of the concept of Barabanov norm was used in [@BR; @BM] to study Lyapunov-maximizing and also Lyapunov-minimizing measures.

Extremal norms for certain locally constant cocycles over sofic shifts have been studied in the papers [@PEDJ; @CGP].

The main purpose of this paper is to establish existence of extremal norms in a far more general setting.

The main result
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We now describe the hypotheses on the automorphism $\Phi$ and the underlying dynamics $T$ from which we will prove the existence of extremal norms. We first describe them informally, leaving the precise definitions for later sections. First, we assume that $T \colon X \to X$ is a transitive *hyperbolic homeomorphism* of a compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product property. Examples include subshifts of finite type and Anosov diffeomorphisms.

Second, we assume that the vector bundle ${\mathbb{E}}$ has a Hölder structure, and that the automorphism $\Phi$ respects this structure. In the case of trivial vector bundles, this means that the matrix function $F$ in formula is Hölder continuous.

Third, we assume that the automorphism $\Phi$ is *fiber-bunched*. In crude terms, this means that the non-conformality 