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Recurrence and fixed points of surface homeomorphisms. (English) Zbl 0634.58023
We prove that if f is a homeomorphism of the annulus which is homotopic to the identity and has a compact invariant chain transitive set L then either f has a fixed point or every point of L moves uniformly in one direction: clockwise or counterclockwise. If f is area preserving then the annulus itself is a chain transitive set so, in the presence of a boundary twist condition, one obtains a fixed point. The same techniques apply to homeomorphisms of the torus T 2. In this setting we show that if f is homotopic to the identity, preserves Lebesgue measure and has mean translation 0 then it has at least one fixed point.
Reviewer: J.Franks

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
Full Text: DOI
[1] DOI: 10.2307/1968772 · Zbl 0063.06074 · doi:10.2307/1968772
[2] DOI: 10.2307/2319484 · Zbl 0355.52007 · doi:10.2307/2319484
[3] DOI: 10.1112/jlms/s2-13.3.486 · Zbl 0342.60049 · doi:10.1112/jlms/s2-13.3.486
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[6] DOI: 10.1007/BF01393824 · Zbl 0516.58017 · doi:10.1007/BF01393824
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