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The paper is devoted to the isotropic realizability of a regular gradientfield u or a more general vector field b, namely the existence of a continuouspositive function $\sigma$ such that $\sigma$b is divergence free in R d or inan open set of R d. First, we prove that under some suitable positivitycondition satisfied by u, the isotropic realizability of u holds either in R dif u does not vanish, or in the open sets {c j < u < c j+1 } if the c j are thecritical values of u (including inf R d u and sup R d u) which are assumed tobe in finite number. It turns out that this positivity condition is notsufficient to ensure the existence of a continuous positive invariant measure$\sigma$ on the torus when u is periodic. Then, we establish a new criterium ofthe existence of an invariant measure for the flow associated with a regularperiodic vector field b, which is based on the equality b $\times$ v = 1 in Rd. We show that this gradient invertibility is not related to the classicalergodic assumption, but it actually appears as an alternative to get theasymptotics of the flow.

Isotropic realizability of fields and reconstruction of invariant  measures under positivity properties. Asymptotics of the flow by a  non-ergodic approach