is a neighborhood $\mathcal{V}$ of $X$ such that $R(Y) \subset U$ for all $Y \in \mathcal{V}$.

Next, we explain the kind of hyperbolicities we are dealing.

\[def1\] A *dominated splitting* over a compact invariant set $\Lambda$ of $X$ is a continuous $DX_t$-invariant splitting $T_{\Lambda}M =
  E \oplus F$ with $E_x \neq \{0\}$, $F_x \neq \{0\}$ for every $x \in \Lambda$ and such that there are positive constants $K, \lambda$ satisfying $$\begin{aligned}
\label{eq:def-dom-split}
    \|DX_t|_{E_x}\|\cdot\|DX_{-t}|_{F_{X_t(x)}}\|<Ke^{-{\lambda}t}, \ \textrm{for all} \ x \in \Lambda, \ \textrm{and
      all} \,\,t> 0.
  \end{aligned}$$

A compact invariant set $\Lambda$ is said to be *partially hyperbolic* if it exhibits a dominated splitting $T_{\Lambda}M = E \oplus F$ such that subbundle $E$ is uniformly contracted. In this case $F$ is the *central subbundle* of $\Lambda$.

A compact invariant set $\Lambda$ is said to be *singular-hyperbolic* if it is partially hyperbolic and the action of the tangent cocycle expands volume along the central subbundle, i.e., $$\begin{aligned}
\label{eq:def-vol-exp}
      \vert \det (DX_t\vert_{F_x}) \vert > C e^{{\lambda}t},
      \forall t>0, \ \forall \ x \in \Lambda.
    \end{aligned}$$

The following definition was given as a particular case of singular hyperbolicity.

\[def:sec-exp\] A *sectional hyperbolic set* is a singular hyperbolic one such that for every two-dimensional linear 