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Multibrot sets

Multibrot sets are bounded sets found in the complex plane for members of the general monic univariate polynomial family of recursions

For an integer d, these sets are connectedness loci for the Julia sets built from the same formula. The full cubic connectedness locus has also been studied; here one considers the two-parameter recursion z ↦ z 3 + 3 k z + c {\displaystyle z\mapsto z^{3}+3kz+c}, whose two critical points are the complex square roots of the parameter k. A parameter is in the cubic connectedness locus if both critical points are stable. For general families of holomorphic functions, the boundary of the Mandelbrot set generalizes to the bifurcation locus.

The Multibrot set is obtained by varying the value of the exponent d. The article has a video that shows the development from d = 0 to 7, at which point there are 6 i.e. (d − 1) {\displaystyle (d-1)} lobes around the perimeter. In general, when d is a positive integer, the central region in each of these sets is always an epicycloid of (d − 1) {\displaystyle (d-1)} cusps. A similar development with negative integral exponents results in (1 − d) {\displaystyle (1-d)} clefts on the inside of a ring, where the main central region of the set is a hypocycloid of (1 − d) {\displaystyle (1-d)} cusps.