Combinatorial dynamics and entropy in dimension one. 2nd ed.

*(English)*Zbl 0963.37001
Advanced Series in Nonlinear Dynamics. 5. Singapore: World Scientific. xvi, 415 p. (2000).

The first edition of this book appeared in 1993 (see the review in Zbl 0843.58034). The major change in the second edition is the addition of two appendices in form of Chapters 5 and 6. The first of these appendices is about the graph maps. Many parts of the combinatorial theory for interval maps can be, and have been, generalized to tree maps or even further, to graph maps. The second appendix is about the rotation theory. It generalizes the theory that is described in Chapter 3 for circle maps of degree 1 to a more general setting.

So, the contents of the second edition is as follows: Chapter 1: Preliminaries; Chapter 2: Interval maps; Chapter 3: Circle maps; Chapter 4: Entropy; Chapter 5: Graph maps; Chapter 6: Rotation theory. The central theme of the Chapters 2 and 3 is the study of forcing relations which exist between periodic orbits of one-dimensional maps. The notion of patterns is defined and widely used. In Chapter 4, the topological entropy of interval maps and circle maps is defined and there are several sections on calculating the entropy.

The book provides an extended bibliography – 398 items – and is an excellent source of information about the combinatorial dynamics in one-dimensional dynamical systems and can be highly recommended for those interested in this field. The book is well and carefully written and organized.

So, the contents of the second edition is as follows: Chapter 1: Preliminaries; Chapter 2: Interval maps; Chapter 3: Circle maps; Chapter 4: Entropy; Chapter 5: Graph maps; Chapter 6: Rotation theory. The central theme of the Chapters 2 and 3 is the study of forcing relations which exist between periodic orbits of one-dimensional maps. The notion of patterns is defined and widely used. In Chapter 4, the topological entropy of interval maps and circle maps is defined and there are several sections on calculating the entropy.

The book provides an extended bibliography – 398 items – and is an excellent source of information about the combinatorial dynamics in one-dimensional dynamical systems and can be highly recommended for those interested in this field. The book is well and carefully written and organized.

Reviewer: Alois Klíč (Praha)

##### MSC:

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

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

37E10 | Dynamical systems involving maps of the circle |

37E15 | Combinatorial dynamics (types of periodic orbits) |

37E25 | Dynamical systems involving maps of trees and graphs |

37E45 | Rotation numbers and vectors |

37A05 | Dynamical aspects of measure-preserving transformations |

37B40 | Topological entropy |