论文标题
薄膜台球反射伪群的密度
Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup
论文作者
论文摘要
Hypersurfaces的反射通过在规范符号形式的方向线上作用于定向线的空间。我们考虑一个任意的$ c^{\ infty} $ - 平滑的超曲面$γ\ subset \ mathbb r^{n+1} $，它是全局严格凸出的封闭式hypersurface，或者是hypersurface的胚芽。我们处理由$γ$及其小变形的反射的组成比产生的假群。在这种情况下，当$γ$是一个全球凸出的超曲面时，我们表明，后者的伪群在相位缸子域之间的哈密顿二型二型二型二型二键群体中密集：方向线的空间与$γ$横流相交的空间。当$γ$是细菌时，我们证明了该案例类似的本地结果。上述变形参数上上述组成差异的衍生物是由RON Perline计算出的Hamiltonian Vector场。为了证明主要结果，我们发现它们生成的Lie代数并证明其$ C^{\ infty} $ - 在汉密尔顿矢量域的Lie代数中密度。我们还证明了Riemannian歧管中的Hypersurfaces上述结果的类似物。
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.
