Nonsingular star flows satisfy Axiom A and the no-cycle condition

We give an affirmative answer to a problem of Liao and Mañé which asks whether, for a nonsingular flow to loose the Ω-stability, it must go through a critical-element-bifurcation. More precisely, a vector field S on a compact boundaryless manifold is called a star system if S has a C1 neighborhood \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{U}$\end{document} in the set of C1 vector fields such that every singularity and every periodic orbit of every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X\in\mathcal{U}$\end{document} is hyperbolic. We prove that any nonsingular star flow satisfies Axiom A and the no cycle condition.