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Recurrence and fixed points of surface homeomorphisms

Published online by Cambridge University Press:  10 December 2009

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Abstract

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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 T2. 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.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

Atkinson, G.. Recurrence of cocycles and random walks. J. Lond. Math. Soc. 13 (1976), 486488.Google Scholar
Brown, M.. A new proof of Brouwer's lemma on translation arcs. Houston J. Math. 10 (1984), 3541.Google Scholar
Conley, C.. Isolated Invariant Sets and the Morse Index. CBMS Regional Conf. Series in Math. 38. AMS, Providence, RI (1978).Google Scholar
Conley, C. & Zehnder, E.. The Birkhoff-Lewis fixed point theorem and a conjecture of Arnold. Invent. Math. 73 (1983), 3349.Google Scholar
Handel, M.. To appear.Google Scholar
Oxtoby, J.. Diameters of arcs and the gerrymandering problem. Amer. Math. Monthly 84 (1977), 155162.Google Scholar
Oxtoby, J. & Ulam, S.. Measure preserving homeomorphisms and metrical transitivity. Ann. Math. 42 (1941), 874920.Google Scholar