By Richard A. Holmgren

Discrete dynamical structures are basically iterated capabilities. Given the convenience with which pcs can do generation, it truly is now attainable for somebody with entry to a private laptop to generate appealing photographs whose roots lie in discrete dynamical structures. pictures of Mandelbrot and Julia units abound in guides either mathematical and never. the maths at the back of the photographs are attractive of their personal correct and are the topic of this article. the extent of the presentation is acceptable for complex undergraduates with a yr of calculus in the back of them. scholars within the author's classes utilizing this fabric have come from a variety of disciplines; many were majors in different disciplines who're taking arithmetic classes out of basic curiosity. strategies from calculus are reviewed as worthy. Mathematica courses that illustrate the dynamics and that would relief the coed in doing the routines are integrated in an appendix.

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**Example text**

If p is a number between f(a) and f(b), then there exists c in [a, b] such that f(c) = p. That is, if f(a) S p S f(b) or f(b) S p S f(a) , then there is c in [a, b] such that f(c) = p. A proof of the Intermediate Value Theorem can be found in introductory analysis textbooks such as those listed in the references. We note that the condition that f be continuous is necessary. We demonstrate this in the following example. 12. for x < 1. Then f(O) = 1 3, for x 2': 1 and f(2) = 3. Even though 2 is between f(O) and f(2) there is no number c in [0,3] such that f(c) = 2.

In the exercises, we ask the reader to find two points that are fixed at 0 after exactly three iterations of h. Given sufficient patience, it is possible to find points that are fixed by h after any number of iterations. Before we look at an example of a point that is eventually periodic but not eventually fixed, we introduce a precise definition of these terms. 6. Let f be a function. The point x is an eventually fixed point of f if there exists N such that r+ 1 (x) = r (x) whenever n 2: N . The point x is eventually periodic with period k if there exists N such that r+k(x) = r(x) whenever n 2: N.

Prove r1(f(A)) :::> A and show by example that it may happen that f-1(f(A)) =I A. b) Let B be a subset of the codomain of the function f . 4 Suppose f : [-1 , IJ -t [-1 , IJ is defined so that f(x) = ~x. a) Find the range of f. b) Find the range of f 0 f. c) Find the range of fn for all natural numbers n . 5 a) Find a function f : [0,2J continuous but not onto. -t [2,4J so that f is one-to-one and 18 2. A Quick Look at Functions b) Find a function f but not one-to-one. : [0,2] -+ [2,4] so that f is onto and continuous c) Find a function f but not continuous.