Critically fixed anti-rational maps, tischler graphs, and their applications

Abstract

We are mainly concerned with maps which take the form of the complex conjugate of a rational map and where all critical points are fixed points, which are known as critically fixed anti-rational maps. These maps have a well understood combinatorial model by a planar graph. We make progress in answering three questions. Using this combinatorial model by planar graph, can we generate all critically fixed anti-rational maps from the most basic example, z -> z 2? Under repeated pullback by a critically fixed anti-rational map, does a simple closed curve off the critical set eventually land and stay in a finite set of homotopy classes of simple closed curves off the critical set? This is known as the global curve attractor problem and has been an area of interest since its introduction by Pilgrim in 2012. Lastly, anti-rational maps can be used to model the physical phenomenon of gravitational lensing, which is where the image of a far away light source is distorted and multiplied by large masses between the light source and the observer. Maximal lensing configurations are where n masses generate 5n - 5 lensed images of a single light source. There are very few known examples of maximal lensing configurations, all generated by Rhie in 2003. Can we use these combinatorial models to inspire new examples of maximal lensing configurations? In this dissertation we show one can generate all critically fixed anti-rational maps from the most basic example, z -> z 2 by a repeated 'blow-up' procedure. We also show that all critically fixed anti-rational maps with 4 or 5 critical points have a finite global curve attractor. Lastly we establish a connection between maximal lensing maps and Tischler graphs and generate new examples of maximal lensing maps.