4.2  Julia and Mandelbrot Sets

    An impetus for studying complex analysis is the comparison of properties of real numbers and functions with their complex counterparts. In this section we take a look at Newton's method for finding solutions to the equation  .  Then, by examining the more general topic of iteration, we will plunge into a breathtaking world of color and imagination. The mathematics surrounding this topic has generated a great deal of popular attention in the past few years.

    Recall from calculus that Newton's method method proceeds by starting with a function and an initial "guess"    as a solution to  .  We then generate a new guess    by the computation  .  Using   in place of  ,  this process is repeated, giving us  .  Thus we obtain a sequence of points  , where  .  The points    are called the iterates of  .  For functions defined on the real numbers, this method gives remarkably good results so that the sequence    often converges to a solution of    rather quickly.  In the late 1800's the british mathematician Arthur Cayley investigated the question as to whether Newton's method can be applied to complex functions.  He wrote a paper giving an analysis for how this method works for quadratic polynomials and indicated his intention to publish a subsequent paper for cubic polynomials.  Unfortunately, Cayley died before producing this paper. As you will see, the extension of Newton's method to the complex domain and the more general question of iteration are quite complicated.

Example 4.7.  Trace out the next five iterates of Newton's method given an initial guess of    as a solution to the equation  ,  where  .  

Solution.  For any guess z for a solution, Newton's method gives as the next guess the number  .  Table 4.1 gives the required iterates, rounded to five decimal places.  Figure 4.2 shows the relative positions of these points on the z plane.  Note that the points are so close together that they appear to coincide, and that the value for agrees to five decimal places with the actual solution .  

            Table 4.1  The iterates of for Newton's method applied to .

            Figure 4.2  The iterates of for Newton's method applied to .

Explore Solution 4.7.

    The complex version of Newton's method also appears to work quite well.  Recall, however, that with functions defined on the reals, not every initial guess produces a sequence that converges to a solution.  Example 4.8 shows that the same is true in the complex case.

Example 4.8.  Show that Newton's method fails for the function    if the initial guess is a real number.

Solution.  From Example 4.7 we know that, for any guess z as a solution of  ,  the next guess at a solution is  .  We let be any real number and be the sequence of iterations produced by the initial seed .  If for any k,  ,  the procedure terminates, as will be undefined.  If all the terms of the sequence are defined, an easy induction argument shows that all the terms of the sequence are real.  Because the solutions of    are  ,  the sequence cannot possibly converge to either solution.  In the exercises we ask you to explore in detail what happens when is in the upper or lower half-plane.

    The case for cubic polynomials is more complicated than that for quadratics.  Fortunately, we can get an idea of what's going on by doing some experimentation with computer graphics.  We begin with the cubic polynomial .  (Recall that the roots of this polynomial are at .)  We associate a color with each root (blue, red, and green, respectively).  We form a rectangular region , which contains the three roots of , and partition this region into equal rectangles  .  We then choose a point    at the center of each rectangle and for each of these points we apply the following algorithm.

1.  With  ,  compute  .  Continue computing successive iterates of this initial point until we either are within a certain preassigned tolerance (say, ) of one of the roots of   ,  or until the number of iterations has exceeded a preassigned maximum.

2.  If Step 1 leaves us within  of one of the roots of , we color the entire rectangle    with the color associated with that root.  
Otherwise, we assume that the initial point    does not converge to any root, and we color the entire rectangle yellow.

    Note that this algorithm doesn't prove anything.  In Step 2, there is no a priori reason to justify the assumption mentioned, nor is there any necessity for an initial point to have its sequence of iterates converging to one of the roots of , just because a particular iteration is within  of that root.  Finally, the fact that one point in a rectangle behaves in a certain way does not imply that all the points in that rectangle behave in a like manner.  Nevertheless, we can use this algorithm as a basis for mathematical explorations.  Indeed, computer experiments such as the one described have contributed to a lot of exciting mathematics during the past 30 years.  The color plates located on the inside front and back covers of this book illustrate the results of applying our algorithm to various functions.  Color plate 1 shows the results for the cubic polynomial  .  The points in the blue, red, and green regions are those "initial guesses" that will converge to the roots   -1,  respectively.  (The roots themselves are located in the middle of the three largest colored regions.)  The complexity of this picture becomes apparent when you observe that, wherever two colors appear to meet, the third color emerges between them.  But then, a closer inspection of the area where this third color meets one of the other colors reveals again a different color between them.  This process continues with an infinite complexity.

