Georgia Institute of Technology College of Sciences

We provide several explicit examples of 3D quasiperiodic linear skew-products with simple Lyapunov spectrum, that is with 3 different Lyapunov multipliers, for which the corresponding Oseledets bundles are measurable but not continuous, colliding in a measure zero dense set.

Theoretical aspects: If a smooth dynamical system on a compact invariant set is structurally stable, then it has the shadowing property, that is, any pseudo (or approximate) orbit has a true orbit nearby. In fact, the system has the Lipschitz shadowing property, that is, the distance between the pseudo and true orbit is at most a constant multiple of the local error in the pseudo orbit. S. Pilyugin and S. Tikhomirov showed the converse of this statement for discrete dynamical systems, that is, if a discrete dynamical system has the Lipschitz shadowing property, then it is structurally stable. In this talk this result will be reviewed and the analogous result for flows, obtained jointly with S. Pilyugin and S. Tikhomirov, will be described. Numerical aspects: This is joint work with Brian Coomes and Huseyin Kocak. A rigorous numerical method for establishing the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous system of ordinary differential equations is presented. Given a suitable approximate connecting orbit and assuming that a certain associated linear operator is invertible, the existence of a true connecting orbit near the approximate orbit and for a nearby parameter value is proved provided the approximate orbit is sufficiently ``good''. It turns out that inversion of the operator is equivalent to the solution of a boundary value problem for a nonautonomous inhomogeneous linear difference equation. A numerical procedure is given to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse (the latter determines how ``good'' the approximating orbit must be).

In this talk we will discuss recent work, obtained in collaboration with Jean Bourgain, on new global well-posedness results along Gibbs measure evolutions for the radial nonlinear wave and Schr\"odinger equations posed on the unit ball in two and three dimensional Euclidean space, with Dirichlet boundary conditions. We consider initial data chosen according to a Gaussian random process associated to the Gibbs measures which arise from the Hamiltonian structure of the equations, and results are obtained almost surely with respect to these probability measures. In particular, this renders the initial value problem supercritical in the sense that there is no suitable local well-posedness theory for the corresponding deterministic problem, and our results therefore rely essentially on the probabilistic structure of the problem. Our analysis is based on the study of convergence properties of solutions. Essential ingredients include probabilistic a priori bounds, delicate estimates on fine frequency interactions, as well as the use of invariance properties of the Gibbs measure to extend the relevant bounds to arbitrarily long time intervals.

This talk is devoted to quasi-periodic Schrödinger operators beyond theAlmost Mathieu, with more general potentials and interactions. The linksbetween the spectral properties of these operators and the dynamicalproperties of the associated quasi-periodic linear skew-products rule thegame. In particular, we present a Thouless formula and some consequencesof Aubry duality. This is a joint work with Joaquim Puig~