$R$ becomes $\begin{pmatrix} 0 & c \\ b & d \end{pmatrix}$. The angle between $v^u$ and $v^s$ cannot be too small, hence we can give bounds to $b$, $c$, $d$ depending only on $C$ and $\delta$. Then $|\operatorname{tr}R H^\ell| = |d| \rho(H)^{- \ell} < 2$ for sufficiently large $\ell$.

Given the $1$-rich measure $\sigma$ with $\|\sigma\|_\infty \le C$, let $\kappa>0$ be as in definition \[d.rich\]. Fix an integer $$\ell \ge \max (\ell_0, 2 / \kappa ).$$ By proposition \[p.unif spectral\], there exists $k' \in {\mathbb{N}}$ such that if $E \in {\mathbb{G}}$ is an elliptic matrix with $\|E\| \le C^{\ell+2}$ then $$\frac{1}{p} \log \|E^p\| < \frac{(\ell+2) \delta}{2} \quad \forall p \ge k'.$$ Fix $p \ge k'$ large enough so that defining $k = (\ell+2) p$, we have $$\frac{1}{k}\log\|H^k\| < \frac{\delta}{2} \quad
\text{for all $H\in {\mathbb{G}}$ s.t.~$\|H\| \le C$ and $\rho(H)<e^{\delta/4}$.}$$ Let us verify that $k$ has the stated properties.

*First case:* We will first prove the proposition in the case where $\omega$ is a Dirac measure $\delta_H$ on some $H \in {\mathbb{G}}$, with $\|H\|\le C$. We will exhibit a Lebesgue space $(X,\mu)$, a map $A\colon X\to{\mathbb{G}}$ with $A_* \mu = \delta_H + \sigma$, and a dynamics $T\in {\mathit{Aut}}(X,\mu)$ such that $\Lambda_k 