Color Plate 1.  Newton's method applied to  .

Extra Example 1.  Investigate Newton's method for finding the roots of  .
Given the initial seed    determine if the sequence    converges to one of the roots  .  

Explore Extra Solution 1.

    There appear to be no yellow regions with any area in Color plate 1, indicating that at least most initial guesses at a solution to    will produce a sequence that converges to one of the three roots.  Color plate 2 demonstrates that this outcome does not always occur.  It shows the results of applying the preceding algorithm to the polynomial  .  The yellow area shown is often referred to as the rabbit.  It consists of a main body and two ears.  Upon closer inspection (Color plate 3) you can see that each of the ears consists of a main body and two ears.  Color plate 2 is an example of a fractal image.  Mathematicians use the term fractal to indicate an object that have this kind of recursive structure.

Color Plate 2.  The rabbit fractal of  .  

Color Plate 3.  A zoom of the rabbit fractal of  .  

 

    In 1918 the French mathematicians Gaston Julia and Pierre Fatou noticed the fractal phenomenon when exploring iterations of functions not necessarily connected with Newton's method.  Beginning with a function  f(z)  and a point  ,  they computed the iterates  , , , ,  and investigated properties of the sequences  .  Their findings did not receive a great deal of attention, in part because computer graphics were not available at this time.  With the recent proliferation of computers, it is not surprising that these investigations were revived in the 1980's.  Detailed studies of Newton's method and the more general topic of iteration were undertaken by a host of mathematicians including Curry, Douady, Garnett, Hubbard, Mandelbrot, Milnor and Sullivan.  We now turn our attention to some of their results by focusing on the iterates produced by quadratics of the form  .  You will be surprised at the startling pictures that graphical iterates of such a simple functions produce.

Example 4.9.  For  ,  analyze all possible iterations when  ,  that is, for the function defined by  .

Solution.  We leave as an exercise the claim that, if  ,  the sequence will converge to 0;  if  ,  the sequence will be unbounded;  and if  ,  the sequence will either oscillate around the unit circle or converge to 1.

    For the function    defined by  ,  and an initial seed  ,  the set of iterates given by  ,  ,  etc., is also called the orbits of    generated by  .  We let    denote the set of points with bounded orbits for  .  Example 4.9 shows that    is the closed unit disk .  The boundary of    is known as the Julia set for the function  .  Thus the Julia set for    is the unit circle  .  It turns out that    is a nice simple set only when  .  Otherwise,     is fractal.  Color plate 4 shows .  The variation in colors indicate the length of time it takes for points to become "sufficiently unbounded" according to the following algorithm, which uses the same notation as our algorithm for iterations via Newton's method.

Color Plate 4.  The Julia set for  .

1.  Compute  .  Continue computing successive iterates of this initial point until the absolute value of one of the iterations exceeds a certain bound (say, L), or until the number of iterations has exceeded a preassigned maximum.
2.  If Step 1 leaves us with an iteration whose absolute value exceeds L, we color the entire rectangle    with a color indicating the number of iterations needed before this value was attained (the more iterations required, the darker the color).  Otherwise, we assume that the orbits of the initial point    do not diverge to infinity, and we color the entire rectangle black.

    Note, again, that this algorithm doesn't prove anything.  It merely guides the direction of our efforts to do rigorous mathematics.

    Color plate 5 shows the Julia set for the function  ,  where  .  The boundary of this set is different from the boundaries of the other sets we have seen, in that it is disconnected.  Julia and Fatou independently discovered a simple criterion that can be used to tell when the Julia set for    is connected or disconnected.  We state their result, but omit the proof, as it is beyond the scope of this text.

Color Plate 5.  A disconnected Julia set for  .  

Theorem 4.9.  The boundary of    is connected if and only if  .  In other words, the Julia set for    is connected if and only if the orbits of  0  are bounded.

Proof.

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.

