Exercise 5.  Show that the Julia set for      is connected.  

Solution 5.

See text and/or instructor's solution manual.

Given the function     and the initial seed    we compute the iterates as follows:  

          ,     ,     ,  ...  .  

Hence, the orbit of      generated by      is the set   .  

The Julia set for    is connected by Theorem 4.9 because the orbit of  0  under    is a bounded set.

We are done.   

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell