zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 (M,d) be a metric space and f a real-valued Borel function on X. For ε>0, set

D ε f(x)=ε -1 sup| f ( x ) - f ( y ) | : d ( x , y ) ε·

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

1 μ(B) B |f(x)-Av B f|dμxCr1 μ(B) kB D ε f(x)dμx

for any ball B of radius r in M, where kB is the ball with the same center and radius kr and Av B f is the μ-average of f over B. If μ is the Lebesgue measure on 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:
28-02Research monographs (measure and integration)
28A80Fractals
28A78Hausdorff and packing measures
42-02Research monographs (Fourier analysis)
30-02Research monographs (functions of one complex variable)
54-02Research monographs (general topology)
54E35Metric spaces, metrizability
54E40Special maps on metric spaces