Instability for a priori unstable Hamiltonian systems: a dynamical approach (Q765075)

The paper considers an a priori unstable 3DOF integrable Hamiltonian system of the form \[ \hat h = \tfrac12(I_1^2+I_2^2) + I_3 + \cos(2\pi\theta_1). \] For a specific perturbation \(H = \hat h + \mu f\), and for \(\mu\) small enough, it is shown that this system has a drifting solution with optimal time of instability. In particular, \(\lim_{t \to \pm \infty} I_2(t) = \pm \infty\) and \(|I_2(\tau) - I_2(0)| \geq 1\) for \(\tau \leq C \mu^{-1} \ln \mu^{-1}\). The proof is based on the construction of symbolic dynamics corresponding to the random iteration of a family of twist maps of the annulus.