Local conjugacy classes for analytic torus flows

Abstract

If a real-analytic flow on the multidimensional torus close enough to linear has a unique rotation vector which satisfies an arithmetical condition Y, then it is analytically conjugate to linear. We show this by proving that the orbit under renormalization of a constant Y-vector field attracts all nearby orbits with the same rotation vector.

Local rigidity of certain solvable group actions on tori

“…Under assumption that all orbits with the same rotation vectors, local rigidity was extended to higher dimension by KAM method. For instance, Dias [4] showed real-analytic perturbations of a linear flow on a torus is analytically conjugate to linear flow if the rotation vector satisfies a more general Diophantine type condition. Karaliolios [10] proved a local rigidity property for Diophantine translations on tori under the weaker condition that at least one orbit admits a Diophantine rotation vector.…”

Reducibility of quasi-periodically forced circle flows

“…Remark 2. The renormalization approach developed here is similar to, but technically more involved than, the renormalization approach to the construction of invariant tori for Hamiltonian and other vector fields [10,11,12,14,17,18], reducibility of skew-product flows on T d × SL(2, R) [15] and construction of lower-dimensional tori for Hamiltonian flows [13]. The small divisors encountered in these problems are produced by frequencies ν ∈ Z d that lie in the "resonant" regions, outside of certain "non-resonant" cones, surrounding some resonant planes, perpendicular to ω.…”