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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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The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.

Harmonic analysis. --- Harmonic functions. --- Functions, Harmonic --- Laplace's equations --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Harmonic analysis. Fourier analysis --- Harmonic analysis --- Fourier analysis --- Harmonic functions --- Analyse harmonique --- Analyse de Fourier --- Fonctions harmoniques --- Fourier Analysis --- Fourier, Transformations de --- Euclide, Espaces d' --- Bessel functions --- Differential equations, Partial --- Fourier series --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Banach algebras --- Time-series analysis --- Analysis, Fourier --- Fourier analysis. --- Basic Sciences. Mathematics --- Analysis, Functions --- Analysis, Functions. --- Calculus --- Mathematical analysis --- Mathematics --- Fourier, Transformations de. --- Euclide, Espaces d'. --- Potentiel, Théorie du --- Fonctions harmoniques. --- Potential theory (Mathematics) --- Analytic continuation. --- Analytic function. --- Banach algebra. --- Banach space. --- Bessel function. --- Borel measure. --- Boundary value problem. --- Bounded operator. --- Bounded set (topological vector space). --- Cartesian coordinate system. --- Cauchy–Riemann equations. --- Change of variables. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Complex plane. --- Conformal map. --- Conjugate transpose. --- Continuous function (set theory). --- Continuous function. --- Convolution. --- Differentiation of integrals. --- Dimensional analysis. --- Dirichlet problem. --- Disk (mathematics). --- Distribution (mathematics). --- Equation. --- Euclidean space. --- Existential quantification. --- Fourier inversion theorem. --- Fourier series. --- Fourier transform. --- Fubini's theorem. --- Function (mathematics). --- Function space. --- Green's theorem. --- Hardy's inequality. --- Hardy–Littlewood maximal function. --- Harmonic function. --- Hermitian matrix. --- Hilbert transform. --- Holomorphic function. --- Homogeneous function. --- Inequality (mathematics). --- Infimum and supremum. --- Interpolation theorem. --- Interval (mathematics). --- Lebesgue integration. --- Lebesgue measure. --- Linear interpolation. --- Linear map. --- Linear space (geometry). --- Line–line intersection. --- Liouville's theorem (Hamiltonian). --- Lipschitz continuity. --- Locally integrable function. --- Lp space. --- Majorization. --- Marcinkiewicz interpolation theorem. --- Mean value theorem. --- Measure (mathematics). --- Mellin transform. --- Monotonic function. --- Multiplication operator. --- Norm (mathematics). --- Operator norm. --- Orthogonal group. --- Paley–Wiener theorem. --- Partial derivative. --- Partial differential equation. --- Plancherel theorem. --- Pointwise convergence. --- Poisson kernel. --- Poisson summation formula. --- Polynomial. --- Principal value. --- Quadratic form. --- Radial function. --- Radon–Nikodym theorem. --- Representation theorem. --- Riesz transform. --- Scientific notation. --- Series expansion. --- Singular integral. --- Special case. --- Subharmonic function. --- Support (mathematics). --- Theorem. --- Topology. --- Total variation. --- Trigonometric polynomial. --- Trigonometric series. --- Two-dimensional space. --- Union (set theory). --- Unit disk. --- Unit sphere. --- Upper half-plane. --- Variable (mathematics). --- Vector space. --- Fourier, Analyse de --- Potentiel, Théorie du. --- Potentiel, Théorie du --- Espaces de hardy