one addressed in [@B], where the map $T$ on the base is fixed and the cocycle $A$ is perturbed. The following results are obtained in [@B]: For any ergodic $T\in{\mathit{Aut}}(X,\mu)$, there is a residual set ${\mathcal{R}}_T \subset L^\infty (X,{\mathbb{G}})$ such that if $A\in{\mathcal{R}}_T$ then either the cocycle $(T,A)$ is uniformly hyperbolic or ${\mathit{LE}}(A,T)=0$. If, in addition, $X$ is a compact Hausdorff space and $\mu$ is a Borel measure then there exists a residual subset ${\mathcal{R}}'_T \subset C^0 (X,{\mathbb{G}})$ with the same properties. (See also [@BV] and [@ArB] for extensions and related results.)

\[d.measdich\] Let us say that a bounded map $A \colon X \to {\mathbb{G}}$ *satisfies the measurable dichotomy* if:

-   either there is a constant $\lambda>1$ such that $\|A(x_n) \cdots A(x_1)\| > \lambda^n$ for every $x_1$, …, $x_n \in X$ (in particular, for every $T\in{\mathit{Aut}}(X,\mu)$ the cocycle $(A,T)$ is uniformly hyperbolic);

-   or the set of $T\in {\mathit{Aut}}(X,\mu)$ for which ${\mathit{LE}}(A,T)=0$ is residual in the weak topology.

Our next goal would be to express that *“most” $A$’s satisfy the measurable dichotomy*. We are able to prove that fact if we restrict ourselves to $C^1$ maps $A \colon X \to {\mathbb{G}}$ on some connected manifold $X$. In fact, we give 