Introdução à Dinâmica Simplética
Pré-requisito: Differential geometry and (basic notions of) symplectic geometry.
The course is an introduction to the subject of Hamiltonian dynamical systems. One important question to consider is the existence of periodic orbits of such systems. The dynamics near a periodic orbit can be described in terms of local symplectomorphisms. The Poincaré-Birkhoff theorem, which asserts that an area-preserving twist map of the annulus must have at least two distinct fixed points, leads to a natural generalization on the existence of fixed points of symplectomorphisms of compact symplectic manifolds, the Arnold conjecture. This conjecture has led to many interesting results in the subject. We plan to discuss these and related topics which include:
- area-preserving diffeomorphisms; the Poincaré-Birkhoff theorem; generating functions;
- group of symplectomorphisms; the flux conjecture;
- Arnold conjecture; the Conley-Zehnder theorem;
- Morse homology.
Time permitting we will also discuss some of the following topics:
- Morse theory for circle-valued functions and closed one-forms;
- Floer homology;
- other symplectic invariants: capacities.
[HZ11] HOFER. H., ZEHNDER, E. – Symplectic invariants and Hamiltonian dynamics, Modern Birkhauser Classics, Birkhauser Verlag, Basel, 2011, Reprint of the 1994 edition.
[MS95] McDu, D., Salamon, D. – Introduction to symplectic topology, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995, Oxford Science Publications.
* Ementa básica. O professor tem autonomia para efetuar qualquer alteração.