<inline-formula><tex-math id="M1">\begin{document}$C^1$\end{document}</tex-math></inline-formula> weak Palis conjecture for nonsingular flows

Abstract

This paper focuses on generic properties of continuous dynamical systems. We prove \begin{document}$C^1$\end{document} weak Palis conjecture for nonsingular flows: Morse-Smale vector fields and vector fields admitting horseshoes are open and dense among \begin{document}$C^1$\end{document} nonsingular vector fields. Our arguments contain three main ingredients: linear Poincare flow, Liao's selecting lemma and the adapting of Crovisier's central model. Firstly, by studying the linear Poincare flow, we prove for a \begin{document}$C^1$\end{document} generic vector field away from horseshoes, any non-trivial nonsingular chain recurrent class contains a minimal set which is partially hyperbolic with 1-dimensional center with respect to the linear Poincare flow. Secondly, to understand the neutral behaviour of the 1-dimensional center, we adapt Crovisier's central model. The difficulties are that we can not build invariant plaque family of any time, the periodic point of a flow is not periodic for the discrete time map. Through delicate analysis of the center manifold of a periodic orbit near the partially hyperbolic set, we manage to yield nice periodic points such that their stable manifolds and unstable manifolds are well-placed for transverse intersection.