$$\begin{aligned}
    r_+^1(L_1L_2)\ge r_+^1(L_1) r_+^1(L_2) \quad\text{and}\quad
    r_-^1(L_1L_2)\le r_-^1(L_1)r_-^1(L_2).
  \end{aligned}$$ Moreover, if the operators are strictly ${\EuScript{J}}$-separated, then the inequalities are strict.

\[rmk:J-mon-spec\] Another important property about the singular values of a ${\EuScript{J}}$-separated operator $L$ is that $$r_+^1 = r_+ \ge 1 (> 1) \quad\text{and}\quad r_-^1 = r_- \le 1 (< 1)$$ if, and only if, $L$ is (strictly) ${\EuScript{J}}$-monotone.

This property will be used a lot of times in our proofs.

### Lyapunov exponents

By Oseledec’s Ergodic Theorem [@Os68], there exist a full probability set $X$ such that for every $x \in Y$ there is an invariant decomposition $$\begin{aligned}
T_xM = \langle X\rangle \oplus E_{1}(x) \oplus \cdots \oplus E_{l(x)}(x)\end{aligned}$$ and numbers $\chi_1 < \cdots < \chi_l$ corresponding to the limits $$\begin{aligned}
  \chi_j = \lim\limits_{t \to +\infty} \frac{1}{t} \log \Vert DX_t(x) \cdot v\Vert,\end{aligned}$$ for every $v \in E_i(x)\setminus \{0\}, i = 1, \cdots, l(x)$.

In this setting, Wojtkowski [@Wojtk01] proved that the logarithm of the pseudo-Euclidean singular values $0 \leq r_q^- \leq \cdots \leq r_1^- \leq r_1^+ \leq \cdots \leq r_p^+$ of $DX_t$ are $\mu$-integrable, and obtained estimates of the Lyapunov exponents related to the singular eigenvalues of strictly ${\EuScript{J}}$-separated maps.

[@Wojtk01 Corollary 3.7] \[thm:lyap-exp-sing-val\] For $1 \leq k_1 \leq p$ and 