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This book proves an analogue of William Thurston's celebrated hyperbolic Dehn surgery theorem in the context of complex hyperbolic discrete groups, and then derives two main geometric consequences from it. The first is the construction of large numbers of closed real hyperbolic 3-manifolds which bound complex hyperbolic orbifolds--the only known examples of closed manifolds that simultaneously have these two kinds of geometric structures. The second is a complete understanding of the structure of complex hyperbolic reflection triangle groups in cases where the angle is small. In an accessible and straightforward manner, Richard Evan Schwartz also presents a large amount of useful information on complex hyperbolic geometry and discrete groups. Schwartz relies on elementary proofs and avoids "ations of preexisting technical material as much as possible. For this reason, this book will benefit graduate students seeking entry into this emerging area of research, as well as researchers in allied fields such as Kleinian groups and CR geometry.

CR submanifolds. --- Dehn surgery (Topology). --- Three-manifolds (Topology). --- CR submanifolds --- Dehn surgery (Topology) --- Three-manifolds (Topology) --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Cauchy-Riemann submanifolds --- Submanifolds, CR --- Low-dimensional topology --- Topological manifolds --- Surgery (Topology) --- Manifolds (Mathematics) --- Arc (geometry). --- Automorphism. --- Ball (mathematics). --- Bijection. --- Bump function. --- CR manifold. --- Calculation. --- Canonical basis. --- Cartesian product. --- Clifford torus. --- Combinatorics. --- Compact space. --- Conjugacy class. --- Connected space. --- Contact geometry. --- Convex cone. --- Convex hull. --- Coprime integers. --- Coset. --- Covering space. --- Dehn surgery. --- Dense set. --- Diagram (category theory). --- Diameter. --- Diffeomorphism. --- Differential geometry of surfaces. --- Discrete group. --- Double coset. --- Eigenvalues and eigenvectors. --- Equation. --- Equivalence class. --- Equivalence relation. --- Euclidean distance. --- Four-dimensional space. --- Function (mathematics). --- Fundamental domain. --- Geometry and topology. --- Geometry. --- Harmonic function. --- Hexagonal tiling. --- Holonomy. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Horosphere. --- Hyperbolic 3-manifold. --- Hyperbolic Dehn surgery. --- Hyperbolic geometry. --- Hyperbolic manifold. --- Hyperbolic space. --- Hyperbolic triangle. --- Hypersurface. --- I0. --- Ideal triangle. --- Intermediate value theorem. --- Intersection (set theory). --- Isometry group. --- Isometry. --- Limit point. --- Limit set. --- Manifold. --- Mathematical induction. --- Metric space. --- Möbius transformation. --- Parameter. --- Parity (mathematics). --- Partial derivative. --- Partition of unity. --- Permutation. --- Polyhedron. --- Projection (linear algebra). --- Projectivization. --- Quotient space (topology). --- R-factor (crystallography). --- Real projective space. --- Right angle. --- Sard's theorem. --- Seifert fiber space. --- Set (mathematics). --- Siegel domain. --- Simply connected space. --- Solid torus. --- Special case. --- Sphere. --- Stereographic projection. --- Subgroup. --- Subsequence. --- Subset. --- Tangent space. --- Tangent vector. --- Tetrahedron. --- Theorem. --- Topology. --- Torus. --- Transversality (mathematics). --- Triangle group. --- Union (set theory). --- Unit disk. --- Unit sphere. --- Unit tangent bundle.

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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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Charles Favre and Thomas Gauthier present new mathematical research in the field of arithmetic dynamics. Specifically, the authors study one-dimensional algebraic families of pairs given by a polynomial with a marked point. Combining tools from arithmetic geometry and holomorphic dynamics, they prove an 'unlikely intersection' statement for such pairs, thereby demonstrating strong rigidity features for them. They further describe one-dimensional families in the moduli space of polynomials containing infinitely many postcritically finite parameters, proving the dynamical André-Oort conjecture for curves in this context, originally stated by Baker and DeMarco.

