Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

Periodic Points of Quadratic Polynomials

October 1998

In a chapter from The Beauty of Fractals, Adrien Douady (who coined the term Mandelbrot set) recollects:

... in 1980, whenever I told my friends that I was just starting with J.H. Hubbard a study of polynomials of degree 2 in one complex variable (and more specifically those of the form zz2 + c), they would all stare at me and ask: Do you expect to find anything new?

However, Douady and Hubbard made a major contribution to the understanding of the Mandelbrot set and its role in the Description of dynamics of iterative processes. As the Fundamental Theorem of Algebra clearly indicates, the complex plane rather than the real line is the proper place for the study of polynomials. And, in hindsight, the study of even simple polynomials of degree 2 in the complex plain has thrown a new light on the behavior of more general iterative processes.

Consider the simplest parabola f(x) = x2 with x real. Starting with a point x0 we form a sequence, x0, x1 = f(x0), x2 = f(x1), x3 = f(x2), and so on, where each consecutive term is obtained by applying the function f to its predecessor. (This is a general definition of an iterative process that works for any function f.)

For x0 = 1/2 we successively get, x1 = 1/4, x2 = 1/16, x3 = 1/256, ... The terms become smaller and eventually converge to x = 0. The same is, of course, true for any starting point x0 that satisfies |x0| < 1. 0 is an attractive fixed point for f(x) = x2. Likewise, for |x0| > 1, the terms grow without bound towards infinity (). Two points satisfy |x0| = 1, x0 = ±1. Obviously, f(1) = 1 so that 1 is a fixed point of f(x). Also, with x0 = -1, xk = 1, for k > 0. There is nothing to be added in the case of the real parameter x.

Consider now the same function in the complex plane: f(z) = z2. z = 0 is still an attractive fixed point to which all iterations converge that start with z0, |z0| < 1. Also, as before, if |z0| > 1, the iterations grow without bound. However, interesting things happen when |z0| = 1. All such points are expressed as eit = cos(t) + i sin(t), where t is the radian measure of the corresponding central angle. When the number is squared, t doubles. Take, for example, z0 = e2/3, so that z03 = 1. We have, z1 = e2*2/3, z2 = e4*2/3 = e2/3 = z0. The two points z0 and z1 form a cycle of period 2. With z0 = e2/5, z1 = e2*2/5, z2 = e4*2/5, z3 = e8*2/5 = e3*2/5, z4 = e6*2/5 = e2/5 = z0, a cycle of period 4. Since modulo 7 doubling of 1 leads to the sequence 1, 2, 4, 8 = 1 (mod 7), starting with z0 = e2/7 gives rise to a cycle of period 3. All numbers corresponding to a fractional part of the whole circle belong to one cycle or another. In 1975, Li and Yorke proved that once there is a cycle of order 3, there are cycles of any other order. (You may try finding such cycles for a few periods other than 2,3,4. Also there exists another cycle of period 3. Can you find it?)

It is natural to think that the behavior of iterations that start with z0 = eit, where t is not a fractional part of 2 is chaotic and generate the set dense on the circle. This is not always the case. Look at the number 22-n!. The sum, when shifted a digit at a time (which is equivalent to multplying it by 2) fills in the set with the only accumulation points at 0 and the negative powers of 2.

For a function f, its filled-in Julia set Kf is defined as the set of starting points z0 for which the iterations {zk} remain bounded. The boundary of Kf is known as the Julia set, Jf. The Julia set of f(z) = z2 is the unit circle, its filled-in Julia set is the unit disk (the unit circle plus its interior.)

Unexpectedly, a great variety of shapes is obtained by a small variation in the basic function. Mandelbrot began with fc(z) = z2 + c. It was proven yet by Julia and Fatou before 1920 that Julia sets are either connected or totally disconnected (something like the Cantor set, where, for any two points, it is possible to find two subsets both open and closed in the set, each containing only one of the points.) For fc, the usual notation for the Julia set is (a simplified) Jc. In 1980, Mandelbrot has discovered the set M of parameter values c for which Julia sets are connected. This set that now bears his name may also be defined as the set of c's for which iterations {zk} starting with z0 = 0 remain bounded. This is due to another theorem of Julia and Fatou that links critical points (points where the derivative of the function vanishes) of a function with the behavior of iterations. The only critical point of fc is z = 0. This is why the point 0 appears in the definition of the Mandelbrot set.

The main purpose of the Mandelbrot set is to index Julia sets corresponding to various values of the parameter c. When c belongs to the Mandelbrot set, Jc is connected. For c outside M, Jc is totally disconnected and known as the fractal dust. Both M and Jc are visualized with a simple algorithm that assigns a color value to a pixel depending on how fast it was found out whether iterations for that pixel escape to infinity. Since for c inside M, the iterations remain bounded, pixels corresponding to the Mandelbrot set consume the greatest amount of the computational time.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

However, iterations inside M evolve differently depending on the value of c. (Behavior of the iterations is related to the appearance of the Julia sets Jc.) For example, for c inside the big cardioid, the iterations converge. For c inside the big circle to the left of the cardioid, the iterations converge to a cycle of period 2. For c inside each wart attached to the cardioid, the iterations converge to a periodic cycle whose period is determined by the corresponding wart. The applet helps you visualize the behavior of the iterates. To select c, click inside the applet. Press the "Iterate" button to see the progress of the iterative process.

The wart that contains the point (-0.1, .75) results in 3-cycles. The point (-0.1, 0.85) that belongs to a smaller wart attached to the previous one, leads to a 6-cycle, very much as in the period doubling we observed for real-valued iterations. For c = (-0.735, 0.165) the iterations converge rather slowly in a delightfully chaotic manner. Starting with c = (-0.5, 0.575), the program draws a nice 5-star. (If the image of the Mandelbrot set disappears, for example, after scrolling the page, click the "Clear" button to bring it back.


  1. Chaos and Fractals, R.L.Devaney and Linda Keen, eds., AMS, 1989.
  2. G.Chang and T.W.Sederberg, Over And Over Again, MAA, 1997
  3. D.Davis, The Nature and Power of Mathematics, Princeton University Press, 1993
  4. T.Y.Li and J.A.Yorke, "Period Three Implies Chaos", Amer. Math. Monthly, 82, 985-992.
  5. B.Mandelbrot, The Fractal Geometry of Nature, W.H.Freeman and Co., NY, 1977.
  6. H.-O.Peitgen, P.H.Richter, The Beauty of Fractals, Springer-Verlag, 1986
  7. The Science of Fractal Images, H.-O.Peitgen, D.Saupe, eds., Springer-Verlag, 1988

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