$1$-rich then there exists $T\in{\mathit{Aut}}(X,\mu)$ for which ${\mathit{LE}}(A,T)$ is small.

-   In § \[ss.measures\], we collect abstract lemmas on perturbation of measures and maps. Then, in § \[ss.convolution\], we relate the convolution measure $\nu^{*N}$ that appears in the definition of richness with a dynamical construction.

-   In § \[ss.endoftheproof\], we conclude the proof. We apply the result from § \[ss.convolution\] to obtain an induced cocycle whose push-forward to ${\mathbb{G}}$ is a measure which is close to an $1$-rich one. A specific argument of continuity is used to allow the use of the results of § \[ss.existence\] despite the fact that we are dealing with a measure that is only close to a $1$-rich one.

Least exponent of order $k$ {#ss.lambdak}
---------------------------

We begin introducing some notation: If $A\colon (X,\mu)\to {\mathbb{G}}$ is a bounded measurable map, $k \in {\mathbb{N}}$, and $T\in{\mathit{Aut}}(X,\mu)$, let $$\Lambda_k (A,T) = \frac{1}{k} \int_X \log \|A^k_T\| \; d\mu \, .$$ Observe that by subadditivity of $\int_X \log \|A^k_T\| d \mu$ we have ${\mathit{LE}}(A,T) \leq \Lambda_k(A,T)$.

We define the *least exponent of order $k$* of $A$ $$\Lambda_k (A) = \inf_{T \in {\mathit{Aut}}(X,\mu)} \Lambda_k (A,T).$$

The continuity property of $\Lambda_k$ states as follows:

\[l.cont\] Given $k\in {\mathbb{N}}$, $C>1$ and $\delta>0$, 