Thurston Theory for a Family of Chebyshev Polynomials and Cosine Maps

Abstract

A cornerstone of complex dynamics is William Thurston's topological characterization of rational functions, a dynamical result that provides a way to understand when topological objects are realized as geometric objects. From the point of view of iteration, these topological objects are finite degree branched maps of the topological sphere S^2 and the geometric objects are holomorphic maps of the Riemann sphere hat{mathbb{C}}, both of which are postcritically finite (i.e., the set of points in the orbit of the critical points is finite). We apply this framework to study a one-parameter family of modified Chebyshev polynomials from a dynamical and nondynamical perspective. Our interest in this family comes from the property that it approximates a one-parameter cosine family. This ties into a natural question that has arisen: can Thurston's characterization be extended to entire transcendental maps? In this setting, the analog of postcritically finite maps are postsingularly finite maps on the complex plane mathbb{C}, but for our cosine family, these notions coincide. Our work is based on the major breakthrough of Hubbard, Schleicher, and Shishikura in their characterization of exponential maps. We adapt their techniques for our cosine family to prove a partial characterization of postsingularly finite topological cosine maps.