S^2上离散幂零群作用的不动点
Fixed points of discrete nilpotent group actions on S^2
摘要：
本文证明了：对于任意不小于2的整数k，在C^1-拓扑中，存在2-球面S^2的单位映射的开邻域ν k，使得：如果G是Diff^1（S^2）的幂零子群，其幂零长度为k，由ν k中的元素生成，则S^2上的自然作用有非空不动点集.此外，如果G-作用具有有限非平凡轨道，则G-作用至少有两个不动点。
英文摘要：
We prove that for each integer k of at least 2, there is an open neigborhood ν_k of the identity map of the 2-sphere S^2, in C^1-topology such that: if G is a nilpotent subgroup of Diff^1(S^2) with length k of nilpotency, generated by elements in ν_k, then the natural action on S^2 has non-empty fixed point set. Moreover, the G-action has at least two fixed points if the action has a finite non-trivial orbit.
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