Connectivity properties of Julia sets of Weierstrass elliptic functions

We discuss the connectivity properties of Julia sets of Weierstrass elliptic ℘ functions, accompanied by examples. We give sufficient conditions under which the Julia set is connected and show that triangular lattices satisfy this condition. We also give conditions under which the Fatou set of ℘ contains a toral band and provide an example of an order two elliptic function on a square lattice whose Julia set is a Cantor set

Blaschke products and parameter spaces

Advisors: Alastair Fletcher.Committee members: Sien Deng; Zhuan Ye.In this thesis, we discuss certain aspects of complex dynamics. We will introduce the important concepts in iteration theory, discuss examples of families of holomorphic mappings, and their dynamics. In particular, we will discuss the family of quadratic functions, the family of Mobius mappings of the disk, and a certain sub-class of Blaschke products. We will show how the dynamics depend on the parameters.M.S. (Master of Science

Green's function and anti-holomorphic dynamics on a torus

We give a new, simple proof of the fact recently discovered by C.-S. Lin and C.-L. Wang that the Green function of a torus has either three or five critical points, depending on the modulus of the torus. The proof uses anti-holomorphic dynamics. As a byproduct we find a one-parametric family of anti-holomorphic dynamical systems for which the parameter space consists only of hyperbolic components and analytic curves separating them.Comment: 17 pages, 3 figures (some details added, some overall revision

Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals

We investigate the existence of the meromorphic extension of the spectral zeta function of the Laplacian on self-similar fractals using the classical results of Kigami and Lapidus (based on the renewal theory) and new results of Hambly and Kajino based on the heat kernel estimates and other probabilistic techniques. We also formulate conjectures which hold true in the examples that have been analyzed in the existing literature

Rigidity and absence of line fields for meromorphic and Ahlfors islands maps

In this note, we give an elementary proof of the absence of invariant line fields on the conical Julia set of an analytic function of one variable. This proof applies not only to rational as well as transcendental meromorphic functions (where it was previously known), but even to the extremely general setting of Ahlfors islands maps as defined by Adam Epstein. In fact, we prove a more general result on the absence of invariant_differentials_, measurable with respect to a conformal measure that is supported on the (unbranched) conical Julia set. This includes the study of cohomological equations for $\log|f'|$, which are relevant to a number of well-known rigidity questions.Comment: 17 page

Heights and totally pp-adic numbers

We study the behavior of canonical height functions $\widehat{h}_f$, associated to rational maps $f$, on totally $p$-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of $\widehat{h}_f$ on the maximal totally $p$-adic field if the map $f$ has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset $X$ in the compositum of all number fields of degree at most $d$ such that $f(X)=X$ for some non-linear polynomial $f$. This answers a question of W. Narkiewicz from 1963.Comment: minor changes: rewording and reference update