Georgia Institute of Technology College of Sciences

We present a methodology to rigorously validate a given approximation of a quasi-periodic Lagrangian torus of a symplectic map. The approach consists in verifying the hypotheses of a-posteriori KAM theory based of the parameterization method (following Rafael de la Llave and collaborators). A crucial point of our imprementation is an analytic Lemma that allows us to control the norm of periodic functions using their discrete Fourier transform. An outstanding consequence of this approach it that the computational cost of the validation is assymptotically equivalent of the cost of the numerical computation of invariant tori using the parametererization method. We pretend to describe some technical aspects of our implementation. This is a work in progress joint with Jordi-Lluis Figueras and Alejandro Luque.

The Standard Map is a discrete time area-preserving dynamical system and is one of the simplest of such systems to exhibit chaotic dynamics. Traditional studies of the Standard Map have employed symmetric forcing functions that do not induce a net flux. Although the dynamics of these maps is rich there are many systems which cannot be modeled with these restrictions. In this talk we will explore the dynamics of the Standard Map when the forcing is asymmetric and induces a positive flux on the system. We will introduce new numerical methods to study these dynamics and give an overview of how transport in the system changes under these new forces.

We show that all time changes of the horocycle flow on compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This provides an answer to a question of A. Katok and J.-P. Thouvenot on the spectral nature of time changes of horocycle flows. Our proofs rely on positive commutator methods for self-adjoint operators and the unique ergodicity of the horocycle flow. www.mat.uc.cl/~rtiedra/download/Horocycles_Bordeaux_2014.pdf <http://www.mat.uc.cl/%7Ertiedra/download/Horocycles_Bordeaux_2014.pdf <http://www.mat.uc.cl/~rtiedra/download/Horocycles_Bordeaux_2014.pdf>>

We consider an atomic model of deposition of materials over a quasi-periodic medium. The atoms of the deposited material interact with the medium (a quasi-periodic interaction) and with their nearest neighbors (a harmonic interaction). This is a quasi-periodic version of the well known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show that there are always perturbative expansions. We also prove a KAM theorem in a-posteriori form. We show that if there is an approximate solution of the equilibrium equation satisfying non-degeneracy conditions, we can adjust one parameter and obtain a true solution which is close to the approximate solution. The proof is based on an iterative method of the KAM type. The iterative method is not based on transformation theory as the most usual KAM theory, but it is based on a novel technique of supplementing the equilibrium equation with another equation that factors the linearization of the equilibrium equilibrium equation.