论文标题
同时分形，加农炮 - 塔尔斯顿地图和大地测量流
Cohomology fractals, Cannon-Thurston maps, and the geodesic flow
论文作者
论文摘要
辅助分形是与双曲线三序列中的共同体学自然相关的图像。我们通过将射线追踪到固定的视觉半径来实时生成这些图像，以实时垂直，不完整和闭合的双曲线三体。我们在尝试说明Cannon-Thurston地图而不使用矢量图形的同时发现了共同体分形。当共同体学类对纤维双重时，我们证明了这两个之间的对应关系。这使我们能够通过将共同分形图像与Cannon-Thurston地图的现有图片进行比较来验证实现。 在一系列实验中，我们探讨了随着视觉半径的增加，共同分形的限制行为。在这些实验的激励下，我们证明了同胞分形的值是正态分布的，但标准偏差有所不同。实际上，辅助分形不会收敛到极限的功能。取而代之的是，我们表明极限是无穷大的球体上的一个分布，仅取决于歧管和协同学类别。
Cohomology fractals are images naturally associated to cohomology classes in hyperbolic three-manifolds. We generate these images for cusped, incomplete, and closed hyperbolic three-manifolds in real-time by ray-tracing to a fixed visual radius. We discovered cohomology fractals while attempting to illustrate Cannon-Thurston maps without using vector graphics; we prove a correspondence between these two, when the cohomology class is dual to a fibration. This allows us to verify our implementations by comparing our images of cohomology fractals to existing pictures of Cannon-Thurston maps. In a sequence of experiments, we explore the limiting behaviour of cohomology fractals as the visual radius increases. Motivated by these experiments, we prove that the values of the cohomology fractals are normally distributed, but with diverging standard deviations. In fact, the cohomology fractals do not converge to a function in the limit. Instead, we show that the limit is a distribution on the sphere at infinity, only depending on the manifold and cohomology class.
