\in X} \log \| \Phi^n_x \|
= \sup_{x \in X} \limsup_{n \to \infty} \frac{1}{n} \log \| \Phi^n_x \| \, .$$ (We use “$\operatorname*{linf}$” to denote a limit that is also an infimum.) These equalities follow from general results on “subadditive ergodic optimization”: see [@Morris_Mather Appendix A].

A trivial upper bound for the maximal Lyapunov exponent, which depends on the chosen Finsler norm, is given by: $$\label{e.starting_point}
\beta(\Phi) \le  \log  \sup_{x \in X} \| \Phi_x \| \, .$$ If equality holds then $\| \mathord{\cdot} \|$ is called an *extremal norm* for $\Phi$. More precisely, the norm is so “tight” that there is *no* vector $u \neq 0$ in ${\mathbb{E}}$ whose expansion factor $\|\Phi(u)\| / \|u\|$ exceeds the maximum asymptotic expansion rate $e^{\beta(\Phi)}$. In particular, if $\beta(\Phi) \le 0$ then the extremal norm is a (non-strict) Lyapunov function for $\Phi$.

Extremal norms first appeared in the 1960 paper [@RS] by Rota and Strang, who considered the particular setting of one-step cocycles (details are given below), but apparently were not considered in our level of generality before.

The existence of an extremal norm is far from automatic[^2], and has strong consequences. In this paper we construct extremal norms for a large and natural 