\Phi_{T^{n-1}x} \circ \cdots \circ \Phi_{Tx} \circ \Phi_x\| \, ,$$ if it exists, is called the *(first) Lyapunov exponent* of $\Phi$ at the point $x$. The Lyapunov exponent is obviously independent of the choice of the Finsler norm. If $\mu$ is a $T$-invariant Borel probability measure for $T$, then the Lyapunov exponent $\chi_1(\Phi, x)$ exists for $\mu$-almost every $x\in X$; this is a well-known consequence of Kingman’s subadditive ergodic theorem; see e.g. [@Krengel]. Let us denote $\chi_1(\Phi, \mu) \coloneqq \int \chi_1(\Phi,\mathord{\cdot}) \, d\mu$. If the measure $\mu$ is ergodic then $\chi_1(\Phi, x)  =  \chi_1(\Phi,\mu)$ for $\mu$-almost every $x\in X$.

In this paper we are interested in the *maximal Lyapunov exponent*, defined as: $$\label{e.def_beta}
\beta(\Phi) \coloneqq \sup_{\mu \in {\mathcal{M}}_T} \chi_1(\Phi, \mu) \, ,$$ where ${\mathcal{M}}_T$ denotes the set of all $T$-invariant Borel probability measures. The supremum is always attained by an ergodic measure – this follows from upper semicontinuity of $\chi_1(\Phi, \mathord{\cdot})$ with respect to the weak-star topology, and the fact that ${\mathcal{M}}_T$ is a compact convex set whose extreme points are exactly the ergodic measures. Let us mention that the maximal Lyapunov exponent can also be characterized in more elementary terms as follows: $$\label{e.beta_other}
\beta(\Phi) = \operatorname*{linf}_{n \to \infty} \frac{1}{n} \sup_{x 