On the quadratic mapping z→z²-μ for complex μ and z: The fractal structure of its M set, and scaling

For each complex μ, denote by F(μ) the largest bounded set in the complex plane that is invariant under the action of the mapping z→z2-μ. Mandelbrot 1980, 1982 (Chap. 19) reported various remarkable properties of the M set (the set of those values of the complex μ for which F(μ) contains domains) and of the closure M* of M. The goals of the present work are as follows. A) To restate some previously reported properties of F(μ), M and M* in new ways, and to report new observations. B) To deduce some known properties of the mapping f for real μ and z, with με{lunate}f- 1 4, 2[and zε{lunate}]- 1 2- 1 2√1+4μ, 1 2+ 1 2√1+4μ[. In many ways, the properties of the transformation f are easier to grasp in the complex plane than in an interval. (This exemplifies the saying that "when one wishes to simplify a theory, one should complexify the variables",) C) To serve as introduction to some recent pure mathematical work triggered by Mandelbrot 1980. Further pure mathematical work is strongly urged. © 1983.