Journal of the
Korean Mathematical Society JKMS

Let $\gamma$ be a hyperbolic closed orbit of a $C^1$ vector field $X$ on a compact boundaryless Riemannian manifold $M$, and let $C_X(\gamma)$ be the chain component of $X$ which contains $\gamma$. We say that $C_X(\gamma)$ is $C^1$ robustly shadowable if there is a $C^1$ neighborhood $\mathcal{U}$ of $X$ such that for any $Y\in\mathcal{U}$, $C_Y(\gamma_Y)$ is shadowable for $Y_t$, where $\gamma_Y$ denotes the continuation of $\gamma$ with respect to $Y$. In this paper, we prove that any $C^1$ robustly shadowable chain component $C_X(\gamma)$ does not contain a hyperbolic singularity, and it is hyperbolic if $C_X(\gamma)$ has no non-hyperbolic singularity.

Keywords: chain component, dominated splitting, homoclinic class, hyperbolicity, robust shadowability, uniform hyperbolicity, vector field

MSC numbers: 37C, 37D