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1. INTRODUCTION TO FRACTALS 5 Figure 9. The procedure is repeated for each of the remaining triangles. of the triangles constituting Sn grows exponentially. Thus, S is far too small for a 2-dimensional object and far too large for a 1-dimensional one, so its dimension has to be strictly between 1 and 2. Another way to construct a fractal set is to start with some simple continuous map f : Rn → Rn and look at the set of points that do not escape to infinity under any number of iterations, i.e., the set of points for which the sequence x, f(x), f 2 (x), ... remains bounded (here and below f n (x) = f(f(... (f(x)) . . . )) is the n’th iterate of f). Let us start with f(x) = 5x(1 − x), x ∈ R. The graph of f is just a parabola looking down: Note that f(x) = 5x(1−x) 5x when x 0, so f n (x) 5nx in this 1 0 1 Figure 10. The graph of f(x) = 5x(1 − x). case, and, therefore, f n (x) → −∞. If x 1, then f(x) 0 and, by the previous remark, we still have f n (x) → −∞. Thus, the points x for which f n (x) does not tend to −∞ must stay in the interval [0, 1] after arbitrarily many iterations. Such points do exist. For example, the fixed point 0 of f satisfies this property, and so do all its pre-images. To find the set of points with bounded iterates exactly, note that each such point should lie in any iterated pre-image f −n ([0, 1]). Moreover, since f −1 ([0, 1]) [0, 1], these pre-images form a nested sequence of closed sets. They can be easily seen on the graph. The set f −1 ([0, 1]) is the union of 2 intervals [0,x1] and [x2, 1]: The second pre-image f −2 ([0, 1]) is a union of four intervals: and so on. The final