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. When the number is squared, t doubles. The two points z 0 and z 1 form a cycle of period 2.
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. Also there exists another cycle of period 3. This is not always the case. The boundary of K f is known as the Julia set , J f . Unexpectedly, a great variety of shapes is obtained by a small variation in the basic function.