A Cantor Julia Set for a general elliptic function

Let Λ=[λ,λi] be the square lattice with invariants g_2(Λ)=16 and g_3(Λ)=0 and let ℘ denote the Weierstrass elliptic function defined in terms of Λ.  Let

f(z) = (℘(z) + 1)/℘(z).

Then f is an elliptic function, periodic with respect to Λ, with an attracting fixed point at approximately 1.29.  The poles of f are located at (λ+λi)/2+Λ ~= .92+.92i + Λ.

The purple points in the picture below are in the Fatou set of f, and the blue points are potentially in the Julia set of f.

[Graphics:HTMLFiles/index_2.gif]

The pictures below are colorings of the Fatou set of f depending on the number of iterations it takes to get to within .11 of the attracting fixed point.  The attracting fixed point is located in the center of the pink disk in the first picture.  The points potentially in the Julia set are colored red.

We successively zoom in on the pole located at approximately .92+.92i

[Graphics:HTMLFiles/index_3.gif]

[Graphics:HTMLFiles/index_4.gif]

[Graphics:HTMLFiles/index_5.gif]

[Graphics:HTMLFiles/index_6.gif]

[Graphics:HTMLFiles/index_7.gif]

The following is a zoom on a different area.

[Graphics:HTMLFiles/index_8.gif]

Graphics by J. Hawkins and L. Koss


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