Table of contents

CONTENTS Introduction § 1. Attractors of semigroups in a normed space § 2. Examples of non-stationary differential equations having attractors § 3. An invariant manifold § 4. Examples of equations having a regular attractor § 5. Semigroups of operators depending on a parameter § 6. The existence of a Lyapunov function for a perturbed semigroup § 7. The behaviour of local unstable manifolds near a critical point References

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. Schröder domains § 1.9. Superattracting cycles. Böttcher 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 Teichmüller spaces § 2.6. A-domains of the parameter space § 2.7. The Mandelbrot set References

CONTENTS Introduction Chapter I. Infinite-dimensional elliptic operators § 1. Lévy's Laplacian § 2. The second-order differential operator generated by Lévy's Laplacian § 3. Differential operators of any even order generated by Lévy's Laplacian § 4. Infinite-dimensional elliptic differential expressions Chapter II. Infinite-dimensional symmetric elliptic operators § 1. The symmetrized Lévy's Laplacian on functions from the domain of the Laplace-Lévy operator § 2. Lévy's Laplacian on functions from the domain of the symmetric Laplace-Lévy operator § 3. Formally self-adjoint elliptic expressions Chapter III. The solution of boundary-value problems for elliptic equations for functionals defined on function spaces § 1. The Dirichlet problem for the Laplace-Lévy and Poisson-Lévy equations § 2. The Dirichlet problem for the Schrödinger-Lévy equation § 3. The Riquier problem for a polyharmonic equation Chapter IV. The solubility of boundary-value problems for infinite-dimensional elliptic equations § 1. Equations with constant coefficients § 2. Self-adjoint equations Appendix. An application of Lévy type expressions to obtain the characteristics of processes described by parabolic equations with random coefficients References