Locally Sierpinski Julia Sets of Weierstrass Elliptic P Functions

Locally Sierpinski Julia Sets of Weierstrass Elliptic ℘ Functions

We define a locally Sierpinski Julia set to be a Julia set of an elliptic function which is a Sierpinski curve in each fundamental domain for the lattice. In order to construct examples, we give sufficient conditions on a lattice for which the corresponding Weierstrass elliptic ℘ function is locally connected and quadratic-like, and we use these results to prove the existence of locally Sierpin...

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 w...

Proof of a Folklore Julia Set Connectedness Theorem and Connections with Elliptic Functions

We prove the following theorem about Julia sets of the maps fn,p,γ(z) = z n + γ zp , for integers n, p ≥ 2, γ ∈ C by using techniques developed for the Weierstrass elliptic ℘ function and adapted to this setting. Folklore connectedness theorem: If fn,p,γ has a bounded critical orbit, then J(fn,p,γ) is connected. This is related to connectivity results by the author and others about J(℘), where ...

A Fundamental Dichotomy for Julia Sets of a Family of Elliptic Functions

We investigate topological properties of Julia sets of iterated elliptic functions of the form g = 1/℘, where ℘ is the Weierstrass elliptic function, on triangular lattices. These functions can be parametrized by C − {0}, and they all have a superattracting fixed point at zero and three other distinct critical values. We prove that the Julia set of g is either Cantor or connected, and we obtain...

Connectivity Properties of Julia Sets of Weierstrass Elliptic Functions First Final Draft