Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c. As discovered by Feigenbaum, such a mapping exhibits a repetition of form at infinitely many scales. Drawing on universal estimates in hyperbolic geometry, this work gives an analysis of the limiting forms that can occur and develops a rigidity criterion for the polynomial f. This criterion supports general conjectures about the behavior of rational maps and the structure of the Mandelbrot set. The course of the main argument entails many facets of modern complex dynamics. Included are foundational results in geometric function theory, quasiconformal mappings, and hyperbolic geometry. Most of the tools are discussed in the setting of general polynomials and rational maps.

Renormalization (Physics) --- Polynomials. --- Dynamics. --- Mathematical physics. --- Analytic function. --- Attractor. --- Automorphism. --- Bernhard Riemann. --- Bounded set. --- Branched covering. --- Cantor set. --- Cardioid. --- Chain rule. --- Coefficient. --- Combinatorics. --- Complex manifold. --- Complex plane. --- Complex torus. --- Conformal geometry. --- Conformal map. --- Conjecture. --- Connected space. --- Covering space. --- Cyclic group. --- Degeneracy (mathematics). --- Dense set. --- Diagram (category theory). --- Diameter. --- Differential geometry of surfaces. --- Dihedral group. --- Dimension (vector space). --- Dimension. --- Disjoint sets. --- Disk (mathematics). --- Dynamical system. --- Endomorphism. --- Equivalence class. --- Equivalence relation. --- Ergodic theory. --- Euler characteristic. --- Filled Julia set. --- Geometric function theory. --- Geometry. --- Hausdorff dimension. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Hyperbolic geometry. --- Implicit function theorem. --- Injective function. --- Integer matrix. --- Interval (mathematics). --- Inverse limit. --- Julia set. --- Kleinian group. --- Limit point. --- Limit set. --- Linear map. --- Mandelbrot set. --- Manifold. --- Markov partition. --- Mathematical induction. --- Maxima and minima. --- Measure (mathematics). --- Moduli (physics). --- Monic polynomial. --- Montel's theorem. --- Möbius transformation. --- Natural number. --- Open set. --- Orbifold. --- Periodic point. --- Permutation. --- Point at infinity. --- Pole (complex analysis). --- Polynomial. --- Proper map. --- Quadratic differential. --- Quadratic function. --- Quadratic. --- Quasi-isometry. --- Quasiconformal mapping. --- Quotient space (topology). --- Removable singularity. --- Renormalization. --- Riemann mapping theorem. --- Riemann sphere. --- Riemann surface. --- Rigidity theory (physics). --- Scalar (physics). --- Schwarz lemma. --- Scientific notation. --- Special case. --- Structural stability. --- Subgroup. --- Subsequence. --- Symbolic dynamics. --- Tangent space. --- Theorem. --- Uniformization theorem. --- Uniformization. --- Union (set theory). --- Unit disk. --- Upper and lower bounds.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book concerns areas of ergodic theory that are now being intensively developed. The topics include entropy theory (with emphasis on dynamical systems with multi-dimensional time), elements of the renormalization group method in the theory of dynamical systems, splitting of separatrices, and some problems related to the theory of hyperbolic dynamical systems.Originally published in 1993.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Ergodic theory. --- Topological dynamics. --- Analytic continuation. --- Automorphism. --- Bifurcation theory. --- Borel–Cantelli lemma. --- Calculation. --- Cauchy's integral formula. --- Central limit theorem. --- Change of variables. --- Character group. --- Characterization (mathematics). --- Conditional entropy. --- Conditional probability. --- Continuous function (set theory). --- Cyclic group. --- Derivative. --- Determinant. --- Diffeomorphism. --- Differential equation. --- Dimension (vector space). --- Dimension. --- Dynamical system. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Equation. --- Ergodicity. --- Even and odd functions. --- Existential quantification. --- Feigenbaum constants. --- Frenet–Serret formulas. --- Fubini's theorem. --- Functional equation. --- Fundamental class. --- Fundamental lemma (Langlands program). --- Geodesic. --- Gibbs measure. --- Ground state. --- Haar measure. --- Hadamard's inequality. --- Hamiltonian mechanics. --- Hilbert space. --- Hyperbolic point. --- Indicator function. --- Infimum and supremum. --- Intrinsic metric. --- Invariant measure. --- Invariant subspace. --- Inverse function. --- Lebesgue measure. --- Lebesgue space. --- Linear map. --- Linearization. --- Liouville's theorem (Hamiltonian). --- Lorenz system. --- Manifold. --- Mathematical induction. --- Measure (mathematics). --- One-parameter group. --- Ordinary differential equation. --- Periodic function. --- Periodic point. --- Periodic sequence. --- Permutation. --- Perturbation theory (quantum mechanics). --- Phase space. --- Piecewise. --- Poincaré recurrence theorem. --- Probability distribution. --- Probability measure. --- Probability theory. --- Recurrence relation. --- Renormalization group. --- Riemannian manifold. --- Rotation number. --- Schrödinger equation. --- Scientific notation. --- Semigroup. --- Semilattice. --- Sign (mathematics). --- Square-integrable function. --- Statistical mechanics. --- Stochastic. --- Subalgebra. --- Subgroup. --- Submanifold. --- Subsequence. --- Subset. --- Summation. --- Symbolic dynamics. --- Symplectic geometry. --- Tangent space. --- Theorem. --- Theory. --- Transitive relation. --- Unit tangent bundle. --- Unitary operator. --- Variable (mathematics). --- Vector bundle. --- Vector field.