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Fixed points of measure preserving torus homeomorphisms. (English) Zbl 0722.58027
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.

MSC:
37A99 Ergodic theory
28D05 Measure-preserving transformations
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References:
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