called *richness*, that guarantees the existence of maps $T$ such that ${\mathit{LE}}(A,T)=0$. In fact, we will prove (theorem \[t.meas\]) that *if $A$ is rich then the generic $T \in {\mathit{Aut}}(X,\mu)$ satisfies ${\mathit{LE}}(A,T)=0$.* The richness condition is explicit and involves only the measure $\nu = A_* \mu$ (see definition \[d.rich\]). It provides some “abundance” of matrices in the support of $\nu$ that makes it possible to find elliptic products, “mix directions”, and make the exponents vanish after a perturbation of the dynamics.

Richness involves absolute continuity. So it turns out that *for the richness condition to be checkable, we have to ask some differentiability of $A$*. Since we are working in the measurable category, that restriction may be considered as a drawback. However, it is perhaps inevitable that some higher regularity must be asked from $A$. We *conjecture* indeed (see § \[ss.openquestions\]) that *there exists a map* (that assumes finitely many values only) *such that the integrated exponent ${\mathit{LE}}(A,T)$ is bounded from below by a constant $\lambda_0>0$ for all $T\in {\mathit{Aut}}(X,\mu)$, and nevertheless $A$ assumes an elliptic value on a set of positive measure.*

\[r.when A varies\] Our problem of studying ${\mathit{LE}}(A,T)$ as a function of $T$ is parallel to the 