Oral Exam: Beyond the Classification of Critically Fixed Anti-Rational Maps
Chris McKay (Dept. of Mathematical Sciences, MSU)
11/23/2020 4:10-5:15pm WebEx Meeting
We outline the connections between anti-rational maps, anti-Thurston maps, Tischler graphs and gravitational lensing. We then dive into 3 open questions I plan to explore in my thesis. The first is known as the global curve attractor problem. For a critically fixed anti-rational map is there a finite set of simple closed curves, A_0, such that the orbit of every simple closed curve under pullback eventually stays in A_0 up to homotopy? We discuss the existing results in the orientation-preserving case and why the orientation-reversing case adds new complexity. The second problem asks if the process of blowing-up arcs to generate new maps which are (Thurston) equivalent to rational maps can be extended to anti-rational maps. Lastly, can we classify the Tischler graphs corresponding to maximal lensing maps? That is, Tischler graphs of anti-rational maps of degree n with 5n-5 fixed points which have simple poles with positive real residue.