## Abstract

Some examples of filled-in Julia sets. For a complex number *c*, the filled-in Julia set is the set of those points in ? whose orbits under the function *z* ^{2} + *c* do not approach infinity. The upper left image is the filled-in Julia set where *c* = −0.123 + 0.745 *i*, known as Douady’s rabbit; in this case there is an attracting 3-cycle. The upper right corresponds to *c* = 0.32 + 0.043 *i*, which has an attracting 11-cycle. At the lower left is the Julia set for a different function, a bifurcation of the quadratic map *rz*(1−*z*) at *r* = 3, discussed in Chapter 7. The image at lower right is the Mandelbrot set, which encodes the collection of *c* for which the Julia set of *z* ^{2} + *c* is connected.

## Keywords

Unit Circle Periodic Point Inverse Iteration Siegel Disk Attractive Fixed Point
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## Copyright information

© Springer Science+Business Media, LLC 2010