Renormalization and 3-Manifolds Which Fiber over the Circle (AM-142) ：Renormalization and 3-Manifolds Which Fiber over the Circle (AM-142) ( Annals of Mathematics Studies )

Publication subTitle ：Renormalization and 3-Manifolds Which Fiber over the Circle (AM-142)

Publication series ：Annals of Mathematics Studies

Author： McMullen Curtis T.;;;

Publisher： Princeton University Press‎

Publication year： 2014

E-ISBN: 9781400865178

P-ISBN(Paperback): 9780691011530

Subject： O189.11 topological space (topological space)

Keyword：数学

Language： ENG

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Description

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.

Chapter

2 Rigidity of hyperbolic manifolds

2.1 The Hausdorff topology

2.2 Manifolds and geometric limits

2.3 Rigidity

2.4 Geometric inflexibility

2.5 Deep points and differentiability

2.6 Shallow sets

3 Three-manifolds which fiber over the circle

3.1 Structures on surfaces and 3-manifolds

3.2 Quasifuchsian groups

3.3 The mapping class group

3.4 Hyperbolic structures on mapping tori

3.5 Asymptotic geometry

3.6 Speed of algebraic convergence

3.7 Example: torus bundles

4 Quadratic maps and renormalization

4.1 Topologies on domains

4.2 Polynomials and polynomial-like maps

4.3 The inner class

4.4 Improving polynomial-like maps

4.5 Fixed points of quadratic maps

4.6 Renormalization

4.7 Simple renormalization

4.8 Infinite renormalization

5 Towers

5.1 Definition and basic properties

5.2 Infinitely renormalizable towers

5.3 Bounded combinatorics

5.4 Robustness and inner rigidity

5.5 Unbranched renormalizations

6 Rigidity of towers

6.1 Fine towers

6.2 Expansion

6.3 Julia sets fill the plane

6.4 Proof of rigidity

6.5 A tower is determined by its inner classes

7 Fixed points of renormalization

7.1 Framework for the construction of fixed points

7.2 Convergence of renormalization

7.3 Analytic continuation of the fixed point

7.4 Real quadratic mappings

8 Asymptotic structure in the Julia set

8.1 Rigidity and the postcritical Cantor set

8.2 Deep points of Julia sets

8.3 Small Julia sets everywhere

8.4 Generalized towers

9 Geometric limits in dynamics

9.1 Holomorphic relations

9.2 Nonlinearity and rigidity

9.3 Uniform twisting

9.4 Quadratic maps and universality

9.5 Speed of convergence of renormalization

10 Conclusion

Appendix A. Quasiconformal maps and flows

A.1 Conformal structures on vector spaces

A.2 Maps and vector fields

A.3 BMO and Zygmund class

A.4 Compactness and modulus of continuity

A.5 Unique integrability

Appendix B. Visual extension

B.1 Naturality, continuity and quasiconformality

B.2 Representation theory

B.3 The visual distortion

B.4 Extending quasiconformal isotopies

B.5 Almost isometries

B.6 Points of differentiability

B. 7 Example: stretching a geodesic

Bibliography

Index

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