TY  - BOOK
AU  - McMullen,Curtis T.
TI  - Renormalization and 3-manifolds which fiber over the circle
T2  - Annals of Mathematics Studies
SN  - 9781400865178
AV  - QA613 .M42 1996eb
U1  - 514/.3 20
PY  - 1996///
CY  - Princeton, New Jersey
PB  - Princeton University Press
KW  - Three-manifolds (Topology)
KW  - Differentiable dynamical systems
KW  - Variétés topologiques à 3 dimensions
KW  - Dynamique différentiable
KW  - MATHEMATICS
KW  - Topology
KW  - bisacsh
KW  - Geometry
KW  - Analytic
KW  - fast
KW  - Algebraic topology
KW  - Analytic continuation
KW  - Automorphism
KW  - Beltrami equation
KW  - Bifurcation theory
KW  - Boundary (topology)
KW  - Cantor set
KW  - Circular symmetry
KW  - Combinatorics
KW  - Compact space
KW  - Complex conjugate
KW  - Complex manifold
KW  - Complex number
KW  - Complex plane
KW  - Conformal geometry
KW  - Conformal map
KW  - Conjugacy class
KW  - Convex hull
KW  - Covering space
KW  - Deformation theory
KW  - Degeneracy (mathematics)
KW  - Dimension (vector space)
KW  - Disk (mathematics)
KW  - Dynamical system
KW  - Eigenvalues and eigenvectors
KW  - Factorization
KW  - Fiber bundle
KW  - Fuchsian group
KW  - Fundamental domain
KW  - Fundamental group
KW  - Fundamental solution
KW  - G-module
KW  - Geodesic
KW  - Harmonic analysis
KW  - Hausdorff dimension
KW  - Homeomorphism
KW  - Homotopy
KW  - Hyperbolic 3-manifold
KW  - Hyperbolic geometry
KW  - Hyperbolic manifold
KW  - Hyperbolic space
KW  - Hypersurface
KW  - Infimum and supremum
KW  - Injective function
KW  - Intersection (set theory)
KW  - Invariant subspace
KW  - Isometry
KW  - Julia set
KW  - Kleinian group
KW  - Laplace's equation
KW  - Lebesgue measure
KW  - Lie algebra
KW  - Limit point
KW  - Limit set
KW  - Linear map
KW  - Mandelbrot set
KW  - Manifold
KW  - Mapping class group
KW  - Measure (mathematics)
KW  - Moduli (physics)
KW  - Moduli space
KW  - Modulus of continuity
KW  - Möbius transformation
KW  - N-sphere
KW  - Newton's method
KW  - Permutation
KW  - Point at infinity
KW  - Polynomial
KW  - Quadratic function
KW  - Quasi-isometry
KW  - Quasiconformal mapping
KW  - Quasisymmetric function
KW  - Quotient space (topology)
KW  - Radon-Nikodym theorem
KW  - Renormalization
KW  - Representation of a Lie group
KW  - Representation theory
KW  - Riemann sphere
KW  - Riemann surface
KW  - Riemannian manifold
KW  - Schwarz lemma
KW  - Simply connected space
KW  - Special case
KW  - Submanifold
KW  - Subsequence
KW  - Support (mathematics)
KW  - Tangent space
KW  - Teichmüller space
KW  - Theorem
KW  - Topology of uniform convergence
KW  - Trace (linear algebra)
KW  - Transversal (geometry)
KW  - Transversality (mathematics)
KW  - Triangle inequality
KW  - Unit disk
KW  - Unit sphere
KW  - Upper and lower bounds
KW  - Vector field
KW  - Electronic books
N1  - Includes bibliographical references and index; Cover; Title; Copyright; Contents; 1 Introduction; 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 maps4.3 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 points7.2 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 spacesA. 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
N2  - 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 quan
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