Gravitational lensing and critically fixed anti-rational maps

Abstract: Studying the dynamics of anti-rational maps, i.e., complex conjugates of rational maps, is a subject closely related to holomorphic dynamics, with intriguing connections to problems in gravitational lensing. In particular, the lens equation for a single-plane gravitational lens made up of N point masses is known to be a fixed point equation for an anti-rational map of degree N. These fixed points are apparent images of a single (point) light source, and it is known from work of Rhie (2003) and Khavinson and Neumann (2006) that for N>1 there can be at most 5N-5 such images, and that this bound is sharp.

Originally motivated by the goal of classifying maximal lensing configurations, i.e., configurations for which the bound 5N-5 is attained, we recently succeeded in giving a complete classification of anti-rational maps for which all critical points are fixed, through simple topological models associated with certain planar graphs. We will explain this classification, the main ideas in the proof, and how this yields a partial classification and new examples of maximal lensing configurations. Finally, we will discuss some open problems and questions.

general relativity and quantum cosmologycomplex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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