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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:
  Alpern S., Prasad V.S.:A combinatorial proof of the Conley-Zehnder-Franks theorem on a fixed point for torus homeomorphisms, preprint (1989) · Zbl 0781.58013  Arnold V.I.:Fixed points of symplectic diffeomorphisms, Proc. Sympos. Pure Math., Vol. 28, Amer. Math. Soc., Providence, R.I. (1976), 66  Brown M.:A new proof of Brouwer’s lemma on translation arcs, Houston J. of Math. 10 (1984), 35–41 · Zbl 0551.57005  Conley C., Zehnder E.:The Birkhoff-Lewis fixed point theorem and a conjecture of Arnold, Invent. Math 73 (1983), 33–49 · Zbl 0516.58017 · doi:10.1007/BF01393824  Fahti A. An orbit closing proof of Brouwer’s lemma on translation arcs, L’enseignement Math. 33 (1987), 315–322 · Zbl 0649.54022  Franks J.:Recurrence and fixed points of a surface homeomorphism, Ergodic theory and dynamical systems 8* (1988) · Zbl 0634.58023  Franks J.:Generalisations of the Poincaré-Birkhoff theorem, Ann. of Math. 188 (1988), 139–151 · Zbl 0676.58037 · doi:10.2307/1971464  Nikishin N.:Fixed points of diffeomorphisms on the two sphere that preserve area, Funkts. Anal. i Prilozhen 8 (1974), 84–85 · Zbl 0323.60067 · doi:10.1007/BF02028320  Oxtoby S., Ulam S.:Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. 42 (1941), 874–920 · Zbl 0063.06074 · doi:10.2307/1968772  Simon C.P.:A bound for the fixed point index of an area-preserving map with applications to mechanics, Invent. Math. 26 (1974), 187–200 · Zbl 0331.55006 · doi:10.1007/BF01418948
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