Fractal geometry: mathematical foundations and applications.

*(English)*Zbl 0689.28003
Chichester etc.: John Wiley & Sons (ISBN 0-471-92287-0). xxii, 288 p. (1990).

First some quotes from preface: ”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 whoe encounter fractals in mathematics or science. Although basically a mathematics book, it attempts to provide an intuitive as well as 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.” ”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.”

The first part, called ”Foundations”, deals with that area of geometric measure theory where measures, in particular Hausdorff measures, are used to find and describe geometric properties of very general subsets of Euclidean spaces. It is closely connected with the earlier book of the author, ”The geometry of fractal sets” (1985; Zbl 0587.28004). But while that book developed the theories of Besicovitch, Marstrand and others in detail, the present book avoids most complicated proofs trying, and in my opinion also succeeding, to reveal the basic ideas. This part consists of 8 chapters. It presents Hausdorff measures and dimension, and also other dimensions, like box-counting and packing dimensions. It shows methods for calculating these dimensions in terms of general measures, their potentials and Fourier transforms. Then local tangential and density properties are discussed. The last three chapters of Part I present the basic equalities and inequalities for the dimensions of orthogonal, Cartesian products, and intersections.

Part II, ”Applications and examples”, consists of 10 chapters each treating a rather different topic. They include self-similar and self- affine fractal constructions, number-theoretic fractals, graphs of nowhere differentiable functions, real and complex dynamical systems and their attractors, various random fractals, multifractal measures and a glance at physical applications.

I think Falconer has gained his aims very well. The book is delightful, accessible to a wide audience, and pleasant to read. It gives good introduction to many branches of mathematics connected with fractals.

The first part, called ”Foundations”, deals with that area of geometric measure theory where measures, in particular Hausdorff measures, are used to find and describe geometric properties of very general subsets of Euclidean spaces. It is closely connected with the earlier book of the author, ”The geometry of fractal sets” (1985; Zbl 0587.28004). But while that book developed the theories of Besicovitch, Marstrand and others in detail, the present book avoids most complicated proofs trying, and in my opinion also succeeding, to reveal the basic ideas. This part consists of 8 chapters. It presents Hausdorff measures and dimension, and also other dimensions, like box-counting and packing dimensions. It shows methods for calculating these dimensions in terms of general measures, their potentials and Fourier transforms. Then local tangential and density properties are discussed. The last three chapters of Part I present the basic equalities and inequalities for the dimensions of orthogonal, Cartesian products, and intersections.

Part II, ”Applications and examples”, consists of 10 chapters each treating a rather different topic. They include self-similar and self- affine fractal constructions, number-theoretic fractals, graphs of nowhere differentiable functions, real and complex dynamical systems and their attractors, various random fractals, multifractal measures and a glance at physical applications.

I think Falconer has gained his aims very well. The book is delightful, accessible to a wide audience, and pleasant to read. It gives good introduction to many branches of mathematics connected with fractals.

Reviewer: P.Mattila

##### MSC:

28A75 | Length, area, volume, other geometric measure theory |

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

54H20 | Topological dynamics (MSC2010) |