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Fractured fractals and broken dreams. Self-similar geometry through metric and measure. (English) Zbl 0887.54001
Oxford Lecture Series in Mathematics and its Applications. 7. Oxford: Clarendon Press. ix, 212 p. (1997).
The main idea of the book is given by the authors in the Preface: “Imagine that a perfectly self-similar fractal set is generated by a simple iteration and then subjected to bending, twisting, breaking, or corrosion. What remains of the initial structure?”
Following this idea, the authors extend the notion of self-similarity to that of BPI-space (“big pieces of itself”). Roughly speaking, a metric space \((M,d)\) is a BPI-space if it is, in some sense, regular and every two balls in \((M,d)\) contain “big” subsets (big in comparison with the radii) which are bi-Lipschitz equivalent (Def. 1.5 and Chapter 6). The BPI-equivalence is an equivalence relation on the class of BPI-spaces (Def. 1.6 and Chapter 7).
The book consists of twenty chapters. As the authors say in Chapter 10, “Rest stop”, Chapters 1-9 provide basic tools, more or less related to the above notions. Further, to compare BPI-spaces which are not BPI-equivalent, the authors consider maps between such spaces. More precisely, they define the interesting notion of looking down: Let \(M\) and \(N\) be two spaces of the same dimension; the space \(M\) looks down on \(N\) if there is a closed subset \(A\) of \(M\) and a Lipschitz map \(f:A\to N\) such that \(f(A)\) has positive measure. Proposition 11.10 says that \(\mathbb{R}^n\) is minimal for looking down, i.e. if \(\mathbb{R}^n\) looks down on a metric space \(M\), then \(\mathbb{R}^n\) and \(M\) are BPI-equivalent.
The book contains a great variety of concepts, examples, results, and open problems. It is impossible to give here even an idea of all of them. Let us only mention the titles of some chapters from the second part of the book (after the “Rest stop”): 13. Sets made from nested cubes; 14. Big pieces of bilipschitz mappings; 17. Deformations of BPI-spaces; 19. Some sets that are far from BPI.
Although the subject is very delicate and rather complicated, the presentation is both intuitive and precise.
Some small remarks: (1) The style of the book is rather nontypical, in particular, almost all the statements are “Lemmas”. (2) The notation is sometimes missleading; for instance, \(d\) is dimension and metric at the same time, \(N\) is a metric space and a natural number. (3) The definition of the Hausdorff metric (p. 52) is different from (and non-equivalent to) the commonly used, though these two metrics are topologically equivalent.

54-02 Research exposition (monographs, survey articles) pertaining to general topology
28A80 Fractals
28A78 Hausdorff and packing measures