Connectivity of Julia sets for Weierstrass elliptic functions on square lattices
For a meromorphic function \(f\) in the complex plane and a positive integer \(n\) denote the \(n\)-th iterate of \(f\) by \(f^n\). The Fatou set \(F(f)\) of \(f\) is defined as the set of all \(z \in \mathbb{C}\) such that \((f^n)\) is defined and normal in a neighbourhood of \(z\), and the Julia set \(J(f)\) of \(f\) is given by \(J(f) = \mathbb{C}_\infty \setminus F(f)\). If \(z_0 \in \mathbb{C}\) is such that \(f'(z_0)=0\) or if \(z_0\) is a multiple pole of \(f\), then \(w_0=f(z_0)\) is called a critical value of \(f\). In this paper the author shows that the Julia set of a Weierstrass elliptic function \(\wp_\Omega\) on a square lattice \(\Omega\) is connected by proving that every Fatou component of \(\wp_\Omega\) contains at most one critical value. Furthermore, he considers the Julia sets of a family of rational functions which arise from the Laurent series of \(\wp_\Omega\).
scientific article; zbMATH DE number 6038829
scientific article