The dynamics of rational transforms: the topological picture

Abstract

CONTENTS List of basic notation Introduction Chapter I. The dynamics of an individual endomorphism § 1.1. The hyperbolic metric § 1.2. Analytic transforms of hyperbolic Riemann surfaces § 1.3. Montel's theorem. The Fatou set and the Julia set § 1.4. The simplest properties of the Julia set § 1.5. Ramified coverings. The Riemann-Hurwitz formula § 1.6. Components of the Fatou set § 1.7. Quasi-conformal maps. The measurable Riemann theorem § 1.8. Attracting cycles. Schroder domains § 1.9. Superattracting cycles. Bottcher domains § 1.10. Neutral rational cycles. The Leau flower § 1.11. Neutral irrational cycles. Siegel discs § 1.12. Arnol'd-Herman rings § 1.13. The density of repelling cycles in § 1.14. Further properties of : the density of inverse images, mixing § 1.15. The absence of wandering components of the Fatou set § 1.16. Rational endomorphisms satisfying axiom A § 1.17. Iterates of polynomials § 1.18. Endomorphisms whose critical point orbits are absorbed by cycles § 1.19. On the measure of the Julia set § 1.20. The Newton iterative process Chapter II. Holomorphic families of rational endomorphisms § 2.1. The -lemma and -stability § 2.2. Structural stability is a generic property § 2.3. The behaviour of orbits of critical points § 2.4. The family § 2.5. Classes of quasi-conformal conjugacy and Teichmuller spaces § 2.6. A-domains of the parameter space § 2.7. The Mandelbrot set References