##
**Fractals everywhere.**
*(English)*
Zbl 0691.58001

Boston, MA etc.: Academic Press, Inc. xii, 394 p. £28.00; $ 39.95 (1988).

This is a beautiful book about fractals based on a course called “Fractal Geometry” which has been taught in the School of Mathematics at Georgia Institute of Technology for undergraduate and graduate students.

After a short introduction in Chapter 2 the main basic ideas especially the space of fractals are introduced. Chapter 3 deals with transformations on metric spaces, changing coordinates, the concept of iterated function systems and how they can define a fractal. Two algorithms, the “Chaos Game” and a Deterministic Algorithm, for computing pictures of fractals are presented. Chapter 4 is devoted to dynamics on fractals. The ideas of addresses of points on fractals, nearby addresses, nearby points, dynamical systems on metric spaces, orbits on fractals, repulsive cycles and equivalent dynamical systems are given. After introducing the shadow of a deterministic dynamical system the shadowing theorem shows the meaningfulness of inaccurably computed orbits. These ideas finally were used for the last subsection of chapter 4 to define chaotic dynamics on fractals. In Chapter 5 the concepts of fractal dimension are defined. Various properties of it are developed. After an experimental determination of the fractal dimension the Hausdorff-Besicovitch dimension is introduced. Chapter 6 deals with fractal interpolation. Here computational algorithms and existence theorems are presented which show how to construct fractal interpolation functions. After introducing hidden variable fractal interpolation functions as shadows of graphs of three-dimensional fractal paths these ideas are extended to the notion of space filling curves.

Chapter 7 is devoted to Julia sets. The escape time algorithm and repelling functions are explained. This chapter ends with an application of Julia sets to biological modelling. Chapter 8 deals with parameter spaces and Mandelbrot sets. A computer graphical technique for producing images of the latter is given.

Chapter 9 is devoted to measures on fractals and measures in general. It is shown that special contraction mappings on the space of normalized Borel measures on compact metric spaces lead to measures which live on fractals. After proving that integrals with respect to these measures can be evaluated with the help of Elton’s ergodic theorem the book ends with the application of these measures to computer graphics.

The book contains not only a vast number of examples and exercises but also a vast number of very instructive pictures. Above all the coloured ones under these are very impressing. These exercises and figures give a very good illustration and supplementation to the definitions, lemmas and theorems contained in the book and are of great help for the understanding of the ideas of the new language called fractal geometry.

This very fine book is not only devoted to mathematicians but also for engineers as well as for scientists. Because of the excellent presentation it seems to be a very good help for teachers for preparing a course on this topic and also for persons who want to study by themselves this beautiful part of mathematics.

After a short introduction in Chapter 2 the main basic ideas especially the space of fractals are introduced. Chapter 3 deals with transformations on metric spaces, changing coordinates, the concept of iterated function systems and how they can define a fractal. Two algorithms, the “Chaos Game” and a Deterministic Algorithm, for computing pictures of fractals are presented. Chapter 4 is devoted to dynamics on fractals. The ideas of addresses of points on fractals, nearby addresses, nearby points, dynamical systems on metric spaces, orbits on fractals, repulsive cycles and equivalent dynamical systems are given. After introducing the shadow of a deterministic dynamical system the shadowing theorem shows the meaningfulness of inaccurably computed orbits. These ideas finally were used for the last subsection of chapter 4 to define chaotic dynamics on fractals. In Chapter 5 the concepts of fractal dimension are defined. Various properties of it are developed. After an experimental determination of the fractal dimension the Hausdorff-Besicovitch dimension is introduced. Chapter 6 deals with fractal interpolation. Here computational algorithms and existence theorems are presented which show how to construct fractal interpolation functions. After introducing hidden variable fractal interpolation functions as shadows of graphs of three-dimensional fractal paths these ideas are extended to the notion of space filling curves.

Chapter 7 is devoted to Julia sets. The escape time algorithm and repelling functions are explained. This chapter ends with an application of Julia sets to biological modelling. Chapter 8 deals with parameter spaces and Mandelbrot sets. A computer graphical technique for producing images of the latter is given.

Chapter 9 is devoted to measures on fractals and measures in general. It is shown that special contraction mappings on the space of normalized Borel measures on compact metric spaces lead to measures which live on fractals. After proving that integrals with respect to these measures can be evaluated with the help of Elton’s ergodic theorem the book ends with the application of these measures to computer graphics.

The book contains not only a vast number of examples and exercises but also a vast number of very instructive pictures. Above all the coloured ones under these are very impressing. These exercises and figures give a very good illustration and supplementation to the definitions, lemmas and theorems contained in the book and are of great help for the understanding of the ideas of the new language called fractal geometry.

This very fine book is not only devoted to mathematicians but also for engineers as well as for scientists. Because of the excellent presentation it seems to be a very good help for teachers for preparing a course on this topic and also for persons who want to study by themselves this beautiful part of mathematics.

Reviewer: Karl Doppel (Wien)

### MSC:

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |

26A18 | Iteration of real functions in one variable |

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

28A80 | Fractals |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |

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

37F45 | Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) |

37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |

11K55 | Metric theory of other algorithms and expansions; measure and Hausdorff dimension |