##
**Combinatorial dynamics and entropy in dimension one.**
*(English)*
Zbl 0843.58034

Advanced Series in Nonlinear Dynamics. 5. Singapore: World Scientific. xi, 329 p. (1993).

The book is a very comprehensive introduction to the combinatorial aspects of one-dimensional dynamics, written by authors who were instrumental in the development of the field. The aspects considered are broadly based around a study of the periodic orbits of maps and deal principally with maps which are only required to be continuous. This is a complementary viewpoint to that advanced in the book by W. de Mélo and S. Van Strien [“One-dimensional dynamics”, Springer (1993; Zbl 0791.58003)] which uses more geometric techniques to study maps with greater regularity and dealing with absolutely continuous invariant measures and renormalization etc.

The book contains three main chapters which deal respectively with interval maps, circle maps and topological entropy. The central theme of the first two chapters is the study of forcing relations which exist between periodic orbits of one-dimensional maps. The most well-known result of this type in Sharkovskii’s theorem which assigns a total ordering to the natural numbers with the property that if map of the interval has an orbit of a given period, then it also has periodic orbits of all periods lower down in the ordering. The book starts out by proving this theorem and goes on the show how it can be extended in many ways, giving for example a more refined relation between the cycle types which exist for a map of the interval. In the case of maps of the circle, the book considers separately forcing relations for maps of degree \(-1\), 0 and 1 and then deals with the remaining cases.

The final chapter defines the topological entropy of interval maps and circle maps giving a very full study of its properties. There are sections on calculating the entropy of a piecewise monotone map; the minimum entropy of a map with a given periodic orbit type; and the continuity of topological entropy in certain classes of maps.

As a whole, the book is carefully written and contains a very detailed account of a body of material along with some new results. The book will serve as a valuable reference for those interested in the combinatorial aspects of one-dimensional dynamical systems.

The book contains three main chapters which deal respectively with interval maps, circle maps and topological entropy. The central theme of the first two chapters is the study of forcing relations which exist between periodic orbits of one-dimensional maps. The most well-known result of this type in Sharkovskii’s theorem which assigns a total ordering to the natural numbers with the property that if map of the interval has an orbit of a given period, then it also has periodic orbits of all periods lower down in the ordering. The book starts out by proving this theorem and goes on the show how it can be extended in many ways, giving for example a more refined relation between the cycle types which exist for a map of the interval. In the case of maps of the circle, the book considers separately forcing relations for maps of degree \(-1\), 0 and 1 and then deals with the remaining cases.

The final chapter defines the topological entropy of interval maps and circle maps giving a very full study of its properties. There are sections on calculating the entropy of a piecewise monotone map; the minimum entropy of a map with a given periodic orbit type; and the continuity of topological entropy in certain classes of maps.

As a whole, the book is carefully written and contains a very detailed account of a body of material along with some new results. The book will serve as a valuable reference for those interested in the combinatorial aspects of one-dimensional dynamical systems.

Reviewer: A.Quas (Cambridge)

### MSC:

37E99 | Low-dimensional dynamical systems |

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

37A99 | Ergodic theory |

28D20 | Entropy and other invariants |

54C70 | Entropy in general topology |