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This paper deals with the long time asymptotics X(t, x)/t of the flow Xsolution to the autonomous vector-valued ODE: X (t, x) = b(X(t, x)) for t $\in$R, with X(0, x) = x a point of the torus Y d := R d /Z d. We assume that thevector field b reads as the product $\rho$ $\Phi$, where $\rho$ : Y d$\rightarrow$ [0, $\infty$) is a non negative regular function and $\Phi$ : Y d$\rightarrow$ R d is a non vanishing regular vector field. In this work, thesingleton condition means that the rotation set C b composed of the averagevalues of b with respect to the invariant probability measures for the flow Xis a singleton {$\zeta$}, or equivalently, that lim t$\rightarrow$$\infty$ X(t,x)/t = $\zeta$ for any x $\in$ Y d. This combined with Liouville's theoremregarded as a divergence-curl lemma, first allows us to obtain the asymptoticsof the flow X when b is a current field. Then, we prove a general perturbationresult assuming that $\rho$ is the uniform limit in Y d of a positive sequence($\rho$ n) n$\in$N satisfying for any n $\in$ N, $\rho$ $\le$ $\rho$ n and C$\rho$n$\Phi$ is a singleton {$\zeta$ n }. It turns out that the limit set C beither remains a singleton, or enlarges to the closed line set [0, lim n$\zeta$ n ] of R d. We provide various corollaries of this perturbation resultinvolving or not the classical ergodic condition, according to the positivityor not of some harmonic means of $\rho$. These results are illustrated bydifferent examples which show that the perturbation result is limited to thescalar perturbation of $\rho$, and which highlight the alternative satisfied bythe rotation set C b. Finally, we prove that the singleton condition allows usto homogenize in any dimension the linear transport equation induced by theoscillating velocity b(x/$\epsilon$) beyond any ergodic condition satisfied bythe flow X.

Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications