Minimality of time-t minimal flows - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:19:27Z http://mathoverflow.net/feeds/question/87448 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87448/minimality-of-time-t-minimal-flows Minimality of time-t minimal flows Alejandro 2012-02-03T16:34:46Z 2012-02-03T16:34:46Z <p>This question is mainly motivated by the question <a href="http://mathoverflow.net/questions/84749/transitivity-of-a-flow-and-its-time-1-map" rel="nofollow">http://mathoverflow.net/questions/84749/transitivity-of-a-flow-and-its-time-1-map</a></p> <p>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 $M$.</p> <p>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?</p> <p>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 $t\in\mathbb{R}$? </p>