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In this paper, we study various aspects of the ODE's flow $X$ solution to theequation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ 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 inany dimension of all logical connections between the following seven conditionsinvolving the field $b$: (i) the everywhere asymptotics of the flow $X$, (ii)the almost-everywhere asymptotics of the flow $X$, (iii) the globalrectification of the vector field $b$ in $Y_d$, (iv) the ergodicity of the flowrelated to an invariant probability measure which is absolutely continuous withrespect to Lebesgue's measure, (v) the unit set condition for Herman's rotationset $C_b$ composed of the means of $b$ related to the invariant probabilitymeasures, (vi) the unit set condition for the subset $D_b$ of $C_b$ composed ofthe means of $b$ related to the invariant probability measures which areabsolutely continuous with respect to Lebesgue's measure, (vii) thehomogenization of the linear transport equation with oscillating data and theoscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The mainand surprising result of the paper is that the almost-everywhere asymptotics ofthe flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ isassumed to be non empty, and that the two conditions turn to be equivalent tothe homogenization of the transport equation when $b$ is divergence free. Incontrast, using an elementary approach based on classical tools of PDE'sanalysis, we extend the two-dimensional results of Oxtoby and Marchetto to any$d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow mayhold without satisfying the everywhere asymptotics of the flow.

Asymptotics of ODE's flows everywhere or almost-everywhere in the torus:from rotation sets to homogenization of transport equations