Fixed points of measure preserving torus homeomorphisms
Theorem: Assume a homeomorphism of the 2-torus is homotopic to the identity and preserves a measure \(\mu\) which is positive on open sets. If it has a lift f: \({\mathbb{R}}^ 2\to {\mathbb{R}}^ 2\) with vanishing mean translation, i.e. \(\int (f(x)-x)d\mu (x)=0,\) then it has at least two fixed points. In particular 2 of the 3 fixed points conjectured by Arnold for diffeomorphisms preserving Lebesgue measure also exist for homeomorphisms. As a corollary we derive an analogous fixed point theorem for the annulus which is a variant of Birkhoff's last geometric theorem. The proof is based on a generalization of Brouwer's lemma on translation arcs and the ``shift decomposition formula'' presented in this paper. This formula also applies to the n-torus showing that f is chain transitive on all of \({\mathbb{R}}^ n\) if it has vanishing mean translation.
scientific article; zbMATH DE number 4190763