set of critical elements of $X$ is the union of the singularities and the periodic orbits of $X$, and will be denoted by ${\operatorname{Crit}}(X)$.

We recall that an hyperbolic set $\Lambda$ for a flow $X_t$ is an invariant subset of $M$ with a decomposition $T_\Lambda M= E^s\oplus E^X \oplus E^u$ of the tangent bundle over $\Lambda$ which is a continuous splitting, where $E^X$ is the direction of the vector field, the subbundles are invariant under the derivative $DX_t$ of the flow $$\begin{aligned}
  DX_t\cdot E^*_x=E^*_{X_t(x)},\quad
  x\in\Lambda, \quad t\in{{\mathbb R}},\quad *=s,X,u;\end{aligned}$$ $E^s$ is uniformly contracted by $DX_t$ and $E^u$ is uniformly expanded: there are $K,\lambda>0$ so that $$\begin{aligned}
\label{eq:def-hyperbolic}
\|DX_t\mid_{E^s_x}\|\le K e^{-\lambda t},
  \quad
  \|DX_{-t} \mid_{E^u_x}\|\le K e^{-\lambda t},
  \quad x\in\Lambda, \quad t\in{{\mathbb R}}.\end{aligned}$$

We say that a point $x \in  M$ is [*nonwandering*]{} for $X$ provided for every neighborhood $U$ of $x$ there is $t > 0$ such that $X_t(x) \cap U \neq \emptyset$, i.e., there exists a point $y \in U$ with $X_t(y) \in U$. We denote by $\Omega(X)$ the non-wandering set of $X$.

An ${\varepsilon}$-chain from $x_0$ to $x_l$ for $X$ is a sequence $\{x_0, x_1, \cdots, x_l\}$ such that for all $0 \leq j \leq l$, the distance 