薄膜台球反射伪群在哈密顿辛同胚伪群中的稠密性
Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup
AI总结：
本文研究了超曲面反射生成的伪群在哈密顿微分同胚伪群中的稠密性，通过计算形变参数导数生成的李代数并证明其稠密性，确立了全局与局部超曲面情形下的核心结论。
摘要：
超曲面的反射通过典范辛形式在有向直线空间上以辛同胚作用。我们考虑任意 $C^{\infty}$-光滑超曲面 $\gamma\subset\mathbb R^{n+1}$，它或者是全局严格凸闭超曲面，或者是超曲面芽。我们处理由 $\gamma$ 的反射及其微小形变的反射的复合比生成的伪群。当 $\gamma$ 是全局凸超曲面时，我们证明后者伪群在相柱子区域间的哈密顿微分同胚伪群中稠密：该相柱空间由与 $\gamma$ 横截相交的有向直线构成。当 $\gamma$ 为芽时，我们证明了类似的局部结果。上述复合差关于形变参数的导数是由 Ron Perline 计算的哈密顿向量场。为证明主要结果，我们求出由它们生成的李代数，并证明其在哈密顿向量场的李代数中是 $C^{\infty}$-稠密的。我们还对黎曼流形中的超曲面证明了上述结果的类似结论。
英文摘要：
Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary $C^{\infty}$-smooth hypersurface $γ\subset\mathbb R^{n+1}$ that is either a global strictly convex closed hypersurface, or a germ of hypersurface. We deal with the pseudogroup generated by compositional ratios of reflections from $γ$ and of reflections from its small deformations. In the case, when $γ$ is a global convex hypersurface, we show that the latter pseudogroup is dense in the pseudogroup of Hamiltonian diffeomorphisms between subdomains of the phase cylinder: the space of oriented lines intersecting $γ$ transversally. We prove an analogous local result in the case, when $γ$ is a germ. The derivatives of the above compositional differences in the deformation parameter are Hamiltonian vector fields calculated by Ron Perline. To prove the main results, we find the Lie algebra generated by them and prove its $C^{\infty}$-density in the Lie algebra of Hamiltonian vector fields. We also prove analogues of the above results for hypersurfaces in Riemannian manifolds.
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