Generic properties of vector fields identical on a compact set and codimension one partially hyperbolic dynamics
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929579195629568 |
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| author | Gan, Shaobo Xi, Ruibin Yang, Jiagang Zheng, Rusong |
| author_facet | Gan, Shaobo Xi, Ruibin Yang, Jiagang Zheng, Rusong |
| contents | 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 $Γ$ of $X_0$, we consider the closed subset $\mathscr{X}^r(M,Γ)$ of $\mathscr{X}^r(M)$, consisting of all $C^r$ vector fields which coincide with $X_0$ on $Γ$. Study of such a set naturally arises when one needs to perturb a system while keeping part of the dynamics untouched.
A vector field $X\in\mathscr{X}^r(M,Γ)$ is called $Γ$-avoiding Kupka-Smale, if the dynamics away from $Γ$ is Kupka-Smale. We show that a generic vector field in $\mathscr{X}^r(M,Γ)$ is $Γ$-avoiding Kupka-Smale. In the $C^1$ topology, we obtain more generic properties for $\mathscr{X}^1(M,Γ)$. With these results, we further study codimension one partially hyperbolic dynamics for generic vector fields in $\mathscr{X}^1(M,Γ)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03629 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic properties of vector fields identical on a compact set and codimension one partially hyperbolic dynamics Gan, Shaobo Xi, Ruibin Yang, Jiagang Zheng, Rusong Dynamical Systems 37C10, 37C27 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 $Γ$ of $X_0$, we consider the closed subset $\mathscr{X}^r(M,Γ)$ of $\mathscr{X}^r(M)$, consisting of all $C^r$ vector fields which coincide with $X_0$ on $Γ$. Study of such a set naturally arises when one needs to perturb a system while keeping part of the dynamics untouched. A vector field $X\in\mathscr{X}^r(M,Γ)$ is called $Γ$-avoiding Kupka-Smale, if the dynamics away from $Γ$ is Kupka-Smale. We show that a generic vector field in $\mathscr{X}^r(M,Γ)$ is $Γ$-avoiding Kupka-Smale. In the $C^1$ topology, we obtain more generic properties for $\mathscr{X}^1(M,Γ)$. With these results, we further study codimension one partially hyperbolic dynamics for generic vector fields in $\mathscr{X}^1(M,Γ)$, 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. |
| title | Generic properties of vector fields identical on a compact set and codimension one partially hyperbolic dynamics |
| topic | Dynamical Systems 37C10, 37C27 |
| url | https://arxiv.org/abs/2411.03629 |