Exercise 9 (a).  Find a value for  c  that is in the Mandelbrot set such that its negative,  -c,  is not in the Mandelbrot set.  

Solution 9 (a).

See text and/or instructor's solution manual.

There are many examples.

The Mandelbrot set is

               

Given the function   .  

        For the initial seed    we compute the iterates as follows:  

          ,     ,     ,  ...  .  

Hence, the orbit of      generated by      is the set     which is a bounded set.

Given the function   .  

        For the initial seed    we compute the iterates as follows:  

          ,     ,     ,     ,  ...  .   

Hence, the orbit of      generated by      is the set     which is an unbounded set.

The number  -2,  is in the Mandelbrot set, but its negative,  2,  is not in the Mandelbrot set.  

We are done.   

Aside.  We can use Mathematica to explore this situation.

This solution is complements of the authors.

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