Exercise 7.  Prove that if    is in the Mandelbrot set, then its conjugate    is also in the Mandelbrot set.  

Thus, the Mandelbrot set is symmetric about the x-axis.   

Hint.   Use mathematical induction.  

Solution 7.

See text and/or instructor's solution manual.

        Suppose   ,   and let    be the orbit of  0  under  .  

By definition of  M, the be the orbit of  0  under      is a bounded set.

Hence, there is some real number  N  such that      for all  k.  

        Let    be the iterates of      under   .   Calculation reveals that

                    .  

Assume that for some    that      then we will also have  .     Calculation reveals that

                    .  

Therefore, by mathematical induction we have      for  all  k.  

We can use the same real number  N  for which      for all  k,  and we have      for all  k.  

Hence, the orbit of  0  determined by      is a bounded set.

Therefore    is also in the Mandelbrot set.    

This solution is complements of the authors.

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