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Introduction to the Modern Theory of Dynamical Systems

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A new capacity for plurisubharmonic functions

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

The theory of complex analytic sets is part of the modern geometric theory of functions of several complex variables. Traditionally, the presentation of the foundations of the theory of analytic sets is introduced in the algebraic language of ideals in Noetherian rings as, for example, in the books of Herve [23] or Gunning-Rossi [19]. However, the modern methods of this theory, the principal directions and applications, are basically related to geometry and analysis (without regard to the traditional direction which is essentially related to algebraic geometry). Thus, at the beginning of this survey, the geometric construction of the local theory of analytic sets is presented. Its foundations are worked out in detail in the book of Gunning and Rossi [19] via the notion of analytic cover which together with analytic theorems on the removal of singularities leads to the minimum of algebraic apparatus necessary in order to get the theory started.

Complex Analytic Sets

In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2 Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 21, 23, 24, and 26 The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy In contrast, structural and the related topological stability seem to be fairly widespread phenomena We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits The first two classes are of a purely topological nature; they are discussed in the present chapter The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject This will require some space so we put that material into a separate chapter The two chapters are intimately connected

Introduction to the Modern Theory of Dynamical Systems: PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS

The field of complex dynamics in higher dimension was initiated in the 1920’s by Fa-tou. It was motivated by studies in Newton’s method, celestial mechanics and functional equations. Recently, new methods from pluripotential theory have been introduced to the subject. These techniques have produced many new interesting results. We give an introduction to this subject and a summary of the most relevant developments.

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Complex dynamics in higher dimensions

Let X be a complex projective manifold and let f be a dominating rational map from X onto X. We show that the topological entropy h(f) of f is bounded from above by the logarithm of its maximal dynamical degree.

Nessim Sibony

Papers

Complex dynamics in higher dimensions

Une borne supérieure pour l'entropie topologique d'une application rationnelle

Quelques problemes de prolongement de courants en analyse complexe

Oka's inequality for currents and applications

Une classe de domaines pseudoconvexes