6 HILLEL FURSTENBERG Figure 11. The set f −1 ([0, 1]). 1 0 1 x1 x2 Figure 12. The set f −2 ([0, 1]). intersection ∞ n=1 f −n ([0, 1]) has pretty much the same structure as the middle third Cantor set. The pictures become even more interesting if we consider the quadratic mapping on the complex plane. Let fc(z) = z2 + c, (c ∈ C). The set of points z ∈ C such that the sequence z, fc(z), fc 2 (z),... does not tend to ∞ is called the filled Julia set of the mapping fc. Several filled Julia sets corresponding to different values of c are shown below.

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