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 Quadratic map applet 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 of the spokes emanate. Zoom in on this point 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. Magnify the corresponding junction points in these filled Julia sets.
Results: Compare the shape of the magnified Mandelbrot set near the junction point with the corresponding magnifications of the filled Julia set. 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.