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Renormalization and 3-Manifolds Which Fiber over the Circle (AM-142)
Curtis T. McMullen

Curtis T. McMullen, Winner of the 1998 Fields Medal, International Congress of Mathematicians

Paperback | 1996 | $87.50 | £73.95 | ISBN: 9780691011530
253 pp. | 7 3/4 x 10
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Many parallels between complex dynamics and hyperbolic geometry have emerged in the past decade. Building on work of Sullivan and Thurston, this book gives a unified treatment of the construction of fixed-points for renormalization and the construction of hyperbolic 3- manifolds fibering over the circle.

Both subjects are studied via geometric limits and rigidity. This approach shows open hyperbolic manifolds are inflexible, and yields quantitative counterparts to Mostow rigidity. In complex dynamics, it motivates the construction of towers of quadratic-like maps, and leads to a quantitative proof of convergence of renormalization.

Reviews:

"A comprehensive study of a theory which brings into parallel two recent and very deep theorems, involving geometry and dynamics. These are Thurston's theorem on the existence of hyperbolic metrics on three-manifolds which fiber over the circle with pseudo-Anosov monodromy, and Sullivan's theorem on the convergence of the renormalization map for real quadratic mappings. . . . The book is very dense in results and the style is superb."--Mathematical Reviews

Table of Contents:

1 Introduction 1
2 Rigidity of hyperbolic manifolds 11
3 Three-manifolds which fiber over the circle 41
4 Quadratic maps and renormalization 75
5 Towers 95
6 Rigidity of towers 105
7 Fixed points of renormalization 119
8 Asymptotic structure in the Julia set 135
9 Geometric limits in dynamics 151
10 Conclusion 175
Appendix A. Quasiconformal maps and flows 183
Appendix B. Visual extension 205
Bibliography 241
Index 251


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