Local Rigidity of Diophantine translations in higher dimensional tori

Abstract

We prove a theorem asserting that, given a Diophantine rotation $\alpha$ in a torus $\T ^{d} \equiv \R ^{d} / \Z ^{d}$, 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 K.A.M. 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.

16 pages

References in corpus (2)

• Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups
• Normal form à la Moser for diffeomorphisms and generalization of Rüssmann's translated curve theorem to higher dimension

Cited by in corpus (5)

• Fibered rotation vector and hypoellipticity for quasiperiodic cocycles in compact Lie groups
• A priori bounds for GIETs, affine shadows and rigidity of foliations in genus 2
• On Stable and Unstable Behaviours of Certain Rotation Segments
• Local Rigidity for Simultaneous Diophantine Translations on Tori of Arbitrary Dimension
• Cohomological rigidity and the Anosov-Katok construction