Sebastian Hensel - Rotation sets for maps of tori

Abstract

Talk

We do some preliminary stuff on Rotation number. Can we generalise this for the torus? Well, we could just do the same procedure. We find out that a rotation of a torus can be understood as as vector, parametrising rotation in one or another direction.

Example

We define the Volle Zeltabbildung $\varphi$. Then we define a homeo of the torus as $f(x,y)=(x,y+\varphi (x))$. The “rotation number” of this depends on the point chosen. It can be something in $0\times [0,1]$. We then define a corresponding $g(x,y)=(x,+\varphi (y),y)$

The map $h=g\circ f$ then has some really complicated. The square get deformed into something that looks like a space filling curve.

Definition

$f\in Home{o}_0(T)$. The rotation set $Rot(f)$ is the set of all vectors that are rotation numbers for some point. More formally, we can define translation sets. These are equal up to integral translation.

If we have a fixed point, then all integral $v$ will be in our translation sets/rotation sets If we have a periodic point, then we will have rational points in the rotaton sets. Conjugation by a $Home{o}_0$ map has the same rotation set. Conjugating a homeo by a linear-homeomorphism has the effect of applying the linear homeo to the rotation set.

The rotation set sits in the plane not the torus, as the subsets might be bigger than the torus.

Theorem

$rot(F)$ is convex, and $rot(F)=\lim \frac{1}{n}{F}^{n}Q$ in the Hausdorf. topology.

We introduce some theorems from Hausdorf topology.

Lemma

$\pi :{\mathbb{R}}^2\to T$ the standard projection. $D\subseteq {\mathbb{R}}^2$ path-connected s.t. $\pi |D$ injective. Then $D$ is $\sqrt{2}$ convex.

Proof of the theorem above: We want to show that the limit in the proof exists (i.e. the subset converges against some subset). Once we show that the limit exists a slightly challenging argument shows that the limit converges to the rotation number. We then make a Fekete-type argument as this proof uses a divisibility argument.

Theorem (Franks)

Suppose $f\in Home{o}_0(T)$, $F$ a llft, If the interior of the rotation set of $F$ contains a rational point, than $f$ has a periodic points. (Rotation sets can be a point a line segment or it has interior)

Theorem (Le Calvez, Sauzet)

Let $f$ be an identity-isotopic homeomorphism where the lift has not fixed points. Then there is a simple clased curve $\gamma$ on $T$ which lifts to a topological line $\tilde{\gamma }$ bounds a halfplane $P$ s.t. $F(P)\subseteq int(P)$. The idea is here, that if a homeo has no fixed points, then it is a translation

This is an equivariant version of

Theorem (Brouwer plane translation thm)

$F\in Home{o}^{+}({\mathbb{R}}^2)$ with no fixed pts. Then any point $x$ lies on a line $\tilde{\gamma }$ which is moved to the right. i.e. the plane is foliated.

Proof of franks: Its enough to show that if $0$ lies in the interior of the rotation set then $F$ has a fixed point. Suppose not. La Calvez tells us there is a homeo with a funny curve $\gamma$ on the Torus which is shifted to the right (in the universal covering). By conjugation, we move this curve to a nice straight line. We take a fundamental domain of the torus on the right of $\tilde{\gamma }$.

Is there any theory for surfaces with nnegative euler characteristic?