Generic properties of vector fields identical on a compact set and codimension one partially hyperbolic dynamics

Let \(\mathscr{X}^r(M)\) be the set of \(C^r\) vector fields on a boundaryless compact Riemannian manifold \(M\). Given a vector field \(X_0\in\mathscr{X}^r(M)\) and a compact invariant set \(\Gamma\) of \(X_0\), we consider the closed subset \(\mathscr{X}^r(M,\Gamma)\) of \(\mathscr{X}^r(M)\), cons...

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Veröffentlicht in:arXiv.org 2024-11
Hauptverfasser: Gan, Shaobo, Xi, Ruibin, Yang, Jiagang, Zheng, Rusong
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Sprache:eng
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Zusammenfassung:Let \(\mathscr{X}^r(M)\) be the set of \(C^r\) vector fields on a boundaryless compact Riemannian manifold \(M\). Given a vector field \(X_0\in\mathscr{X}^r(M)\) and a compact invariant set \(\Gamma\) of \(X_0\), we consider the closed subset \(\mathscr{X}^r(M,\Gamma)\) of \(\mathscr{X}^r(M)\), consisting of all \(C^r\) vector fields which coincide with \(X_0\) on \(\Gamma\). Study of such a set naturally arises when one needs to perturb a system while keeping part of the dynamics untouched. A vector field \(X\in\mathscr{X}^r(M,\Gamma)\) is called \(\Gamma\)-avoiding Kupka-Smale, if the dynamics away from \(\Gamma\) is Kupka-Smale. We show that a generic vector field in \(\mathscr{X}^r(M,\Gamma)\) is \(\Gamma\)-avoiding Kupka-Smale. In the \(C^1\) topology, we obtain more generic properties for \(\mathscr{X}^1(M,\Gamma)\). With these results, we further study codimension one partially hyperbolic dynamics for generic vector fields in \(\mathscr{X}^1(M,\Gamma)\), giving a dichotomy of hyperbolicity and Newhouse phenomenon. As an application, we obtain that \(C^1\) generically in \(\mathscr{X}^1(M)\), a non-trivial Lyapunov stable chain recurrence class of a singularity which admits a codimension 2 partially hyperbolic splitting with respect to the tangent flow is a homoclinic class.
ISSN:2331-8422