Complex Dynamics:Julia Sets and the Mandelbrot Set

Mark McClure
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Mark McClure: University of North Carolina at Asheville

Chapter 11 in Mathematica in Action, 2010, pp 277-300 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/978-0-387-75477-2_12

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