class of vector bundle automorphisms.

Previous results {#ss.known}
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Consider the case of a $1$-dimensional vector bundle ${\mathbb{E}}$, with an arbitrary Finsler norm $\| \mathord{\cdot} \|$. Given $\Phi \in {\mathrm{Aut}}({\mathbb{E}},T)$, there exists a unique continuous function $f \colon X \to {\mathbb{R}}$ such that $$\label{e.operator}
u \in {\mathbb{E}}_x \quad \Rightarrow \quad \|\Phi(u)\|_{Tx} = e^{f(x)} \| u\|_x \, .$$ Note that in this case the maximal Lyapunov exponent $\beta(\Phi)$ equals: $$\label{e.beta_f}
\beta (f) \coloneqq \sup_{\mu \in {\mathcal{M}}_T} \int f \, d\mu \, .$$ Any other Finsler norm ${\mathopen{| {\kern -1.5pt} | {\kern -1.5pt} |} {\cdot}
\mathclose{| {\kern -1.5pt} | {\kern -1.5pt} |}}$ is of the form: $${\mathopen{| {\kern -1.5pt} | {\kern -1.5pt} |} {u}
\mathclose{| {\kern -1.5pt} | {\kern -1.5pt} |}}_x = e^{h(x)} \| u\|_x \, ,$$ for some continuous function $h \colon X \to {\mathbb{R}}$. Then ${\mathopen{| {\kern -1.5pt} | {\kern -1.5pt} |} {\cdot}
\mathclose{| {\kern -1.5pt} | {\kern -1.5pt} |}}$ is a extremal norm if and only if $h$ satisfies the “cohomological inequality”: $$f + h \circ T  - h \le \beta(f) \, .$$ Such a function $h$ is called a *subaction* for $(T,f)$. Existence of subactions can fail dramatically: see e.g. [@Bousch_Jenkinson [§]{}3] and [@Garibaldi_book Appendix]. However, if the 