Applications of Differential Chains to Complex Analysis and Dynamics

TL;DR

This thesis explores the use of differential chains to extend complex analysis theorems and analyze dynamical systems, providing new tools for integration over complex curves and understanding flow behaviors on manifolds.

Contribution

It introduces novel applications of differential chains to generalize classical theorems and analyze flows, including new results in complex analysis and dynamical systems theory.

Findings

Generalized Cauchy theorems to non-rectifiable curves

Proved asymptotic cycles are differential chains

Established equality of differential chains for ergodic measures

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.