Types of Behavior with toral bands.

If two of the critical values of a Weierstrass elliptic function lie in one component of the Fatou set then the Fatou set will have a toral band.

There are four possibilities for the location of the third critical value:
(1) it lies in the Julia set.
(2) it belongs to a different cycle of Fatou components than the first two critical values.
(3) it belongs to the same cycle of Fatou components as the first two, but lies in a different Fatou component.
(4) it lies in the same Fatou component as the other two critical values.

We show examples of the first three possibilities.

■ The third critical value lies in the Julia set.

In this example, g2=26.5626 and g3=-26.2672, and the lattice is real rectangular.  There is an attracting fixed point at 1.5566, which is colored yellow, and points colored purple iterate to this point.  The Julia set is colored teal.  The critical values are colored red.

■ The third critical value belongs to a different cycle of Fatou components than the first two critical values.

In this example, g2=27.85 and g3=-28.338, and the lattice in this case is real rhombic.  There is a real superattracting fixed point at -3.047, and the points colored dark blue are attracted to this fixed point.  The two complex critical values head to an attracting fixed point at 1.542, and the points colored teal iterate here.  The points in the Julia set are colored red.

■  The third critical value belongs to the same cycle of Fatou components as the first two,but lies in a different Fatou component.

Here, g2=-1.451-4.984i and g3 = -2.136-0.801i.  There is an attracting two-cycle at {-.422+.517i, -.537+2.25i} which is colored yellow, and points colored purple head to this cycle.  The Julia set is colored teal.  The critical values are colored red.

Rhombic Example with R in the Julia set.

■ Here we exhibit a Weierstrass elliptic function on a rhombic lattice such that the real critical value is a prepole and hence the entire real axis lies in the Julia set.

When g2=.7 and g3=7.1512427 the lattice is real rhombic.  The complex critical values head to  attracting fixed points at -.652+.964i and -.652-.964i, colored yellow in the pictures below.  Points colored purple head to the fixed point at -.652+.964i , and points colored blue head to the fixed point at -.652-.964i.  The points colored teal are in the Julia set.  In this example, , and thus the real critical value is a prepole.

Below we zoom in to the real critical point, located at 1.08.

Created by Mathematica  (July 24, 2003)