Fuzzy Mandelbrot sets

Abstract In this paper, we present a new approach to the Mandelbrot set via fuzzy set theory. We introduce and characterize the fuzzy Mandelbrot set by describing its membership function that determines the degrees of belonging of the points to this set under the iteration even if the orbit of the point is not limited. Then we investigate the basic topological properties of the fuzzy Mandelbrot set and its strong α-cut such as closedness, openness, boundness, connectedness and simply connectedness.