MATHEMATICS / Geometry / Algebraic. --- Affine plane. --- Affine space. --- Affine transformation. --- Algebraic closure. --- Algebraic curve. --- Algebraic equation. --- Algebraic extension. --- Algebraic surface. --- Algebraic variety. --- Algebraically closed field. --- Analysis. --- Analytic function. --- Analytic geometry. --- Approximation. --- Arithmetic dynamics. --- Asymmetric graph. --- Ball (mathematics). --- Bifurcation theory. --- Boundary (topology). --- Cantor set. --- Characterization (mathematics). --- Chebyshev polynomials. --- Coefficient. --- Combinatorics. --- Complex manifold. --- Complex number. --- Computation. --- Computer programming. --- Conjugacy class. --- Connected component (graph theory). --- Continuous function (set theory). --- Coprime integers. --- Correspondence theorem (group theory). --- Counting. --- Critical graph. --- Cubic function. --- Datasheet. --- Disk (mathematics). --- Divisor (algebraic geometry). --- Elliptic curve. --- Equation. --- Equidistribution theorem. --- Equivalence relation. --- Euclidean topology. --- Existential quantification. --- Fixed point (mathematics). --- Function space. --- Geometric (company). --- Graph (discrete mathematics). --- Hamiltonian mechanics. --- Hausdorff dimension. --- Hausdorff measure. --- Holomorphic function. --- Inequality (mathematics). --- Instance (computer science). --- Integer. --- Intermediate value theorem. --- Intersection (set theory). --- Inverse-square law. --- Irreducible component. --- Iteration. --- Jordan curve theorem. --- Julia set. --- Limit of a sequence. --- Line (geometry). --- Metric space. --- Moduli space. --- Moment (mathematics). --- Montel's theorem. --- P-adic number. --- Parameter. --- Pascal's Wager. --- Periodic point. --- Polynomial. --- Power series. --- Primitive polynomial (field theory). --- Projective line. --- Quotient ring. --- Rational number. --- Realizability. --- Renormalization. --- Riemann surface. --- Ring of integers. --- Scientific notation. --- Set (mathematics). --- Sheaf (mathematics). --- Sign (mathematics). --- Stone–Weierstrass theorem. --- Subharmonic function. --- Support (mathematics). --- Surjective function. --- Theorem. --- Theory. --- Topology. --- Transfer principle. --- Union (set theory). --- Unit disk. --- Variable (computer science). --- Variable (mathematics). --- Zariski topology. --- Polynomials. --- Dynamics. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Algebra --- Algebraic geometry. --- Mathematics.

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The authors offer an introduction to mathematical analysis with applications in economic theory & econometrics.

Economics, Mathematical. --- Mathematical analysis. --- Econometrics. --- Approximation. --- Axiom of choice. --- Banach space. --- Bijection. --- Bounded function. --- Budget set. --- Calculation. --- Cardinality. --- Cauchy sequence. --- Central limit theorem. --- Combination. --- Compact space. --- Complete metric space. --- Concave function. --- Conditional expectation. --- Continuous function (set theory). --- Continuous function. --- Contraction mapping. --- Contradiction. --- Convex analysis. --- Convex set. --- Countable set. --- Dense set. --- Differentiable function. --- Dimension (vector space). --- Dimension. --- Division by zero. --- Dynamic programming. --- Empty set. --- Equation. --- Equivalence class. --- Estimator. --- Existential quantification. --- Finite set. --- Fixed-point theorem. --- Function (mathematics). --- Hahn–Banach theorem. --- Independence (probability theory). --- Indicator function. --- Inequality (mathematics). --- Infimum and supremum. --- Intermediate value theorem. --- Karush–Kuhn–Tucker conditions. --- Law of large numbers. --- Lebesgue measure. --- Limit of a sequence. --- Limit superior and limit inferior. --- Linear algebra. --- Linear function. --- Linear map. --- Linear subspace. --- Loss function. --- Markov chain. --- Mathematical optimization. --- Mathematics. --- Maximal element. --- Measurable function. --- Measure (mathematics). --- Metric space. --- Monotonic function. --- Normed vector space. --- Null set. --- Open set. --- Optimization problem. --- Parameter. --- Pareto efficiency. --- Partially ordered set. --- Preference (economics). --- Preference relation. --- Probability distribution. --- Probability space. --- Probability theory. --- Probability. --- Quantity. --- Random variable. --- Rational number. --- Real number. --- Scientific notation. --- Sequence. --- Set (mathematics). --- Simple function. --- Special case. --- Stochastic process. --- Stone–Weierstrass theorem. --- Subsequence. --- Subset. --- Summation. --- Surjective function. --- Theorem. --- Topological space. --- Topology. --- Uncountable set. --- Uniform continuity. --- Uniform distribution (discrete). --- Union (set theory). --- Upper and lower bounds. --- Utility. --- Variable (mathematics). --- Vector space. --- Zorn's lemma.