Title Renormalization and 3-manifolds which fiber over the circle / by Curtis T. McMullen

Published Princeton, New Jersey : Princeton University Press, 1996 ©1996

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Description 1 online resource (264 pages) : illustrations, tables Series Annals of Mathematics Studies ; Number 142 Annals of mathematics studies ; no. 142. Contents 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 Summary 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 Analysis Algebraic topology Analytic continuation Automorphism Beltrami equation Bifurcation theory Boundary (topology) Cantor set Circular symmetry Combinatorics Compact space Complex conjugate Complex manifold Complex number Complex plane Conformal geometry Conformal map Conjugacy class Convex hull Covering space Deformation theory Degeneracy (mathematics) Dimension (vector space) Disk (mathematics) Dynamical system Eigenvalues and eigenvectors Factorization Fiber bundle Fuchsian group Fundamental domain Fundamental group Fundamental solution G-module Geodesic Geometry Harmonic analysis Hausdorff dimension Homeomorphism Homotopy Hyperbolic 3-manifold Hyperbolic geometry Hyperbolic manifold Hyperbolic space Hypersurface Infimum and supremum Injective function Intersection (set theory) Invariant subspace Isometry Julia set Kleinian group Laplace's equation Lebesgue measure Lie algebra Limit point Limit set Linear map Mandelbrot set Manifold Mapping class group Measure (mathematics) Moduli (physics) Moduli space Modulus of continuity Möbius transformation N-sphere Newton's method Permutation Point at infinity Polynomial Quadratic function Quasi-isometry Quasiconformal mapping Quasisymmetric function Quotient space (topology) Radon-Nikodym theorem Renormalization Representation of a Lie group Representation theory Riemann sphere Riemann surface Riemannian manifold Schwarz lemma Simply connected space Special case Submanifold Subsequence Support (mathematics) Tangent space Teichmüller space Theorem Topology of uniform convergence Topology Trace (linear algebra) Transversal (geometry) Transversality (mathematics) Triangle inequality Unit disk Unit sphere Upper and lower bounds Vector field Bibliography Includes bibliographical references and index Notes In English Print version record Subject Three-manifolds (Topology) Differentiable dynamical systems. MATHEMATICS -- Topology. MATHEMATICS -- Geometry -- Analytic. Differentiable dynamical systems Three-manifolds (Topology) Form Electronic book ISBN 9781400865178 1400865174 0691011540 9780691011547

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