The subject grew out of a math olympiad problem: The president and an assassin are in a room with reflecting walls. Assassin's bullets would hit the president even after any number of ricochets off the walls. Thus, the assassin has at his disposal infinitely many deadly possibilities. The presidential security detail can instantaneously position themselves at arbitrary spots in the room, and hence at the spots where they would intercept optimal subsets of deadly bullets. Can a finite number of security guards protect the prez? I will talk about mathematical ramifications of this olympiad question.

We will show that linear quasi-periodic systems with coefficients in SL(2,R) and with arbitrarily many frequencies are reducible under an arithmetical condition on the frequency and on the rotation number which is weaker than the diophantine condition and similar to the usual Brjuno condition. This gives an extension of a theorem by Eliasson. (joint work with S. Marmi)

I shall discuss about the results recently published in

Gelfreich, V.; Simó, C.; Vieiro, A., Dynamics of 4D symplectic maps near a double resonance. Physica D: Nonlinear Phenomena, 243(1):92--110 (Holland): 2012.

In this work we consider a family of 4D symplectic mappings near a doubly resonant elliptic fixed point. Around a weak double resonance (a junction of two resonances of different orders, both being larger than 4) the dynamics can be described in terms of a simple (in general non-integrable) Hamiltonian model. The main results are:

1) We show that the non-integrability of the normal form is expected because of the generic splitting of the invariant manifolds associated with a normally hyperbolic invariant cylinder.

2) Using an interpolating vector field of the truncated normal form we provide a strong numerical evidence of the fact that the normally hyperbolic invariant cylinder of the 4D standard-like map is not analytic and it has the regularity that can be expected in the generic case.

3) Finally, we use a 4D generalisation of the standard map to illustrate the difference between a truncated normal form (interpolating Hamiltonian) and the full standard-like map. In particular, we (numerically) evaluate the volume of a 4D parallelotope defined by 4 vectors tangent to the stable and unstable manifolds respectively. In good agreement with the general theory, this volume is exponentially small with respect to a small parameter and we derive an empirical asymptotic formula which suggests amazing similarity to its 2D analog.