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.

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.

Reviewer: V.Jiménez López (Murcia)

##### MSC:

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 |