Iterates

Robert P. Munafo, 2002 May 29.

The values of Z_n that are calculated during an iteration process. The iterates form a sequence of points Z_n (also called the point's orbit or the critical orbit of the point's Julia set), with one member for each positive integer n. The sequence is defined by the recurrance relation:

Z₀ = 0

Z_n+1 = Z_n² + C

where C is the point for which the iteration is being performed.

If the values of Z_n diverge to infinity by getting progressively larger and larger, the point C is not in the Mandelbrot set.

If the values converge on a single value or a finite repeating set of N values, the point is in the Mandelbrot Set and is said to have period N. The set of N values is the limit cycle.

If the values follow a chaotic, non-repeating pattern and never diverge to infinity the point is in the Mandelbrot Set and also on the boundary. Not all points on the boundary have chaotic iteration, however. The Misiurewicz points are the best examples. See also accuracy.

See the page on algorithms for more information about how to write a Mandelbrot program.

Julia sets can be plotted via the inverse iteration method.