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 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.