Applications of Differential Chains to Complex Analysis and Dynamics

Abstract

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.