One-dimensional dynamics.

*(English)*Zbl 0791.58003
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 25. Berlin: Springer-Verlag. xiii, 605 p. (1993).

By the early seventies the theory of hyperbolic dynamical systems was largely complete. Since then, more and more effort was put into the investigation of non-hyperbolic systems. Those include ones with elliptic behavior as well as chaotic ones. There is still no general non- hyperbolic theory but the number of examples which were understood as been growing. In the area of dynamical systems in one real dimension the progress has been particularly widespread. We may be approaching the point when a general theory in one dimension will appear, and this book is a significant step towards such a theory.

Trying to describe the present state of knowledge about one-dimensional systems is a formidable task. The techniques tend to be very complicated, and some crucial works are understood only by few. The authors’ success in creating a complete and coherent image of the subject is spectacular. The book would be accessible to an advanced undergraduate student, but anybody after reading it might get a firm grasp of all important ideas and techniques in the field. Many complete proofs are included which are sometimes better than in original papers. For specialists in the area, the book may prove to be an indispensable guide to the literature on the subject. This is due to its clear structure, numerous historical remarks and a careful bibliographical list. I keep it within reach of my hand together with a LaTeX manual and Webster’s dictionary.

Of the two major branches of dynamical systems, more attention is given to chaotic ones. Circle maps are restricted to diffeomorphisms and treated in Chapter I. Chapter II is devoted to topological theory of chaotic systems on the interval. The exposition of this subject is the most coherent I have ever seen. A proof of weak monotonicity of the kneading invariant in the logistic family is also included. Chapter III presents results about systems which are hyperbolic or “somewhat hyperbolic”. Chapter IV introduces a strong and modern technique for dealing with non-hyperbolic cases, and namely the distortion of cross- ratios. Some general results due to this method, like non-existence of wandering intervals, are presented. Chapter V is concerned with the issue of invariant measures, beginning with hyperbolic cases and proceeding to non-hyperbolic ones. Chapter VI presents Sullivan’s theory of renormalization for unimodal mappings. This chapter is the most detailed account of Sullivan’s work available in literature.

Trying to describe the present state of knowledge about one-dimensional systems is a formidable task. The techniques tend to be very complicated, and some crucial works are understood only by few. The authors’ success in creating a complete and coherent image of the subject is spectacular. The book would be accessible to an advanced undergraduate student, but anybody after reading it might get a firm grasp of all important ideas and techniques in the field. Many complete proofs are included which are sometimes better than in original papers. For specialists in the area, the book may prove to be an indispensable guide to the literature on the subject. This is due to its clear structure, numerous historical remarks and a careful bibliographical list. I keep it within reach of my hand together with a LaTeX manual and Webster’s dictionary.

Of the two major branches of dynamical systems, more attention is given to chaotic ones. Circle maps are restricted to diffeomorphisms and treated in Chapter I. Chapter II is devoted to topological theory of chaotic systems on the interval. The exposition of this subject is the most coherent I have ever seen. A proof of weak monotonicity of the kneading invariant in the logistic family is also included. Chapter III presents results about systems which are hyperbolic or “somewhat hyperbolic”. Chapter IV introduces a strong and modern technique for dealing with non-hyperbolic cases, and namely the distortion of cross- ratios. Some general results due to this method, like non-existence of wandering intervals, are presented. Chapter V is concerned with the issue of invariant measures, beginning with hyperbolic cases and proceeding to non-hyperbolic ones. Chapter VI presents Sullivan’s theory of renormalization for unimodal mappings. This chapter is the most detailed account of Sullivan’s work available in literature.

Reviewer: G.Swiatek (Stony Brook)

##### MSC:

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37E10 | Dynamical systems involving maps of the circle |

37E05 | Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth) |

37E20 | Universality and renormalization of dynamical systems |

37A99 | Ergodic theory |

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |