Experiment: Junction Points

Goal: Your goal in this experiment is to view a remarkable relationship between the structure of the Mandelbrot set near special c-values and the shape of the corresponding filled Julia sets.

Procedure: Using the program below (or the Julia set computer Lab 6 in CAS 330), for at least 5 different decorations attached directly to the main cardioid magnify the central portion (the junction point) of the antenna from which all the spokes emanate. Zoom in on this points several times using the magnify feature of the program. Now, using the mouse, click on this point to draw the filled Julia set for this c-value.

The junction point of the 2/5 bulb

Results: Compare the shape of the magnified Mandelbrot set near the junction point with the corresponding filled Julia set. Magnify the Julia set if necessary. Are there any similarities between these two images? Explain in an essay. In your essay include the c-values you worked with as well as several pictures explaining your observations.

Notes and Questions: The similarity between magnified portions of the Mandelbrot set and the corresponding filled Julia sets holds only near certain c-values such as the central junctions of the antenna. These are c-values for which 0 is eventually periodic. Such c-values are called Misiurewicz points.

The Mandelbrot set

Choose which portion of the set you want to display by changing the corners of the window. The set is plotted using the escape criterion (see Chapter 16 and 17 of the book). To visualize the whole set use the region (-2.1,-1.2)-(.7,1.2).

Iterates = 100;
\(\(Mandelbrot = 
        Compile[{{c, _Complex}}, 
          Length[FixedPointList[#\^2 + c &, 0, Iterates, 
              SameTest -> \((Abs[#] > 
                      2.0 &)\)]]];\)\)

CrealInf = -2.1;
CrealSup =  .7;
CimagInf = -1.2;
CimagSup = 1.2;

    
$TextStyle = {FontFamily -> "Times", FontSize -> 14}; 
    MandelbrotSet = 
      DensityPlot[
        Mandelbrot[x + I y], {x, CrealInf, CrealSup}, {y, CimagInf, 
          CimagSup}, Mesh -> False,  Frame -> True, 
        Axes -> False, PlotPoints -> 200, 
        AspectRatio -> Automatic, 
        FrameLabel -> {"Re(c)", "Im(c)"}, 
        PlotLabel -> "The c-plane", RotateLabel -> False, 
        ColorFunction -> (If[# ≥ 1, RGBColor[0, 0, 0], 
                Hue[#]] &), ImageSize -> 400];

Filled Julia sets

The first cell compiles the function and the second one draws the picture. Choose a value of c to plot the corresponding filled Julia set.

Iterates = 100;
\(\(Julia = 
        Compile[{{z, _Complex}, {c, _Complex}}, 
          Length[FixedPointList[#\^2 + c &, z, Iterates, 
              SameTest -> \((Abs[#] > 
                      2.0 &)\)]]];\)\)

c = - .7 - 0.28 I;

    
RealZInf = -1.6;
RealZSup = 1.6;
ImagZInf = -1.2;
ImagZSup = 
        1.2;


$TextStyle = {FontFamily -> "Times", 
          FontSize -> 14};
FilledJuliaSet = 
        DensityPlot[
          Julia[x + I y, c], {x, RealZInf, RealZSup}, {y, ImagZInf, 
            ImagZSup}, Mesh -> False,  Frame -> True, 
          Axes -> False, PlotPoints -> 200, 
          AspectRatio -> Automatic, 
          FrameLabel -> {"Re(z)", "Im(z)"}, 
          RotateLabel -> False, 
          PlotLabel -> "Filled Julia set for c =" <> ToString[c], 
          ColorFunction -> (If[# ≥ 1, RGBColor[0, 0, 0], 
                  Hue[#]] &), 
          ImageSize -> 400];
Clear[c];