A fundamental dichotomy for Julia sets of a family of elliptic functions

Abstract

Abstract. 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 examples of each type.

Single and double toral band Fatou components in meromorphic dynamics

“…Elliptic functions with double toral bands and Cantor Julia sets have been wellstudied [13,16,18,20,21]; we begin by showing double toral bands do not need to be invariant so need not be associated to Cantor Julia sets.…”

“…For (1), every real number must iterate to the attracting fixed point by Proposition 3.6. For ( 2) and (3), we have that 0 ≤ |f Λ,b (z)| < 1 for all z ∈ R (see [20,21]). Using Proposition 3.6, f Λ,b must have exactly one fixed point on R which is attracting.…”

“…Recall that J(f ) is a Cantor Julia set if J(f ) is a compact, totally disconnected, perfect subset of Ĉ. Up to now, double toral bands, which often yield Cantor Julia sets, have been studied the most ( [16], [20], [21], [13]), and there are examples of elliptic functions with double toral bands where J(f ) is not Cantor [22]. It has also been shown for ℘ Λ , that if Λ is a square lattice or triangular lattice then J(℘ Λ ) is connected [8,16].…”

Untitled

“…Elliptic functions with double toral bands and Cantor Julia sets have been wellstudied [13,16,18,20,21]; we begin by showing double toral bands do not need to be invariant so need not be associated to Cantor Julia sets.…”

“…Recall that J(f ) is a Cantor Julia set if J(f ) is a compact, totally disconnected, perfect subset of Ĉ. Up to now, double toral bands, which often yield Cantor Julia sets, have been studied the most ( [16], [20], [21], [13]), and there are examples of elliptic functions with double toral bands where J(f ) is not Cantor [22]. It has also been shown for ℘ Λ , that if Λ is a square lattice or triangular lattice then J(℘ Λ ) is connected [8,16].…”