Existence of a vector field with a finite number of limit cycles. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:27:53Z http://mathoverflow.net/feeds/question/28659 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28659/existence-of-a-vector-field-with-a-finite-number-of-limit-cycles Existence of a vector field with a finite number of limit cycles. Petya 2010-06-18T20:07:42Z 2010-06-18T23:04:03Z <p>The following question is related to the Seifert conjecture.</p> <p>Let \$M\$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on \$M\$ contains a vector field with a finite number of (stable) limit cycles (closed trajectories)? Is it easy to construct? </p> http://mathoverflow.net/questions/28659/existence-of-a-vector-field-with-a-finite-number-of-limit-cycles/28668#28668 Answer by rpotrie for Existence of a vector field with a finite number of limit cycles. rpotrie 2010-06-18T21:11:13Z 2010-06-18T23:04:03Z <p>In any dimension bigger or equal to \$4\$, the answer is yes. See <a href="http://www.jstor.org/pss/1970973" rel="nofollow">here</a>.</p> <p>In dimension 3, the question is adressed <a href="http://www.springerlink.com/content/v7055uqt2l8j687x/" rel="nofollow">here</a>. In fact, in this paper neccessary and sufficient conditions to be homotopic to a non singular Morse-Smale flow are given. Morse-Smale means to have finitely many non degenerate closed orbits and that those are al the non-wandering set. In dimension 3 there are restrictions to satisfy that property, however, <a href="http://www.jstor.org/pss/1970458" rel="nofollow">this paper</a> proves that one has in the homotopy class vector fields whose minimal sets consist of finitely many periodic orbits (maybe degenerate). </p> <p>The reference is K. Yano, The homotopy class of non singular Morse Smale vector fields on 3 manifolds, Inventiones Math. (1985).</p>