A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds.

*(English)*Zbl 0632.58014The authors study area-preserving maps of smooth manifolds and prove the following result. Theorem: Let \(h: M^ 2\to M^ 2\) be an orientation preserving homeomorphism of a smooth orientable two-manifold \(M^ 2\), which preserves area. If p is an isolated fixed point of h, then the index of h at p is less than or equal to 1.

This theorem may be viewed as a generalization of C. P. Simon’s one [see Invent. Math. 26, 187-200 (1974; Zbl 0331.55006)], which deals with \(C^ k\) diffeomorphisms. The method used in this paper is necessarily completely different from Simon’s, and the theorem of this paper can be applied to derive the same conclusions as in Simon’s paper under less restrictive assumptions.

This theorem may be viewed as a generalization of C. P. Simon’s one [see Invent. Math. 26, 187-200 (1974; Zbl 0331.55006)], which deals with \(C^ k\) diffeomorphisms. The method used in this paper is necessarily completely different from Simon’s, and the theorem of this paper can be applied to derive the same conclusions as in Simon’s paper under less restrictive assumptions.

Reviewer: Ding Tongren

##### MSC:

58C30 | Fixed-point theorems on manifolds |

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\textit{S. Pelikan} and \textit{E. E. Slaminka}, Ergodic Theory Dyn. Syst. 7, 463--479 (1987; Zbl 0632.58014)

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##### References:

[1] | DOI: 10.1007/BF01418948 · Zbl 0331.55006 · doi:10.1007/BF01418948 |

[2] | Brown, Houston J. Math. 10 pp 35– (1984) |

[3] | Brown, Houston J. Math. none pp none– (none) |

[4] | DOI: 10.1016/0040-9383(75)90020-8 · Zbl 0321.55025 · doi:10.1016/0040-9383(75)90020-8 |

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