RTNS08 (Recent Trends in Nonlinear Science 2008)

COURSES

• Karl-Goswin Grosse-Erdmann (Mons-Hainaut University)
  Chaotic infinite dimensional dynamics
• Rafael de la Llave (Texas University at Austin)
  Stability and diffusion in Hamiltonian systems
• Mikhail Lyubich (Stony Brook University)
  Introduction to holomorphic dynamics

CONTENTS

Chaotic infinite dimensional dynamics
Karl-Goswin Grosse-Erdmann (Mons-Hainaut University)

Abstract. Contrary to common belief, chaos is not restricted to non-linear maps. However, in order to encounter linear chaos one has to go to infinite dimensions. In the course we shall present an introduction to the study of chaotic linear maps as it has developed over the last 25 years.

1. Topological dynamics: dense orbits, chaotic maps, Birkhoff transitivity criterion, examples
2. Hypercyclic operators: definition, basis examples, finite vs. infinite dimension, chaotic linear maps
3. Hypercyclicity Criterion: hypercyclicity criterion, eigenvalue criterion, Herrero's problem, equivalent conditions
4. Classes of chaotic operators: composition operators, shift operators, differentiation operators
5. Some theoretical highlights: Ansari's theorem, the Ansari-Bernal theorem, hypercyclic subspaces,...

Stability and diffusion in Hamiltonian systems
Rafael de la Llave (Texas University at Austin)

Abstract.

1. KAM Theory. Russman conditions. (KAM theory without action angle variables, A. Gonzalez, A. Jorba, R. de la Llave, J. Villanueva (2005) Nonlinearity; KAM theory with parameters, A. Gonzalez, A. Haro, R. de la Llave; Quasi-periodic solutions, H. Broer, A. Huitema, M. Sevryuk, Lecture notes in Mathematics)
2. Normally hyperbolic invariant manifolds.
3. Lambda Lemma/Scatering map/Exchange Lemma. (Persistence of normally hyperbolic manifolds, A. Haro, R. de la Llave; Lectures on partial hyperbolicity, Y. pesin (2004) EMS; Geometric properties of the scattering map to a normally hyperbolic invariant manifold, A. Delshams, R. de la Llave, T. M.Seara (2007) Advances in Mathematics)
4. Mechanisms of instability. (Geometric mechanisms of instability in Hamiltonian systems, A. Delshams, M. Gidea, R. de la Llave, T. M-Seara (2007) Montreal ASI)
5. Numerical methods. (A. Haro, R. de la Llave (2007) SIAM Journal Applied Dynamical Systems (2007); Fast and low storage algorithms for computation of quasi-periodic solutions, R. Calleja, G. Huguet, R. de la Llave, Y. Sire (2007))

Introduction to holomorphic dynamics
Mikhail Lyubich (Stony Brook University) Abstract. We will start from scratch with the definition of the Julia and Fatou sets of a polynomial, and continue with basic properties of dynamics in the complex plane: classification of periodic points and Fatou components, external rays and topological models for Julia sets, quasiconformal maps and Sullivan's No Wandering Domains Theorem. We will then proceed to the parameter plane with a definition and basic properties of the Mandelbrot set, and will discuss the fundamental dynamical-parameter relation. Finally, we will take a look at the baby Mandelbrot sets, and will explain how the idea of renormalization justifies their existence.