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Applications of Differential Chains to Complex Analysis and Dynamics

Authors
  • Pugh, Harrison
Type
Preprint
Publication Date
Dec 26, 2010
Submission Date
Dec 26, 2010
Identifiers
arXiv ID: 1012.5542
Source
arXiv
License
Yellow
External links

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.

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