Revisiting the asymptotics of the flow for some dynamical systems on the torus

In this paper we study the large time asymptotics of the flow of a dynamical system $X'=b(X)$ posed in the $d$-dimensional torus. Rather than using the classical unique ergodicity condition which is not fulfilled if $b$ vanishes at different points, we only assume that the set of the averages of $b$ with respect to the invariant probability measures for the flow is reduced to a singleton. We also rewrite the Liouville theorem which holds for any invariant probability measure $\mu$, namely $\mu\,b$ is divergence free, as a divergence-curl formula satisfied by any regular periodic function. The combination of these two tools turns out to be a new approach to get the asymptotics for some flows. This allows us to obtain the desired asymptotics in any dimension when $b = a\,\xi$ with $a$ a possibly vanishing periodic nonnegative function and $\xi$ a nonzero vector in $R^d$, or when $b = A\nabla v$ with $A$ a periodic nonnegative symmetric matrix-valued function and $v$ a periodic function.