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Intended for researchers in Riemann surfaces, this volume summarizes a significant portion of the work done in the field during the years 1966 to 1971.

Riemann surfaces --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Surfaces, Riemann --- Functions --- Congresses --- Differential geometry. Global analysis --- RIEMANN SURFACES --- congresses --- Congresses. --- MATHEMATICS / Calculus. --- Affine space. --- Algebraic function field. --- Algebraic structure. --- Analytic continuation. --- Analytic function. --- Analytic set. --- Automorphic form. --- Automorphic function. --- Automorphism. --- Beltrami equation. --- Bernhard Riemann. --- Boundary (topology). --- Canonical basis. --- Cartesian product. --- Clifford's theorem. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Complex multiplication. --- Conformal geometry. --- Conformal map. --- Coset. --- Degeneracy (mathematics). --- Diagram (category theory). --- Differential geometry of surfaces. --- Dimension (vector space). --- Dirichlet boundary condition. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Euclidean space. --- Existential quantification. --- Explicit formulae (L-function). --- Exterior (topology). --- Finsler manifold. --- Fourier series. --- Fuchsian group. --- Function (mathematics). --- Generating set of a group. --- Group (mathematics). --- Hilbert space. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Hyperbolic geometry. --- Hyperbolic group. --- Identity matrix. --- Infimum and supremum. --- Inner automorphism. --- Intersection (set theory). --- Intersection number (graph theory). --- Isometry. --- Isomorphism class. --- Isomorphism theorem. --- Kleinian group. --- Limit point. --- Limit set. --- Linear map. --- Lorentz group. --- Mapping class group. --- Mathematical induction. --- Mathematics. --- Matrix (mathematics). --- Matrix multiplication. --- Measure (mathematics). --- Meromorphic function. --- Metric space. --- Modular group. --- Möbius transformation. --- Number theory. --- Osgood curve. --- Parity (mathematics). --- Partial isometry. --- Poisson summation formula. --- Pole (complex analysis). --- Projective space. --- Quadratic differential. --- Quadratic form. --- Quasiconformal mapping. --- Quotient space (linear algebra). --- Quotient space (topology). --- Riemann mapping theorem. --- Riemann sphere. --- Riemann surface. --- Riemann zeta function. --- Scalar multiplication. --- Scientific notation. --- Selberg trace formula. --- Series expansion. --- Sign (mathematics). --- Square-integrable function. --- Subgroup. --- Teichmüller space. --- Theorem. --- Topological manifold. --- Topological space. --- Uniformization. --- Unit disk. --- Variable (mathematics). --- Riemann, Surfaces de --- RIEMANN SURFACES - congresses --- Fonctions d'une variable complexe --- Surfaces de riemann

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In 1920, Pierre Fatou expressed the conjecture that--except for special cases--all critical points of a rational map of the Riemann sphere tend to periodic orbits under iteration. This conjecture remains the main open problem in the dynamics of iterated maps. For the logistic family x- ax(1-x), it can be interpreted to mean that for a dense set of parameters "a," an attracting periodic orbit exists. The same question appears naturally in science, where the logistic family is used to construct models in physics, ecology, and economics. In this book, Jacek Graczyk and Grzegorz Swiatek provide a rigorous proof of the Real Fatou Conjecture. In spite of the apparently elementary nature of the problem, its solution requires advanced tools of complex analysis. The authors have written a self-contained and complete version of the argument, accessible to someone with no knowledge of complex dynamics and only basic familiarity with interval maps. The book will thus be useful to specialists in real dynamics as well as to graduate students.

Geodesics (Mathematics) --- Polynomials. --- Mappings (Mathematics) --- Maps (Mathematics) --- Functions --- Functions, Continuous --- Topology --- Transformations (Mathematics) --- Algebra --- Geometry, Differential --- Global analysis (Mathematics) --- Mathematics --- Absolute value. --- Affine transformation. --- Algebraic function. --- Analytic continuation. --- Analytic function. --- Arithmetic. --- Automorphism. --- Big O notation. --- Bounded set (topological vector space). --- C0. --- Calculation. --- Canonical map. --- Change of variables. --- Chebyshev polynomials. --- Combinatorics. --- Commutative property. --- Complex number. --- Complex plane. --- Complex quadratic polynomial. --- Conformal map. --- Conjecture. --- Conjugacy class. --- Conjugate points. --- Connected component (graph theory). --- Connected space. --- Continuous function. --- Corollary. --- Covering space. --- Critical point (mathematics). --- Dense set. --- Derivative. --- Diffeomorphism. --- Dimension. --- Disjoint sets. --- Disjoint union. --- Disk (mathematics). --- Equicontinuity. --- Estimation. --- Existential quantification. --- Fibonacci. --- Functional equation. --- Fundamental domain. --- Generalization. --- Great-circle distance. --- Hausdorff distance. --- Holomorphic function. --- Homeomorphism. --- Homotopy. --- Hyperbolic function. --- Imaginary number. --- Implicit function theorem. --- Injective function. --- Integer. --- Intermediate value theorem. --- Interval (mathematics). --- Inverse function. --- Irreducible polynomial. --- Iteration. --- Jordan curve theorem. --- Julia set. --- Limit of a sequence. --- Linear map. --- Local diffeomorphism. --- Mathematical induction. --- Mathematical proof. --- Maxima and minima. --- Meromorphic function. --- Moduli (physics). --- Monomial. --- Monotonic function. --- Natural number. --- Neighbourhood (mathematics). --- Open set. --- Parameter. --- Periodic function. --- Periodic point. --- Phase space. --- Point at infinity. --- Polynomial. --- Projection (mathematics). --- Quadratic function. --- Quadratic. --- Quasiconformal mapping. --- Renormalization. --- Riemann sphere. --- Riemann surface. --- Schwarzian derivative. --- Scientific notation. --- Subsequence. --- Theorem. --- Theory. --- Topological conjugacy. --- Topological entropy. --- Topology. --- Union (set theory). --- Unit circle. --- Unit disk. --- Upper and lower bounds. --- Upper half-plane. --- Z0.

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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.

Differential dynamical systems --- Drie-menigvuldigheden (Topologie) --- Three-manifolds (Topology) --- Trois-variétés (Topologie) --- Differentiable dynamical systems. --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Low-dimensional topology --- Topological manifolds --- 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. --- Differentiable dynamical systems --- 515.16 --- 515.16 Topology of manifolds --- Topology of manifolds