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This question is mainly motivated by the question Transitivity of a flow and its time-1 map

Let $M$ be a closed smooth manifold and $\Phi\colon\mathbb{R}\times M\to M$ be a smooth minimal flow, i.e. given any $x\in M$ its $\Phi$-orbit $\lbrace\Phi^t(x) : t\in\mathbb{R}\rbrace$ is dense in $M$.

Question 1: Is it true that there exists at least one $t\in\mathbb{R}$ such that the time-$t$ diffeomorphism $\Phi^t\colon M\to M$ is minimal?

Question 2: Assuming $\Phi$ is not conjugate to a suspension flow (i.e. there is no closed codimension-1 submanifold everywhere transverse to the flow), then is it true that $\Phi^t\colon M\to M$ is a minimal diffeomorphism for any $t\in\mathbb{R}$?

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