  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.
