Quasi-periodicity in relative quasi-periodic tori

TL;DR

This paper investigates the structure of relative quasi-periodic tori in dynamical systems with symmetry, providing a new approach based on group-invariant lifts to describe their geometry and flow properties.

Contribution

It introduces an alternative method to analyze relative quasi-periodic tori using commuting, group-invariant lifts, establishing a complete structural description under certain hypotheses.

Findings

The structure of relative quasi-periodic tori is a principal torus bundle.

Fibers of the bundle are tori with dimension exceeding the reduced torus by at most the group rank.

The approach ensures the tori have minimal dimension and ergodic flow.

Abstract

At variance from the cases of relative equilibria and relative periodic orbits of dynamical systems with symmetry, the dynamics in relative quasi-periodic tori (namely, subsets of the phase space that project to an invariant torus of the reduced system on which the flow is quasi-periodic) is not yet completely understood. Even in the simplest situation of a free action of a compact and abelian connected group, the dynamics in a relative quasi-periodic torus is not necessarily quasi-periodic. It is known that quasi-periodicity of the unreduced dynamics is related to the reducibility of the reconstruction equation, and sufficient conditions for it are virtually known only in a perturbation context. We provide a different, though equivalent, approach to this subject, based on the hypothesis of the existence of commuting, group-invariant lifts of a set of generators of the reduced torus. Under this hypothesis, which is shown to be equivalent to the reducibility of the reconstruction equation, we give a complete description of the structure of the relative quasi-periodic torus, which is a principal torus bundle whose fibers are tori of a dimension which exceeds that of the reduced torus by at most the rank of the group. The construction can always be done in such a way that these tori have minimal dimension and carry ergodic flow.