Questions and extensions

• why are the critical points so important for the theory?

• develop a program to display the Julia curves in more detail.

• include an option in the program to explore the Mandelbrot set.

• for some ranges of c the contours shown in Figure 6 pinch off and become separated. Is there any particular significance to this phenomenon, or of the region in the Mandelbrot set in which it does not occur?

• the curves in Figures 8 and 9 represent only part of the theme developed in the reference to Jensen, Kadanoff and Procaccia. Modify the program to calculate the ratios of the peripheral chords according to this article; investigate the validity of the correlations implied by their two stage de Bruijn matrix.

• quadratic iterations give a comprehensive introduction to the general theory, but further varieties of designs can be obtained by varying the iteration equations.

• the diagram of fixed points as a function of $\mu$ for real iterations would be much more informative if relative frequencies were shown in the chaotic bands, which can be done nicely by coloring the display.

• the windows between bands of the $\mu$ vs fixed-point curves correspond to the nodules surrounding the Mandelbrot set. Examining both diagrams at greater magnification should reveal details of this correspondence.

• fixed points of the higher iterates correspond to cycles of doubled period, but this is no longer true once the chaotic region has been reached. Look up some of the literature concerning possible periods and how to determine them.

• modify the program to follow iterates of points from various regions of the complex plane and designated parameters from the Mandelbrot set. What limitations on the trajectory are implied by the finiteness of the fixed point representation, and thus on the details which the program can explore?

• use the contour program in <PLOT> to build up the Mandelbrot set by finding where iterates of z²-c have unit derivative.