Techniques in fractal geometry. (English) Zbl 0869.28003

Falconer, Kenneth

Chichester: John Wiley & Sons. xvii, 256 p. (1997).

This is third book of the author concerning fractal geometry. It is devoted to powerful techniques in this area of the recent time. After reviewing mathematical background as well as the basics of fractal geometry as dimension notions (Hausdorff, packing and box dimension) and IFS a detailed enumeration of techniques for studying dimensions is given. Rather special seems the chapter about the bounded distortion principle for cookie-cutter sets, but it helps to present the following thermodynamic formalism as simple as possible. The main parts are the chapters which explain the use of the ergodic, the renewal and martingale theorems in fractal geometry. The author shifts reader’s attention from fractal sets to fractal measure. Topics like tangent measures, there is some overlap to the recent book of P. Mattila [“Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability” (1995; Zbl 0819.28004)], dimensions of measures and an excursion through multifractal attest it. Finally, the book summarizes techniques arising from the study of different differential equations and related fractal structures (for examples attractors, fractal boundary or fractal domains).

Reviewer: H.Haase (Greifswald)

MSC:

28A80	Fractals
28-02	Research exposition (monographs, survey articles) pertaining to measure and integration
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
28A78	Hausdorff and packing measures

Keywords:

packing dimension; Hausdorff dimension; tangent measure; multifractal; iterated function system; fractal geometry; box dimension; IFS; cookie-cutter sets; thermodynamic formalism; ergodic; renewal; martingale; fractal sets; fractal measure; differential equations; attractors; fractal boundary; fractal domains

Citations:

Zbl 0819.28004