  In this paper, we study various aspects of the ODE's flow $X$ solution to the
equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus
$Y_d$, where $b$ is a regular $\mathbb{Z}^d$-periodic vector field from
$\mathbb{R}^d$ in $\mathbb{R}^d$.We present an original and complete picture in
any dimension of all logical connections between the following seven conditions
involving the field $b$: (i) the everywhere asymptotics of the flow $X$, (ii)
the almost-everywhere asymptotics of the flow $X$, (iii) the global
rectification of the vector field $b$ in $Y_d$, (iv) the ergodicity of the flow
related to an invariant probability measure which is absolutely continuous with
respect to Lebesgue's measure, (v) the unit set condition for Herman's rotation
set $C_b$ composed of the means of $b$ related to the invariant probability
measures, (vi) the unit set condition for the subset $D_b$ of $C_b$ composed of
the means of $b$ related to the invariant probability measures which are
absolutely continuous with respect to Lebesgue's measure, (vii) the
homogenization of the linear transport equation with oscillating data and the
oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main
and surprising result of the paper is that the almost-everywhere asymptotics of
the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is
assumed to be non empty, and that the two conditions turn to be equivalent to
the homogenization of the transport equation when $b$ is divergence free. In
contrast, using an elementary approach based on classical tools of PDE's
analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any
$d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may
hold without satisfying the everywhere asymptotics of the flow.
