our results.

\[s.technical\] contains the proofs of several subsidiary results, therefore making the paper self-contained.

In \[s.examples\] we exhibit some “pathological” examples, including an example that fits in the setting of our main results, but where no Riemannian extremal norm exists.

The fiber-bunched setting {#s.setting}
=========================

In this , we fix the basic setting for our theorems. Namely, we define and state the basic properties of Hölder vector bundles, intrinsically hyperbolic homeomorphisms, fiber-bunching, holonomies, and irreducibility. Our approach is influenced by [@BGV; @Viana; @KalSad], and we tried to make it as general as possible. We also obtain some new regularity results that are essential for the main theorems of the paper. However, to make the presentation more fluid, we postpone most proofs to \[s.technical\].

The Hölder exponent {#ss.theta}
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From now on, assume that $(X,{\mathrm{d}})$ is a compact metric space. We also fix $\theta > 0$ such that the algebra of $\theta$-Hölder functions on $X$ is *normal*, that is, given any two disjoint compact subsets of $X$, there exists a $\theta$-Hölder function that takes values in the interval $[0,1]$ and equals $0$ on one set and $1$ on the other. This assumption is automatically satisfied if $\theta \le 1$. If 