is an *automorphism of ${\mathbb{E}}$ covering $T$* if the diagram $$\begin{tikzcd}
{\mathbb{E}}\arrow[r,"\Phi"] \arrow[d,swap,"\pi"]   & {\mathbb{E}}\arrow[d,"\pi"] \\
X   \arrow[r,swap,"T"]                      & X
\end{tikzcd}$$ commutes and moreover the restriction of $\Phi$ to each fiber ${\mathbb{E}}_x \coloneqq \pi^{-1}(x)$ is a linear isomorphism $\Phi_x$ onto the fiber ${\mathbb{E}}_{Tx}$. The set of such automorphisms is denoted ${\mathrm{Aut}}({\mathbb{E}},T)$. The simplest situation is when the vector bundle is trivial, say ${\mathbb{E}}= X \times {\mathbb{R}}^d$. Then $\Phi$ takes the form $$\label{e.cocycle}
\Phi(x,u) = (T(x),F(x)u) \, ,$$ for some continuous map $F \colon X \to {\mathit{GL}}(d,{\mathbb{R}})$. The pair $(T,F)$ is called a *(linear) cocycle*.

A *Finsler norm*[^1] on ${\mathbb{E}}$ is a continuous map $\| \mathord{\cdot} \| \colon {\mathbb{E}}\to {\mathbb{R}}$ whose restriction to each fiber ${\mathbb{E}}_x$ is a norm. If $L$ is a linear map from a fiber ${\mathbb{E}}_x$ to another fiber ${\mathbb{E}}_y$, then we define the operator norm: $$\label{e.def_operator_norm}
\|L\|_{y \gets x} \coloneqq \sup_{\substack{u \in {\mathbb{E}}_x \\ u\neq 0}} \frac{\|L(u)\|}{\|u\|} \, .$$ When no confusion is likely to arise we denote this simply by $\|L\|$.

Fix an automorphism $\Phi$ covering $T$ and a Finsler norm $\| \mathord{\cdot} \|$. Given $x \in X$, the limit $$\chi_1(\Phi, x) \coloneqq \lim_{n \to +\infty} \frac{1}{n} \log \|\Phi^n_x\|
= \lim_{n \to +\infty} \frac{1}{n} \log \| 