# Robustly shadowable chain transtive sets and hyperbolicity

## TL;DR

This paper establishes that for $C^1$-vector fields on manifolds, chain transitive sets are hyperbolic precisely when they are robustly shadowable, linking stability and hyperbolicity in dynamical systems.

## Contribution

It proves an equivalence between hyperbolicity and robust shadowability for chain transitive sets in $C^1$-vector fields.

## Key findings

- Chain transitive sets are hyperbolic iff they are robustly shadowable.
- Robust shadowability characterizes hyperbolic structure in dynamical systems.
- The result applies to $C^1$-vector fields on compact manifolds.

## Abstract

We say that a compact invariant set $\Lambda$ of a $C^1$-vector field $X$ on a compact boundaryless Riemannian manifold $M$ is robustly shadowable if it is locally maximal with respect to a neighborhood $U$ of $\Lambda$, and there exists a $C^1$-neigborhood $\mathcal{U}$ of $X$ such that for any $Y \in \mathcal{U}$, the continuation $\Lambda_Y$ of $\Lambda$ for $Y$ and $U$ is shadowable for $Y_t$. In this paper, we prove that any chain transitive set of a $C^1$-vector field on $M$ is hyperbolic if and only if it is robustly shadowable.