    In 1980, the Polish born mathematician Benoit Mandelbrot used computer graphics to study the following set:  

               

Extra Example 2.  Plot the Julia set.

Explore Extra Solution 2.

    The set M has come to be known as the Mandelbrot Set.  Color plate 6 shows its intricate nature. Technically, the Mandelbrot set is not fractal because it is not self-similar (although it may look that way).  However, it is infinitely complex.  Color plate 7 shows a zoom over the upper portion of the set shown in Color plate 6.  Likewise, Color plate 8 zooms in on the upper portion of Color plate 7.  In Color plate 8 you can see the emergence of another structure very similar to the Mandelbrot set that we began with.  Although it isn't an exact replica, if you zoomed in on this set at almost any spot, you would eventually see yet another "Mandelbrot clone" and so on ad infinitum!  In the remainder of this section we look at some of the properties of this amazing set.

Color Plate 6.  The Mandelbrot set M.

Color Plate 7.  A zoom of the upper portion of the Mandelbrot set M.

Color Plate 8.  A zoom of the upper portion of the Color Plate 7.

Example 4.11.  Show that is a subset of the Mandelbrot set  M.

Solution.  Let    be the orbits of 0 generated by  ,  where  .  Then

              

We show that    is bounded, and, in particular, we show that    for all n by mathematical induction.  Clearly    if  .  We assume that    for some value of    (our goal is to show ).  Now,  

            
            
            

    In the exercises, we ask you to show that, if  ,  then c is not in the set M.  Thus the Mandelbrot set depicted in Color plate 6 contains the disk   and is contained in the disk .  

        We can use other methods to determine which points belong to M.  To do so, we need some additional vocabulary.

Definition 4.6 (Fixed Point).  The point    is a fixed point for the function  .

Definition 4.7 (Attracting Point).  The point    is an attracting point for the function .

    Theorems 4.10 and 4.11 explain the significance of these terms.

Theorem 4.10.  Suppose that   is an attracting fixed point for the function .  Then there is a disk    about    such that the iterates of all the points in    are dawn toward the point    in the sense that, if  ,  then  .  In fact, if is the iterate of  ,  then  .

Proof.

Proof of Theorem 4.10 is in the book.
Complex Analysis for Mathematics and Engineering

    In 1905, Pierre Fatou showed that, if the function    defined by    has attracting fixed points, then the orbits of 0 determined by    must converge to one of them.  Because a convergent sequence is bounded, this condition implies that c must belong to M.  In the exercises we ask you to show that the main cardioid-shaped body of M in Color plate 6 is composed of those points c for which    has attracting fixed points.  You will find Theorem 4.10 to be a useful characterization of those points.

Theorem 4.11.  The function    defined by    has attracting fixed points iff    or  ,  where the square root designates the principal square root function.

Proof.

Proof of Theorem 4.11 is in the book.
Complex Analysis for Mathematics and Engineering

Definition 4.8 (n-Cycle).  An n-cycle for a function    is a set    of  n  complex numbers such that    for  ,  and  .  

Definition 4.9 (Attracting n-Cycle).  An n-cycle    for a function    is said to be attracting if  ,  where    is the
composition of    with itself  n  times.  For example, if  ,  then .

Example 4.12.  Example 4.10 shows that    is a 2-cycle for the function  .  It is not an attracting 2-cycle because    and  .  Hence  ,  so that  .  

Explore Solution 4.12.

    In the exercises, we ask you to show that, if is an attracting n-cycle for a function f, then not only does satisfy  ,  but also that  ,  for  .  

    It turns out that the large disk to the left of the cardioid in Color plate 6 consists of those points c for which    has a 2-cycle.  The large disks above and below the main cardioid disk are the points c for which    has a 3-cycle.

    Continuing with this scheme, we see that the idea of  n-cycles explains the appearance of the "buds" that you see on Color plate 6.  It does not, however, begin to do justice to the enormous complexity of the entire set.  Even Color plates 7 and 8 are mere glimpses into its awesome beauty.  In the exercises, we suggest several references for projects that you could pursue for a more detailed study of topics relating to those covered in this section.

Extra Example 3.  Plot the Mandelbrot set.

Explore Extra Solution 3.