J.P. Locquet, J. Perret, et al.
SPIE Optical Science, Engineering, and Instrumentation 1998
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.
J.P. Locquet, J. Perret, et al.
SPIE Optical Science, Engineering, and Instrumentation 1998
Charles A Micchelli
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