The initial version, proved by Birkhoff (for a single fixed point in 1912, as below in 1935), is as follows:
Theorem. Let be an area-preserving, orientation preserving homeomorphism of the compact annulus. If it rotates the two circles bounding in opposite directions, then it has at least two distinct fixed points.
This is also called Poincaré’s Last Theorem as it was stated by Poincaré in 1905. Since then a lot of authors have extended this result, including Kerekjarto, Franks, Carter, Yoccoz and Le Calvez. Building on their methods, Marc Bonino has explained the following purely topological version:
Theorem (Marc Bonino). Let be an orientation preserving homeomorphism of the annulus. One of the following holds:
- There are two distinct fixed points;
- There is a Jordan curve homotopic to the annulus such that contains only fixed points;
- There is a simple curve joining the two components of the boundary of such that meets at most one (local) side of and contains only fixed points.
I find it striking that the two hypothesis of conservativity and twist, which look of a different nature, are replaced in the above by two very similar conditions.
There is an example of Bestivna and Handel (ETDS 1992) which shows that the above cannot be readily generalized on the torus. More precisely, there is a homeomorphism of the 2-torus, isotopic to the identity, with no fixed point and such that each homologically non-trivial closed curve meets its image.