This works investigates the Lyapunov–Oseledets spectrum of transfer operator cocycles associated to one-dimensional random paired tent maps depending on a parameter
ε
\varepsilon
, quantifying the strength of the leakage between two nearly invariant regions. We show that the system exhibits metastability, and identify the second Lyapunov exponent
λ
2
ε
\lambda _2^\varepsilon
within an error of order
ε
2
|
log
ε
|
\varepsilon ^2|\log \varepsilon |
. This approximation agrees with the naive prediction provided by a time-dependent two-state Markov chain. Furthermore, it is shown that
λ
1
ε
=
0
\lambda _1^\varepsilon =0
and
λ
2
ε
\lambda _2^\varepsilon
are simple, and the only exceptional Lyapunov exponents of magnitude greater than
−
log
2
+
O
(
log
log
1
ε
/
log
1
ε
)
-\log 2+ O\Big (\log \log \frac 1\varepsilon \big /\log \frac 1\varepsilon \Big )
.