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A discussion of the properties of conformal mappings in the complex plane, closely related to the study of fractals and chaos. Indeed, the book ends in a detailed study of the famous Mandelbrot set, which describes very general properties of such mappings. Focusing on the analytic side of this contemporary subject, the text was developed from a course taught over several semesters and aims to help students and instructors to familiarize themselves with complex dynamics. Topics covered include: conformal and quasi-conformal mappings, fixed points and conjugations, basic rational iteration, classification of periodic components, critical points and expanding maps, some applications of conformal mappings, the local geometry of the Fatou set, and quadratic polynomials and the Mandelbrot set.
Discusses the properties of conformal mappings in the complex plane, a contemporary subject that is closely connected to the study of fractals and chaos. The text was developed out of a course taught over several semesters, with a focus on helping students and instructors to familiarize themselves with complex dynamics. 28 illustrations.
I. Conformal and Quasiconformal Mappings.- 1. Some Estimates on Conformal Mappings.- 2. The Riemann Mapping.- 3. Montel’s Theorem.- 4. The Hyperbolic Metric.- 5. Quasiconformal Mappings.- 6. Singular Integral Operators.- 7. The Beltrami Equation.- II. Fixed Points and Conjugations.- 1. Classification of Fixed Points.- 2. Attracting Fixed Points.- 3. Repelling Fixed Points.- 4. Superattracting Fixed Points.- 5. Rationally Neutral Fixed Points.- 6. Irrationally Neutral Fixed Points.- 7. Homeomorphisms of the Circle.- III. Basic Rational Iteration.- 1. The Julia Set.- 2. Counting Cycles.- 3. Density of Repelling Periodic Points.- 4. Polynomials.- IV. Classification of Periodic Components.- 1. Sullivan’s Theorem.- 2. The Classification Theorem.- 3. The Wolff-Denjoy Theorem.- V. Critical Points and Expanding Maps.- 1. Siegel Disks.- 2. Hyperbolicity.- 3. Subhyperbolicity.- 4. Locally Connected Julia Sets.- VI. Applications of Quasiconformal Mappings.- 1. Polynomial-like Mappings.- 2. Quasicircles.- 3. Herman Rings.- 4. Counting Herman Rings.- 5. A Quasiconformal Surgical Procedure.- VII. Local Geometry of the Fatou Set.- 1. Invariant Spirals.- 2. Repelling Arms.- 3. John Domains.- VIII. Quadratic Polynomials.- 1. The Mandelbrot Set.- 2. The Hyperbolic Components of—.- 3. Green’s Function of—c.- 4. Green’s Function of—.- 5. External Rays with Rational Angles.- 6. Misiurewicz Points.- 7. Parabolic Points.- Epilogue.- References.- Symbol Index.