Iterates

Robert P. Munafo, 2023 Aug 5.

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:

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.

Diverging to Infinity

In the recurrence relation (shown above), if Z and C are both real numbers, then the result of each iteration will be a real number, and if the iteration "diverges to infinity", that will involve a large positive value of Z_n that keep getting larger as n increases. It never "diverges to negative infinity".

For the Mandelbrot Set, Z and C are usually non-real, and "diverging to infinity" means that the magnitude of Z_n gets progressively larger. In the usual interpretation of complex numbers as points on a plane, there is no single "infinity" to which the iteration diverges, because the angle will keep changing.

However this interpretation of the complex plane is not optimal for generalising the quadratic iteration Z -> Z² + C to complex numbers; the Riemann sphere is more applicable. For practical purposes this means that all "overflows" of either the real or imaginary component are interpreted as meaning that the value of Z has diverged.

Dwell Limit

Typically in practice one needs to limit how many times the calculation Z_n+1 = Z_n² + 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.

Myrberg Version

Pekka Myrberg studied two equivalent recurrence relations, on the real axis only, in the late 1950s and early 1960s. A 1962 paper [PM1] uses the recurrence Z_n+1 = 1 + C Z_n² (see Myrberg polynomials), significant for being the first to work out the value of the coordinate commonly called the "Feigenbaum point", to 7 digits. His 1963 paper [PM2] uses the recurrence Z_n+1 = Z_n² - C, and is the first to publish coordinates of the real-axis mu-atom nuclei for periods 3 and 4, with 9-digit and 11-digit accuracy respectively.

Related Methods

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.