are sent into contracting directions. This forces the Lyapunov exponents to drop close to zero.

From theorem \[t.meas\] and proposition \[p.rich\], we obtain the following result:

\[t.measclassif\] Let $X$ be a compact connected manifold (possibly $1$-dimensional, possibly with boundary), and $\mu$ be a volume measure. Let $A\colon X\to{\mathbb{G}}$ be a $C^1$ map. Then:

-   either there is a *closed* interval $I \subset {\mathbb{P}}^1$ such that $A(x) \cdot I \subset I$ for every $x\in X$;

-   or ${\mathit{LE}}(A,T)=0$ for the generic $T\in {\mathit{Aut}}(X,\mu)$ (w.r.t. the weak topology).

It is easy to describe all the possible behaviors of the Lyapunov exponent under alternative (i) in the theorem – see § \[ss.addendum\]. In fact, for an open and dense set of maps $A$ (i) implies uniform hyperbolicity for any dynamics, so we obtain the result mentioned in the abstract:

\[c.measdich\] Let $1 \le r \le \infty$ and let $X$ be a compact connected $C^r$-manifold. Then the set of $A \in C^r(X,{\mathbb{G}})$ that satisfy the measurable dichotomy is $C^r$-open and dense.

We remark that $L^\infty$ and $C^0$ versions of corollary \[c.measdich\] are also true:

\[p.easy dich\] Let $(X,\mu)$ be a non-atomic Lebesgue space. There exists a residual subset ${\mathcal{R}}\subset L^\infty(X,{\mathbb{G}})$ such that measurable dichotomy 