Let $M$ be a compact smooth manifold, $V$ be a vector field on $M$ and $\phi_t$ be the corresponding flow. Let $\{ U_i \}_{i = 1, \cdots n}$ be an open cover of $M$. My question is that if it is possible to write $\phi_t$ as a composition $\phi_t = \phi_{n,t} \circ \cdots \circ \phi_{1,t}$ such that $\phi_{i,t}$ is supported in $U_i$? That is, $\phi_{i,t}$ is identify outside $U_i$.
More generally, if $M$ is just a smooth manifold, $V$ is a complete vector field, and $\{ U_i \}_{i \in \mathbb{N}}$ is a locally finite open cover. Is there a decomposition $\phi_t = \prod_{i \in \mathbb{N}} \phi_{i,t}$ such that $\phi_{i,t}$ is supported in $U_i$?