Abstract:
We prove a theorem asserting that, given a Diophantine
rotation $\alpha $ in a torus $\mathbb{T} ^{d} \equiv \mathbb{R} ^{d} / \mathbb{Z} ^{d}$,
any perturbation, small enough in the $C^{\infty}$ topology,
that does not
destroy all orbits with rotation vector $\alpha$ is actually
smoothly conjugate to the rigid rotation. The proof relies
on a KAM scheme (named after Kolmogorov – Arnol'd – Moser),
where at each step the existence of an invariant measure with rotation
vector $\alpha$ assures that we can linearize the equations
around the same rotation $\alpha$. The proof of the convergence of
the scheme is carried out in the $C^{\infty}$ category.
Keywords:KAM theory, quasi-periodic dynamics, Diophantine translations, local rigidity.
References