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A Subfamily of Julia Sets

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The Julia set of a complex rational function is the self-similar set of points for which the orbits of are bounded ( is the iterate of ). These sets were named after the the French mathematician Gaston Julia, who published his results in 1918. One subfamily of Julia sets with interesting topology and dynamics uses functions of the type

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where , and starts the iterations at the point . Sets homeomorphic to the Sierpinski carpet appear, as well as Cantor and Mandelpinski necklaces, and several other structures. These sets have interesting properties related to critical points and connectedness. They are a source of open problems.

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Contributed by: Enrique Zeleny (January 2015)
Open content licensed under CC BY-NC-SA

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A site with a series of papers about this family, mainly by Robert L. Devaney, can be found at http://math.bu.edu/people/bob/papers.html.

Reference

[1] R. L. Devaney, A First Course in Chaotic Dynamical Systems, Westview Press, 1992.

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