Georgia Institute of Technology College of Sciences

We will discuss KAM results for symplectic and presymplectic maps. Firstly, we will study geometric properties of a symplectic dynamical system which will allow us to prove a KAM theorem in a-posteriori format. Then, a corresponding theorem for a parametric family of symplectic maps will be presented. Finally, using similar method, we will extend the theorems to presymplectic maps. These results appear in the work of Alishah, de la Llave, Gonzalez, Jorba and Villanueva.

We finish our discussion on Oseledets Theorem by proving the convergence of the filtration. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.

We present the proof of the Shannon-McMillan-Breiman Theorem. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.

We will present a proof of the subadditive ergodic theorem. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.