Recurrence and fixed points of surface homeomorphisms
Recurrence and fixed points of surface homeomorphisms
复制标题
表面同胚的递归点和不动点
DOI：
10.1017/s0143385700009366
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发表时间：
1988
影响因子：
0.9
通讯作者：
J. Franks
中科院分区：
文献类型：
--
作者：
J. Franks
摘要 我们证明，如果 f 是与恒等式同伦的环的同胚，并且具有紧不变链传递集 L，则 f 要么有不动点，要么 L 中的每个点沿一个方向均匀移动：顺时针或逆时针。如果 f 是保面积的，则环本身是链传递集，因此，在存在边界扭曲条件的情况下，可以获得一个不动点。相同的技术适用于环面 T2 的同胚。在这一设置中，我们表明，如果 f 与恒等式同伦，保留勒贝格测度且均值平移为 0，则它至少有一个不动点。
Abstract 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.