Kenneth Falconer - Fractal Geometry: Mathematical Foundations and Applications

This edition first published 2014

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ISBN: 978-1-119-94239-9

I am frequently asked questions such as What are fractals?, What is fractal dimension?, How can one find the dimension of a fractal and what does it tell us anyway? or How can mathematics be applied to fractals? This book endeavours to answer some of these questions.

The main aim of the book is to provide a treatment of the mathematics associated with fractals and dimensions at a level which is reasonably accessible to those who encounter fractals in mathematics or science. Although basically a mathematics book, it attempts to provide an intuitive as well as a mathematical insight into the subject.

The book falls naturally into two parts. Part I is concerned with the general theory of fractals and their geometry. Firstly, various notions of dimension and methods for their calculation are introduced. Then geometrical properties of fractals are investigated in much the same way as one might study the geometry of classical figures such as circles or ellipses: locally, a circle may be approximated by a line segment, the projection or shadow of a circle is generally an ellipse, a circle typically intersects a straight line segment in two points (if at all) and so on. There are fractal analogues of such properties, usually with dimension playing a key role. Thus, we consider, for example, the local form of fractals and projections and intersections of fractals.

Part II of the book contains examples of fractals, to which the theory of the first part may be applied, drawn from a wide variety of areas of mathematics and physics. Topics include self-similar and self-affine sets, graphs of functions, examples from number theory and pure mathematics, dynamical systems, Julia sets, random fractals and some physical applications.

There are many diagrams in the text and frequent illustrative examples. Computer drawings of a variety of fractals are included, and it is hoped that enough information is provided to enable readers with a knowledge of programming to produce further drawings for themselves.

It is hoped that the book will be a useful reference for researchers, by providing an accessible development of the mathematics underlying fractals and showing how it may be applied in particular cases. The book covers a wide variety of mathematical ideas that may be related to fractals and, particularly in Part II, provides a flavour of what is available rather than exploring any one subject in too much detail. The selection of topics is to some extent at the author's whimthere are certainly some important applications that are not included. Some of the material dates back to early twentieth century, whilst some is very recent.

Notes and references are provided at the end of each chapter. The references are by no means exhaustive; indeed, complete references on the variety of topics covered would fill a large volume. However, it is hoped that enough information is included to enable those who wish to do so to pursue any topic further.

It would be possible to use the book as a basis for a course on the mathematics of fractals, at postgraduate or, perhaps, final-year undergraduate level, and exercises are included at the end of each chapter to facilitate this. Harder sections and proofs are marked with an asterisk and may be omitted without interrupting the development.

An effort has been made to keep the mathematics to a level that can be understood by a mathematics or physics graduate and, for the most part, by a diligent final-year undergraduate. In particular, measure theoretic ideas have been kept to a minimum, and the reader is encouraged to think of measures as mass distributions on sets. Provided that it is accepted that measures with certain (intuitively almost obvious) properties exist, there is little need for technical measure theory in our development.

Results are always stated precisely to avoid the confusion which would otherwise result. Our approach is generally rigorous, but some of the harder or more technical proofs are either just sketched or omitted altogether. (However, a few harder proofs that are not available in that form elsewhere have been included, in particular those on sets with large intersection and on random fractals.) Suitable diagrams can be a help in understanding the proofs, many of which are of a geometric nature. Some diagrams are included in the book; the reader may find it helpful to draw others.

Chapter 1 begins with a rapid survey of some basic mathematical concepts and notation, for example, from the theory of sets and functions, which are used throughout the book. It also includes an introductory section on measure theory and mass distributions which, it is hoped, will be found adequate. The section on probability theory may be helpful for the chapters on random fractals and Brownian motion.

With the wide variety of topics covered, it is impossible to be entirely consistent in the use of notation, and inevitably, sometimes, there has to be a compromise between consistency within the book and standard usage.

In the past few years, fractals have become enormously popular as an art form, with the advent of computer graphics, and as a model of a wide variety of physical phenomena. Whilst it is possible in some ways to appreciate fractals with little or no knowledge of their mathematics, an understanding of the mathematics that can be applied to such a diversity of objects certainly enhances one's appreciation. The phrase the beauty of fractals is often heardit is the author's belief that much of their beauty is to be found in their mathematics.