Transcendental Thurston Theory and Dynamical Approximations

Abstract

A holomorphic map is said to be postsingularly finite (PSF) if it has finitely many singular values and every singular orbit is finite. If every singular value is a critical value, a postsingularly finite map is also called postcritically finite (PCF). As shown by Douady and Hubbard in their study of the Mandelbrot set, these maps are crucial in understanding the structure of parameter spaces, and can often be determined by a finite amount of combinatorial data. Thurston's theory considers a certain class of continuous maps with a finite postsingular set and asks when such a map is equivalent to a holomorphic PSF map. While this theory was initially introduced by William Thurston for topological analogs of rational maps on the sphere, the last few decades have seen a push to generalize this theory to topological analogs of entire maps on the plane, starting with the work of Hubbard, Schleicher and Shishikura on topological versions of exponential functions. In this thesis, we explore the relationship between finite and infinite degree Thurston theory, and use this relationship to establish dynamically meaningful approximations for PSF entire functions by PCF polynomials.