with finitely many connected components and $\overline{U} \neq {\mathbb{P}}^1$, such that $\overline{A(U)} \subset U$ for each $A \in \Sigma$.

Structure of the paper
----------------------

In section \[s.proofmain\] we prove theorem \[t.meas\]. In section \[s.consequences\] we prove proposition \[p.rich\] and then theorem \[t.measclassif\]. Section \[s.continuous\] deals with the continuous case and contains the proof of theorem \[t.cont\] (that uses again theorem \[t.meas\] and proposition \[p.rich\]). The proof of theorem \[t.discrete\] is given in section \[s.discrete\] and it is independent of the other results. In the appendix we present some technical results that are used throughout the paper.

Proof of theorem \[t.meas\] {#s.proofmain}
===========================

In all this section, $(X,\mu)$ denotes a non-atomic Lebesgue space (not necessarily with $\mu(X)=1$).

The following are roughly the main steps of the proof:

-   In § \[ss.lambdak\], we show that given $A\colon (X,\mu) \to {\mathbb{G}}$, the *existence* of some dynamics $T\in{\mathit{Aut}}(X,\mu)$ for which ${\mathit{LE}}(A,T)$ is small depends only on the push-forward measure $A_* \mu$ in ${\mathbb{G}}$. That is very useful, because it allows us to address the questions in Lebesgue spaces $(X,\mu)$ and maps $A\colon (X,\mu) \to {\mathbb{G}}$ that are convenient for our constructions.

-   In § \[ss.existence\], we show that if a measure $\nu=A_* \mu$ is 