of the linear maps $\Phi_x$ is small when compared to the hyperbolicity rates of $T$. The precise condition involves the Hölder exponent of the automorphism, so that more regular automorphisms are allowed to be less conformal. In the case that $T$ and $\Phi$ are differentiable, fiber-bunching means that the projectivization of $\Phi$ is a *partially hyperbolic* diffeomorphism.

Actually, for $d \ge 3$ we need to assume a stronger form of fiber-bunching.

Our last assumption is *irreducibility*, meaning that $\Phi$ admits no nontrivial regular subbundle, where regular means as regular as the automorphism itself. We remark that this condition is satisfied for typical fiber-bunched automorphisms: it holds on an open and dense subset of infinite codimension.

The main result of this paper is that *under the conditions above, extremal norms exist*. See \[c.extremal\] for a precise statement.

In the case where the base dynamics $T$ is a subshift of finite type, we are able to improve our main result and obtain an extremal norm with a further property akin to the Barabanov property: see \[ss.Barabanov\].

Classical Barabanov norms are usually non-Riemannian (that is, they do not come from inner products), and it is easy to produce examples[^4]. On the other hand, 