Getting acquainted with fractals
This book provides an answer to the question of what type of mathematics is behind the famous fractal pictures. One of the main features of fractal objects is their (often non-integral) dimension. The first chapter presents the foundations of the theory of Hausdorff measure and dimension; it shows ways to produce fractal sets by means of “deleting and replacing” and it gives ideas of how to compute the dimension of such objects. The second chapter is related to the second main feature of fractals, namely their self-similarity. Iterative function systems are studied here. Starting with a finite number of similarities, by iteration we obtain a fractal set as the limit image. A rigorous theoretical treatment is complemented by many representative examples showing how a proper choice of iterated mappings with carefully chosen parameters may lead to nice pictures (including island archipelago, tree, Barnsley fern and grass).
In the third chapter, the theory of iteration of complex polynomials, resulting in Julia sets and the Mandelbrot set, is developed. Again, examples of impressive pictures based on precise formulas illustrate the theory. The exposition is written as a rigorous mathematical text with precise definitions, theorems and some proofs. It is readable with skills at the level of an undergraduate student. However, it is not intended as a textbook. In particular, the presentation is not self-contained and difficult proofs are referred to. For readers interested in the full depth of the theory it is recommended to supplement their studies with other sources. The purpose of this book is just to yield a bridge between the fractal pictures and serious mathematics and this is successfully achieved.