Resumen de A Rigidity Property of Perturbations of n Identical Harmonic Oscillators

Massimo Villarini

• Let Xε:S2n-1→TS2n-1 be a smooth perturbation of X0, the vector field associated to the dynamical system defined by n identical uncoupled harmonic oscillators constrained to their 1-energy level. We are dealing with the case when any orbit of every Xε is closed: while in general is false that the vector fields of the perturbation are orbitally equivalent to the unperturbed X0 (Villarini in Ergod Theory Dyn Syst 39:1–32, 2019), we prove that this rigidity behaviour is indeed true if each Xε restricted to a codimension 2 sphere in S2n-1 is orbitally conjugated to a subsystem of X0 made by n-1 harmonic oscillators. In other words: to have a non-rigid, or truly non-linear, perturbation of X0 at least two harmonic oscillators must be destroyed by the perturbation. We use this rigidity result to prove a linearization theorem for real analytic multicentres. Finally we give an example of a real analytic perturbation of X0 showing discontinuous changing of integer invariants of the vector fields of the perturbation.