in our setting, there is more flexibility as the norm is allowed to depend on the basepoint. So one could wonder if the Finsler extremal norms in our main result could be taken Riemannian. Unfortunately, that is not the case: we construct an explicit example in \[ss.Riemann\].

Consequences
------------

As a consequence of our result on the existence of extremal norms, we can show that the maximal Lyapunov exponent is a locally Lipschitz function on the space of strongly bunched irreducible automorphisms (see \[p.Wirth\] for a more precise statement), thus extending a result of Wirth [@Wirth].

We are also able to obtain several general properties of strongly bunched automorphisms $\Phi$ (not necessarily irreducible):

-   Their growth obeys certain uniform bounds: see \[t.polynomial\].

-   They obey the *subordination principle*: if $\mu$ and $\nu$ are invariant probability measures such that $\nu$ is Lyapunov maximizing in the sense that $\chi_1(\Phi,\nu) = \beta(\Phi)$, and $\operatorname{supp}\mu \subseteq \operatorname{supp}\nu$, then $\mu$ is Lyapunov maximizing as well: see \[t.subordination\]. This property is far from being tautological, even in the commutative setting; in fact it was introduced in this setting by Bousch [@Bousch_Walters].

-   The maximal Lyapunov exponent $\beta(\Phi)$ can be approximated by Lyapunov exponents of measures supported 