The topological entropy of the subdivision map of a finite subdivision rule restricted to the 1-skeleton of its model subdivision complex, which we call textbf{core entropy}, is examined. We consider core entropy for finite subdivision rules realizing quadratic Misiurewicz polynomials and matings of such polynomials. It is shown that for a non-restrictive class of finite subdivision rules realizing quadratic Misiurewicz polynomials, core entropy equals Thurston's core entropy. We also show that the core entropy of formal and degenerate matings of Misiurewicz polynomials is determined by Thurston's core entropy of the mated polynomials. / Doctor of Philosophy / Imagine taking a programmable calculator, inputting a number, and repeatedly pushing one of the buttons which corresponds to one of the calculator's built-in functions. For example, starting by inputting 0.5 and hitting the "x2" button over and over, or starting with 1.47 and repeatedly pressing the "sin(x)" button. The calculator may eventually return numbers that get closer and closer to a specific value, it may repeatedly cycle through some collection of specific numbers, it may not exhibit a clear pattern at all. It is of interest to understand, in some average sense, when, how often, and in what manner these patterns are exhibited and, in a quantitative fashion, compare how complicated the patterns are for different buttons on the calculator corresponding to different functions. For example, is the "x2" button, in some average sense, more or less "complex", in terms of the patterns exhibited by the above procedure, than the "sin(x)" button? Modeling or simulating physical phenomena such as particle motion or the orbits of collections of celestial bodies often entails the use of computer programs. These computer programs carry out calculations which often involves repeated application of various pre-programmed functions. Repeatedly pushing a button on a calculator can be viewed as a simplified version of what goes on with the calculations that a computer carries out in simulating physical phenomena. Understanding how to compare the patterns exhibited by simple, fundamental collections of functions makes for a good starting point for understanding the models that represent various physical phenomena. This work contributes to this endeavor by investigating a quantity which measures the complexity of some fundamental functions.

Combinatorial methods are utilized to examine preimage iterations of topologically glued polynomials. In particular, this paper addresses using finite subdivision rules and Hubbard trees as tools to model the dynamic behavior of mated quadratic functions. Several methods of construction of invariant structures on modified degenerate matings are detailed, and examples of parameter-based families of matings for which these methods succeed (and fail) are given. / Ph. D.

Nearly Euclidean Thurston maps are simple generalizations of rational Lattes maps. A Thurston map is called nearly Euclidean if its local degree at each critical point is 2 and it has exactly four postcritical points. We investigate when such a map has the property that the associated pullback map on Teichmuller space is constant. We also show that no Thurston map of degree 2 has constant pullback map. / Ph. D.