Scaling of Mandelbrot sets generated by critical point preperiodicity (Q1072675)

In numerical studies of the dynamics of families of rational maps f(\(\mu\),z), which depend holomorphically on a complex parameter \(\mu\), it appears that remarkably precise copies of the standard Mandelbrot diagram M (for the family \(z^ 2+\mu)\) occur in parts of the corresponding plots for the general families. The authors give a general case when this will occur: if for \(\mu =0\) a critical point of f(0,z) falls after a finite number of steps on an unstable fixed point of f(0,z), then for \(\mu\) near 0 an infinity of more and more accurate copies of M occurs. The scaling of this situation is worked out.