Local conjugacy classes for analytic torus flows

TL;DR

This paper proves that real-analytic torus flows near linear ones are analytically conjugate to linear flows if they meet certain arithmetical conditions, using renormalization techniques to demonstrate orbit attraction.

Contribution

It establishes a new criterion for conjugacy of near-linear torus flows based on arithmetical conditions and renormalization dynamics.

Findings

Flows with rotation vector Y are conjugate to linear flows under condition Y.

Renormalization attracts all nearby orbits with the same rotation vector.

The result extends understanding of stability 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.