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Complex Dynamics:Julia Sets and the Mandelbrot Set

  • Mark McClure

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Mark McClure
    • 1
  1. 1.University of North Carolina at AshevilleAshevilleUSA

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