Volume preserving diffeomorphisms as Poincare maps for volume preserving flows

Let $q:M\to M$ be a volume-preserving diffeomorphism of a smooth manifold $M$. We study the possibility to present $q$ as the Poincaré map, corresponding to a volume-preserving vector field on $\mathbb{T}\times M$, $\mathbb{T} = \mathbb{R}/\mathbb{Z}$.

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