Follow the rabbit

The rabbit polynomial, or Douady's rabbit, is the quadratic polynomial f(z) = z2 + c, where c ≈ -0.122561 + 0.744862 i is a zero of the cubic polynomial c3 + 2c2 + c + 1. The critical points are 0 and ∞. The postcritial set is {0, c, c2+c, ∞}. Since there are four postcritical points and the critical points are simple, f is a NET map.

The Julia set for f is shown in the figure below. The filled Julia set is also called Douady's rabbit.

Since f is a NET map, f is given by a NET map presentation. This is described in detail in the paper Origami, affine maps, and complex dynamics. A NET map presentation diagram for f is shown below.

This figure encodes the information of the NET map presentation. Except for the circle around the vertex (2,1), this information is also given in the file rabbit.input, which is the input file for the program NETmap for this example. The parallelogram has vertices (0,0), 2 λ1, λ2, and 2 λ1 + λ2, where λ1 = (0,-1) and λ2 = (2,1). In addition there are green line segments joining (0,0) to (1,0), λ1 to (1,-1), λ2 to itself, and λ1 + λ2 to itself. The circle around the vertex λ2 indicates that it is the translation vector b. (The translation vector is either (0,0), λ1, λ2, or λ1 + λ2.) This isn't part of the input to NETmap because NETmap considers all four possible input vectors where that matters to the output.

When you run the program NETmap, it asks for inputs of the root of the file name of the input file, a bound N for the absolute values of the numerators and denominators of slopes p/q for which it will compute the slopes of the preimages, and the minimum and maximum x-coordinates, xmin and xmax, for the figure showing the half spaces. Here are the output files, with input N=50, xmin=-2, and xmax=1. The program creates the graphics files as postscript files, but here we have converted them to pdf files. For detailed explanations of the output files, see the program documentation in the menu on the left.