Title: Topological Entropy, Thurston Sets, and the Mandelbrot Set
Abstract: Given a family of dynamical systems, which numbers are realized as the topological entropy of some member of the family, and how does the answer to this question inform our understanding of the family?  A way to get at this question is to study the Thurston Set -- a subset of the complex plane defined in terms of the collection of all topological entropies realized by the family.  I and collaborators proved a variety of results about the geometrical and topological structures of Thurston sets for families of quadratic polynomials.  In particular, I will discuss results (joint with G. Tiozzo and C. Wu) about how core entropy varies over the Mandelbrot set.  Specifically, we proved that the collection of all Galois conjugates of norm at least 1 of the exponential of core entropy, together with the unit circle, varies continuously in the Hausdorff topology with external angle for the Mandelbrot set. On the other hand, the Galois conjugates with norm < 1 exhibit "Persistence" along principal veins in the Mandelbrot set.