$d(x_{j-1}, x_j) < {\varepsilon}$.

We define the [*chain recurrent set*]{} of $X$ by $R(X) = \{x \in M; \textrm{there \ is \ an} \ {\varepsilon}-\textrm{chain \ from} \ x \ \textrm{to} \ x\}$. We say that two points are [*chain equivalent*]{} provided, given ${\varepsilon}> 0$, there is an ${\varepsilon}$-chain from $x$ to $y$ and from $y$ to $x$. It is known that this is an equivalence relation, the equivalence classes are called [*chain components*]{} of $R(X)$ and, for flows, the components are actually the connected components of $R(X)$. If $X$ admits a single chain component on an invariant set $\Lambda$, we say that $X$ is [*chain transitive*]{} on $\Lambda$. See [@Rob99], for instance.

From $C^1$ Pugh’s closing lemma we have $\Omega(X) \subset P_*(X) \subset R(X)$.

\[rmk:wen-preperiod\] As proved by Wen [@wen00], $C^1$ preperiodic sets do not explode under $C^1$ perturbations, i.e., for any $0 \leq i \leq d$ and for any neighborhood $U$ of $P_*(X)$, there is a $C^1$ neighborhood $\mathcal{V}$ of $X$ such that $P_*(Y) \subset U$, for any $Y \in \mathcal{V}$.

Indeed, for example in [@shub87], we can see that the recurrent set do not explode under $C^0$ perturbations, i.e., if $U$ is a neighborhood of $R(X)$ there 