$1 \leq k_2 \leq q$

$$\begin{aligned}
\chi^-_1 + \cdots + \chi^-_{k_1} \leq \sum_{i=1}^{k_1} \int \log r^-_i d\mu \ \textrm{and} \ \chi^+_1 + \cdots + \chi^+_{k_2} \geq \sum_{i=1}^{k_2} \int \log r^+_i d\mu.\end{aligned}$$

This result will be very useful in proof of Theorem \[mthm:strong-homog-to-quad\].

Some applications
=================

In this section, we present some applications of this theory related to some kind of hyperbolicities, as partial and singular ones. In particular, we provide another proof of [@GanLiWen2005 Theorem A].

### Some results about partial and sectional hyperbolicity from ${\EuScript{J}}$-separation

The author, together with V. Araújo, proved in [@ArSal2012] the following useful theorem which relates partial hyperbolicity and ${\EuScript{J}}$-separated sets for a flow.

[@ArSal2012 Theorem A] \[mthm:Jseparated-parthyp\] A maximal invariant subset $\Lambda$ of a trapping region $U$ whose singularities are hyperbolic is a partially hyperbolic set for a flow $X_t$ if, and only if, there is a $C^1$ field ${\EuScript{J}}$ of non-degenerate quadratic forms with constant index, equal to the dimension of the stable subspace of $\Lambda$, such that $X_t$ is a non-negative strictly ${\EuScript{J}}$-separated flow on $U$.

This result will be useful in our applications of Theorem \[mthm:strong-homog-equiv\].

In the sequence, we can give another proof of next result from [@GanLiWen2005].

[@GanLiWen2005 Theorem 