Combinations of complex dynamical systems. (English) Zbl 1045.37028

Pilgrim, Kevin M.

Lecture Notes in Mathematics 1827. Berlin: Springer (ISBN 3-540-20173-4/pbk). ix, 118 p. (2003).

The author deals with combinations of branched coverings of the two sphere, which occur in the classification of certain complex dynamical systems. It tries to bring known combination procedures for rational maps in a common (and more general) framework.
The main contents of this monograph are the following. A process to combine coverings is described. Then the author investigates the uniqueness of these combinations. It follows a converse construction, decomposition of coverings and their uniqueness are treated. Also, a result on the number of combinatorial classes of annulus maps over a tree is presented.
It has to be said that this monograph is very technical. However, a lot of explanations, motivations and examples are given. Moreover, it is well organized and clearly written. Nonetheless it still remains to be very technical.

Reviewer: Peter Raith (Wien)

MSC:

37F20	Combinatorics and topology in relation with holomorphic dynamical systems
37-02	Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37E99	Low-dimensional dynamical systems

Keywords:

complex dynamical system; combination; mating; tameness; tree; multicurve; decomposition; twist theorem