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 $\mathcal{U}$ in the set of C1 vector fields such that every singularity and every periodic orbit of every $X\in\mathcal{U}$ is hyperbolic. We prove that any nonsingular star flow satisfies Axiom A and the no cycle condition.

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