criterium is obtained:

\[p.rich\] Let $M$ be a compact manifold (possibly $1$-dimensional, possibly with boundary, possibly not connected), with a smooth volume measure $\mu$, and let $A\colon M \to {\mathbb{G}}$ be a $C^1$ map. Assume there are points $p_1$, …, $p_k \in M$ such that the matrix $A(p_k) \cdots A(p_1)$ is elliptic and moreover $A$ is not locally constant at at least one of the $p_i$’s. Then $A_* \mu$ is rich.

Our main theorem is:

\[t.meas\] Let $(X,\mu)$ be a non-atomic Lebesgue space. Let $A \colon X \to {\mathbb{G}}$ be a bounded measurable map such that the measure $A_* \mu$ is rich. Then there is a residual set ${\mathcal{R}}_A \subset {\mathit{Aut}}(X,\mu)$ such that ${\mathit{LE}}(A,T)=0$ for all $T \in {\mathcal{R}}_A$.

The following is an informal outline of the proof. Richness implies the existence of products that send any chosen direction into any other. Perturbing the dynamics in the weak topology we can make most of the orbits periodic. Also, we do that perturbation so that a very small “rich part” of the space is kept invariant. Then another well chosen perturbation makes most of the orbits spend a few iterates in the rich region in such a way that expanded directions 