length $2$ that send the expanding direction of $H$ exactly to the contracting one. Then we construct a dynamics so that orbits spend a long time in the (hyperbolic) $\omega$-part of the space, then spend two iterates in the $\sigma$-part, then return to the $\omega$-part. This makes $\Lambda_k(\omega+\sigma)$ small for $k$ large enough.

-   To reduce the general case to the previous one, we (essentially) approximate a given $\omega$ by a linear combination of Dirac measures, and use lemmas \[l.cont\] and \[l.convex\].

Given a matrix $H \in {\mathbb{G}}$, denote by $\rho(H) \in [1,\infty)$ its spectral radius. (That is, $\rho(H) = \max(|\lambda_1|,|\lambda_2|)$ if $\lambda_1$, $\lambda_2$ are the eigenvalues of $H$.)

*Definition of $k$:* Given $C>1$ and $\delta>0$, there is $\ell_0 \in {\mathbb{N}}$ with the following properties: If $H, R \in {\mathbb{G}}$ are matrices such that

-   $\|H\| \le C$, $\|R\| \le C^2$;

-   $H$ is a hyperbolic matrix and moreover $\rho(H) \ge e^{\delta/4}$;

-   $R(e^u) = e^s$, where $e^u$ and $e^s \in {\mathbb{P}}^1$ denote respectively the expanding and contracting eigendirections of $H$;

then the matrix $R H^\ell$ is elliptic for every $\ell \ge \ell_0$. To prove this fact, take a basis $\{v^u, v^s\}$ of unit eigenvectors of $H$. In this basis, 