  In this article we investigate rigidity properties of integrable
area-preserving twist maps of the cylinder. More specifically, we prove that if
a deformation of the standard integrable map preserves rotational invariant
circles (i.e., homotopically non-trivial invariant curves of the cylinder on
which the dynamics is conjugate to a rotation) for any rational rotation number
in an open interval (without any further assumption on its size), then the
deformation must be trivial. Analogue phenomena for integrable Tonelli
Hamiltonian flows in higher dimensions are discussed.
