Renormalization and 3-manifolds which fiber over the circle.

*(English)*Zbl 0860.58002
Annals of Mathematics Studies. 142. Princeton, NJ: Princeton Univ. Press. vii, 253 p., $ 55.00; £43.00/hbk (1996).

This is a sequel to the author’s masterly work entitled: “Complex Dynamics and Renormalization” [Annals of Mathematical Studies. 135 (1995; Zbl 0822.30002)] and is written with the same lucidity and care which characterized the earlier work. In this volume the author presents an approach to renormalization and to the geometrization of 3-manifolds which fiber over the circle. Geometric limits and rigidity together with harmonic analysis on hyperbolic 3-space are the essential tools of the trade.

The book has chapters with the following headings: 1. Introduction, 2. Rigidity of hyperbolic manifolds, 3. Three-manifolds which fiber over the circle, 4. Quadratic maps and renormalization, 5. Towers, 6. Rigidity of towers, 7. Fixed points of renormalization, 8. Asymptotic structure in the Julia set, 9. Geometric limits in dynamics and 10. Conclusion.

The two appendices are headed: A. Quasiconformal maps and flows and B. Visual extension.

The bibliography has 102 entries and lists all the recent work in this newly developing field.

This is an important work on complex dynamics that is very much addressed to the specialist in the field and as such the needs of the more general readers are largely ignored, that is, very little help is given for such readers to locate the sources of prerequisites for reading the book or to motivate them.

The book has chapters with the following headings: 1. Introduction, 2. Rigidity of hyperbolic manifolds, 3. Three-manifolds which fiber over the circle, 4. Quadratic maps and renormalization, 5. Towers, 6. Rigidity of towers, 7. Fixed points of renormalization, 8. Asymptotic structure in the Julia set, 9. Geometric limits in dynamics and 10. Conclusion.

The two appendices are headed: A. Quasiconformal maps and flows and B. Visual extension.

The bibliography has 102 entries and lists all the recent work in this newly developing field.

This is an important work on complex dynamics that is very much addressed to the specialist in the field and as such the needs of the more general readers are largely ignored, that is, very little help is given for such readers to locate the sources of prerequisites for reading the book or to motivate them.

Reviewer: C.S.Sharma (London)

##### MSC:

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

37F99 | Dynamical systems over complex numbers |

30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |