For each complex μ, denote by (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 set (the set of those values of the complex μ for which (μ) contains domains) and of the closure ^* of . The goals of the present work are as follows. A) To restate some previously reported properties of (μ), and ^* in new ways, and to report new observations. B) To deduce some known properties of the mapping f for real μ and z, with μf-, 2[and z]--√1+4μ, +√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.