Example 4.10.  Show that the Julia set for    is connected.

Solution.  We apply Theorem 4.9 and compute the orbits of 0 for  .  We have  ,  ,  ,  and  .  Thus the orbits of 0  are the sequence  ,  which is clearly a bounded sequence.  Thus, by Theorem 4.9, the Julia set for    is connected.

Explore Solution 4.10.

We apply Theorem 4.8 and compute the orbits of  0  for  .  

Thus, the orbits of  0  is the sequence    which is clearly a bounded sequence.  Thus, by Theorem 4.14, the Julia set for    is connected.

(c) 2006 John H. Mathews, Russell W. Howell