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Some novel types of fractal geometry. (English) Zbl 0970.28001
Oxford Mathematical Monographs. Oxford: Clarendon Press. x, 164 p. £49.95 (2001).

Mandelbrot introduced the term fractal in 1975 as a general term for various irregular geometric forms. He also gave a definition: a set is a fractal if its topological dimension is strictly less than its Hausdorff dimension, but he soon took it back realizing, quite correctly, that it leaves out many natural fractal type objects. Up to now there is no generally accepted definition of fractal (and in my opinion there should not be) but the term has become extremely widely used and different people give different meanings to it. Mostly it has meant a subset of a Euclidean space which is different from classical objects, such as smooth surfaces, rectifiable curves, and rectifiable sets.

What is the novelty in the fractal geometries which Stephen Semmes discusses in his book? Various answers can be given to this. Much of the story of the book happens in rather general metric spaces where there are no smooth objects to compare with and give a meaning to fractality. But there are properties; for example some metric spaces have “decent calculus” similar to the Euclidean case, and some don’t have this. Another theme which continues throughout the book is the discussion of novel fractal-type counterexamples, of Laakso, Bourdon and Pajot, and others, to various questions about geometry of and analysis on metric spaces.

Much of the material of the book is connected with the recent developments of analysis on metric spaces, by Hajłasz, Heinonen, Koskela, and others. One of the basic facts is the Poincaré inequality. There are various ways what to mean by this, but Semmes mainly uses the following: Let $\left(M,d\right)$ be a metric space and $f$ a real-valued Borel function on $X$. For $\epsilon >0$, set

${D}_{\epsilon }f\left(x\right)={\epsilon }^{-1}sup\left\{|f\left(x\right)-f\left(y\right)|:d\left(x,y\right)\le \epsilon \right\}·$

If $\mu$ is a positive Borel measure on $X$, one says that the pair $\left(M,\mu \right)$ has Poincaré inequalities if there exist positive constants $C$ and $k$ such that for all $\epsilon >0$

$\frac{1}{\mu \left(B\right)}{\int }_{B}|f\left(x\right)-A{v}_{B}f|d\mu x\le Cr\frac{1}{\mu \left(B\right)}{\int }_{kB}{D}_{\epsilon }f\left(x\right)d\mu x$

for any ball $B$ of radius $r$ in $M$, where $kB$ is the ball with the same center and radius $kr$ and $A{v}_{B}f$ is the $\mu$-average of $f$ over $B$. If $\mu$ is the Lebesgue measure on ${ℝ}^{n}$, this reduces to the classical Poincaré inequality. Then “decent calculus” on a space could mean that it has Poincaré inequalities. Heisenberg groups and many more general Carnot-Carathéodory spaces are important examples of these, and they are also discussed in the book.

Another central topic is the BPI-property of metric spaces; BPI stands for “big pieces of itself”. Roughly speaking BPI means that in any two balls there are considerable portions which are bilipschitz equivalent with natural Lipschitz constants coming from the scales. This was already central in the book “Fractured fractals and broken dreams. Self-similar geometry through metric and measure” (1997; Zbl 0887.54001) by G. David and S. Semmes. The present book is to a large extent a discussion on many natural questions involving “decent calculus”, BPI and various closely related topics. It touches a lot of analysis developed in Euclidean spaces during the last decades in connection with, for example, quasiconformal mappings and harmonic analysis. Reading it one can get taste of various developments. And maybe also about what will be; much of the book flows through questions, and conjectures, and former conjectures which have been disproved (very often by Laakso). The book is really more a discussion than a text-book in a traditional “definition – theorem – proof” style. It is an inspiring book. It may not be a book for many standard graduate courses, and it may not be a reference book for several decades. But it is an excellent book to give insight on what is presently happening in this area and what could be done in near future.

##### MSC:
 28-02 Research monographs (measure and integration) 28A80 Fractals 28A78 Hausdorff and packing measures 42-02 Research monographs (Fourier analysis) 30-02 Research monographs (functions of one complex variable) 54-02 Research monographs (general topology) 54E35 Metric spaces, metrizability 54E40 Special maps on metric spaces