Assouad dimension and fractal geometry. (English) Zbl 1467.28001
Cambridge Tracts in Mathematics 222. Cambridge: Cambridge University Press (ISBN 978-1-108-47865-6/hbk; 978-1-108-77845-9/ebook). xvi, 269 p. (2021).
In the excellent book of K. Falconer [Fractal geometry. Mathematical foundations and applications. 3rd ed. Hoboken, NJ: John Wiley & Sons (2014; Zbl 1285.28011)] on fractal geometry Hausdorff, packing and box-counting dimension are introduces, but Assouad dimension is not mentioned. This Cambridge tract in mathematics on Assouad dimension and fractal geometry aims to be and is in fact a good companion to the book of Falconer.
In the first part of the book, the basic theory is developed. Assouad dimension of sets and measures, its dual, the lower dimension, and the corresponding dimension spectra are introduced. Basic properties of these quantities are studied. In addition the relationship between Assouad dimension and weak tangents is described. In the second part of the book examples are studied. Self-similar, self-affine and self-conformal fractals as well as limit sets of Kleinian groups are considered. Furthermore products, projections, intersections, distance sets of fractals and Kakeya sets serve as examples. The third part of the book is devoted to applications. The Assouad dimension originated in the context of Assouad’s embedding theorem [P. Assouad, C. R. Acad. Sci., Paris, Sér. A 288, 731–734 (1979; Zbl 0409.54020)]. Beside the application of Assouad dimension in embedding theory applications in number theory, probability theory and even functional analysis are given. At the end of the book we find an extensive list of open problems for further research.
The book is very well written and illustrated. The reader gets to know an almost complete spectrum of resent results and historical developments concerning Assouad dimension. The author seems to be a leading expert in this field of research.
Due to the lack of countable and Lipschitz stability of the Assouad dimension (and even monotony of the lower dimension) these notions are not as important as Hausdorff dimension. Nevertheless the book is a must read for everybody working in dimension theory. A student, interested in fractals and dimension theory, should first read the book of Falconer.
In the first part of the book, the basic theory is developed. Assouad dimension of sets and measures, its dual, the lower dimension, and the corresponding dimension spectra are introduced. Basic properties of these quantities are studied. In addition the relationship between Assouad dimension and weak tangents is described. In the second part of the book examples are studied. Self-similar, self-affine and self-conformal fractals as well as limit sets of Kleinian groups are considered. Furthermore products, projections, intersections, distance sets of fractals and Kakeya sets serve as examples. The third part of the book is devoted to applications. The Assouad dimension originated in the context of Assouad’s embedding theorem [P. Assouad, C. R. Acad. Sci., Paris, Sér. A 288, 731–734 (1979; Zbl 0409.54020)]. Beside the application of Assouad dimension in embedding theory applications in number theory, probability theory and even functional analysis are given. At the end of the book we find an extensive list of open problems for further research.
The book is very well written and illustrated. The reader gets to know an almost complete spectrum of resent results and historical developments concerning Assouad dimension. The author seems to be a leading expert in this field of research.
Due to the lack of countable and Lipschitz stability of the Assouad dimension (and even monotony of the lower dimension) these notions are not as important as Hausdorff dimension. Nevertheless the book is a must read for everybody working in dimension theory. A student, interested in fractals and dimension theory, should first read the book of Falconer.
Reviewer: Jörg Neunhäuserer (Goslar)
MSC:
| 28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |
| 28A80 | Fractals |
