# Iterates

Robert P. Munafo, 2012 Apr 20.

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 recurrence relation:

Z_{0} = 0

Z_{n+1} = Z_{n}^{2} + 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.

Typically in practice one needs to limit how many times the
calculation Z_{n+1} = Z_{n}^{2} + C is performed, using
a maximum dwell value of some kind.
See the page on algorithms for more information about how to write a
Mandelbrot program.

Much more informative pictures can be produced with the extra information provided by the distance estimator algorithm.

Julia sets can be plotted via the inverse iteration method.

See also inverse Mandelbrot iteration.

revisions: 20020529 oldest version on record; 20111208 add link to DEM/M; 20120420 link to maximum dwell

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2012. Mu-ency index