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Iteration of Rational Functions: Complex Analytic Dynamical Systems (Graduate Texts in Mathematics)

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Many people outside mathematics know of Julia sets, Mandelbrot sets, attractors, chaos, and fractals. They might have encountered them in color renditions as moving pictures that serve as computer screen savers. The book is about the underlying mathematics.

The rational functions (i.e., quotients of polynomials) form a relatively simple class of functions of a single complex variable. One of the operations we do on functions f is substitution: Starting with a complex number z (i.e., a point in the plane), repeated application of f yields a sequence of complex numbers (points), z, f(z), f(f(z)), etc. The resulting sets have a surprisingly rich structure, and they serve as models for dynamics that arises in different subjects. They are the subject of this book. I will let the reader discover the precise definitions in the book: The sets are defined from some basic geometric and analytic properties of these sequences. The set of points z where the sequence has one of two exclusive analytic properties divides the plane into two parts, an `inside' and an `outside', if you like. Well, the mathematical definition is more complicated than that, but the idea is roughly as I said.

The French mathematician Gaston Julia (1893-1978) was the first to make precise the sets that we now know from a host of fun computer experiments. And it is interesting that Julia invented the sets in 1919 without the benefit of computer programs.

Indeed the subject stayed in the background in the mathematical fashions until it became possible to easily visualize the possibilities with simple computer experiments. As a result, the interest in Julia's work had a more renaissance, and its connections to exciting properties in non-linear dynamics turned into an exciting subject in pure and applied mathematics.

This beautifully written book is primarily about the mathematical side of the subject, but it is full of hands-on-examples that illustrate the theory, and that are easy to follow for students. The exercises are lovely, and make the book eminently attractive for the teaching of the subject to students who will need very little more in the way of background than a familiarity with the complex numbers.
Palle Jorgensen, October 2004.