Lusternik-Schnirelmann理论与闭合Reeb轨道
Lusternik-Schnirelmann Theory and Closed Reeb Orbits
摘要：
我们发展了一个变体的Lusternik-Schnirelmann理论的移位算子在等变Floer和辛同调。我们的关键结果是当周期轨道是孤立的时，在移位算子的作用下，谱不变量是严格递减的。作为应用，我们证明了标准切触球、球的单位余切丛和其它切触流形上的简单闭Reeb轨道的新的重数结果。我们还证明了下Conley-Zehnder指标具有一定的递归性质，并从不同的角度重新证明了Long和Zhu的公共跳跃定理的一个变体。这是多重性结果证明中的第二个组合要素。
英文摘要：
We develop a variant of Lusternik-Schnirelmann theory for the shift operator in equivariant Floer and symplectic homology. Our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated. As an application, we prove new multiplicity results for simple closed Reeb orbits on the standard contact sphere, the unit cotangent bundle to the sphere and some other contact manifolds. We also show that the lower Conley--Zehnder index enjoys a certain recurrence property and revisit and reprove from a different perspective a variant of the common jump theorem of Long and Zhu. This is the second, combinatorial ingredient in the proof of the multiplicity results.
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