Family of Fuzzy Mandelblog Sets

In this paper, we consider the family of parameterized Mandelbrot-like sets generated as any point c∈C∖{0} of the complex plane belongs to any member of this family for a real parameter t≥1, provided that its corresponding orbit of 0 does not escape to infinity under iteration fcn0=fcn−102+logct; otherwise, it is not a member of this set. This classically means there is only a binary membership possibility for all points. Here, we call this type of fractal set a Mandelblog set, and then we introduce a membership function that assigns a degree to each c to be an element of a fuzzy Mandelblog set under the iterations, even if the orbits of the points are not limited. Moreover, we provide numerical examples and gray-scale graphics that illustrate the membership degrees of the points of the fuzzy Mandelblog sets under the effects of iteration parameters. This approach enables the formation of graphs for these fuzzy fractal sets by representing points that belong to the set as white pixels, points that do not belong as black pixels, and other points, based on their membership degrees, as gray-toned pixels. Furthermore, the membership function facilitates the direct proofs of the symmetry criteria for these fractal sets.