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This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Lattés map has been made more inclusive, and the écalle-Voronin theory of parabolic points is described. The résidu itératif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated. Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field.

Functions of complex variables --- Holomorphic mappings --- Riemann surfaces --- Fonctions d'une variable complexe --- Applications holomorphes --- Riemann, surfaces de --- Holomorphic mappings. --- Mappings, Holomorphic --- Functions of complex variables. --- Riemann surfaces. --- Surfaces, Riemann --- Functions --- Functions of several complex variables --- Mappings (Mathematics) --- Complex variables --- Elliptic functions --- Functions of real variables --- Absolute value. --- Addition. --- Algebraic equation. --- Attractor. --- Automorphism. --- Beltrami equation. --- Blaschke product. --- Boundary (topology). --- Branched covering. --- Coefficient. --- Compact Riemann surface. --- Compact space. --- Complex analysis. --- Complex number. --- Complex plane. --- Computation. --- Connected component (graph theory). --- Connected space. --- Constant function. --- Continued fraction. --- Continuous function. --- Coordinate system. --- Corollary. --- Covering space. --- Cross-ratio. --- Derivative. --- Diagram (category theory). --- Diameter. --- Diffeomorphism. --- Differentiable manifold. --- Disjoint sets. --- Disjoint union. --- Disk (mathematics). --- Division by zero. --- Equation. --- Euler characteristic. --- Existential quantification. --- Exponential map (Lie theory). --- Fundamental group. --- Harmonic function. --- Holomorphic function. --- Homeomorphism. --- Hyperbolic geometry. --- Inequality (mathematics). --- Integer. --- Inverse function. --- Irrational rotation. --- Iteration. --- Jordan curve theorem. --- Julia set. --- Lebesgue measure. --- Lecture. --- Limit point. --- Line segment. --- Linear map. --- Linearization. --- Mandelbrot set. --- Mathematical analysis. --- Maximum modulus principle. --- Metric space. --- Monotonic function. --- Montel's theorem. --- Normal family. --- Open set. --- Orbifold. --- Parameter space. --- Parameter. --- Periodic point. --- Point at infinity. --- Polynomial. --- Power series. --- Proper map. --- Quadratic function. --- Rational approximation. --- Rational function. --- Rational number. --- Real number. --- Riemann sphere. --- Riemann surface. --- Root of unity. --- Rotation number. --- Schwarz lemma. --- Scientific notation. --- Sequence. --- Simply connected space. --- Special case. --- Subgroup. --- Subsequence. --- Subset. --- Summation. --- Tangent space. --- Theorem. --- Topological space. --- Topology. --- Uniform convergence. --- Uniformization theorem. --- Unit circle. --- Unit disk. --- Upper half-plane. --- Winding number.

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This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

Differential equations, Elliptic. --- Quasiconformal mappings. --- Mappings, Quasiconformal --- Conformal mapping --- Functions of complex variables --- Geometric function theory --- Mappings (Mathematics) --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear --- Differential equations, Partial --- Adjoint equation. --- Analytic function. --- Analytic proof. --- Banach space. --- Beltrami equation. --- Boundary value problem. --- Bounded mean oscillation. --- Calculus of variations. --- Cantor function. --- Cartesian product. --- Cauchy–Riemann equations. --- Central limit theorem. --- Characterization (mathematics). --- Complex analysis. --- Complex plane. --- Conformal geometry. --- Conformal map. --- Conjugate variables. --- Continuous function (set theory). --- Coordinate space. --- Degeneracy (mathematics). --- Differential equation. --- Directional derivative. --- Dirichlet integral. --- Dirichlet problem. --- Disk (mathematics). --- Distribution (mathematics). --- Elliptic operator. --- Elliptic partial differential equation. --- Equation. --- Equations of motion. --- Euler–Lagrange equation. --- Explicit formulae (L-function). --- Factorization. --- Fourier transform. --- Fubini's theorem. --- Geometric function theory. --- Geometric measure theory. --- Geometry. --- Harmonic conjugate. --- Harmonic function. --- Harmonic map. --- Harmonic measure. --- Hilbert transform. --- Holomorphic function. --- Homeomorphism. --- Hyperbolic geometry. --- Hyperbolic trigonometry. --- Invertible matrix. --- Jacobian matrix and determinant. --- Julia set. --- Lagrangian (field theory). --- Laplace's equation. --- Limit (mathematics). --- Linear differential equation. --- Linear equation. --- Linear fractional transformation. --- Linear map. --- Linearization. --- Lipschitz continuity. --- Locally integrable function. --- Lusin's theorem. --- Mathematical optimization. --- Mathematics. --- Maxima and minima. --- Maxwell's equations. --- Measure (mathematics). --- Metric space. --- Mirror symmetry (string theory). --- Moduli space. --- Modulus of continuity. --- Monodromy theorem. --- Monotonic function. --- Montel's theorem. --- Operator (physics). --- Operator theory. --- Partial derivative. --- Partial differential equation. --- Poisson formula. --- Polynomial. --- Quadratic function. --- Quasiconformal mapping. --- Quasiconvex function. --- Quasisymmetric function. --- Renormalization. --- Riemann sphere. --- Riemann surface. --- Riemannian geometry. --- Riesz transform. --- Riesz–Thorin theorem. --- Sign (mathematics). --- Sobolev space. --- Square-integrable function. --- Support (mathematics). --- Theorem. --- Two-dimensional space. --- Uniformization theorem. --- Upper half-plane. --- Variable (mathematics). --- Weyl's lemma (Laplace equation).