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
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2014 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.