Christopher McKay's Ph.D. Defense in Mathematics (Dept. of Mathematical Sciences, MSU)

04/11/2022

Abstract:

A critically fixed anti-rational map is the complex conjugate of the quotient of two polynomials such that all critical points are fixed points. Associated to such a map is a specific invariant graph called the Tischler graph. It is natural to ask, given a graph satisfying some set of properties, is this graph associated to a critically fixed anti-rational map? Geyer extended Thurston’s characterization of rational maps to tell us when a given topological Tischler graph is associated to a critically fixed anti-rational map. Using this theorem, we see that an obstruction can prevent a graph from representing such a map. These obstructions are represented by simple closed curves. In this talk we will introduce how to construct Tischler graphs and how to spot a topological Tischler graph which is not associated to a critically fixed anti-rational map. We will then study the dynamics of simple closed curves under pullback. We will show that critically fixed anti-rational maps with 4 or 5 critical points have a finite global curve attractor, that is that every simple closed curve under pullback eventually lands and stays in a finite set of homotopy classes of simple closed curves. We will also exploit the Tischler graph to show that all critically fixed anti-rational maps can be generated by a sequence of “blow-ups” of our most basic example, $z \mapsto \bar{z}^2$.