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.

Outline

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,...

Outline

1. KAM Theory. Russman conditions.
2. Normally hyperbolic invariant manifolds.
3. Lambda Lemma/Scatering map/Exchange Lemma.
4. Mechanisms of instability.
5. Numerical methods.

References

• A. Gonzalez, A. Jorba, R. de la Llave, J. Villanueva, "KAM theory without action angle variables", (2005) Nonlinearity.
• A. Gonzalez, A. Haro, R. de la Llave, "KAM theory with parameters".
• H. Broer, A. Huitema, M. Sevryuk, "Quasi-periodic solutions", Lecture notes in Mathematics.
• A. Haro, R. de la Llave, "Persistence of normally hyperbolic manifolds".
• Y. pesin, "Lectures on partial hyperbolicity" (2004) EMS.
• A. Delshams, R. de la Llave, T. M.Seara, "Geometric properties of the scattering map to a normally hyperbolic invariant manifold", (2007) Advances in Mathematics.
• A. Delshams, M. Gidea, R. de la Llave, T. M-Seara, "Geometric mechanisms of instability in Hamiltonian systems", (2007) Montreal ASI.
• A. Haro, R. de la Llave, "A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity", SIAM Journal Applied Dynamical Systems, (2007)
• R. Calleja, G. Huguet, R. de la Llave, Y. Sire, "Fast and low storage algorithms for computation of quasi-periodic solutions", (2007)

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.