Julia Sets and the Mandelbrot Set

Exercises

1. (a) Find the cycles of the cardioids of several midget Mandelbrot sets in each spoke of the antenna above the 3-cycle disc. Can you find a pattern? Here is the answer.

(b) Find the cycles of the cardioids of several midget Mandelbrot sets in each spoke of the antenna above the 4-cycle disc. Can you find a pattern? Here is the answer.

(c) Find the cycles of the cardioids of several midget Mandelbrot sets in each spoke of the antenna above the 5-cycle disc. Can you find a pattern? Here is the answer.

(d) Putting all this together, can you find a general pattern? Here is the answer.

2. For the quadratic iteration z -> z2 + c we know

(i) Julia sets are either connected or are Cantor sets (the Dichotomy theorem),

(ii) the Julia set for c is connected if and only if the iterates of z0 = 0 remain bounded, and

(iii) the Mandelbrot set is the set of those c with connected Julia sets, that is, those c for which the iterates of z0 = 0 remain bounded.

The iterates start with z0 = 0 because that is the critical point of z2 + c and the connectivity of the Julia sets is determined by the iterates of the critical point.

In this problem we study the iteration of z -> (z2)/2 + (z3)/3 + c.

This function has two critical points, z0 = 0 and z0 = -1. In this exercise we explore the effect of two critical points on the shapes of the Julia sets. In particular, does the Dichotomy theorem still hold?

The left picture shows the c for which the iterates of z0 = -1 remain bounded, the middle picture shows the c for which the iterates of both z0 = -1 and z0 = 0 remain bounded, the right picture shows the c for which the iterates of z0 = 0 remain bounded. Call these sets M(-1), M(-1,0), and M(0).

M(-1) M(-1,0) M(0)

(a) Take points lying outside both M(-1) and M(0) (and consequently, also outside M(-1,0)). Are the Julia sets Cantor sets? Here is the answer.

(b) Take points lying inside M(-1,0) (and consequently, also inside M(-1) and M(0)). Are the Julia sets connected? Here is the answer.

(c) Take points in M(-1) but not in M(0). Are the Julia sets connected or Cantor sets? Here is the answer.

(d) Take points in M(0) but not in M(-1). Are the Julia sets connected or Cantor sets? Here is the answer.

(e) Take one example from each of parts (a) - (d). Follow the iterates of both critical points. What pattern can you infer? Here is the answer.

Return to Mandelbrot set and Julia Set Lab.