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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.