##
**Fractals everywhere. Revised with the assistance of Hawley Rising III. Answer key by Hawley Rising III.
2nd ed.**
*(English)*
Zbl 0784.58002

Boston, MA: Academic Press Professional. xiv, 532 p. (1993).

Among numerous books on fractal geometry that have appeared in the last decade, “Fractals everywhere” by Michael F. Barnsley is a very specific and impressive one. The presentation is very intuitive. Ingenious pictures and exercises considerably help the reader to understand the ideas thoroughly.

The first nine chapters of the present, second edition coincide almost exactly with the first edition; slight differences are mentioned in the Foreword. [For review of the first edition (1988) see Zbl 0691.58001.] The new part consists of chapter X and Selected Answers. Chapter X clarifies and summarizes the author’s “ philosophy of fractals”.

Originally, fractals were defined as sets whose Hausdorff dimension is greater than topological dimension [B. Mandelbrot, The fractal geometry of nature (1982; Zbl 0504.28001)]. However, many authors soon resigned this definition, since “it excluded a number of sets which ought to be regarded as fractals” [K. Falconer, Fractal geometry (1990; Zbl 0689.28003)]. In chapter II of the book under review the author states that “... it is too soon to be formal about the exact meaning of a ‘fractal’...” (see p. 33) and for the first eight chapters he admits that every compact nonempty subset of a given metric space \(X\) is a fractal. The reviewer must confess that this approach is amazing for her! In Chapter III (pp. 79 and 81) the author nearly defines (but finally refuses to define) a deterministic fractal as the attractor of an iterated function system (IFS). Unfortunately, he does not mention the following theorem which links the above two approaches (comp. Corollary 9.14 p. 134 in ‘Fractal Geometry’ by K. Falconer): Any compact subset of \(R^ n\) can be approximated arbitrarily closely by the attractor of an IFS. Chapter VI deals with fractal interpolation functions, i.e. functions whose graphs are the attractors of some IFS’s. Finally, in chapter IX, fractals are the attractors of Markov operators associated with IFS’s with probabilities. Thus, in chapters I–IX, the author gives several possible definitions of fractals; though they are not equivalent, all of them reflect presented in chapter X general idea of the fractal system, the associated fractals, and objects which can be approximated by these fractals.

The reviewer would suggest the following minor corrections:

1. On page 34 (def. 7.1) the notation \(A+\varepsilon\) for the parallel set of \(A\) at distance \(\varepsilon\) is misleading and inconsistent with the commonly used \((A_ \varepsilon\) or \(A+\varepsilon \cdot B_ 0\), where \(B_ 0\) is the unit ball).

2. The usage of quotation marks is often unjustified (e.g., pp. 22, 34, 34).

3. The notation \(\bigcup^ n_{n=1}\) is confusing (see e.g. pp. 80, 81, 91, 95).

4. On pages 110 and 111 (Th. 11.1), \(w_{n_ p}\) should be replaced by \(w_{n,p}\), since \(n_ p\) looks like a subsequence.

5. On page 243, \(w_ n\) should be replaced by \(M_ n\).

6. On page 349, the condition “if continuous” is superflous, because \(f\) is a (weak) contraction.

The first nine chapters of the present, second edition coincide almost exactly with the first edition; slight differences are mentioned in the Foreword. [For review of the first edition (1988) see Zbl 0691.58001.] The new part consists of chapter X and Selected Answers. Chapter X clarifies and summarizes the author’s “ philosophy of fractals”.

Originally, fractals were defined as sets whose Hausdorff dimension is greater than topological dimension [B. Mandelbrot, The fractal geometry of nature (1982; Zbl 0504.28001)]. However, many authors soon resigned this definition, since “it excluded a number of sets which ought to be regarded as fractals” [K. Falconer, Fractal geometry (1990; Zbl 0689.28003)]. In chapter II of the book under review the author states that “... it is too soon to be formal about the exact meaning of a ‘fractal’...” (see p. 33) and for the first eight chapters he admits that every compact nonempty subset of a given metric space \(X\) is a fractal. The reviewer must confess that this approach is amazing for her! In Chapter III (pp. 79 and 81) the author nearly defines (but finally refuses to define) a deterministic fractal as the attractor of an iterated function system (IFS). Unfortunately, he does not mention the following theorem which links the above two approaches (comp. Corollary 9.14 p. 134 in ‘Fractal Geometry’ by K. Falconer): Any compact subset of \(R^ n\) can be approximated arbitrarily closely by the attractor of an IFS. Chapter VI deals with fractal interpolation functions, i.e. functions whose graphs are the attractors of some IFS’s. Finally, in chapter IX, fractals are the attractors of Markov operators associated with IFS’s with probabilities. Thus, in chapters I–IX, the author gives several possible definitions of fractals; though they are not equivalent, all of them reflect presented in chapter X general idea of the fractal system, the associated fractals, and objects which can be approximated by these fractals.

The reviewer would suggest the following minor corrections:

1. On page 34 (def. 7.1) the notation \(A+\varepsilon\) for the parallel set of \(A\) at distance \(\varepsilon\) is misleading and inconsistent with the commonly used \((A_ \varepsilon\) or \(A+\varepsilon \cdot B_ 0\), where \(B_ 0\) is the unit ball).

2. The usage of quotation marks is often unjustified (e.g., pp. 22, 34, 34).

3. The notation \(\bigcup^ n_{n=1}\) is confusing (see e.g. pp. 80, 81, 91, 95).

4. On pages 110 and 111 (Th. 11.1), \(w_{n_ p}\) should be replaced by \(w_{n,p}\), since \(n_ p\) looks like a subsequence.

5. On page 243, \(w_ n\) should be replaced by \(M_ n\).

6. On page 349, the condition “if continuous” is superflous, because \(f\) is a (weak) contraction.

Reviewer: M.Moszyńska (Warszawa)

### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |

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

28A80 | Fractals |

28A78 | Hausdorff and packing measures |

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

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

30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |