微分链在复分析和动力学中的应用
Applications of Differential Chains to Complex Analysis and Dynamics
摘要：
本文共分为三个部分。在第一部分中，我们介绍了J.Harrison的微分链理论。在第二部分中，我们应用这些工具推广了复分析中的柯西定理。我们现在可以在非可求长曲线和远离所讨论的全纯函数的奇点的无发散向量场上进行积分，而不需要在其上进行积分的分段光滑路径。在第三部分中，我们着重讨论了它在动力学中的应用，特别是紧致黎曼流形上的流动。证明了渐近环是微分链，并且对于遍历测度，渐近环作为微分链等价于与向量场和遍历测度相关的微分链.第一部分是简要的，但第二和第三部分包含新的结果。
英文摘要：
This thesis is divided into three parts. In the first part, we give an introduction to J. Harrison's theory of differential chains. In the second part, we apply these tools to generalize the Cauchy theorems in complex analysis. Instead of requiring a piecewise smooth path over which to integrate, we can now do so over non- rectifiable curves and divergence-free vector fields supported away from the singularities of the holomorphic function in question. In the third part, we focus on applications to dynamics, in particular, flows on compact Riemannian manifolds. We prove that the asymptotic cycles are differential chains, and that for an ergodic measure, they are equal as differential chains to the differential chain associated to the vector field and the ergodic measure. The first part is expository, but the second and third parts contain new results.
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