A property of vector fields without singularity in \({\mathcal G}^1 (M)\) (Q2709599)

The author proves the following three assertions. NEWLINENEWLINENEWLINETheorem A. If \(X\) is in \({\mathcal G}^1(M)\) and has no singularities, then for any \(x\in\Sigma(X)\), \(\overline{\text{orbit}(x)}\cap \overline{\text{per}(X)}\neq \emptyset.\) NEWLINENEWLINENEWLINETheorem B. Let \(X\) be in \({\mathcal G}^1(M)\) and have no singularities. If \(\overline{\text{per}_i(X)}\cap \overline{\text{per}_j(X)}= \emptyset\) for all \(0\leq i<j\leq n-1),\) then \(\overline{\text{per}(X)}\) is hyperbolic. NEWLINENEWLINENEWLINETheorem C. (\(\Omega\)-stability conjecture for flows). If \(X\) is \(\Omega\)-stable, then \(\Omega(X)\) is hyperbolic and the periodic points are dense in \(\Omega(X).\) NEWLINENEWLINENEWLINEHere \(M\) is an \(n\)-dimensional compact smooth manifold without boundary, \({\mathcal X}^1(M)\) is the set of \(C^1\) vector fields on \(M\) with the \(C^1\) topology, \(\overline{\text{per}(X)}\) is the closure of the set of periodic orbits of \(X,\) \(\overline{\text{per}_j(X)}\) is the closure of the set of hyperbolic periodic orbits with index \(j\) (index is the dimension of the stable subspace). Finally, \({\mathcal G}^1(M)\) stands for the set of \(X\in{\mathcal X}^1(M)\) which has a neigborhood \({\mathcal U}\) such that if \(Y\in{\mathcal U}\) then all the periodic orbits and singularities of \(Y\) are hyperbolic.