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Dimension theory in dynamical systems: contemporary views and applications. (English) Zbl 0895.58033
Chicago Lectures in Mathematics. Chicago: The University of Chicago Press. xi, 304 p. (1998).
The book under review, written by a leading expert in the field, brings to the fore a new area of research which, since the appearance of Benoit Mandelbrot’s famous book [‘The fractal geometry of nature’ (W. H. Freeman Co., Boston) (1982; Zbl 0504.28001)], has gained considerable popularity among specialists: interrelations between dimension theory (especially fractal geometry) and dynamical systems.
The purpose of the book, as emphasized by its division into two parts [“Carathéodory dimension characteristics” (Chapters 1-4) and “Applications to dimension theory and dynamical systems” (Chapters 5-8)], is twofold. In Part I, a “basic language” is developed for this discipline. This is done by introducing \(C\)-structures, a generalization of a classical Carathéodory construction in dimension theory. They allow to consider a lot of “natural” definitions of “dimension” for sets and measures (such as Hausdorff, box and \(q\)-dimensions) under a unified view. Another advantage of this approach is that some standard invariants from the theory of dynamical systems (namely topological and measure-theoretic entropy) can be introduced from a purely dimensional theoretic perspective. Part II illustrates how dimension theory can be usefully applied to the study of dynamical systems (and conversely). Among other topics, it includes: a description of the geometric structure of Cantor-type constructions by using symbolic dynamics; a mathematical analysis of some dimension characteristics of dynamical systems such as correlation and information dimensions; formulas of dimension of well-known invariant sets (Julia sets, solenoids, horseshoes) for hyperbolic dynamical systems; and a proof of the Eckmann-Ruelle conjecture (establishing the existence of pointwise dimension for almost every point with respect to a hyperbolic invariant measure).
While the book is written in a lucid, rigorous and (essentially) self-contained way, the newcomer may found it a bit hard. Hence the reviewer recommends a previous acquaintance with both “parent” disciplines to a potential reader.

37A99 Ergodic theory
28D05 Measure-preserving transformations
28A80 Fractals
54H20 Topological dynamics (MSC2010)
37B99 Topological dynamics
28D20 Entropy and other invariants
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
54F45 Dimension theory in general topology
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration