a complete classification in that case (see theorem \[t.measclassif\]) that permits us to: 1) obtain the measurable dichotomy for an open and dense set of maps $A$ (corollary \[c.measdich\]), 2) describe precisely which maps do not satisfy the dichotomy and explicit all the possible behaviors of the Lyapunov exponents for these maps (proposition \[p.addendum\]).

In this regard, we also show that if we consider only maps $A$ that assume finitely many values, then “most” of them will satisfy dichotomy (see theorem \[t.discrete\]). However, that case is unrelated to richness, and actually we lose any explicit condition for zero exponents.

In the continuous setting, we show that under certain conditions on the map $A$ and also on the volume-preserving homeomorphism $T$, it is possible to perturb $T$ and make the exponent drop or (under a stronger assumption) vanish. We obtain those results as consequences of the afore-mentioned measurable results. For that we use Alpern’s technique [@Alpern; @Lusin] of approximating measurable automorphisms by homeomorphisms.

At last, we obtain a dichotomy result concerning homeomorphisms.

\[d.contdich\] We say that a continuous map *$A \colon X \to {\mathbb{G}}$ satisfies the continuous dichotomy* if for the generic $T\in{\mathit{Homeo}}(X,\mu)$:

-   either the cocycle $(A,T)$ is uniformly hyperbolic;

