uniformly perfect group (every element can be written by a product of at most $c$ commutators). Furthermore $\tilde{H}$ is uniformly perfect and every element is a product of two elements.

I didn’t write down everything in this section.

3. Periodic orbits

Theorem

The rotation number is zero if and only if $f$ has fixed points

Proof: Right to left is pretty easy. For the other direction we lift to the translation number. An orbit must then be bounded. This means there is a accumulation point $\sup {\tilde{f}}^n(x)$ in the orbit of the lift. It can be shown that this point is fixed, by shifting the whole sequence by $\tilde{f}$ and noticing that the limit stays the same. (Not really, since the accumulation point might be the limit of a subsequence which is not invariant by shifting but this issue can be fixed using some pictures).

Lemma

The rotation numbers are rational iff there are periodic points.

Proof: This kinda follows from the previous theorem. In one direction, we take $f^n$. If there is a fixed point, then $f$ has periodic points and the rotation number (which is multiplicative) times $n$ is a whole number. The other direction is similarly simple. (Johannes actually wrote down more than that but I didn’t look to closely. e.g. wandering points have been involved)

4. Infinite orbits

Let $G$ be a group acting on $S^1$. We are interested in the closures of orbits $\overline{Gx}$.

Theorem

Assume there are no finite orbits. Then there is a non-empty closed invariant $K\subseteq S^1$ s.t. $K$ is minimal. Meaning, for each $x\in K$ we have $\overline{Gx}=K$ (“Orbits of points in K are dense in K”). Furthermore $K$ is uniquely minimal (meaning for every $x\in S^1$ our $K$ sits inside $\overline{Gx}$)

Using this theorem and some additional information, we get that $K$ is a Cantor set, i.e. a closed set with empty interious and no isolated. (but might have positive measure)

Meaning, it is possible for the closure of an orbit to look like a cantor set.

Definition

We define the notion of Semi-conjugate system. This is conjugation but only in one direction.

Could you explain everything from the claim onward to me?

Could you communicate what kind of culture we have here? Some of us need a grade and it is not clear how they should prepare and present the talk. Should we present a talk with 90 minutes of content or should we present a talk that is 90 minutes long and incorporate discussion initiated by students and teachers.