量子施泰因罗德幂与哈密顿映射
Quantum Steenrod powers and Hamiltonian maps
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中文总结 AI 辅助
该研究在闭辛流形上的哈密顿动力学中获新成果，如特定哈密顿微分同胚与流形性质关系、周期轨道与简单周期点关系等，还建立一般康利猜想新情形，证明依赖积分哈密顿弗洛尔理论等。
中文摘要 AI 辅助
我们在一般的闭辛流形$(M,\omega)$上的哈密顿动力学中证明了一系列新结果，包括：1. 若$M$允许一个哈密顿微分同胚，它要么是伪旋转要么具有有限阶，则$M$是几何单有理的，解决了麦克杜夫 - 萨拉蒙列表中问题24的一个变体。2. 若一个哈密顿量在非挠同调类中有一个周期轨道，那么它有无限多个简单周期点，在特定条件下同样结论成立，这两个结果补充了霍弗 - 策恩德猜想的已知情形。3. 若一个哈密顿微分同胚有一个辛退化最大值，那么它有无限多个简单周期点，解决了源于金兹堡和居雷尔工作的一个开放问题。我们还建立了一般康利猜想的新情形：一般哈密顿微分同胚有无限多个周期点。证明依赖于第一和第四作者开发的积分哈密顿弗洛尔理论包的系统应用、等变弗洛尔同调中的库内特同构以及对量子幂映射的新定量分析。
英文摘要
We prove a series of new results in Hamiltonian dynamics on a general closed symplectic manifold $(M, ω)$, including: 1. If $M$ admits a Hamiltonian diffeomorphism which is either a pseudo-rotation or has finite order, then $M$ is geometrically uniruled. This resolves a variant of Problem 24 in McDuff--Salamon's list, which predicts an obstruction to the existence of such special Hamiltonian diffeomorphisms in terms of genus zero numerical invariants. 2. If a Hamiltonian possesses a periodic orbit in a non-torsion homology class, then it has infinitely many simple periodic points. The same conclusion holds if the manifold is not geometrically uniruled and the diffeomorphism is minimal for rational Floer homology. These two general results complement the known cases of the Hofer--Zehnder conjecture. 3. If a Hamiltonian diffeomorphism possesses a symplectically degenerate maximum, then it has infinitely many simple periodic points. This resolves an open question which stems from the work of Ginzburg and Gürel. We also establish new cases of the generic Conley conjecture: infinitely many periodic points for generic Hamiltonian diffeomorphisms. The proofs rely on a systematic application of the integral Hamiltonian Floer theory package developed by the first and fourth author, a Künneth isomorphism in equivariant Floer homology, and new quantitative analysis of quantum power maps, which is of independent interest.
