*star flows* and its relation with infinitesimal Lyapunov functions.

We recall that a point $p \in M$ is said to be a $C^1$ preperiodic point of $X$ if for any $C^1$ neighborhood $\mathcal{V}$ of $X$ and any neighborhood $U \subset M$ of $p$, there is $g\in \mathcal{V}$ and $q \in U$ such that $q \in {\operatorname{Per}}(g)$. We denote this set by $P_*(X)$, and it is easy to see that it is closed and $X$-invariant.

We can assume that the periodic points are hyperbolic and we can define a $C^1 i$-preperiodic point $p$ of $X$, $0 \leq i \leq d$, if there are sequences $X_n$ of flows and $p_n$ of periodic points of $X_n$ with index $i$ such that $$\begin{aligned}
\lim\limits_{n \to \infty} X_n = X \ \textrm{and} \ \lim\limits_{n \to \infty} p_n = p.\end{aligned}$$ We denote by $P_*^i(X)$ the set of $C^1 i$-preperiodic of $X$ and, then, $P_*(X) = \cup_{i=0}^{d} P_*^i(X)$.

Precisely, it is proved the following.

\[cor:star-flow\] A vector field $X \in {\mathfrak{X}^{1}(M)}$ is star if, and only if, satisfies all of the next properties:

1.  there is a neighborhood $U$ of $P_*(X)$ and a field of quadratic forms ${\EuScript{J}}$ with index $0 \leq {\operatorname{ind}}\leq \dim(M) - 1$ defined 