A Rigidity Property of Perturbations of n Identical Harmonic Oscillators

Abstract

Let $$X_{\varepsilon } : S^{2n-1} \rightarrow T S^{2n-1}$$ be a smooth perturbation of $$X_0$$ , 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 $$X_{\varepsilon }$$ is closed: while in general is false that the vector fields of the perturbation are orbitally equivalent to the unperturbed $$X_0$$ (Villarini in Ergod Theory Dyn Syst 39:1–32, 2019), we prove that this rigidity behaviour is indeed true if each $$X_{\varepsilon }$$ restricted to a codimension 2 sphere in $$S^{2n-1}$$ is orbitally conjugated to a subsystem of $$X_0$$ made by $$n-1$$ harmonic oscillators. In other words: to have a non-rigid, or truly non-linear, perturbation of $$X_0$$ 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 $$X_0$$ showing discontinuous changing of integer invariants of the vector fields of the perturbation.