Dynamics in one complex variable.
3rd ed.

*(English)*Zbl 1085.30002
Annals of Mathematics Studies 160. Princeton, NJ: Princeton University Press (ISBN 0-691-12488-4/pbk; 0-691-12487-6/hbk). viii, 304 p. (2006).

This is the third edition of a textbook on complex dynamics which has become a standard reference in the field. The basic outline has not changed from the first edition so that we refer to its review [Dynamics in one complex variable. Introductory lectures. Wiesbaden: Vieweg. (1999; Zbl 0946.30013)].

However, many improvements and additions have been made. These changes are best summarized by quoting from the preface to the third edition: A brief historical survey has been added in §4.1, the definition of Lattès map has been made more inclusive in §7.4, the Écalle-Voronin theory of parabolic points is described in §10.12, the résidu itératif is studied in §12.9, the material on two complex variables in Appendix D has been expanded, and recent results on effective computability have been added in Appendix H. The list of references has also been updated and expanded.

The book was called “an excellent introduction to complex dynamics” in the review of the second edition. The thorough revision that has been done has made it even better.

However, many improvements and additions have been made. These changes are best summarized by quoting from the preface to the third edition: A brief historical survey has been added in §4.1, the definition of Lattès map has been made more inclusive in §7.4, the Écalle-Voronin theory of parabolic points is described in §10.12, the résidu itératif is studied in §12.9, the material on two complex variables in Appendix D has been expanded, and recent results on effective computability have been added in Appendix H. The list of references has also been updated and expanded.

The book was called “an excellent introduction to complex dynamics” in the review of the second edition. The thorough revision that has been done has made it even better.

Reviewer: Walter Bergweiler (Kiel)

##### MSC:

30-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable |

30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |