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Tuesday, November 17, 2015 - 17:00 ,
Location: Skiles 005 ,
Mikel Viana ,
Georgia Tech (Math) ,
Organizer:

In previous talks, we discussed an algorithm (Nash-Moser iteration) to compute invariant whiskered tori for fibered holomorphic maps. Several geometric and number-theoretic conditions are necessary to carry out each step of the iteration. Recently, there has been interest in studying what happens if some of the conditions are removed. In particular, the second Melnikov condition we found can be hard to verify in higher dimensional problems. In this talk, we will use a method due to Eliasson, Moser and Poschel to obtain quasi-periodic solutions which, however, lose an important geometric property relative to the solutions previously constructed.

Tuesday, November 10, 2015 - 17:00 ,
Location: Skiles 005 ,
Rafael de la Llave ,
Georgia Tech ,
Organizer:

In the study of perturbation theories in Dynamical systems one is often interested in solving differential equations involving frequencies satisfying number theoretic properties. We will present some estimates ofsums involving Diophantine frequencies leading to sharp estimates on the differential equations.

Thursday, October 29, 2015 - 17:00 ,
Location: Skiles 006 ,
Mikel Viana ,
Georgia Tech (Math) ,
Organizer:

We consider fibered holomorphic dynamics, generated by a skew product
over an irrational translation of the torus. The invariant object that
organizes the dynamics is an invariant torus. Often one can find an
approximately invariant torus K_0, and we construct an invariant torus,
starting from K_0. The main technique is a KAM iteration in
a-posteriori format. In this talk we give the details of the iterative procedure using the geometric and number-theoretic conditions presented last time.

Tuesday, October 6, 2015 - 17:00 ,
Location: Skiles 005 ,
Mikel de Viana ,
Georgia Tech ,
Organizer:

We consider fibered holomorphic dynamics, generated by a skew product over an irrational translation of the torus. The invariant object that organizes the dynamics is an invariant torus. Often one can find an approximately invariant torus K_0, and we construct an invariant torus, starting from K_0. The main technique is a KAM iteration in a-posteriori format. The asymptotic properties of the derivative cocycle A_K play a crucial role: In this first talk we will find suitable geometric and number-theoretic conditions for A_K. Later, we will see how to relax these conditions.

Tuesday, September 29, 2015 - 17:00 ,
Location: Skiles 005 ,
Rafael de la Llave ,
Georgia Tech (Math) ,
Organizer:

We will review the notion of Whitney differentiability and the Whitney
embedding theorem. Then, we will also review its applications in KAM
theory (continuation of last week's talk).

Tuesday, September 22, 2015 - 17:00 ,
Location: Skiles 254 ,
Rafael de la Llave ,
Georgia Institute of Technology ,
Organizer: Lei Zhang

We will review the notion of Whitney differentiability and the Whitney embedding theorem. Then, we will also review its applications in KAM theory.

Tuesday, September 15, 2015 - 17:00 ,
Location: Skiles 005 ,
Xiaolong He ,
Georgia Tech (Math)/Hunan University ,
Organizer:

We investigate the existence of quasi-periodic solutions for
state-dependent delay
differential equationsusing the parameterization
method, which is different from the usual way-working on the solution
manifold. Under the assumption of finite-time differentiability of
functions and exponential dichotomy, the existence and smoothness of
quasi-periodic solutions are investigated by using contraction
arguments We also develop a KAM
theory to seek analytic quasi-periodic solutions. In contrast with the
finite differentonable theory, this requires adjusting parameters. We
prove that the set of parameters which guarantee the
existence of analytic quasi-periodic solutions is of positive measure.
All of these results are given in an a-posterior form. Namely, given a
approximate solution satisfying some non-degeneracy conditions, there is
a true solution nearby.

Tuesday, September 8, 2015 - 17:00 ,
Location: Skiles 005 ,
Jiaqi Yang ,
Georgia Tech ,
Organizer:

(continuation of last week's seminar): 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.

Tuesday, September 1, 2015 - 17:00 ,
Location: Skiles 005 ,
Jiaqi Yang ,
Georgia Tech ,
Organizer:

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.

Wednesday, April 16, 2014 - 12:05 ,
Location: Skiles ,
Rafael de la Llave ,
Georgia Tech ,
Organizer:

We prove Ruelle's Entropy Inequality for C^1 maps. 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.